THE ULTIMATE TETRATORUS

Michael Elgersma, Minneapolis
Stan Wagon, Macalester College

Platonic Tori

For 4 of the 5 regular polyhedra, there is a simple perfect closed loop — a “Platonic torus”.

cube                       octahedron             dodecahedron             icosahedron

H. Steinhaus (1956):  Is there a closed loop of tetrahedra meeting face-to-face?

S. Świerczkowski (1958): NO.

Main Tool:  Barycentric coordinates: 4×4 matrices (Subscript[M, 1]), (Subscript[M, 2]), (Subscript[M, 3]), (Subscript[M, 4]) reflecting in the four faces.

(Subscript[M, 1])=(({{-1, 0, 0, 0}, {(2/3), 1, 0, 0}, {(2/3), 0, 1, 0}, {(2/3), 0, 0, 1}}))    (Subscript[M, 2])=(({{1, (2/3), 0, 0}, {0, -1, 0, 0}, {0, (2/3), 1, 0}, {0, (2/3), 0, 1}}))    (Subscript[M, 3])=(({{1, 0, (2/3), 0}, {0, 1, (2/3), 0}, {0, 0, -1, 0}, {0, 0, (2/3), 1}}))    (Subscript[M, 4])=(({{1, 0, 0, (2/3)}, {0, 1, 0, (2/3)}, {0, 0, 1, (2/3)}, {0, 0, 0, -1}}))

A tetrahedral chain is a string such as 1234321 where entries are reflection-faces; 
Corresponds to the matrix product  (Subscript[M, 1])·(Subscript[M, 2])·(Subscript[M, 3])·(Subscript[M, 4])·(Subscript[M, 3])·(Subscript[M, 2])·(Subscript[M, 1]).

Main  point: No nontrivial product of the (Subscript[M, i]) can be the identity matrix.

Proof:  Induction on word-length.

Demonstration of Some Chains

Example: Start string with 123. Continue   4 3 2 1 4 3   and repeat 5 times.

Manipulate[
Module[
{
next, 
faces={{2,3,4},{4,3,1},{1,2,4},{3,2,1}},
cols =Lighter@ {Red, Green, Blue},
Mathieson={1,2,4,1,3,2,1,4,3},
Tetrahelix={1,2,3,4},
Wagon330={1,2,3,4,2,1,4,1,3,2,3},
Wagon36A={1,2,3,2,1,2,3,2,4,2,3,2},
Wagon36={1,2,3,1,4,1,3,2,1,2,4,1,2,3,2,1,4,2},
Wagon48={1,2,1,4,3,4},
Elgersma30= {1,2,3,2,1,3,2,4,2,3},
Elgersma60= {1,2,3,4,3},
Wagon150={1,2,3,4,1,2,3,4,1,2,3,4,1,2,3,4,3,1,2, 3,1,2, 3,2,4},
Wagon174 = {1,2,3,4,2,3,4,2,3,2,1,4,2,3,4,3,2,4,1,2,3,2,4,3,2,4,3,2,1,2,4,3,1,4,3,1,4,3,4,2,1,4,3,1,3,4,1,2,4,3,4,1,3,4,1,3,4,2},
Elgersma174={1,2,3,2,4,2,4,3,4,3,1,3,2,3,4,3,2,3,1,3,4,3,4,2,4,2,3,2,1,2,4,3,4,1,4,1,3,1,3,2,3,4,3,1,3,4,3,2,3,1,3,1,4,1,4,3,4,2}
},

If[j==0, k=Max[1,k]];

Which[
special==="Tetrahelix", seq=Tetrahelix,
special==="96 loop", seq=Mathieson,
special==="330 loop", seq=Wagon330,
special==="48 loop", seq=Wagon48,
special==="150 loop", seq=Wagon150,
special==="30 loop", seq=Elgersma30,
special==="36 loop", seq=Wagon36,
special==="36A loop", seq=Wagon36A,
special==="174 loop", seq=Wagon174,
special==="alternate 174 loop", seq=Elgersma174,
special==="60 loop", seq=Elgersma60,
True,
seq=Join[{1,2,3},Select[{M1, M2, M3, M4, M5, M6,M7, M8, M9, M10}, IntegerQ]]
];
 seq1=Flatten[Table[seq, {j}]];seq1=Join[seq1, Take[seq,Min[k, Length[seq]]]];
mats=N[M /@ seq1];mats=FoldList[Dot,mats[[1]],Rest[mats]];
graphicData=Table[{cols[[Mod[i,3,1]]],Table[next=(fromBary[mats[[i]].#1,startV]&)/@IdentityMatrix[4];{Polygon[next[[f]]]},{f,faces}]},{i,Length[mats]}];
Pane[
Column[{
seq1;
Which[
k==0,Style[StringForm["Sequence is `` repeated `` ``.",SequenceForm@@seq,j, If[j≤1 ,"time", "times"]],FontFamily->Times],
k≥1, Style[StringForm["Sequence is `` repeated ``, with `` additional.",SequenceForm@@seq,j,k],FontFamily->Times]
],
Style[StringForm["Discrepancy from perfect closure is ``.",NumberForm[err[mats[[-1]]]; dist[startV, next],3]],FontFamily->"Times"],

gg=Graphics3D[{graphicData}, 
PlotRange->All,ImageSize-> 400,Lighting->"Neutral", Boxed->False,Axes->!True, PlotLabel->{None,dist[next,startV];err[mats[[-1]]]}[[1]], PlotRegion->Automatic, PlotRangePadding->0];
pr=PlotRange[gg];
Show[gg, PlotRange->pr, ImageSize->{400, 400}, SphericalRegion-> True]
},
Alignment->Center],
ImageSize->{600, 420},Alignment->Center]], 
Row[{
"special = None  ",
Control@{{M1,4, "4"},DeleteCases[rr,3], PopupMenu , Appearance->Small,Enabled->special==="None"}, Spacer[8],
Control@{{M2,3, "5"}, rr, PopupMenu ,Enabled->special==="None"},  Spacer[8],
Control@{{M3,"—", "6"}, rr, PopupMenu ,Enabled->special==="None"},  Spacer[8],
Control@{{M4,"—", "7"}, rr, PopupMenu ,Enabled->special==="None"},Spacer[8],
Control@{{M5,"—", "8"}, rr, PopupMenu ,Enabled->special==="None"},Spacer[8],
Control@{{M6,"—", "9"}, rr, PopupMenu ,Enabled->special==="None"},Spacer[8],
Control@{{M7,"—", "10"}, rr, PopupMenu ,Enabled->special==="None"},Spacer[8],
Control@{{M8,"—", "11"}, rr, PopupMenu ,Enabled->special==="None"}}], 
Row[{Control@{{j, 1, "periods"}, 0,30,1, ImageSize->Medium,Appearance->"Labeled"},Spacer[30], Control@{{k,0, "extra"}, 0,14,1, ImageSize->Medium,Appearance->"Labeled"}}], 
{special, {"None",Style["Tetrahelix",Red],"174 loop","alternate 174","150 loop","330 loop","36 loop","30 loop","60 loop","48 loop"}, Setter}, TrackedSymbols->True,
Initialization:>(
rr=Append[Range[4], "—"];

startV=N[{{0,0,(Sqrt[(2/3)])-(1/(2 (Sqrt[6])))},{-(1/(2 (Sqrt[3]))),-(1/2),-(1/(2 (Sqrt[6])))},{-(1/(2 (Sqrt[3]))),(1/2),-(1/(2 (Sqrt[6])))},{(1/(Sqrt[3])),0,-(1/(2 (Sqrt[6])))}}];
fromBary[bary_, pts4_] := Sum[bary[[i]]*pts4[[i]], {i,4}];fromBary[bary_, pts4_] :=bary. pts4;perms = Permutations[{1,2,3,4}];
err[K_] := Max[Flatten[Abs[K-IdentityMatrix[4]]]];
dist[tet1_, tet2_] := Min[Table[Max[Norm/@ ((tet1-tet2[[pp]]))], {pp,perms}]];

{M[1], M[2], M[3], M[4]} =N@{{{-1,0,0,0},{(2/3),1,0,0},{(2/3),0,1,0},{(2/3),0,0,1}},{{1,(2/3),0,0},{0,-1,0,0},{0,(2/3),1,0},{0,(2/3),0,1}},{{1,0,(2/3),0},{0,1,(2/3),0},{0,0,-1,0},{0,0,(2/3),1}},{{1,0,0,(2/3)},{0,1,0,(2/3)},{0,0,1,(2/3)},{0,0,0,-1}}})]

Almost Perfect Tetrahedral Loops Exist

The Świerczkowski result stood for 56 years with no improvement. In 2014 I wondered if one can come close to perfection with a chain having no self-intersections? Can one make a physical hoax loop?

(YES, and I sent a 3D-printed model to S. Świerczkowski.)

Tetrahedral Loop Conjecture (2014).  For any ϵ>0, there is a tetrahedral chain with no collisions and a terminal gap less than ϵ.

Tetrahedral Loop Theorem (M. Elgersma and SW, 2016). The conjecture is true (and constructively so).

The history leading to the Theorem is interesting.

2013: First Hoax, 20% error

First one: length=48       string =(((121434))^8)      Gap is 0.2 (unit sides): gap size is 20%.

Better: (1/100)% error

Length=36   (((123141321    241232142))^2)   ϵ=0.0004  (size of a human hair)

Proton-Sized Error: (10^(-13))

Length =174;   gap size=6·(10^(-13))  (“Mr. Proton”)

Subatomic error: (10^(-18))

Length: 540   gap size = (10^(-18))  (“Mr. Higgs”)

c={1,2,3,4,1,2,3,4,3,4,1,3,2,3,4,1,2,1,3,4,1,2,4,2,1,4,3,1,2,1,4,3,2,3,1,4,3,4,3,2,1,4,3,2,1,2,3,4,1,2,3,4,1,4,1,2,4,3,4,1,2,3,2,4,1,2,3,1,3,2,1,4,2,3,2,1,4,3,4,2,1,4,1,4,3,2,1,4,3,2,1,2,3,4,1,2,3,4,3,4,1,3,2,3,4,1,2,1,3,4,1,2,4,2,1,4,3,1,2,1,4,3,2,3,1,4,3,4,3,2,1,4,3,2,1,4,1,2,3,4,1,2,3,2,3,4,2,1,2,3,4,1,4,2,3,4,1,3,1,4,3,2,4,1,4,3,2,1,2,4,3,2,3,2,1,4,3,2,1,4,1,2,3,4,1,2,3,4,3,4,1,3,2,3,4,1,2,1,3,4,1,2,4,2,1,4,3,1,2,1,4,3,2,3,1,4,3,4,3,2,1,4,3,2,1,2,3,4,1,2,3,4,1,4,1,2,4,3,4,1,2,3,2,4,1,2,3,1,3,2,1,4,2,3,2,1,4,3,4,2,1,4,1,4,3,2,1,4,3,2,1,2,3,4,1,2,3,4,3,4,1,3,2,3,4,1,2,1,3,4,1,2,4,2,1,4,3,1,2,1,4,3,2,3,1,4,3,4,3,2,1,4,3,2,1,4,1,2,3,4,1,2,3,2,3,4,2,1,2,3,4,1,4,2,3,4,1,3,1,4,3,2,4,1,4,3,2,1,2,4,3,2,3,2,1,4,3,2,1,4,1,2,3,4,1,2,3,4,3,4,1,3,2,3,4,1,2,1,3,4,1,2,4,2,1,4,3,1,2,1,4,3,2,3,1,4,3,4,3,2,1,4,3,2,1,2,3,4,1,2,3,4,1,4,1,2,4,3,4,1,2,3,2,4,1,2,3,1,3,2,1,4,2,3,2,1,4,3,4,2,1,4,1,4,3,2,1,4,3,2,1,2,3,4,1,2,3,4,3,4,1,3,2,3,4,1,2,1,3,4,1,2,4,2,1,4,3,1,2,1,4,3,2,3,1,4,3,4,3,2,1,4,3,2,1,4,1,2,3,4,1,2,3,2,3,4,2,1,2,3,4,1,4,2,3,4,1,3,1,4,3,2,4,1,4,3,2,1,2,4,3,2,3,2,1,4,3,2,1,4};

pp=polygonsUp[RotateLeft[c, 50+17]];
im=Graphics3D[{EdgeForm[{AbsoluteThickness[.6], GrayLevel[.25]}], 
{RGBColor[1, .1, .1], pp[[1;;90]]},{Lighter@Lighter@Red, pp[[91;;180]]},
{RGBColor[0, .7,0], pp[[181;;270]]},{RGBColor[.3,1,.3], pp[[271;;360]]},
{RGBColor[0.4,0.4,1], pp[[361;;450]]},{Lighter@Lighter@Lighter@Blue, pp[[451;;540]]}}, 
ImageSize->800,Lighting->"Neutral", Boxed->False, Background->GrayLevel[.8], PlotRangePadding->0, 
ViewPoint->{-2.797221676208145`,-0.1009354572324734`,1.9014107729853036`}, ViewVertical->{-0.29470233357828163`,0.6285074027388725`,0.8251129030996264`}]

b=1234123434132341213412;  π is 4-cycle 1->2->3->4->1;  (OverBar[b]) is the reverse of b

The 540-string =((((b 4 (OverBar[b]))  (Subscript[(b 4 (OverBar[b])), π])(b 4 (OverBar[b]))(Subscript[(b 4 (OverBar[b])), (π^3)])))^3) 

123412343413234121341242143121432314343214321234123414124341232412313214232143421414321432123412343413234121341242143121432314343214321412341232342123414234131432414321243232143214123412343413234121341242143121432314343214321234123414124341232412313214232143421414321432123412343413234121341242143121432314343214321412341232342123414234131432414321243232143214123412343413234121341242143121432314343214321234123414124341232412313214232143421414321432123412343413234121341242143121432314343214321412341232342123414234131432414321243232143214

Subatomic error: (10^(-18))

The barycentric matrix of the string:

mat[i_]:=Transpose[ReplacePart[DiagonalMatrix[{1,1,1,1}],3->ReplacePart[{(2/3),(2/3),(2/3),(2/3)},i->-1]]];
mat[3]//MatrixForm

The matrix for this sequence is very close to the identity.

NumberForm[MatrixForm[N[N[Dot@@ (mat/@IntegerDigits[
]),20]]],2]

Summary

We used Mathematica to check millions of strings, using machine precision, high precision, and infinite precision (symbolic); we also implemented a symbolic algorithm to check for collisions.

We were happy with these sporadic examples: no collisions, error getting small. 

It seemed clear that error->0, but we found no uniform pattern. and so were left with:

Small Gap Conjecture.  For any ϵ>0, there is an embedded chain with gap error under ϵ.

Exponentially Small Gap Conjecture.  For any L, there is an embedded L-chain with gap error under (0.9^L).

I tried to paste together long chains; nothing. In 2016, Michael Elgersma returned to the problem and found some new patterns. The key is the tetrahelix.

The Tetrahelix

The TETRAHELIX: Equally spaced vertices on a spiral; angular step θ=(cos^(-1))(-(2/3)). 
This is 1234 1234 1234 1234…, in terms of face reflections.

Vertices are just:  ((3/10)(Sqrt[3]) cos(n θ), (3/10)(Sqrt[3]) sin(n θ),  n(1/(Sqrt[10])))

The Tetrahelix

The TETRAHELIX: Equally spaced vertices on a spiral; angular step θ=(cos^(-1))(-(2/3)). 
This is 1234 1234 1234 1234…, in terms of face reflections.

Points are simple:  ((3/10)(Sqrt[3]) cos(n θ), (3/10)(Sqrt[3]) sin(n θ),  n(1/(Sqrt[10])))

Art Museum, Mito Japan

The Quadrahelix: L=11

The “quadrahelix” proves the conjecture. Join two tetrahelices at a common tetrahedron.

Below: L=11 red tetrahedra, so 23 in the Vee. The gray one is the “pivot”.

Lval=11;Graphics3D[{pp=polygonsUp[QHNew[Lval]];
{Red,pp[[1;;Lval]]},
{GrayLevel[.8],pp[[Lval+1;;Lval+1]]},
{Green,pp[[Lval+2;;2Lval+1]]}}, Lighting->"Neutral", Boxed->False, PlotRangePadding->0]

Reflect in the green end to get the quadrahelix; 2·23=46 in all. Large gap in this case.

Lval=11;
im=Graphics3D[{pp=polygonsUp[QHNew[Lval]];
{Red,pp[[1;;Lval]]},
{GrayLevel[.8],pp[[Lval+1;;3Lval+2;; 2Lval +1]]},
{Green,pp[[Lval+2;;2Lval+1]]},
{Lighter@Lighter@Blue,pp[[2Lval+2;;3Lval+1]]},
{Lighter@Yellow,pp[[3Lval+3;;4Lval+2]]}}, Lighting->"Neutral", Boxed->False]

View of General Quadrahelix:

DynamicModule[{Lval1}, Manipulate[Lval=If[IntegerQ[Lval1], Lval1,Lval1[[1]]];test=Lval;Lval=Lval1;
s[t_] := Style[ToString[t],FontFamily-> "Times",FontSlant-> "Plain", 30];
s2[t_] := Style[ToString[t],FontSlant-> "Plain", 20];
s2Red[t_] := Style[ToString[t],FontSlant-> "Plain", Red,20];
s1[t_] := Style[ToString[t],FontFamily-> "Times",FontSlant-> "Italic", 30];Column[{Row[Join[{s1@"L"}, s/@ {" = ", Lval,"         Number of tetrahedra = ",  4 Lval+2}]] ,
Row[s/@{"Gap error = ",NumberForm[errorGapShiftedQuad[Lval,20],2]}],
zzz=Show[Print3DChainQuad[Lval, Colors-> {Red, Green, Lighter@Lighter@Blue, Lighter@Yellow}, EdgeStyle->{AbsoluteThickness[.5], Black}],ImageSize->600, PlotRangePadding->0,
ViewPoint-> vp[Lval], ViewVertical->vv[Lval],
Background->GrayLevel[.4]]}], {{Lval1,1;4, Style["L",24, FontSlant->"Italic"]}, {1->s2Red@1,2->s2Red@2,3->s2@3,4->s2@4,5->s2@5,6->s2@6,7->s2Red@7,8->s2@8,9->s2@9,10->s2Red@10,11->s2@11,12->s2@12,13->s2@13,28->s2@28,29->s2Red@29,30->s2@30,40->s2Red@40,70->s2Red@70,182->s2Red@182}, SetterBar}, SaveDefinitions->True,TrackedSymbols->{Lval1} ]]

The Quadrahelix Miracle: L=10

When L=10 (42 tetrahedra) a miracle happens:

Lval=10;Graphics3D[{pp=polygonsUp[QHNew[Lval]];
{Red,pp[[1;;Lval]]},
{GrayLevel[.8],pp[[Lval+1;;Lval+1]]},
{Green,pp[[Lval+2;;2Lval+1]]}}, Lighting->"Neutral", Boxed->False]

The Quadrahelix: L=70

Another miracle when L=70 (so 282 tetrahedra).

The quadrahelix is asymptotically planar: a rhombus.

im=Graphics3D[{pp=polygonsUp[QHNew[Lval=70]];
{Red,pp[[1;;Lval]]},
{GrayLevel[.8],pp[[Lval+1;;3Lval+2;; 2Lval +1]]},
{Green,pp[[Lval+2;;2Lval+1]]},
{Lighter@Lighter@Blue,pp[[2Lval+2;;3Lval+1]]},
{Lighter@Yellow,pp[[3Lval+3;;4Lval+2]]}}, Lighting->"Neutral", Boxed->False,ImageSize->1000]

im=Graphics3D[{pp=polygonsUp[QHNew[Lval=70]];
{Red,pp[[1;;1]]},
{Lighter@Yellow,pp[[4Lval+2;;4Lval+2]]}}, Background->Blue,Lighting->"Neutral", Boxed->False,ImageSize->400]

The Quadrahelix Miracle

When L=10:  the roof (of the red tetrahelix) is (almost) parallel to the floor.

pp=Print3DChainQuad[Lval=10, Background->GrayLevel[.7]];
ppp=Cases[pp, _Polygon, ∞];
PL1=planeThru[startPts=ppp[[2,1]]];
PL2=planeThru[pts={ppp[[2Lval+2,1,1]],ppp[[2Lval+1,1,2]],Mean[ppp[[2Lval+2,1,{2,3}]]]}];
PL3=planeThru@ppp[[2Lval+3,1]];
im=Show[bb=Graphics3D[{{Opacity[.3],FaceForm[];Join[ Take[ppp, {3,2 Lval+1}],Take[ppp, {2 Lval+4,4 Lval+1}]]},
{Opacity[1], Take[ppp, {1,Round[Lval/2]5}]},
{Opacity[1], Take[ppp, {4 Lval,4 Lval+3}]},
{Opacity[1], Take[ppp, {2 Lval+2,2 Lval+3}]/. -> (GrayLevel[.3]) },{}}], 

gg1=Graphics3D[{{Red, AbsolutePointSize[20],Point@pts};
{a,b,c}=ppp[[2Lval+3,1]];(nn0=nn=normal[{a,b,c}]);
mm0=mm=Mean[{a,b,c}]; 


Blue, Opacity[.4],Polygon[{mm +15 ( a-mm), mm+ 15Cross[nn, a-mm],mm -15( a-mm), mm- 5 Cross[nn, a-mm]}]
/. {x_, y_, z_?NumericQ}:>{x,y,z} -.0nn  ,
{a,b,c}=ppp[[2,1]];mm=Mean[{a,b,c}]; (nn=normal[{a,b,c}]);
Cyan, Polygon[{mm +6( a-mm), mm+ 3Cross[nn, a-mm],mm -5 ( a-mm), mm- 2 9 Cross[nn, a-mm]}]
, Tube[{mm0-.4 nn0, mm0+3.4 nn0}]}
],PlotRange-> PlotRange[ bb]1.2,Boxed->False,ImageSize->500,
AxesLabel->{x,y,z}, Background->None,ViewPoint->{-2.0540816871321192`,-0.7584633483744919`,-2.579822042653528`},ViewVertical->{0.09397075554516361`,-0.8456978524725683`,-0.5947314165175651`}]

Wonderful: The floor of the last green tetrahedron is almost parallel to the red floor, and a reflection of it. Flipping everything then gives an almost closed chain!

pp=Print3DChainQuad[Lval=10, Background->GrayLevel[.7]];
ppp=Cases[pp, _Polygon, ∞];
PL1=planeThru[startPts=ppp[[2,1]]];
PL2=planeThru[pts={ppp[[2Lval+2,1,1]],ppp[[2Lval+1,1,2]],Mean[ppp[[2Lval+2,1,{2,3}]]]}];
PL3=planeThru@ppp[[2Lval+3,1]];
im=Show[bb=Graphics3D[{{Opacity[.8],
Join[ Take[ppp, {3,2 Lval+1}],Take[ppp, {2 Lval+4,4 Lval+1}]]},
{Opacity[.8], Take[ppp, {1,Round[Lval/2]5}]},
{Opacity[.8], Take[ppp, {4 Lval,4 Lval+3}]},
{Opacity[.8], Take[ppp, {2 Lval+2,2 Lval+3}]/. -> (GrayLevel[.3]) },
{Opacity[.8], Take[ppp, {4Lval+4,8 Lval+4}]/. -> Yellow},

{}}],

gg1=Graphics3D[{{Red, AbsolutePointSize[20],Point@pts};
{a,b,c}=ppp[[2Lval+3,1]];(nn0=nn=normal[{a,b,c}]);
mm0=mm=Mean[{a,b,c}]; 

Blue, Opacity[.4],Polygon[{mm +15 ( a-mm), mm+ 15Cross[nn, a-mm],mm -15( a-mm), mm- 5 Cross[nn, a-mm]}]
/. {x_, y_, z_?NumericQ}:>{x,y,z} -.0nn ;
{a,b,c}=ppp[[2,1]];mm=Mean[{a,b,c}]; (nn=normal[{a,b,c}]);
Cyan, Polygon[{mm +6( a-mm), mm+ 3Cross[nn, a-mm],mm -5 ( a-mm), mm- 2 9 Cross[nn, a-mm]}]}
],PlotRange-> PlotRange[ bb]1.2,Boxed->False,Lighting->"Neutral", PlotRangePadding->0,ImageSize->700,
AxesLabel->{x,y,z}]

Graphics[Inset[im,{0,0},{0,0},3],Frame->!True, PlotRange->{{.1,2.43}, {.9,2.1}},ImageSize->500]

The Quadrahelix Never Self-Intersects

δ: the total x-y angular change of the first tetrahelix: δ=(L+1)θ, reduced mod 2  π.
ρ: the dihedral angle between two reflection planes.

5 planes give 4 sectors, each has ACUTE (!!) dihedral angle ρ.
Each color lives in its quadrant: No collisions.

Acute Angle Proof: Use dot product of normal vectors to two faces:

cos ρ=(normal vector to red floor)·((Subscript[V, L+3])-(Subscript[V, L+1]))=((Sqrt[6])/5)(1-cos δ)=O((δ^2)).

12°≤ρ<90°

No Collision: A simple determinant shows that red stays on proper side of the red/green plane, etc.

Clear[vp,vv];{vp[_], vv[_]}= {{-1,-2.3,2}, {0 0,1}};
{vp[2], vv[2]}={{-2.436476625192898`,-1.5809907523644915`,1.361635008299429`},{-1.2575593022759337`,-0.5485412269834631`,-0.42982756819051593`}};
{vp[3], vv[3]}= {{2.4953587172326834`,-1.5857876841226364`,-1.2443722486522362`},{0.7010438434352279`,0.14075386215118846`,-0.8871206085133403`}};
{vp[4], vv[4]}={{2.747755021873639`,-0.34500755417750245`,-1.6188922531560468`},{1.4558994856197323`,-0.027887826102026555`,-0.003036588746728503`}};
{vp[5], vv[5]}={{-0.8470694943595101`,-0.699128038337188`,-3.0139165976742497`},{-0.6143738991006478`,-0.6099116349418946`,-0.7482741046064467`}};
{vp[6], vv[6]}={{2.876264615359726`,-1.0392679590580234`,1.4481104832522036`},{0.5485449466481384`,0.46739648637445863`,0.8779234229199029`}};
{vp[7], vv[7]} = {{-2.3625688126863076`,-0.8182879415616459`,-2.0096451054892284`},{-0.2432889135952974`,-0.7491420355651607`,-0.7848122861438762`}};
{vp[8],vv[8]}= {{1.5693071224552035`,-2.7744810366438486`,0.35990294902243497`},{0.9560932817456527`,-0.40839901700076225`,-0.2836650044213117`}};
{vp[9],vv[9]}= {{1.203956023136`,2.7273273424585858`,1.184134900014493`},{0.21214752807183623`,1.2826217743361785`,-0.20558454205607354`}};
{vp[10], vv[10]}={{-2.3625688126863076`,-0.8182879415616459`,-2.0096451054892284`},{-0.2432889135952974`,-0.7491420355651607`,-0.7848122861438762`}};

{vp[13], vv[13]}={{-0.5973267733834999`,-0.15288056776773928`,-3.147987969767081`},{-0.44502159721532025`,-0.3682717763309116`,1.3621930886128617`}};

{vp[15], vv[15]}={{-2.7454490257121833`,-1.380416355760745`,-0.9203044778573286`},{0.6945202918667437`,0.9018611848699623`,-0.17473218785165903`}};

{vp[25], vv[25]}={{0.7802285087015227`,0.32958073103844093`,3.093965096108481`},{-0.5877585403943673`,-0.11259378208821114`,1.5823921509273404`}};

three=.;
Lval=7;Lval=13;Lval=15;

pp=polygonsUp[cc=QuadrahelixShifted[Lval,P4[[15]]]];
threeX={ pp[[Lval+1,1,4,2]],
Mean[{pp[[Lval+1,1,4,3]],pp[[Lval+1,1,4,1]]}], pp[[Lval+2,1,1,3]]};
three=THPoint/@ {-1,0,2};
im=Show[gg=Graphics3D[{Opacity[.5;1],
EdgeForm[{GrayLevel[.7], AbsoluteThickness[2]}],
{Red,pp[[1;;Lval]]},{Green,pp[[Lval+2;;2Lval+1]]},{Blue,pp[[2 Lval+2;;3Lval+1]]},{Yellow,pp[[3Lval+3;;4 Lval+2]]},
{GrayLevel[.4],pp[[{Lval+1, 3Lval+2}]]}}],
biplane=planeThru[biPlanePts=(Which[Lval==13,pp [[2Lval+1,1,Mod[Lval+1,4,1]]],
Lval==10||Lval==40, pp [[2Lval+1,1,Mod[Lval+3,4,1]]]];pp [[2Lval+1,1,{2,1,4,3}[[Mod[Lval,4,1]]]]])];
quadPlane=planeThru[{ pp[[Lval+1,1,4,2]],
Mean[{pp[[Lval+1,1,4,3]],pp[[Lval+1,1,4,1]]}], pp[[Lval+2,1,1,3]]}];
quadPlane2=planeThru[{ pp[[4 Lval+3 - (Lval+1),1,4,2]],
Mean[{pp[[(4 Lval+3 )-(Lval+1),1,4,3]],pp[[(4 Lval+3 )-(Lval+1),1,4,1]]}], pp[[(4 Lval+3 )-(Lval+2),1,1,3]]}];

startPlane= planeThru[pp[[1,1, Mod[Lval+1,4,1]]]];
endPlane= planeThru[pp[[-1,1, Mod[Lval+1,4,1]]]];
specialPlane= planeThru[Which[Lval==7, pp[[-1,1,4]], Lval==13, pp[[-1,1,3]], Lval==10||Lval==40, pp[[Lval+1,1,4]]];
pp[[Lval+1,1, Mod[Lval+1,4,1]]]];

Plot3D[biplane, {x,-600,600}, {y,-60,60},PlotStyle->Magenta],

Plot3D[quadPlane, {x,-5,5}, {y,-5,5},PlotStyle->RGBColor[.5,.5,1],RegionFunction-> (SameSideOfPlane[{#1,#2,#3},three[[1]],biPlanePts]&)],
Plot3D[quadPlane2, {x,-15,15}, {y,-15,15},PlotStyle->RGBColor[.5,1,.5],RegionFunction-> (!SameSideOfPlane[{#1,#2,#3},three[[1]],biPlanePts]&)],
Plot3D[startPlane, {x,-5,5}, {y,-5,5},PlotStyle->Yellow,RegionFunction-> (SameSideOfPlane[{#1,#2,#3},three[[1]],biPlanePts]&)],
Plot3D[endPlane, {x,-5,5}, {y,-5,5},PlotStyle->Pink,RegionFunction-> (!SameSideOfPlane[{#1,#2,#3},three[[1]],biPlanePts]&)],
Plot3D[specialPlane, {x,-115,115}, {y,-115,115},PlotStyle->Yellow,RegionFunction-> (SameSideOfPlane[{#1,#2,#3},three[[1]],biPlanePts]&)];

xExpr=Reduce[quadPlane==biplane][[2]];
mPrint[xExpr];
gg=Graphics3D[{Text[Style[ρ,24], {1/2,1/2}.{{xExpr,y,quadPlane}/. x-> xExpr/. y->0,
Mean[{pp[[Round[1],1,1,1]],pp[[Round[ Lval],1,1,1]]}]}],AbsolutePointSize[15],Orange, Point[tri1=pp[[1,1,Mod[Lval+1,4,3]]]];Point[ tri2=pp[[Lval+1,1,4,1]]];Sphere[#,.1]& /@  pp[[Lval+1,1,4,{1,3}]],
Lighter@Cyan,Tube[Table[{xExpr,y,quadPlane/. x-> xExpr}, {y,-10,10,  1}],.08]}

],

 PlotLabel->( StringForm["L = ``", Lval]; None), ImageSize->750, PlotRange->({{-0.65,4.2275},{-6.435,0.518},{-5.5,1.8203}}),Boxed->False,PlotRangePadding->0,AxesLabel->{x,y,z},Axes->!True,
ViewPoint-> vp[Lval], ViewVertical->vv[Lval], Lighting->"Neutral"]
w1=normal[tri1]//Simplify;w2=normal[tri2]//Simplify;w1.w2;

Proof that Gap -> 0

Gap -> 0   Key: Each tetrahelix step rotates θ=(cos^(-1))(-(2/3)). If (L+1)θ reduced mod 2 π (called δ   ) is near 0, then the tetrahedron L+1 is nearly identical to the 0th. This means the ceiling of tetrahedron L+1 is almost parallel to the ceiling of the “invisible” tetrahedron, which is identical to the floor of tetrahedron number 1.

Getting (L+1)θ to (almost) vanish mod 2   π requires a rational approximation (θ/(2π))∼(k/(L+1)). 
  Classic: Use convergents to the continued fraction of (θ/(2π)):

i=16;θ=ArcCos[-(2/3)];cf=Take[ContinuedFraction[N[(θ/(2 π)),20]],i]/. "0"->0;
ff=Fold[Row[{#2,"+",(1/(#1))}]&,1,Rest[Reverse[cf]]]/. Row[{0,"+",x___}]:>Row[{x}];
Style[ff, {FontFamily->"Comic Sans MS",18,ScriptSizeMultipliers->1}]

con=Convergents[((ArcCos[-(2/3)])/(2. π))];
Row[Join[con[[2;;11]],{"...",con[[14]],"...",con[[21]]}]]/. (67/183)->Style[(67/183),{Red,FontWeight->Bold}]

{N[(θ/(2 π))],(67/183.),N[(θ/(2 π))]-(67/183.)}

Convergence of convergents is fast!

The denominators give the good L+1 values:    
        L=1,2,7,10,29,40,70,182,253,1960,…,26000,…,65390016,…

Error in the convergent (k/(L+1)) is quadratic :          |(θ/(2π))-(k/(L+1))|<(1/(L^2))

Clear denominators:   δ=(Abs[(L+1)θ-2 π k])<((2π)/L)  so  (δ^2)=O((1/(L^2)))

The (L^2) is important since we need the (angular difference) × (overall size) to be small.
Geometry gives:  gap ~ L sin(2π-4ρ)=O(L cos ρ)=O(L (δ^2))=O(L(1/(L^2)))=O((1/L))->0    QED.

L=253, Number of tetrahedra = 1014

Lval=253;
Show[Graphics3D[pp=polygonsUp[QuadrahelixShifted[Lval]];
{Red,pp[[1;;Lval]],Gray, pp[[{Lval+1, 3 Lval+2}]],
{Green,pp[[Lval+2;;2Lval+1]]},
{Blue,pp[[2Lval+2;;3Lval+1]]},
{Yellow,pp[[3Lval+3;;4Lval+2]]}}] , Boxed->False,ImageSize->800, Background->Black,PlotRangePadding->0, ViewPoint->{0.21273538145405596`,1.9960315030899747`,-2.7240781736488344`}, ViewVertical->{-0.7678283206544199`,0.5710400747780772`,-0.9199750309173182`}]

Denominator[Convergents[N[((ArcCos[-(2/3)])/(2 π)),20]]]-1

(Denominator[Convergents[N[((ArcCos[-(2/3)])/(2 π)),200]]]-1)[[-1]]

L=521269338782055651792691214128196053088348030247372007924246566932650514801545115813925856156787510
Gap size is (10^(-100)).   (Remember: I was excited a few years ago to have 20% error.)

In about two minutes one can get a million-digit value of L, so gap <(10^(-1000000)).

Even Better?

Our best old example:  gap (10^(-17)),   540  tetrahedra.

The quadrahelix:           gap (10^(-17)),   (10^17)  tetrahedra.

There is room for improvement!

Open Problem.  Find a sequence of chains with N tetrahedra so that gap size is exponentially small, say (0.9^N), instead of the (1/N) of the quadrahelix.

Acknowledgement

The 3D graphics and symbolic power of Mathematica was essential. The discoveries would have been impossible without it.

Reference: Current (Fall 2017) issue of The Mathematical Intelligencer

Code for Art, Including Print3DChainQuad

uniqueQ[set_, poly_] := Select[set,Sort[#[[1]]]==Sort[poly[[1]]]&]=={poly};
color[p_,c_] := p/. Polygon[z_List, c1___] :> Polygon[z, VertexColors->{c,c,c}];
Chain3DNotClean[L_] :=Graphics3D[polygonsUp[RotateLeft[double@specFormBetter[L],L]]];

Options[Print3DChainQuad]= {PivotColor->White,EdgeStyle->None, ShowFiller->True,FillerOnlyQ->False, Background->Black, FillColor->White, ShiftChain->True, Colors-> {RGBColor[1,.4,.4], RGBColor[.3,1,.3],RGBColor[.3,.3,1], Gray, White }};
Print3DChainQuad[L_, opts___] := (
{shiftQ,esty,back,fillQ,fillOnlyQ, fillcol, cols, pivcol}={ShiftChain,EdgeStyle,Background,ShowFiller,FillerOnlyQ, FillColor, Colors, PivotColor}/. {opts} /. Options[Print3DChainQuad];
pp=Flatten[(pp0=N[polygonsUp[RotateLeft[Quadrahelix[L],If[shiftQ,L,0]]]]/. 
Polygon[z_] :> Polygon/@ z),1] 
(* /. {
{X,x_, y_, z_?NumericQ} /;0<Norm[{x,y,z}-a]<.6 :> (mPrint@z;a),
{X,x_, y_, z_?NumericQ} /;0<Norm[{x,y,z}-b]<.6 :> b,
{X,x_, y_, z_?NumericQ} /;0<Norm[{x,y,z}-c]<.6 :> c}*);
pp = pp /. x_?NumericQ :> Round[10^6 x];
     ppClean=Select[pp, uniqueQ[pp,#]&];
ppClean = ppClean/.  x_?NumericQ :>x/800000.;
col=Append[cols,fillcol] ;
 im=Graphics3D[pp=ppClean;
{
If[fillOnlyQ,{},{
EdgeForm[GrayLevel[.2]],EdgeForm[esty],

{
color[pp[[1;;2L+1]],col[[1]]],
color[pp[[2L+2;;2L+3]],pivcol],
color[pp[[2L+4;;4L+3]],col[[2]]],
color[pp[[4L+4;;6L+3]],col[[3]]],
color[pp[[6L+4;;6L+5]],pivcol],
color[pp[[6L+6;;8L+6]],col[[4]]]
},

j = Mod[L+1,4,1];
inds= j /.{1->{1,1},3->{2,2}, 4-> {3,3}, 2->({2,2};{1,1})};
}],

threeRedFaces=First/@ pp[[{1,2,3}]];threeBlackFaces=First/@ pp[[{-3,-2,-1}]];
mPrint[choices=First@First[SortBy[Table[{t,Norm[Mean[t[[1]]]-Mean[t[[2]]]]},{t,Tuples[{threeRedFaces,threeBlackFaces}]}],Last]]];


(*{{a,b,c}=pp[[inds[[1]],1]];
{a1,b1,c1}=Nearest[pp[[16L+10-2;;16L+10]][[inds[[2]],1]],#][[1]]& /@ {a,b,c};
{a1,b1,c1}=pp[[16L+10-2;;16L+10]][[inds[[2]],1]];
{Red, AbsolutePointSize[10], Red,Point[choices[[1]]],Blue, Point[{a1,b1,c1}; choices[[2]]]};
EdgeForm[],
If[fillQ,zz={
EdgeForm[],
color[#,col[[9]]]&/@ Polygon/@ {{a,b,b1}, {a1, b1, a}},
color[#,col[[9]]]&/@ Polygon/@ {{b,c,c1}, {b1, c1, b}},
color[#,col[[9]]]&/@ Polygon/@ {{c,a,a1}, {c1, a1, c}}},{}]}*)

},
Boxed->False,Axes->!True, AxesLabel->{x,y,"z"},ImageSize->400, PlotRange->All, ViewPoint-> vp[L], ViewVertical->vv[L],
PlotRangePadding->None, Background->back, Lighting->"Neutral"]);

Clear[vp,vv];{vp[_], vv[_]}= {{-1,-2.3,2}, {0 0,1}};
{vp[2], vv[2]}={{-2.436476625192898`,-1.5809907523644915`,1.361635008299429`},{-1.2575593022759337`,-0.5485412269834631`,-0.42982756819051593`}};
{vp[3], vv[3]}= {{2.4953587172326834`,-1.5857876841226364`,-1.2443722486522362`},{0.7010438434352279`,0.14075386215118846`,-0.8871206085133403`}};
{vp[4], vv[4]}={{2.747755021873639`,-0.34500755417750245`,-1.6188922531560468`},{1.4558994856197323`,-0.027887826102026555`,-0.003036588746728503`}};
{vp[5], vv[5]}={{-0.8470694943595101`,-0.699128038337188`,-3.0139165976742497`},{-0.6143738991006478`,-0.6099116349418946`,-0.7482741046064467`}};
{vp[6], vv[6]}={{2.876264615359726`,-1.0392679590580234`,1.4481104832522036`},{0.5485449466481384`,0.46739648637445863`,0.8779234229199029`}};
{vp[7], vv[7]} = {{-2.993845608929946`,-0.9018820134701585`,-0.7165872617273455`},{-1.4547501491926975`,0.1589756191563722`,-0.5779333130297644`}};
{vp[8],vv[8]}= {{1.5693071224552035`,-2.7744810366438486`,0.35990294902243497`},{0.9560932817456527`,-0.40839901700076225`,-0.2836650044213117`}};
{vp[9],vv[9]}= {{1.203956023136`,2.7273273424585858`,1.184134900014493`},{0.21214752807183623`,1.2826217743361785`,-0.20558454205607354`}};
{vp[10], vv[10]}={{2.6459111892023546`,0.9691474528101651`,1.5329407012553835`},{1.335500825338027`,-0.37285831912103706`,0.1879056949327902`}};
{vp[25], vv[25]}={{0.7802285087015227`,0.32958073103844093`,3.093965096108481`},{-0.5877585403943673`,-0.11259378208821114`,1.5823921509273404`}};

{vp[70], vv[70]}={{-1.7850236478169377`,-0.3401899910491482`,-2.6434752404220268`},{1.247625233799548`,-0.5094680168949465`,0.7001126391794809`}};

Print3DChainQuad[9, EdgeStyle->Black, Background->RGBColor[1,1,.6]];

vvv = {ViewPoint, ViewVertical} /. AbsoluteOptions[]

Table[seq=QuadrahelixShiftedNormalized[LL];
ppp=polygonsUp[seq];
(*K=SeqToMatrixLast[seq];*)
{LL,Max[Norm/@( ppp[[1,1,4]]-ppp[[-1,1,4]])]}, {LL, {5,6,7,10}}]

gapErrorQuadrahelixTrueShifted/@ {5,6,7,10}

GraphicsGrid[Partition[Show[
Print3DChainQuad[#, ShowFiller->True,EdgeStyle->{AbsoluteThickness[.8],
Black},ShiftChain->!False,FillColor->RGBColor[1,1,.7],PivotColor->GrayLevel[.4],
Colors->{RGBColor[1,.4,.4], RGBColor[.3,1,.3],RGBColor[.3,.3,1], Lighter@Yellow, White },Background->RGBColor[.8,.8,.8]]]&/@ {5,6,7,10},2],ImageSize-> 350]

Junk

Code for Figure , Three Polyhedron Tori

IcosahedralLoop

zero[n_] := Table[0, {n}];
normal[L_] :=Module[{p1, p2, p3},{p1, p2, p3} = L[[{1,2,3}]]; Cross[(p2-p1), p3-p1]/Norm[Cross[(p2-p1), p3-p1]]]
distance[pt_, plane3_] := Simplify[Abs[normal[plane3]. (pt-plane3[[1]])]]
proj[pt_, plane3_] :=FullSimplify[ pt + distance[pt, plane3] normal[plane3]]
baryProj[X_, pts_] :=(var=x/@ Range[Length@pts];
Simplify@Quiet[var/.First@FindInstance[var.pts ==proj[X,pts]&&Total[var]==1, var]]);

(* this uses verts *) 

matrix[face_] := Table[ 
If[MemberQ[face, i],ReplacePart[ zero[Length[verts]],i->1],
ReplacePart[zero[Length[verts]],(Prepend[Thread[face->2 (baryProj[verts[[i]], verts[[face]]])], i->-1])]], {i,Length[verts]}];
poly=PolyhedronData[name={"Icosahedron", "Tetrahedron", "Cube"}[[1]]];
verts=PolyhedronData[name, "VertexCoordinates"];
faces=PolyhedronData[name, "Faces"][[2,1]];


matrix[face_] := Table[ 
If[MemberQ[face, i],ReplacePart[ zero[Length[verts]],i->1],
rrr=ReplacePart[zero[Length[verts]],(ppp=Prepend[Thread[face->2 (baryProj[verts[[i]], verts[[face]]])], i->-1])]], {i,Length[verts]}];

All reflections for Icosa. (missing some, but I have enough)

matrixFinal[1]={{-1,-(2/3),0,0,0,0,0,(4/3),0,0,0,(4/3)},{0,1,0,0,0,0,0,0,0,0,0,0},{0,(1/3) (1-(Sqrt[5])),-1,0,0,0,0,(2/3) (2+(Sqrt[5])),0,0,0,(1/3) (1-(Sqrt[5]))},{0,(1/3) (3+(Sqrt[5])),0,-1,0,0,0,-((2 (Sqrt[5]))/3),0,0,0,(1/3) (3+(Sqrt[5]))},{0,(4/3),0,0,-1,0,0,-(2/3),0,0,0,(4/3)},{0,(1/3) (1-(Sqrt[5])),0,0,0,-1,0,(1/3) (1-(Sqrt[5])),0,0,0,(2/3) (2+(Sqrt[5]))},{0,(1/3) (3+(Sqrt[5])),0,0,0,0,-1,(1/3) (3+(Sqrt[5])),0,0,0,-((2 (Sqrt[5]))/3)},{0,0,0,0,0,0,0,1,0,0,0,0},{0,(4/3),0,0,0,0,0,(4/3),-1,0,0,-(2/3)},{0,-((2 (Sqrt[5]))/3),0,0,0,0,0,(1/3) (3+(Sqrt[5])),0,-1,0,(1/3) (3+(Sqrt[5]))},{0,(2/3) (2+(Sqrt[5])),0,0,0,0,0,(1/3) (1-(Sqrt[5])),0,0,-1,(1/3) (1-(Sqrt[5]))},{0,0,0,0,0,0,0,0,0,0,0,1}};
matrixFinal[2]={{-1,-(2/3),0,0,0,0,(4/3),(4/3),0,0,0,0},{0,1,0,0,0,0,0,0,0,0,0,0},{0,-((2 (Sqrt[5]))/3),-1,0,0,0,(1/3) (3+(Sqrt[5])),(1/3) (3+(Sqrt[5])),0,0,0,0},{0,(2/3) (2+(Sqrt[5])),0,-1,0,0,(1/3) (1-(Sqrt[5])),(1/3) (1-(Sqrt[5])),0,0,0,0},{0,(4/3),0,0,-1,0,(4/3),-(2/3),0,0,0,0},{0,(4/3),0,0,0,-1,-(2/3),(4/3),0,0,0,0},{0,0,0,0,0,0,1,0,0,0,0,0},{0,0,0,0,0,0,0,1,0,0,0,0},{0,(1/3) (1-(Sqrt[5])),0,0,0,0,(2/3) (2+(Sqrt[5])),(1/3) (1-(Sqrt[5])),-1,0,0,0},{0,(1/3) (1-(Sqrt[5])),0,0,0,0,(1/3) (1-(Sqrt[5])),(2/3) (2+(Sqrt[5])),0,-1,0,0},{0,(1/3) (3+(Sqrt[5])),0,0,0,0,(1/3) (3+(Sqrt[5])),-((2 (Sqrt[5]))/3),0,0,-1,0},{0,(1/3) (3+(Sqrt[5])),0,0,0,0,-((2 (Sqrt[5]))/3),(1/3) (3+(Sqrt[5])),0,0,0,-1}};

matrixFinal[3] = {{-1,-(2/3),0,0,0,0,(4/3),0,0,0,(4/3),0},{0,1,0,0,0,0,0,0,0,0,0,0},{0,(1/3) (1-(Sqrt[5])),-1,0,0,0,(2/3) (2+(Sqrt[5])),0,0,0,(1/3) (1-(Sqrt[5])),0},{0,(1/3) (3+(Sqrt[5])),0,-1,0,0,-((2 (Sqrt[5]))/3),0,0,0,(1/3) (3+(Sqrt[5])),0},{0,(1/3) (1-(Sqrt[5])),0,0,-1,0,(1/3) (1-(Sqrt[5])),0,0,0,(2/3) (2+(Sqrt[5])),0},{0,(4/3),0,0,0,-1,-(2/3),0,0,0,(4/3),0},{0,0,0,0,0,0,1,0,0,0,0,0},{0,(1/3) (3+(Sqrt[5])),0,0,0,0,(1/3) (3+(Sqrt[5])),-1,0,0,-((2 (Sqrt[5]))/3),0},{0,-((2 (Sqrt[5]))/3),0,0,0,0,(1/3) (3+(Sqrt[5])),0,-1,0,(1/3) (3+(Sqrt[5])),0},{0,(4/3),0,0,0,0,(4/3),0,0,-1,-(2/3),0},{0,0,0,0,0,0,0,0,0,0,1,0},{0,(2/3) (2+(Sqrt[5])),0,0,0,0,(1/3) (1-(Sqrt[5])),0,0,0,(1/3) (1-(Sqrt[5])),-1}};
matrixFinal[4] = {{-1,-(2/3),0,(4/3),0,0,0,0,0,0,(4/3),0},{0,1,0,0,0,0,0,0,0,0,0,0},{0,(4/3),-1,-(2/3),0,0,0,0,0,0,(4/3),0},{0,0,0,1,0,0,0,0,0,0,0,0},{0,-((2 (Sqrt[5]))/3),0,(1/3) (3+(Sqrt[5])),-1,0,0,0,0,0,(1/3) (3+(Sqrt[5])),0},{0,(1/3) (1-(Sqrt[5])),0,(2/3) (2+(Sqrt[5])),0,-1,0,0,0,0,(1/3) (1-(Sqrt[5])),0},{0,(1/3) (3+(Sqrt[5])),0,-((2 (Sqrt[5]))/3),0,0,-1,0,0,0,(1/3) (3+(Sqrt[5])),0},{0,(2/3) (2+(Sqrt[5])),0,(1/3) (1-(Sqrt[5])),0,0,0,-1,0,0,(1/3) (1-(Sqrt[5])),0},{0,(1/3) (1-(Sqrt[5])),0,(1/3) (1-(Sqrt[5])),0,0,0,0,-1,0,(2/3) (2+(Sqrt[5])),0},{0,(4/3),0,(4/3),0,0,0,0,0,-1,-(2/3),0},{0,0,0,0,0,0,0,0,0,0,1,0},{0,(1/3) (3+(Sqrt[5])),0,(1/3) (3+(Sqrt[5])),0,0,0,0,0,0,-((2 (Sqrt[5]))/3),-1}};

matrixFinal[5] = {{-1,-(2/3),0,(4/3),0,0,0,0,0,0,0,(4/3)},{0,1,0,0,0,0,0,0,0,0,0,0},{0,(4/3),-1,-(2/3),0,0,0,0,0,0,0,(4/3)},{0,0,0,1,0,0,0,0,0,0,0,0},{0,(1/3) (1-(Sqrt[5])),0,(2/3) (2+(Sqrt[5])),-1,0,0,0,0,0,0,(1/3) (1-(Sqrt[5]))},{0,-((2 (Sqrt[5]))/3),0,(1/3) (3+(Sqrt[5])),0,-1,0,0,0,0,0,(1/3) (3+(Sqrt[5]))},{0,(2/3) (2+(Sqrt[5])),0,(1/3) (1-(Sqrt[5])),0,0,-1,0,0,0,0,(1/3) (1-(Sqrt[5]))},{0,(1/3) (3+(Sqrt[5])),0,-((2 (Sqrt[5]))/3),0,0,0,-1,0,0,0,(1/3) (3+(Sqrt[5]))},{0,(4/3),0,(4/3),0,0,0,0,-1,0,0,-(2/3)},{0,(1/3) (1-(Sqrt[5])),0,(1/3) (1-(Sqrt[5])),0,0,0,0,0,-1,0,(2/3) (2+(Sqrt[5]))},{0,(1/3) (3+(Sqrt[5])),0,(1/3) (3+(Sqrt[5])),0,0,0,0,0,0,-1,-((2 (Sqrt[5]))/3)},{0,0,0,0,0,0,0,0,0,0,0,1}};

matrixFinal[6]={{1,0,0,0,0,0,0,0,0,0,0,0},{-(2/3),-1,0,0,(4/3),0,0,0,(4/3),0,0,0},{(1/3) (3+(Sqrt[5])),0,-1,0,-((2 (Sqrt[5]))/3),0,0,0,(1/3) (3+(Sqrt[5])),0,0,0},{(1/3) (1-(Sqrt[5])),0,0,-1,(2/3) (2+(Sqrt[5])),0,0,0,(1/3) (1-(Sqrt[5])),0,0,0},{0,0,0,0,1,0,0,0,0,0,0,0},{(1/3) (3+(Sqrt[5])),0,0,0,(1/3) (3+(Sqrt[5])),-1,0,0,-((2 (Sqrt[5]))/3),0,0,0},{(1/3) (1-(Sqrt[5])),0,0,0,(1/3) (1-(Sqrt[5])),0,-1,0,(2/3) (2+(Sqrt[5])),0,0,0},{(4/3),0,0,0,-(2/3),0,0,-1,(4/3),0,0,0},{0,0,0,0,0,0,0,0,1,0,0,0},{(2/3) (2+(Sqrt[5])),0,0,0,(1/3) (1-(Sqrt[5])),0,0,0,(1/3) (1-(Sqrt[5])),-1,0,0},{-((2 (Sqrt[5]))/3),0,0,0,(1/3) (3+(Sqrt[5])),0,0,0,(1/3) (3+(Sqrt[5])),0,-1,0},{(4/3),0,0,0,(4/3),0,0,0,-(2/3),0,0,-1}};
matrixFinal[7]={{1,0,0,0,0,0,0,0,0,0,0,0},{-(2/3),-1,0,0,(4/3),(4/3),0,0,0,0,0,0},{(2/3) (2+(Sqrt[5])),0,-1,0,(1/3) (1-(Sqrt[5])),(1/3) (1-(Sqrt[5])),0,0,0,0,0,0},{-((2 (Sqrt[5]))/3),0,0,-1,(1/3) (3+(Sqrt[5])),(1/3) (3+(Sqrt[5])),0,0,0,0,0,0},{0,0,0,0,1,0,0,0,0,0,0,0},{0,0,0,0,0,1,0,0,0,0,0,0},{(4/3),0,0,0,(4/3),-(2/3),-1,0,0,0,0,0},{(4/3),0,0,0,-(2/3),(4/3),0,-1,0,0,0,0},{(1/3) (3+(Sqrt[5])),0,0,0,(1/3) (3+(Sqrt[5])),-((2 (Sqrt[5]))/3),0,0,-1,0,0,0},{(1/3) (3+(Sqrt[5])),0,0,0,-((2 (Sqrt[5]))/3),(1/3) (3+(Sqrt[5])),0,0,0,-1,0,0},{(1/3) (1-(Sqrt[5])),0,0,0,(2/3) (2+(Sqrt[5])),(1/3) (1-(Sqrt[5])),0,0,0,0,-1,0},{(1/3) (1-(Sqrt[5])),0,0,0,(1/3) (1-(Sqrt[5])),(2/3) (2+(Sqrt[5])),0,0,0,0,0,-1}};
matrixFinal[8]={{1,0,0,0,0,0,0,0,0,0,0,0},{-(2/3),-1,0,0,0,(4/3),0,0,0,(4/3),0,0},{(1/3) (3+(Sqrt[5])),0,-1,0,0,-((2 (Sqrt[5]))/3),0,0,0,(1/3) (3+(Sqrt[5])),0,0},{(1/3) (1-(Sqrt[5])),0,0,-1,0,(2/3) (2+(Sqrt[5])),0,0,0,(1/3) (1-(Sqrt[5])),0,0},{(1/3) (3+(Sqrt[5])),0,0,0,-1,(1/3) (3+(Sqrt[5])),0,0,0,-((2 (Sqrt[5]))/3),0,0},{0,0,0,0,0,1,0,0,0,0,0,0},{(4/3),0,0,0,0,-(2/3),-1,0,0,(4/3),0,0},{(1/3) (1-(Sqrt[5])),0,0,0,0,(1/3) (1-(Sqrt[5])),0,-1,0,(2/3) (2+(Sqrt[5])),0,0},{(2/3) (2+(Sqrt[5])),0,0,0,0,(1/3) (1-(Sqrt[5])),0,0,-1,(1/3) (1-(Sqrt[5])),0,0},{0,0,0,0,0,0,0,0,0,1,0,0},{(4/3),0,0,0,0,(4/3),0,0,0,-(2/3),-1,0},{-((2 (Sqrt[5]))/3),0,0,0,0,(1/3) (3+(Sqrt[5])),0,0,0,(1/3) (3+(Sqrt[5])),0,-1}};
matrixFinal[9]={{1,0,0,0,0,0,0,0,0,0,0,0},{-(2/3),-1,(4/3),0,0,0,0,0,0,(4/3),0,0},{0,0,1,0,0,0,0,0,0,0,0,0},{(4/3),0,-(2/3),-1,0,0,0,0,0,(4/3),0,0},{(2/3) (2+(Sqrt[5])),0,(1/3) (1-(Sqrt[5])),0,-1,0,0,0,0,(1/3) (1-(Sqrt[5])),0,0},{(1/3) (3+(Sqrt[5])),0,-((2 (Sqrt[5]))/3),0,0,-1,0,0,0,(1/3) (3+(Sqrt[5])),0,0},{(1/3) (1-(Sqrt[5])),0,(2/3) (2+(Sqrt[5])),0,0,0,-1,0,0,(1/3) (1-(Sqrt[5])),0,0},{-((2 (Sqrt[5]))/3),0,(1/3) (3+(Sqrt[5])),0,0,0,0,-1,0,(1/3) (3+(Sqrt[5])),0,0},{(1/3) (3+(Sqrt[5])),0,(1/3) (3+(Sqrt[5])),0,0,0,0,0,-1,-((2 (Sqrt[5]))/3),0,0},{0,0,0,0,0,0,0,0,0,1,0,0},{(4/3),0,(4/3),0,0,0,0,0,0,-(2/3),-1,0},{(1/3) (1-(Sqrt[5])),0,(1/3) (1-(Sqrt[5])),0,0,0,0,0,0,(2/3) (2+(Sqrt[5])),0,-1}};
matrixFinal[10]={{1,0,0,0,0,0,0,0,0,0,0,0},{-(2/3),-1,(4/3),0,0,0,0,0,(4/3),0,0,0},{0,0,1,0,0,0,0,0,0,0,0,0},{(4/3),0,-(2/3),-1,0,0,0,0,(4/3),0,0,0},{(1/3) (3+(Sqrt[5])),0,-((2 (Sqrt[5]))/3),0,-1,0,0,0,(1/3) (3+(Sqrt[5])),0,0,0},{(2/3) (2+(Sqrt[5])),0,(1/3) (1-(Sqrt[5])),0,0,-1,0,0,(1/3) (1-(Sqrt[5])),0,0,0},{-((2 (Sqrt[5]))/3),0,(1/3) (3+(Sqrt[5])),0,0,0,-1,0,(1/3) (3+(Sqrt[5])),0,0,0},{(1/3) (1-(Sqrt[5])),0,(2/3) (2+(Sqrt[5])),0,0,0,0,-1,(1/3) (1-(Sqrt[5])),0,0,0},{0,0,0,0,0,0,0,0,1,0,0,0},{(1/3) (3+(Sqrt[5])),0,(1/3) (3+(Sqrt[5])),0,0,0,0,0,-((2 (Sqrt[5]))/3),-1,0,0},{(1/3) (1-(Sqrt[5])),0,(1/3) (1-(Sqrt[5])),0,0,0,0,0,(2/3) (2+(Sqrt[5])),0,-1,0},{(4/3),0,(4/3),0,0,0,0,0,-(2/3),0,0,-1}};
matrixFinal[11]=matTest11;
matrixFinal[12]={{-1,0,(2/3) (2+(Sqrt[5])),0,0,0,(1/3) (1-(Sqrt[5])),(1/3) (1-(Sqrt[5])),0,0,0,0},{0,-1,-((2 (Sqrt[5]))/3),0,0,0,(1/3) (3+(Sqrt[5])),(1/3) (3+(Sqrt[5])),0,0,0,0},{0,0,1,0,0,0,0,0,0,0,0,0},{0,0,-(2/3),-1,0,0,(4/3),(4/3),0,0,0,0},{0,0,(4/3),0,-1,0,(4/3),-(2/3),0,0,0,0},{0,0,(4/3),0,0,-1,-(2/3),(4/3),0,0,0,0},{0,0,0,0,0,0,1,0,0,0,0,0},{0,0,0,0,0,0,0,1,0,0,0,0},{0,0,(1/3) (3+(Sqrt[5])),0,0,0,(1/3) (3+(Sqrt[5])),-((2 (Sqrt[5]))/3),-1,0,0,0},{0,0,(1/3) (3+(Sqrt[5])),0,0,0,-((2 (Sqrt[5]))/3),(1/3) (3+(Sqrt[5])),0,-1,0,0},{0,0,(1/3) (1-(Sqrt[5])),0,0,0,(2/3) (2+(Sqrt[5])),(1/3) (1-(Sqrt[5])),0,0,-1,0},{0,0,(1/3) (1-(Sqrt[5])),0,0,0,(1/3) (1-(Sqrt[5])),(2/3) (2+(Sqrt[5])),0,0,0,-1}};
matrixFinal[13]=matTest13;
matrixFinal[14]={{-1,0,0,(1/3) (1-(Sqrt[5])),(2/3) (2+(Sqrt[5])),0,0,0,0,0,(1/3) (1-(Sqrt[5])),0},{0,-1,0,(1/3) (3+(Sqrt[5])),-((2 (Sqrt[5]))/3),0,0,0,0,0,(1/3) (3+(Sqrt[5])),0},{0,0,-1,-(2/3),(4/3),0,0,0,0,0,(4/3),0},{0,0,0,1,0,0,0,0,0,0,0,0},{0,0,0,0,1,0,0,0,0,0,0,0},{0,0,0,(1/3) (3+(Sqrt[5])),(1/3) (3+(Sqrt[5])),-1,0,0,0,0,-((2 (Sqrt[5]))/3),0},{0,0,0,(1/3) (1-(Sqrt[5])),(1/3) (1-(Sqrt[5])),0,-1,0,0,0,(2/3) (2+(Sqrt[5])),0},{0,0,0,(4/3),-(2/3),0,0,-1,0,0,(4/3),0},{0,0,0,-((2 (Sqrt[5]))/3),(1/3) (3+(Sqrt[5])),0,0,0,-1,0,(1/3) (3+(Sqrt[5])),0},{0,0,0,(4/3),(4/3),0,0,0,0,-1,-(2/3),0},{0,0,0,0,0,0,0,0,0,0,1,0},{0,0,0,(2/3) (2+(Sqrt[5])),(1/3) (1-(Sqrt[5])),0,0,0,0,0,(1/3) (1-(Sqrt[5])),-1}};
matrixFinal[15]={{-1,0,0,(1/3) (1-(Sqrt[5])),0,(2/3) (2+(Sqrt[5])),0,0,0,0,0,(1/3) (1-(Sqrt[5]))},{0,-1,0,(1/3) (3+(Sqrt[5])),0,-((2 (Sqrt[5]))/3),0,0,0,0,0,(1/3) (3+(Sqrt[5]))},{0,0,-1,-(2/3),0,(4/3),0,0,0,0,0,(4/3)},{0,0,0,1,0,0,0,0,0,0,0,0},{0,0,0,(1/3) (3+(Sqrt[5])),-1,(1/3) (3+(Sqrt[5])),0,0,0,0,0,-((2 (Sqrt[5]))/3)},{0,0,0,0,0,1,0,0,0,0,0,0},{0,0,0,(4/3),0,-(2/3),-1,0,0,0,0,(4/3)},{0,0,0,(1/3) (1-(Sqrt[5])),0,(1/3) (1-(Sqrt[5])),0,-1,0,0,0,(2/3) (2+(Sqrt[5]))},{0,0,0,(4/3),0,(4/3),0,0,-1,0,0,-(2/3)},{0,0,0,-((2 (Sqrt[5]))/3),0,(1/3) (3+(Sqrt[5])),0,0,0,-1,0,(1/3) (3+(Sqrt[5]))},{0,0,0,(2/3) (2+(Sqrt[5])),0,(1/3) (1-(Sqrt[5])),0,0,0,0,-1,(1/3) (1-(Sqrt[5]))},{0,0,0,0,0,0,0,0,0,0,0,1}};
matrixFinal[16]={{-1,0,0,0,(1/3) (3+(Sqrt[5])),0,0,0,(1/3) (3+(Sqrt[5])),0,-((2 (Sqrt[5]))/3),0},{0,-1,0,0,(1/3) (1-(Sqrt[5])),0,0,0,(1/3) (1-(Sqrt[5])),0,(2/3) (2+(Sqrt[5])),0},{0,0,-1,0,(1/3) (1-(Sqrt[5])),0,0,0,(2/3) (2+(Sqrt[5])),0,(1/3) (1-(Sqrt[5])),0},{0,0,0,-1,(1/3) (3+(Sqrt[5])),0,0,0,-((2 (Sqrt[5]))/3),0,(1/3) (3+(Sqrt[5])),0},{0,0,0,0,1,0,0,0,0,0,0,0},{0,0,0,0,(2/3) (2+(Sqrt[5])),-1,0,0,(1/3) (1-(Sqrt[5])),0,(1/3) (1-(Sqrt[5])),0},{0,0,0,0,-((2 (Sqrt[5]))/3),0,-1,0,(1/3) (3+(Sqrt[5])),0,(1/3) (3+(Sqrt[5])),0},{0,0,0,0,-(2/3),0,0,-1,(4/3),0,(4/3),0},{0,0,0,0,0,0,0,0,1,0,0,0},{0,0,0,0,(4/3),0,0,0,(4/3),-1,-(2/3),0},{0,0,0,0,0,0,0,0,0,0,1,0},{0,0,0,0,(4/3),0,0,0,-(2/3),0,(4/3),-1}};
matrixFinal[17]={{-1,0,0,-((2 (Sqrt[5]))/3),(1/3) (3+(Sqrt[5])),(1/3) (3+(Sqrt[5])),0,0,0,0,0,0},{0,-1,0,(2/3) (2+(Sqrt[5])),(1/3) (1-(Sqrt[5])),(1/3) (1-(Sqrt[5])),0,0,0,0,0,0},{0,0,-1,-(2/3),(4/3),(4/3),0,0,0,0,0,0},{0,0,0,1,0,0,0,0,0,0,0,0},{0,0,0,0,1,0,0,0,0,0,0,0},{0,0,0,0,0,1,0,0,0,0,0,0},{0,0,0,(4/3),(4/3),-(2/3),-1,0,0,0,0,0},{0,0,0,(4/3),-(2/3),(4/3),0,-1,0,0,0,0},{0,0,0,(1/3) (1-(Sqrt[5])),(2/3) (2+(Sqrt[5])),(1/3) (1-(Sqrt[5])),0,0,-1,0,0,0},{0,0,0,(1/3) (1-(Sqrt[5])),(1/3) (1-(Sqrt[5])),(2/3) (2+(Sqrt[5])),0,0,0,-1,0,0},{0,0,0,(1/3) (3+(Sqrt[5])),(1/3) (3+(Sqrt[5])),-((2 (Sqrt[5]))/3),0,0,0,0,-1,0},{0,0,0,(1/3) (3+(Sqrt[5])),-((2 (Sqrt[5]))/3),(1/3) (3+(Sqrt[5])),0,0,0,0,0,-1}};
matrixFinal[18]=matTest18;
matrixFinal[19]={{-1,0,(1/3) (3+(Sqrt[5])),0,0,0,0,-((2 (Sqrt[5]))/3),0,(1/3) (3+(Sqrt[5])),0,0},{0,-1,(1/3) (1-(Sqrt[5])),0,0,0,0,(2/3) (2+(Sqrt[5])),0,(1/3) (1-(Sqrt[5])),0,0},{0,0,1,0,0,0,0,0,0,0,0,0},{0,0,-(2/3),-1,0,0,0,(4/3),0,(4/3),0,0},{0,0,(4/3),0,-1,0,0,-(2/3),0,(4/3),0,0},{0,0,(1/3) (1-(Sqrt[5])),0,0,-1,0,(1/3) (1-(Sqrt[5])),0,(2/3) (2+(Sqrt[5])),0,0},{0,0,(1/3) (3+(Sqrt[5])),0,0,0,-1,(1/3) (3+(Sqrt[5])),0,-((2 (Sqrt[5]))/3),0,0},{0,0,0,0,0,0,0,1,0,0,0,0},{0,0,(2/3) (2+(Sqrt[5])),0,0,0,0,(1/3) (1-(Sqrt[5])),-1,(1/3) (1-(Sqrt[5])),0,0},{0,0,0,0,0,0,0,0,0,1,0,0},{0,0,(4/3),0,0,0,0,(4/3),0,-(2/3),-1,0},{0,0,-((2 (Sqrt[5]))/3),0,0,0,0,(1/3) (3+(Sqrt[5])),0,(1/3) (3+(Sqrt[5])),0,-1}};
matrixFinal[20]={{-1,0,(1/3) (3+(Sqrt[5])),0,0,0,-((2 (Sqrt[5]))/3),0,(1/3) (3+(Sqrt[5])),0,0,0},{0,-1,(1/3) (1-(Sqrt[5])),0,0,0,(2/3) (2+(Sqrt[5])),0,(1/3) (1-(Sqrt[5])),0,0,0},{0,0,1,0,0,0,0,0,0,0,0,0},{0,0,-(2/3),-1,0,0,(4/3),0,(4/3),0,0,0},{0,0,(1/3) (1-(Sqrt[5])),0,-1,0,(1/3) (1-(Sqrt[5])),0,(2/3) (2+(Sqrt[5])),0,0,0},{0,0,(4/3),0,0,-1,-(2/3),0,(4/3),0,0,0},{0,0,0,0,0,0,1,0,0,0,0,0},{0,0,(1/3) (3+(Sqrt[5])),0,0,0,(1/3) (3+(Sqrt[5])),-1,-((2 (Sqrt[5]))/3),0,0,0},{0,0,0,0,0,0,0,0,1,0,0,0},{0,0,(2/3) (2+(Sqrt[5])),0,0,0,(1/3) (1-(Sqrt[5])),0,(1/3) (1-(Sqrt[5])),-1,0,0},{0,0,-((2 (Sqrt[5]))/3),0,0,0,(1/3) (3+(Sqrt[5])),0,(1/3) (3+(Sqrt[5])),0,-1,0},{0,0,(4/3),0,0,0,(4/3),0,-(2/3),0,0,-1}};

icosaLoop=Show[poly,Graphics3D[{Table[Text[Style[i,30], verts[[i]]], {i,8}];PointSize[.02],  
Blue, GraphicsComplex[matrixFinal[5].verts, poly[[1,2]]],
Green,GraphicsComplex[matrixFinal[10].matrixFinal[5].verts, poly[[1,2]]],
Orange, GraphicsComplex[matrixFinal[17].matrixFinal[10].matrixFinal[5].verts, poly[[1,2]]],
Orange, GraphicsComplex[matrixFinal[12].matrixFinal[17].matrixFinal[10].matrixFinal[5].verts, poly[[1,2]]],
{Cyan, GraphicsComplex[matrixFinal[10]. matrixFinal[12].matrixFinal[17].matrixFinal[10].matrixFinal[5].verts, poly[[1,2]]]},
{Magenta, GraphicsComplex[matrixFinal[5].matrixFinal[10].matrixFinal[12].matrixFinal[17].matrixFinal[10].matrixFinal[5].verts, poly[[1,2]]]},
{Magenta, GraphicsComplex[matrixFinal[12].matrixFinal[5].matrixFinal[10].matrixFinal[12].matrixFinal[17].matrixFinal[10].matrixFinal[5].verts, poly[[1,2]]]},
Thickness[.02],Red, Line[verts[[faces[[1]]]]];{}}], AxesLabel->{x,y,z},
ViewVertical->{0,0,1},PlotRange->{{-1,5}, {-2,2}, {-1,5}}, Axes->True, ImageSize-> 700, ViewPoint->{0,-3,0}];

icosaLoop1=N@icosaLoop;

{vv,vp}= {ViewVertical, ViewPoint}/. AbsoluteOptions[]

icosa=Graphics3D[{Red,icosaLoop1[[1,1]],
Blue,icosaLoop1[[1,2,3]],
Red,icosaLoop1[[1,2,5]],Blue,icosaLoop1[[1,2,7]],Red,icosaLoop1[[1,2,9]],
Blue,icosaLoop1[[1,2,10,2]],Red,icosaLoop1[[1,2,11,2]],Blue,icosaLoop1[[1,2,12,2]]},
PlotRange->All, Axes->!True,Boxed->False,  Lighting->"Neutral",
 ViewVertical->{-0.4276282867508321`,0.47780414073640065`,1.1111439069852098`}, ViewPoint->{0.5657724285523273`,-2.696410540071817`,1.1871275241016068`},ImageSize->300,PlotRegion->{{-.05, 1.05}, {-.2, 1.16}}]/. {Red->Lighter@Red, Blue->Lighter@Lighter@Blue}

Code for Dodecahedral Torus

{dx,dy}={1/10 (5+3 Sqrt[5]),1+2/Sqrt[5]};
θ=2 ArcCos[Sqrt[1/10 (5+Sqrt[5])]];
d1=Normal[PolyhedronData[{"Dodecahedron","C2"},"Faces"]];
d2=d1/.Polygon[x_]:>Table[Polygon[{(j+2) dx,i dy,0}+RotationMatrix[i j θ,{0,0,1}].#&/@x],{i,-1,1,2},{j,-1,1,2}];
d3=d1/.Polygon[x_]:>Table[Polygon[2 {dx,i dy,0}+#&/@x],{i,-1,1,2}];
d4=d1/.Polygon[x_]:>Polygon[{4 dx,0,0}+#&/@x];
imDodec=Graphics3D[{Specularity[Blue,10];ffb=FaceForm[Lighter@Lighter@Blue],d1,Lighter@Red,d2,ffb,d3,ffb,d4}, Lighting->"Neutral", Boxed->False, ViewAngle->.4815, ImageSize-> 300,ViewPoint->{0.08742465194189818`,-0.07184018515843538`,-2.997865226795434`},ViewVertical->{1.3928363543030613`,-0.06476881560904248`,0.11932312783878982`},PlotRegion->{{-.22, 1.25},{-.4, 1.4}}];

{vp,vv}={ViewPoint, ViewVertical}/. AbsoluteOptions[];

ii=Show[imDodec, Boxed->False, Axes->False, ImageSize->200]

Octahedral Loop

p0=PolyhedronData["Octahedron"][[1]];
v=PolyhedronData["Octahedron", "Faces"];
p1=GraphicsComplex[p0[[1]]/. {x_, y_, z_} :> RotationMatrix[π/4, {0,0,1}].({x,y,z} ), p0[[2]]];
verts=p1[[1]];
vec=Mean[verts[[{3,5,6}]]];
vec1=Mean[verts[[{3,1,2}]]];
rot = RotationMatrix[π/3,vec];
rot1 = RotationMatrix[π/3,vec{-1,1,1}];
ww=N[2 Sqrt[2]/3,20];

imOct=Graphics3D[{{PointSize[.02], Thickness[.01],Line[verts[[{6,3,5}]]];
Line[verts[[{1,2,3}]]]};
xxx={
{Red,p1}, 
{Blue,p2=GraphicsComplex[p1[[1]]/. {x_, y_, z_} :> rot.({x,y,z} + 2 vec), p1[[2]]]},
{Red,p3= GraphicsComplex[p1[[1]]/. {x_, y_, z_} :> rot.rot.({x,y,z} +4 vec), p1[[2]]]},
{Blue,p4=GraphicsComplex[p1[[1]]/. {x_, y_, z_} :> rot1.({x,y,z} +2 vec1), p1[[2]]]},
{Red, p5=GraphicsComplex[p1[[1]]/. {x_, y_, z_} :> rot1.rot1.({x,y,z} +4 vec1), p1[[2]]]},
{Red,p6=GraphicsComplex[p1[[1]]/. {x_, y_, z_} :> {x,y,z} +{0,0,-2(ww )}, p1[[2]]]},
{Yellow,FaceForm@Blue,p7=GraphicsComplex[p2[[1]]/. {x_, y_, z_} :> ({x,y,-2 ww -z}) , p2[[2]]]},

{Yellow,FaceForm@Blue,p8=GraphicsComplex[p4[[1]]/. {x_, y_, z_} :> ({x,y,-2 ww -z}) , p4[[2]]]}}}
, Axes->Automatic, Boxed->True,AxesLabel->{x,y,z},ImageSize->600, PlotRange->({{-2,2}, {-.8,.8}, {-2.7,.8}}),ViewPoint->vp, ViewVertical->vv,Lighting->"Neutral"]/. Blue->Lighter@Lighter@Blue/. Red->Lighter@Red;
GraphicsRow[{
Show[imOct, Boxed->False, Axes->False],Show[imOct, Boxed->False, Axes->False]}, ImageSize->700, Spacings->-20];

Both

GraphicsRow[{Graphics3D[{{Lighter@Red,Cuboid/@ {{0,0,0}, {1,0,1}, {0,0,2}, {-1,0,1}}},
{Lighter@Blue,Cuboid/@ {{1,0,0}, {1,0,2}, {-1,0,2}, {-1,0,0}}}}, Lighting->"Neutral", Boxed->False, ImageSize->100, PlotRangePadding->.4],
Show[imOct,Boxed->False,Axes->False,ViewPoint->( ViewPoint/.AbsoluteOptions[]),ViewVertical->( ViewVertical/.AbsoluteOptions[])],imDodec, icosa}, ImageSize->1100]

Code for Figure TH

Lval=16;ht=(Lval+2)h;
im1=Show[Show[Graphics3D[{pp=polygonsUp[Take[specFormBetter[Lval],Lval]];
EdgeForm[{Opacity[1],AbsoluteThickness[1.5],Black}],

{Opacity[.8], {Red, pp[[1]], Green, pp[[2]], Blue,pp[[3]], Yellow, pp[[4]],
Red, pp[[5]], Green, pp[[6]], Blue,pp[[7]], Yellow, pp[[8]],
Red, pp[[9]], Green, pp[[10]], Blue,pp[[11]], Yellow, pp[[12]],
Red, pp[[13]], Green, pp[[14]], Blue,pp[[15]], Yellow, pp[[16]]}}, AbsolutePointSize[10], {RGBColor[.4,1,1],Sphere[#,.05 1.5]&/@ TH[Lval]},
{AbsoluteThickness[1], Black, Opacity[1], Line[{  {{0, r,0},{0, -r, 0}},{{r,0, 0},{-r,0, 0}}}],
Line[{  {{0, r,ht},{0, -r, ht}},{{r,0, ht},{-r,0, ht}}}];{}},
AbsolutePointSize[6],Point[{0,0,0}]}]],
ParametricPlot3D[{r Cos[θ], r Sin[θ], 0}, {θ,0, 2π},PlotStyle->{Black,AbsoluteThickness[2]}],
ParametricPlot3D[{r Cos[θ], r Sin[θ], ht}, {θ,0, 2π},PlotStyle->{Black,AbsoluteThickness[2]}];
Graphics3D[{Magenta,Tube[Table[{r Cos[L θ], r Sin[L θ], h L}, {L,0, Lval+2, .1}],.03]}],
ParametricPlot3D[{r Cos[θ], r Sin[θ], z}, {θ,0, 2π}, {z,0,(2+ Lval) h},
PlotStyle->{Blue,Opacity[.03]}, Mesh-> {8,Lval+1}];
Boxed->False, Lighting->"Neutral", ViewVertical->{0,0,1}, ViewPoint->{1,-2,1},ImageSize->140, PlotRange-> {{-.6,.6}, {-.6,.6}, {-.1,ht 1.025}}]/. Red->RGBColor[1,.4,.4]/. Green -> RGBColor[.3,1,.3]/.Blue -> RGBColor[.3,.3,1]/.Yellow -> RGBColor[1,1,.4];
θθ=π/4+.65;im2=Show[im1, ViewPoint->{1  Cos[θθ],-2 Sin[θθ],.3}]

Lval=253;
Show[Graphics3D[pp=polygonsUp[QuadrahelixShifted[Lval]];
{Opacity[1] ,EdgeForm[],{Red,pp[[1;;2]],
{Yellow,pp[[4Lval+1;;4Lval+2]]}}}] , Boxed->False,ImageSize->1100,PlotRangePadding->0,
Background->Black,Lighting->"Neutral",PlotLabel->Style["gap is 1/2000th of a side", White,24]]

The Quadrahelix Gap Approaches 0   DELE

We need to show that the parallelism approaches perfection. This logarithmic chart shows that parallelism gets better as L increases.

Show[ListPlot[data=Table[{Lval,Log10[ArcSin[(1/5) (2 (Sqrt[6])) ((Sin[(1/2) (Lval+1) θ])^2)]]},{Lval,4,2030;20030;2030}],Frame->True, PlotRange->{-5.9,0}, BaseStyle-> 16,PlotLabel->"(Subscript[log, 10]) angle between ceiling and floor, radians", FrameLabel->{"L"," "}],
Graphics[{Red, AbsolutePointSize[11], Point[d1=Select[data, #[[2]]<-4.2&]]}],
GridLines->{Join[{10},First/@ d1],{}}, FrameTicks-> {{{-#,10^-ToString@#}& /@ Range[0,10]/. 10^-"0"->1,{-#,10^-ToString@#}& /@ Range[0,10]/. 10^-"0"->1},{First/@ d1, {}}},
PlotRangePadding->{10,{.2,0.1}}, AspectRatio-> .5, ImageSize->1350, FrameTicksStyle->18,PlotRange-> All, BaseStyle->24]

lens={48,60,30,36, 68, 60, 84, 92, 174, 116, 124, 132, 140, 108, 174, 148, 164, 540};
errs={.186, .165, .015, .00027, .00021, 1.09 10^-5, 1.31 10^-6, 1.02 10^-6,6.22 10^-7, 5.09 10^-11, 2.43 10^-12, 1.12 10^-12, 8.6 10^-13, 8.16 10^-13, 1.23 10^-13 ,7.4 10^-14, 1.05 10^-14, 5. 10^-18};
data=Transpose[{lens, errs}];
Show[ListPlot[data/. {x_,y_} :> {x, Log10@y},PlotRange->All, Frame->True], Plot[-.11 x, {x,1, 550}, PlotStyle->Red], FrameLabel-> {"Number of tetrahedra", "log10 error"}]

Logerror= -x .11
error=10^( -x  .11)
error=10^(.11) ^( -x)
error=(1.288^(-x))

((((1/1.288)))^x)

cache182=Print3DChainQuad[182, Colors-> {Red, Green, Lighter@Lighter@Blue, Lighter@Yellow}, EdgeStyle->{AbsoluteThickness[.5], Black}];

Lval=13;
Show[Print3DChainQuad[Lval, Colors-> {Red, Green, Lighter@Lighter@Blue, Lighter@Yellow}, EdgeStyle->{AbsoluteThickness[.5], Black}],ImageSize->600, PlotRangePadding->0,
ViewPoint-> vp[Lval], ViewVertical->vv[Lval]]

{ViewPoint, ViewVertical}/. AbsoluteOptions[]

Cached 182 and 70

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       {RGBColor[1, 1, 1/3], RGBColor[1, 1, 1/3], RGBColor[1, 1, 1/3]}], 
     Polygon[{{-1.5948062500000002, 0.85726, 2.1015550000000003}, {-1.9761062500000002, -0.19926125000000003, 
        1.5530375}, {-2.5802300000000002, 0.23723375000000002, 2.5565350000000002}}, 
      VertexColors -> {RGBColor[1, 1, 1/3], RGBColor[1, 1, 1/3], RGBColor[1, 1, 1/3]}], 
     Polygon[{{-2.5802300000000002, 0.23723375000000002, 2.5565350000000002}, 
       {-1.9761062500000002, -0.19926125000000003, 1.5530375}, {-2.6694612500000003, 0.8369475000000001, 
        1.46342875}}, VertexColors -> {RGBColor[1, 1, 1/3], RGBColor[1, 1, 1/3], RGBColor[1, 1, 1/3]}], 
     Polygon[{{-1.58001875, 0.7593975000000001, 0.8554787500000001}, {-1.9761062500000002, -0.19926125000000003, 
        1.5530375}, {-2.6694612500000003, 0.8369475000000001, 1.46342875}}, 
      VertexColors -> {RGBColor[1, 1, 1/3], RGBColor[1, 1, 1/3], RGBColor[1, 1, 1/3]}], 
     Polygon[{{-2.6694612500000003, 0.8369475000000001, 1.46342875}, {-1.5948062500000002, 0.85726, 
        2.1015550000000003}, {-1.58001875, 0.7593975000000001, 0.8554787500000001}}, 
      VertexColors -> {RGBColor[1, 1, 1/3], RGBColor[1, 1, 1/3], RGBColor[1, 1, 1/3]}], 
     Polygon[{{-0.7644937500000001, 0.10798375, 1.543285}, {-1.5948062500000002, 0.85726, 2.1015550000000003}, 
       {-1.58001875, 0.7593975000000001, 0.8554787500000001}}, VertexColors -> 
       {RGBColor[1, 1, 1/3], RGBColor[1, 1, 1/3], RGBColor[1, 1, 1/3]}], 
     Polygon[{{-1.9761062500000002, -0.19926125000000003, 1.5530375}, {-1.5948062500000002, 0.85726, 
        2.1015550000000003}, {-0.7644937500000001, 0.10798375, 1.543285}}, 
      VertexColors -> {RGBColor[1, 1, 1/3], RGBColor[1, 1, 1/3], RGBColor[1, 1, 1/3]}], 
     Polygon[{{-1.28560625, -0.41184750000000003, 0.53298}, {-1.9761062500000002, -0.19926125000000003, 
        1.5530375}, {-1.58001875, 0.7593975000000001, 0.8554787500000001}}, 
      VertexColors -> {RGBColor[1, 1, 1/3], RGBColor[1, 1, 1/3], RGBColor[1, 1, 1/3]}], 
     Polygon[{{-1.9761062500000002, -0.19926125000000003, 1.5530375}, {-1.28560625, -0.41184750000000003, 
        0.53298}, {-0.7644937500000001, 0.10798375, 1.543285}}, VertexColors -> 
       {RGBColor[1, 1, 1/3], RGBColor[1, 1, 1/3], RGBColor[1, 1, 1/3]}], 
     Polygon[{{-1.28560625, -0.41184750000000003, 0.53298}, {-0.44397250000000005, 0.50295, 
        0.40145875000000003}, {-1.58001875, 0.7593975000000001, 0.8554787500000001}}, 
      VertexColors -> {RGBColor[1, 1, 1/3], RGBColor[1, 1, 1/3], RGBColor[1, 1, 1/3]}], 
     Polygon[{{-1.58001875, 0.7593975000000001, 0.8554787500000001}, {-0.44397250000000005, 0.50295, 
        0.40145875000000003}, {-0.7644937500000001, 0.10798375, 1.543285}}, 
      VertexColors -> {RGBColor[1, 1, 1/3], RGBColor[1, 1, 1/3], RGBColor[1, 1, 1/3]}], 
     Polygon[{{-0.08269625000000001, -0.62667375, 0.7963375}, {-0.44397250000000005, 0.50295, 
        0.40145875000000003}, {-0.7644937500000001, 0.10798375, 1.543285}}, 
      VertexColors -> {RGBColor[1, 1, 1/3], RGBColor[1, 1, 1/3], RGBColor[1, 1, 1/3]}], 
     Polygon[{{-0.7644937500000001, 0.10798375, 1.543285}, {-1.28560625, -0.41184750000000003, 0.53298}, 
       {-0.08269625000000001, -0.62667375, 0.7963375}}, VertexColors -> {RGBColor[1, 1, 1/3], 
        RGBColor[1, 1, 1/3], RGBColor[1, 1, 1/3]}], 
     Polygon[{{-0.44369000000000003, -0.46503125000000006, -0.38943500000000003}, 
       {-1.28560625, -0.41184750000000003, 0.53298}, {-0.08269625000000001, -0.62667375, 0.7963375}}, 
      VertexColors -> {RGBColor[1, 1, 1/3], RGBColor[1, 1, 1/3], RGBColor[1, 1, 1/3]}], 
     Polygon[{{-0.44397250000000005, 0.50295, 0.40145875000000003}, {-1.28560625, -0.41184750000000003, 
        0.53298}, {-0.44369000000000003, -0.46503125000000006, -0.38943500000000003}}, 
      VertexColors -> {RGBColor[1, 1, 1/3], RGBColor[1, 1, 1/3], RGBColor[1, 1, 1/3]}], 
     Polygon[{{0.6387, 0.019345, 0.0059275000000000005}, {-0.44397250000000005, 0.50295, 0.40145875000000003}, 
       {-0.08269625000000001, -0.62667375, 0.7963375}}, VertexColors -> {RGBColor[1, 1, 1/3], 
        RGBColor[1, 1, 1/3], RGBColor[1, 1, 1/3]}], 
     Polygon[{{-0.44369000000000003, -0.46503125000000006, -0.38943500000000003}, 
       {0.6387, 0.019345, 0.0059275000000000005}, {-0.08269625000000001, -0.62667375, 0.7963375}}, 
      VertexColors -> {RGBColor[1, 1, 1/3], RGBColor[1, 1, 1/3], RGBColor[1, 1, 1/3]}], 
     Polygon[{{-0.44397250000000005, 0.50295, 0.40145875000000003}, {0.6387, 0.019345, 0.0059275000000000005}, 
       {-0.44369000000000003, -0.46503125000000006, -0.38943500000000003}}, 
      VertexColors -> {RGBColor[1, 1, 1/3], RGBColor[1, 1, 1/3], RGBColor[1, 1, 1/3]}]}}, Null}, Null}, 
 Boxed -> False, Axes -> False, AxesLabel -> {x, y, "z"}, ImageSize -> 400, PlotRange -> All, 
 ViewPoint -> {-2.2687728585291667, -1.1662073356338516, -1.9448984977924124}, 
 ViewVertical -> {0.9499382169642279, -0.16690871650522984, 0.26411903434890915}, PlotRangePadding -> None, 
 Background -> GrayLevel[0], Lighting -> "Neutral"];

Print3DChainQuad[182, Colors-> {Red, Green, Lighter@Lighter@Blue, Lighter@Yellow}, EdgeStyle->{AbsoluteThickness[.5], Black}]= cache182;
Print3DChainQuad[70, Colors-> {Red, Green, Lighter@Lighter@Blue, Lighter@Yellow}, EdgeStyle->{AbsoluteThickness[.5], Black}]= cache70;

Code (I should put all this in a package, but it is working ok here)

ρ0[L_] := ArcCos[(1/5) (2 (Sqrt[6])) ((Sin[(((L+1) θ)/2)])^2)];

P4=Permutations[Range[4]];
P3=Permutations[Range[3]];
id=IdentityMatrix[4];
THString[n_] := Take[Flatten@Table[{1,2,3,4}, Quotient[n,4]+1], n];
TetrahedronFromFaces[z_] := DeleteDuplicates@Flatten[z[[1]],1]

QHNew[L_]:=With[{base=Prepend[Delete[Take[Flatten[Table[{1,2,3,4},((2 L)/4)+1]],2 L+1],L+1],4]},Join[base,{Mod[2 L+2,4,1]},Reverse[Rest[base]]]];

Quadrahelix[L_,P_:P4[[15]]] := ((S=Take[Flatten[Table[Range[4],{(L/4)+1}]],L];
base=Join[S,{Mod[L+1,4,1]},Reverse[S]];
Join[base,base/. x_Integer:>P[[x]]]));

QuadrahelixShifted[L_,P_:P4[[15]]] := ((S=Take[Flatten[Table[Range[4],(L/4)+1]],L];
base=Join[S,{Mod[L+1,4,1]},Reverse[S]];
RotateLeft[Join[base,base/. x_Integer:>P[[x]]],L]));

QuadrahelixShiftedNormalized[L_] := Module[{Q}, 
(Q=QuadrahelixShifted[L]) /. Thread[Range[4]-> InversePermutation[Take[Q,4]]]];


(* could define the following using my symbolic matrix K *) errorTrueDistanceQuadrahelixShiftedNormalizedUsingPolygons[L_] := Module[{start, end},
(start= THPoint/@ {-1,0,1,2};
end =DeleteDuplicates@ Flatten[polygonsUp[QuadrahelixShiftedNormalized[L]][[-1,1]],1];
Min[Norm/@ Table[start - end[[P4[[k]]]],{k,1,24}]])];


matrixTHPermQuad[L_/; Mod[L,4]==0,prec_] :=(MatrixPower[N[M[3].M[4].M[1].M[2],prec], Quotient[L,4]])
matrixTHPermQuad1[L_/; Mod[L,4]==0,prec_] :=(MatrixPower[N[M[3].M[2].M[1].M[4],prec], Quotient[L,4]])
matrixTHPermQuad2[L_/; Mod[L,4]==0,prec_] :=(MatrixPower[N[M[1].M[4].M[3].M[2],prec], Quotient[L,4]])

matrixTHPermQuad[L_/; Mod[L,4]==2,prec_] :=(MatrixPower[N[M[3].M[2].M[1].M[4],prec], Quotient[L,4]])

matrixTHPermQuad[L_/; Mod[L,4]==1,prec_] :=(MatrixPower[N[M[4].M[1].M[2].M[3],prec], Quotient[L,4]])
matrixTHPermQuad1[L_/; Mod[L,4]==1,prec_] :=(MatrixPower[N[M[1].M[4].M[3].M[2],prec], Quotient[L,4]])
matrixTHPermQuad2[L_/; Mod[L,4]==1,prec_] :=(MatrixPower[N[M[2].M[1].M[4].M[3],prec], Quotient[L,4]])

matrixTHPermQuad[L_/; Mod[L,4]==3,prec_] :=(MatrixPower[N[M[2].M[3].M[4].M[1],prec], Quotient[L,4]])
matrixTHPermQuad1[L_/; Mod[L,4]==3,prec_] :=(MatrixPower[N[M[1].M[4].M[3].M[2],prec], Quotient[L,4]])
matrixTHPermQuad2[L_/; Mod[L,4]==3,prec_] :=(MatrixPower[N[M[4].M[3].M[2].M[1],prec], Quotient[L,4]])


(* gapErrorQuadrahelixTrueShifted[L_]:=(pp=polygonsUp[QuadrahelixShiftedNormalized[L]];Min[Table[gapTris@@t,{t,Tuples[{Delete[pp[[1,1]],Mod[L,4,1]],Delete[pp[[-1,1]],Mod[L,4,1]]}]}]]);*) 

gapErrorQuadrahelixTrueShifted[L_]:=(sw=If[EvenQ[L], 0, 3];pp=polygonsUp[QuadrahelixShiftedNormalized[L]];Min[Table[gapTris@@t,{t,Tuples[{Delete[pp[[1,1]],Mod[L+sw,4,1]],Delete[pp[[-1,1]],Mod[L+sw,4,1]]}]}]]);


(* these are based on normalized sequences*) 

errorGapShiftedQuad[L_ /; Mod[L,4]==2, prec_:$MachinePrecision]:= (
temp1=N[matrixTH[L-2, prec].M[1].M[2],prec];
temp2=N[matrixTHPermQuad[L, prec].M[3].M[2],prec];
K= temp1.M[3].temp1.temp2.M[1].temp2;
end=(fromBary[#, startUp]& /@ Transpose[K])[[{2,3,4}]];
(free1=Sort[Sort/@ ( (((fromBary[#, startUp]& /@ Transpose[M[1]]))[[#]]& /@ faces[[{1,2,3}]]))]);
(free2=Sort[Sort/@ (((fromBary[#1,startUp]&)/@Transpose[K])[[#]]& /@ faces[[{1,3,4}]])]);
gap=Min@Table[gapTris@@t, {t,Tuples[{free1, free2}]}]);

errorGapShiftedQuad[L_ /; Mod[L,4]==3, prec_:$MachinePrecision]:= (
temp1=N[matrixTH[L-3, prec].M[1].M[2].M[3],prec];
temp2=N[matrixTHPermQuad[L, prec].M[2].M[3].M[4],prec];
temp3=N[matrixTHPermQuad1[L, prec].M[1].M[4].M[3],prec];
temp4=N[matrixTHPermQuad2[L, prec].M[4].M[3].M[2],prec];
K= temp1.M[4].temp2 .  temp3.M[2].temp4;
end=(fromBary[#, startUp]& /@ Transpose[K.M[1]])[[{1,2,3}]];
(free1=Sort[Sort/@ ( (((fromBary[#, startUp]& /@ Transpose[M[4]]))[[#]]& /@ faces[[{1,2,4}]]))]);
(free2=Sort[Sort/@ (((fromBary[#1,startUp]&)/@Transpose[K])[[#]]& /@ faces[[{1,3,4}]])]);
gap=Min@Table[gapTris@@t, {t,Tuples[{free1, free2}]}]);

errorGapShiftedQuad[L_ /; Mod[L,4]==1, prec_:$MachinePrecision ]:= 
(temp1=N[matrixTH[L-1, prec].M[1],prec];
temp2=N[matrixTHPermQuad[L, prec].M[4],prec];
temp3=N[matrixTHPermQuad1[L, prec].M[1],prec];
temp4=N[matrixTHPermQuad2[L, prec].M[2],prec];
K= temp1.M[2].temp2 .  temp3.M[4].temp4;
end=(fromBary[#, startUp]& /@ Transpose[K])[[{1,2,3}]];
(free1=Sort[Sort/@ ( (((fromBary[#, startUp]& /@ Transpose[M[2]]))[[#]]& /@ faces[[{2,3,4}]]))]);
(free2=Sort[Sort/@ (((fromBary[#1,startUp]&)/@Transpose[K])[[#]]& /@ faces[[{2,3,4}]])]);
gap=Min@Table[gapTris@@t, {t,Tuples[{free1, free2}]}]);

errorGapShiftedQuad[L_ /; Mod[L,4]==0, prec_:$MachinePrecision ]:= 
(temp1=N[matrixTH[L, prec],prec];
temp2=N[matrixTHPermQuad[L, prec],prec];
temp3=N[matrixTHPermQuad1[L, prec],prec];
temp4=N[matrixTHPermQuad2[L, prec],prec];
K= temp1.M[1].temp2 .  temp3.M[3].temp4;
end=(fromBary[#, startUp]& /@ Transpose[K])[[faces[[2]]]];
(free1=Sort[Sort/@ ( (((fromBary[#, startUp]& /@ Transpose[M[1]]))[[#]]& /@ faces[[{1,2,3}]]))]);
(free2=Sort[Sort/@ (((fromBary[#1,startUp]&)/@Transpose[K])[[#]]& /@ faces[[{2,3,1}]])]);
gap=Min@Table[gapTris@@t, {t,Tuples[{free1, free2}]}]
);

rhombusPoints3D={{0,0,0},{0,0,1},{-((2 (Sqrt[6]))/5),0,(4/5)},{-((2 (Sqrt[6]))/5),0,-(1/5)}};
rhombusPoints={{0,0},{0,1},{((2 (Sqrt[6]))/5),(4/5)},{((2 (Sqrt[6]))/5),-(1/5)}};
rhombusPointsNeg={{0,0},{0,1},{-((2 (Sqrt[6]))/5),(4/5)},{-((2 (Sqrt[6]))/5),-(1/5)}};

(* a little imperfect sometimes*) 
PermutationMatrix4[p_] :=Table[RotateRight[IdentityMatrix[4][[i]], p[[i]]-i] , {i, Length[p]}];
gapTris[T1_,T2_]:=Min[tt=Table[Max[Table[N[EuclideanDistance[T1[[i]],T2[[p[[i]]]]]],{i,3}]],{p,P3}]]
(* these two below are slow, but fine for checking against faster code below *) 

gapErrorTrueShifted[L_]:=(pp=OTPolygonsShifted[L];Min[Table[gapTris@@t,{t,Tuples[{Delete[pp[[1,1]],Mod[L,4,1]],Delete[pp[[-1,1]],Mod[L,4,1]]}]}]]);
gapErrorTrue[L_]:=(pp=OTPolygons[L];Min[Table[gapTris@@t,{t,Tuples[{Delete[pp[[1,1]],Mod[L,4,1]],Delete[pp[[-1,1]],Mod[L,4,1]]}]}]])

distSq[pt_, face_] := (pl=Simplify[planeThru[face]];
ppp=Expand[-(pl-z)];
norm={Coefficient[ppp,x],Coefficient[ppp,y],Coefficient[ppp,z]};
(* http://mathworld.wolfram.com/Point-PlaneDistance.html*) 

Simplify[Simplify[(((((nnn=norm.pt+(ppp-norm.{x,y,z}))/(Norm[norm]))))^2)]])


keyFactor=(1/u)Factor[Expand[(3 I-5 I (Sqrt[3])+3 (Sqrt[5])+(Sqrt[15])+12 I u) (-3 I-5 I (Sqrt[3])-3 (Sqrt[5])+(Sqrt[15])-12 I u)]];  (* not really a poly *) 
keyFactor=24Factor[(-4-2 I (Sqrt[5]))(1/u)+(3-3 I (Sqrt[5])) +6 u]  (* not really a poly *) ;
keyRoots={(1/12) (-3+5 (Sqrt[3])+I (3 (Sqrt[5])+(Sqrt[15]))),(1/12) (-3-5 (Sqrt[3])+I (3 (Sqrt[5])-(Sqrt[15])))};
keyAngles = {β1,β2}=ArcCos[(1/12)(-3+{1,-1}5(Sqrt[3]))];
root5Rule= {(Sqrt[5])->I (-2-3 z),1/(Sqrt[5])->1/(I (-2-3 z))};   (* turns (Sqrt[5])*)
permSpecial={2,3,4,1};
OT[n_] := double[specFormBetter[n]]
OTShifted[n_] := RotateLeft[double[specFormBetter[n]],n]
OTPolygons[L_]:= polygonsUp[OT[L]]
OTPolygonsShifted[L_]:= polygonsUp[OTShifted[L]];
γ=ArcCos[1/12 (-3+5 (Sqrt[3]))];γ1=ArcCos[1/12 (-3-5 (Sqrt[3]))];

trigRuleL={Sin[(z_Integer+L)θ]:>TrigExpand@ Sin[Expand[(z+L)θ]],
Cos[(z_Integer+L)θ]:>TrigExpand@ Cos[Expand[(z+L)θ]],
Cos[(z_Integer+L)θ]:>TrigExpand@ Cos[Expand[(z+L)θ]],
Sin[(z_Integer+L)θ]:>TrigExpand@ Sin[Expand[(z+L)θ]],
Sin[(z_Integer+2L)θ]:>TrigExpand@ Sin[Expand[(z+2L)θ]],
Cos[(z_Integer+2L)θ]:>TrigExpand@ Cos[Expand[(z+2L)θ]],
Cos[(z_Integer+2L)θ]:>TrigExpand@ Cos[Expand[(z+2L)θ]],
Sin[(z_Integer+2L)θ]:>TrigExpand@ Sin[Expand[(z+2L)θ]],
Sin[(z_Integer+3L)θ]:>TrigExpand@ Sin[Expand[(z+3L)θ]],
Cos[(z_Integer+3L)θ]:>TrigExpand@ Cos[Expand[(z+3L)θ]],
Cos[(z_Integer+3L)θ]:>TrigExpand@ Cos[Expand[(z+3L)θ]],
Sin[(z_Integer+3L)θ]:>TrigExpand@ Sin[Expand[(z+3L)θ]],
Sin[(z_Integer+4L)θ]:>TrigExpand@ Sin[Expand[(z+4L)θ]],
Cos[(z_Integer+4L)θ]:>TrigExpand@ Cos[Expand[(z+4L)θ]],
Cos[(z_Integer+4L)θ]:>TrigExpand@ Cos[Expand[(z+4L)θ]],
Sin[(z_Integer+4L)θ]:>TrigExpand@ Sin[Expand[(z+4L)θ]]};

angleError[L_, prec_:20]:= Min[N[{Abs[2π-Mod[L θ- γ,2π]],Abs[Mod[L θ- γ,2π]],
Abs[2π-Mod[L θ- γ1,2π]],Abs[Mod[L θ- γ1,2π]]}],prec];


angleErrorQuad[L_, prec_:20]:= Min[N[{Abs[2π-Mod[(L+1)θ,2π]],Abs[Mod[(L+1) θ,2π]]},prec]];

Chair={{0,0,0},{0,0,1},{-(Sqrt[(1/5) (2-(Sqrt[3]))]),((5+(Sqrt[3]))/(5 (Sqrt[2]))),(6/5)},{-(Sqrt[(6/5)]),((3 (Sqrt[2]))/5),1+((2 (Sqrt[3]))/5)},{-(Sqrt[(6/5)]),(1/5) (Sqrt[2]) (3-2 (Sqrt[3])),(2/5) (3+(Sqrt[3]))},{-(Sqrt[(2/15)]),(1/15) (9 (Sqrt[2])-8 (Sqrt[6])),(1/5) (3+2 (Sqrt[3]))},{-((1+(Sqrt[3]))/(Sqrt[10])),(1/10) ((Sqrt[2])-5 (Sqrt[6])),((2 (Sqrt[3]))/5)},{-2 (Sqrt[(2/15)]),-((4 (Sqrt[(2/3)]))/5),-(1/5)}};

Δ=EuclideanDistance[Chair[[4]], Chair[[8]]];
ω=EuclideanDistance[Chair[[2]], Chair[[6]]];
SameSideOfPlane[pt1_,pt2_, pts_] := Det[pt1-# & /@ pts]Det[pt2-# & /@ pts]>0;

SameSideOfPlane[pt1_,pt2_, pts_] := Det[pt1-# & /@ pts]Det[pt2-# & /@ pts]>0;

planeThru[{a_, b_, c_}]:=Module[{z}, z/.First@Solve[Reduce[Det[{x,y,z}-# & /@ {a,b,c}]==0],z]];
planeIntersect[{a_, b_, c_},{d_, e_, f_}]:=Module[{}, {x,y,z} /.ToRules@Reduce[z==planeThru[{a,b,c}]&& 
z==planeThru[{d,e,f}]]];

goodValsQH=Rest[Denominator/@ Convergents[N[ArcCos[-2/3]/(2 π), 220]]-1];

goodVals=Join[ Union[{6,36,66, 432, 686, 940,178, 64708},
{4,75,766, 258,512, 1274,1020,1528}],
{4,75,432,686,1274,38707,64708,1268598,1870543,45631109,176411141,233817317,3113400370,6460618058,58842821865,466989321390,532292761314,2745915283354,19325006497121,30360467387359,279722922940343,2994332020919024,87518018383360557,152263541551460114777,278487693609040641936,4433648985644204733741,5818702749655926097676,96337324610581173170592,260516244268242813586381,3129592660080585060604871,4735433720035293001011254,4557006935595881007632767,16599509655614971310296967,194965656256493551869000382,808504637401608269088962697,829661153992819121406892432,5483916377274767219358055645,105964330782397456903654704646,143113746502288198277992291410,1605819099181508112761739945905,18542365352071835965739640099361,1004636212957741024979723083642461,3050993369577366746870648531126107,10482178540174881824314581122803611,50058515691121814676924807360850945,155101345307643880302191670018221495,658106913516821384536830306906390764,3981417215152895734277628220336180710,8222978605229957414482714973347953466,58754738452387668217207120136128070422,54513177062310606537002033383116297666,167781092577008881291211186902360665755,3502684439611823904527391672348996658573,6619412752786140825162318478972047075353,34204538201959735856178944500116849191865,127082011741878485678918532714872299275328,698139415705977782130864375013977400834362,21335290869228884406708145972985944105984793,39024277829391375799202958771247629853256591,227135748000493851356029016110406506055930069,2481525225560363142805859235380602739530133172,58841262164557095429051614717135508576385210747,74355828481646295242045194277145228074795445807,338495620296493207292788982046436505143682555814,1379540221238791655682770553563591893920820760782,4038707095181909845294651841035784580341576299015,88071497283970015671321528867153615482119173717550,702728471121637556559331396956586058483121909902950,138330055254880400930646217583831803366195159583374082,250539617978287888081572906173776719238357615860066689,805265295940052766917276439303017104820104498013368920,9251704607286778006908071203117367697287801409000853800,10307509521205118661906920548594161521346263522874289410,34175329647091190886531957510411967478533823541303283242,78658168815387500434970835569418096478413910605480855895,1651421436635536039006650586677175362308646405385014017326,2228114638452510552153857727426039848570999844048512308400,3938107127317983024599164733248183381227396864050195320606,10104328893088476601352187193922406611025793572148902949613,140171679176085816226312866008829958923054477880885261449834,1309067839234196400378513668337175786198656239639510804939617,60356184708504086743031778607460711348942104795173836574293507,64573835913451936053221297531683821031995208263426288414961642,292080908716090210583831349434029099402230991370877414192310710,15473104637098844613864611199560055321113636544904173260125060455,41727296053817196960134781181320074515018266700973893694245925683,120489870303972253998945291126600132096732157169183054996608521367,503229573866411941250008450721366427033108384907692436920233384304,4409389183740963870799450556614309433978584552760995417458844609358,8315548793615515800348892662507252440924060720614298397997455834412,41122038678278408338094952318011621695427279604550829490434965951820,159623060239573498817930847720907536838537509866921660044728705977681,157897833079535808184980882165223385589661283782255339804235556975275,1513675938032605234138717637457138621191063469757397638661948838695556,35947151627704647925759867064543290398226506261823153335420961962757300,73407979193441901085658451766543719417644075993403704309503872764210157,216364249273024497648801970696017068309657491116095823802435614715408755,1758309432766062222759932914096751844288044351537758671654001766631753306,1154956751001447421999461642808784060272304508575177496149604785244031050122,2280719692946671703964841007960215475769570178974340151867061248378062773356,7998874139274228596116744599603527239425303089849735710422442532144851123498,22870859475877461506384854433659150302778643599150044475549871133300521647261,88076955268617950039609197361524954319847738547226675094614966821689992092455,399235448895717159795462802166798351341839872778309519424813016662481673794662,1513457231818633021752428535532160331395670219806487635507475024648869459713102,1583255719115755842895839402686518015760897781995040499029378045157891686785204,4679968670050144707544107340905196362917465783796568633566231114964652833283511,22703135484195019964455232103534585455829958621091648382746018529812025819314133,317197431925084678983186353314887809952377752459227427775174420286474493661728597,2621312023582528681724384684255108343091025190670479113749361489271283876938406919,19222370326194799304388088995469917550814465555094228606239923629024781212551408872,119980777437697145041740914028228883063724431460742351404170390002038169917867759751,1223080631617057890789173220020067287019552006041530743953906797126642877922470805581,2566142040671812926620087354068363457102828443543803839311983984255323925762809370914,520995570436310648531663589649283557458718110377743877709354529950228838938455482920848,1212234808297162429338388259376237149460871819623343976202744591238193525622045714320760,27083232366303288087265855409432997376793396278240924568631399819272100846825293572765617,30779089160304163181255493573726820266412328343757721796121207708991230464140043266829168,307795538900904582285276674728331798589831694830206113833233451922603960591075372763745515,584811988641505001389297855882936776913251061316654505870345696136216690718010702260661862,803966116855497969224797688054381937593464703181104251542705332821566089533980694917983423,30580201553300368641597367822030444716524540215228933587655754036890493266390636031502961909,93348536893612101863241698842200098024760550052049009266052672776314611978239869484344852575,3519997079490803567293149621449366923342330162858598618514009307827445562293847028245599796577,7133342695875219236449540941740933944709420875769246246294071288431205736565933925975544445730,28204122034921324165659801337889962447914158984894510567075245512887718647284676766656452333839,45872854400319595866714472477705410644225948915226388973759903913547180839209982112972450723624,512088591416919684787721419291847810880543004895455163070905494789541379498376965770314573300360,978304328433519773708728366105990211116860060875683937168051085665535578157543949427656695877096,9953069065716677116217931072976778805652806615120649760318733343309640777132128251949274159416098,117631422281259686609473207599301542212241784277723268796371212282609193834431732651187957165957480,10543018197922372722463297490163222603979202389225163427281302550935619489865334048929705080301707700,10021748859140317070670606276035026550890854358977791419357055984002968975063788933115779224144920189,48723264420359411667055447260193812800446503188176163924307991882645025904989691364249429811072569582}];

r=((3 (Sqrt[3]))/10);
h=(1/(Sqrt[10]));θ=ArcCos[-2/3];

specFormBetter[L_] := (S =Take[Flatten[Table[Range[4], L/4 + 1]], L];
 base=Join[S,{Mod[L+1,4,1]}, Reverse[S]];
base = Join[base, base/. x_Integer:>{2,3,4,1}[[x]]]);

specForm[L_] := (S =Take[Flatten[Table[Range[4], L/4 + 1]], L];
 base=Join[S,{Mod[L+1,4,1]}, Reverse[S]];
base = Join[base, base/. x_Integer:>{2,4,3,1}[[x]]]);


a={0,0,0};
b={1,0,0};
c={1/2, (Sqrt[3])/2,0};

d= {(1/2),(1/(2 (Sqrt[3]))), (Sqrt[2/3])};
{a,b,c,d}=Table[{r Cos[n θ],r Sin[n θ],n h},{n,-1,2}];


startUpFaces={{a,b,c},{a,b,d}, {b,c,d},{a,c,d}};
startUp={a,b,c,d};


MatricesToTetsFacesUp[mats_] := Table[qqq=fromBary[mats[[i]] . #,startUp]& /@ IdentityMatrix@4;Table[qqq[[f]], {f, faces}], {i,Length[mats]}]

MatricesToTetsUp[mats_] := Table[fromBary[mats[[i]].#,startUp]& /@ IdentityMatrix@4, {i,Length[mats]}];

MatricesToTetsSymUp[mats_] := Table[fromBary[mats[[i]].#,startUp]& /@ IdentityMatrix@4, {i,Length[mats]}];

SeqToTets[seq_] := MatricesToTets[SeqToMatrixList[seq]];
SeqToTetsSym[seq_] := MatricesToTetsSym[SeqToMatrixListSym[seq]];

polygonsUp[seq_]:=Polygon/@ MatricesToTetsFacesUp[SeqToMatrixList[seq]];


polygonsUpLikeBefore[seq_]:= Polygon/@ MatricesToTetsFacesUp[SeqToMatrixList[seq]];
polygonsUpLikeBeforeSym[seq_]:= Polygon/@ MatricesToTetsFacesUp[SeqToMatrixListSym[seq]];

Show[Print3DChain[6, EdgeStyle->None,FillerOnlyQ->!True, Background->RGBColor[1,1,.6]] ,  Lighting->"Neutral"]

MAYBE DELETE THE EXTRANEOUS RED??  And or Gray

uniqueQ[set_, poly_] := Select[set,Sort[#[[1]]]==Sort[poly[[1]]]&]=={poly};
color[p_,c_] := p/. Polygon[z_List, c1___] :> Polygon[z, VertexColors->{c,c,c}];
Chain3DNotClean[L_] :=Graphics3D[polygonsUp[RotateLeft[double@specFormBetter[L],L]]];

Options[Print3DChain]= {EdgeStyle->None, ShowFiller->True,FillerOnlyQ->False, Background->Black, FillColor->White, ShiftChain->True, Colors-> {RGBColor[.6,.1,.1],RGBColor[1,.4,.4],  RGBColor[.45,1,.45],RGBColor[0,.4,.0],
RGBColor[0,0,.7], RGBColor[0.65,0.65,1], White,RGBColor[0.2,0.2,.2]}};
Print3DChain[L_, opts___] := (
{shiftQ,esty,back,fillQ,fillOnlyQ, fillcol, cols}={ShiftChain,EdgeStyle,Background,ShowFiller,FillerOnlyQ, FillColor, Colors}/. {opts} /. Options[Print3DChain];
pp=Flatten[(pp0=N[polygonsUp[RotateLeft[double@specFormBetter[L],If[shiftQ,L,0]]]]/. 
Polygon[z_] :> Polygon/@ z),1] 
(* /. {
{X,x_, y_, z_?NumericQ} /;0<Norm[{x,y,z}-a]<.6 :> (mPrint@z;a),
{X,x_, y_, z_?NumericQ} /;0<Norm[{x,y,z}-b]<.6 :> b,
{X,x_, y_, z_?NumericQ} /;0<Norm[{x,y,z}-c]<.6 :> c}*);
pp = pp /. x_?NumericQ :> Round[10^6 x];
     ppClean=Select[pp, uniqueQ[pp,#]&];
ppClean = ppClean/.  x_?NumericQ :>x/800000.;
col=Append[cols,fillcol] ;
 im=Graphics3D[pp=ppClean;
{
If[fillOnlyQ,{},{
EdgeForm[GrayLevel[.2]],EdgeForm[esty],

{
color[pp[[1;;2L+1]],col[[1]]],
color[pp[[2L+2;;4L+3]],col[[2]]],
color[pp[[4L+4;;6L+5]],col[[3]]],
color[pp[[6L+6;;8L+5]],col[[4]]],


color[ pp[[8L+6;;10L+5]],col[[5]]],
color[ pp[[10L+6;;12L+7]],col[[6]]],
color[ pp[[12L+8;;14L+9]],col[[7]]],
color[ pp[[14L+10;;16L+10]],col[[8]]]


},

j = Mod[L+1,4,1];
inds= j /.{1->{1,1},3->{2,2}, 4-> {3,3}, 2->({2,2};{1,1})};

{color[pp[[inds[[1]]]],Red],
Opacity[1],color[ pp[[16L+10-2;;16L+10]][[inds[[2]]]],col[[4]]]};{}

}],

threeRedFaces=First/@ pp[[{1,2,3}]];threeBlackFaces=First/@ pp[[{-3,-2,-1}]];
mPrint[choices=First@First[SortBy[Table[{t,Norm[Mean[t[[1]]]-Mean[t[[2]]]]},{t,Tuples[{threeRedFaces,threeBlackFaces}]}],Last]]];


{a,b,c}=pp[[inds[[1]],1]];
{a1,b1,c1}=Nearest[pp[[16L+10-2;;16L+10]][[inds[[2]],1]],#][[1]]& /@ {a,b,c};
{a1,b1,c1}=pp[[16L+10-2;;16L+10]][[inds[[2]],1]];
{Red, AbsolutePointSize[10], Red,Point[choices[[1]]],Blue, Point[{a1,b1,c1}; choices[[2]]]};
EdgeForm[],
If[fillQ,zz={
EdgeForm[],
color[#,col[[9]]]&/@ Polygon/@ {{a,b,b1}, {a1, b1, a}},
color[#,col[[9]]]&/@ Polygon/@ {{b,c,c1}, {b1, c1, b}},
color[#,col[[9]]]&/@ Polygon/@ {{c,a,a1}, {c1, a1, c}}},{}]},
Boxed->False,Axes->!True, AxesLabel->{x,y,"z"},ImageSize->600, PlotRange->All, ViewPoint-> vp[L], ViewVertical->vv[L],
PlotRangePadding->None, Background->back, Lighting->"Neutral"]);

(*{vp[_], vv[_]}= {{-1,-2.3,2}, {0 ,0,1}};
{vp[2], vv[2]}={{-2.436476625192898`,-1.5809907523644915`,1.361635008299429`},{-1.2575593022759337`,-0.5485412269834631`,-0.42982756819051593`}};
{vp[3], vv[3]}= {{2.4953587172326834`,-1.5857876841226364`,-1.2443722486522362`},{0.7010438434352279`,0.14075386215118846`,-0.8871206085133403`}};
{vp[4], vv[4]}={{-2.8424392429867407`,0.5390177528600011`,1.385640289552348`},{-0.8736910573290675`,0.3947460774766312`,0.9953916223066416`}};
{vp[5], vv[5]}={{1.1193814116109855`,-2.96630784128573`,0.48785555860991825`},{0.7699479310300471`,-0.524897388320674`,0.6994048813748218`}};
{vp[6], vv[6]}={{2.9577770138323527`,1.643347812102079`,0.031033931498950784`},{1.1494477566674526`,-0.3309062741903174`,-0.1900644759966073`}};
{vp[7], vv[7]} = {{-3.2074691094910612`,-0.012082201262409793`,-0.044675855596423404`},{-2.260224654094242`,0.6789186247740917`,-0.4956279394188679`}};
{vp[8],vv[8]}= {{1.5693071224552035`,-2.7744810366438486`,0.35990294902243497`},{0.9560932817456527`,-0.40839901700076225`,-0.2836650044213117`}};
{vp[9],vv[9]}= {{0.3657034698156067`,1.7191681402208556`,2.683416083616998`},{-0.682900961330594`,0.26453167454283627`,0.8369298827602413`}};
{vp[10], vv[10]}={{-2.827979964421134`,-1.4809672655824224`,-0.3150639286017498`},{-0.5711182253590957`,-0.5473905431378594`,-0.7713878893366312`}};
{vp[25], vv[25]}={{0.7802285087015227`,0.32958073103844093`,3.093965096108481`},{-0.5877585403943673`,-0.11259378208821114`,1.5823921509273404`}};

{vp[182], vv[182]}={{-2.2687728585291667`,-1.1662073356338516`,-1.9448984977924124`},{0.9499382169642279`,-0.16690871650522984`,0.26411903434890915`}};*)

matrixTH[L_ /; Mod[L,4]==0] :=(MatrixPower[M[1].M[2].M[3].M[4], Quotient[L,4]])
matrixTHPerm[L_/; Mod[L,4]==0] :=(MatrixPower[M[2].M[3].M[4].M[1], Quotient[L,4]])

matrixTH[L_ /; Mod[L,4]==2] :=(MatrixPower[ M[1].M[2].M[3].M[4], Quotient[L,4]])
matrixTHPerm[L_/; Mod[L,4]==2] :=(MatrixPower[M[2].M[3].M[4].M[1], Quotient[L,4]])

matrixTH[L_ /; Mod[L,4]==1] :=(MatrixPower[M[1].M[2].M[3].M[4], Quotient[L,4]])
matrixTHPerm[L_/; Mod[L,4]==1] :=(MatrixPower[M[2].M[3].M[4].M[1], Quotient[L,4]])

matrixTH[L_ /; Mod[L,4]==3] :=(MatrixPower[M[1].M[2].M[3].M[4], Quotient[L,4]])
matrixTHPerm[L_/; Mod[L,4]==3] :=(MatrixPower[M[2].M[3].M[4].M[1], Quotient[L,4]])



matrixTH[L_ /; Mod[L,4]==0,prec_] :=(MatrixPower[N[M[1].M[2].M[3].M[4],prec], Quotient[L,4]])
matrixTHPerm[L_/; Mod[L,4]==0,prec_] :=(MatrixPower[N[M[2].M[3].M[4].M[1],prec], Quotient[L,4]])

matrixTH[L_ /; Mod[L,4]==2,prec_] :=(MatrixPower[N[ M[1].M[2].M[3].M[4],prec], Quotient[L,4]])
matrixTHPerm[L_/; Mod[L,4]==2,prec_] :=(MatrixPower[N[M[2].M[3].M[4].M[1],prec], Quotient[L,4]])


matrixTH[L_ /; Mod[L,4]==1,prec_] :=(MatrixPower[N[M[1].M[2].M[3].M[4],prec], Quotient[L,4]])
matrixTHPerm[L_/; Mod[L,4]==1,prec_] :=(MatrixPower[N[M[2].M[3].M[4].M[1],prec], Quotient[L,4]])

matrixTH[L_ /; Mod[L,4]==3,prec_] :=(MatrixPower[N[ M[1].M[2].M[3].M[4],prec], Quotient[L,4]])
matrixTHPerm[L_/; Mod[L,4]==3,prec_] :=(MatrixPower[N[M[2].M[3].M[4].M[1],prec], Quotient[L,4]])


errorFull[L_ /; Mod[L,4]==2, prec_:$MachinePrecision]:= (K=(temp1=N[matrixTH[L-2, prec],prec]) .M[1].M[2].M[3].M[2].M[1].N[Inverse[temp1],prec].(temp2=N[(matrixTHPerm[L-2,prec]),prec]) .M[2].M[3].M[4].M[3].M[2].N[Inverse[temp2],prec];
Max[K.K-IdentityMatrix[4]])

errorFull[L_ /; Mod[L,4]==0, prec_:$MachinePrecision]:= (K=(temp3=N[matrixTH[L,prec],prec]) .M[1].N[Inverse[temp3],prec].(temp4=N[(matrixTHPerm[L,prec]),prec]) .M[2].N[Inverse[temp4],prec];
Max[K.K-IdentityMatrix[4]])


errorFull[L_ /; Mod[L,4]==1, prec_:$MachinePrecision ]:= (K=(temp1=N[matrixTH[L-1, prec],prec]) .M[1].M[2].M[1].N[Inverse[temp1],prec].(temp2=N[(matrixTHPerm[L-1,prec]),prec]) .M[2].M[3].M[2].N[Inverse[temp2],prec];
Max[K.K-IdentityMatrix[4]])

errorFull[L_ /; Mod[L,4]==3, prec_:$MachinePrecision]:= (K=(temp3=N[matrixTH[L-3,prec],prec]) .M[1].M[2].M[3].M[4].M[3].M[2].M[1].N[Inverse[temp3],prec].(temp4=N[(matrixTHPerm[L-3,prec]),prec]).M[2].M[3].M[4].M[1].M[4].M[3].M[2].N[Inverse[temp4],prec];
Max[K.K-IdentityMatrix[4]]);

errorGap[L_ /; Mod[L,4]==2, prec_:$MachinePrecision]:= (K=(temp1=N[matrixTH[L-2, prec],prec]) .M[1].M[2].M[3].M[2].M[1].N[Inverse[temp1],prec].(temp2=N[(matrixTHPerm[L-2,prec]),prec]) .M[2].M[3].M[4].M[3].M[2].N[Inverse[temp2],prec];
end=fromBary[#, startUp]& /@ Transpose[K.K];
start=startUp;
Max[Norm/@(end-start)]);


errorGap[L_ /; Mod[L,4]==0, prec_:$MachinePrecision]:= (K=(temp3=N[matrixTH[L,prec],prec]) .M[1].N[Inverse[temp3],prec].(temp4=N[(matrixTHPerm[L,prec]),prec]) .M[2].N[Inverse[temp4],prec];
end=fromBary[#, startUp]& /@ Transpose[K.K];
start=startUp;
Max[Norm/@(end-start)]);


errorGap[L_ /; Mod[L,4]==1, prec_:$MachinePrecision ]:= (K=(temp1=N[matrixTH[L-1, prec],prec]) .M[1].M[2].M[1].N[Inverse[temp1],prec].(temp2=N[(matrixTHPerm[L-1,prec]),prec]) .M[2].M[3].M[2].N[Inverse[temp2],prec];
end=fromBary[#, startUp]& /@ Transpose[K.K];
start=startUp;
Max[Norm/@(end-start)]);

errorGap[L_ /; Mod[L,4]==3, prec_:$MachinePrecision]:= (K=(temp3=N[matrixTH[L-3,prec],prec]) .M[1].M[2].M[3].M[4].M[3].M[2].M[1].N[Inverse[temp3],prec].(temp4=N[(matrixTHPerm[L-3,prec]),prec]).M[2].M[3].M[4].M[1].M[4].M[3].M[2].N[Inverse[temp4],prec];
end=fromBary[#, startUp]& /@ Transpose[K.K];
start=startUp;
Max[Norm/@(end-start)]);


(*errorGapShifted[L_ /; Mod[L,4]==2, prec_:$MachinePrecision]:= ((temp1=N[matrixTH[L-2, prec],prec]);
K= M[3].M[2].M[1].N[Inverse[temp1],prec].
(temp2=N[(matrixTHPerm[L-2,prec]),prec]) .M[2].M[3].M[4].M[3].M[2].N[Inverse[temp2],prec].temp1.M[1].M[2];
end=(fromBary[#, startUp]& /@ Transpose[K.K])[[{2,3,4}]];
start=startUp[[{2,3,4}]];
Max@Table[Norm[end[[i]]-start[[i]]], {i,3}]);*)

errorGapShifted[L_ /; Mod[L,4]==2, prec_:$MachinePrecision]:= (temp1=N[matrixTH[L-2, prec],prec];
K= M[3].M[2].M[1].N[Inverse[temp1],prec].
(temp2=N[(matrixTHPerm[L-2,prec]),prec]) .M[2].M[3].M[4].M[3].M[2].N[Inverse[temp2],prec].temp1.M[1].M[2];
end=(fromBary[#, startUp]& /@ Transpose[K.K])[[{2,3,4}]];
(free1=Sort[Sort/@ ( (((fromBary[#, startUp]& /@ Transpose[M[1]]))[[#]]& /@ faces[[{1,2,3}]]))]);
(free2=Sort[Sort/@ (((fromBary[#1,startUp]&)/@Transpose[K.K])[[#]]& /@ faces[[{1,3,4}]])]);
gap=Min@Table[gapTris@@t, {t,Tuples[{free1, free2}]}]);



(*errorGapShifted[L_ /; Mod[L,4]==3, prec_:$MachinePrecision]:= ((temp3=N[matrixTH[L-3,prec],prec]);

K=M[4].M[3].M[2].M[1].N[Inverse[temp3],prec].(temp4=N[(matrixTHPerm[L-3,prec]),prec]).M[2].M[3].M[4].M[1].M[4].M[3].M[2].N[Inverse[temp4],prec]. temp3.M[1].M[2].M[3];
end=(fromBary[#, startUp]& /@ Transpose[K.K])[[{1,2,3}]];
start=startUp[[{1,2,3}]];
Max[Norm/@(end-start)]);*)


errorGapShifted[L_ /; Mod[L,4]==3, prec_:$MachinePrecision]:=(
 temp3=N[matrixTH[L-3,prec],prec];
K=M[4].M[3].M[2].M[1].N[Inverse[temp3],prec].(temp4=N[(matrixTHPerm[L-3,prec]),prec]).M[2].M[3].M[4].M[1].M[4].M[3].M[2].N[Inverse[temp4],prec]. temp3.M[1].M[2].M[3];
end=(fromBary[#, startUp]& /@ Transpose[K.K.M[1]])[[{1,2,3}]];
(free1=Sort[Sort/@ ( (((fromBary[#, startUp]& /@ Transpose[M[4]]))[[#]]& /@ faces[[{1,2,4}]]))]);
(free2=Sort[Sort/@ (((fromBary[#1,startUp]&)/@Transpose[K.K])[[#]]& /@ faces[[{1,3,4}]])]);
gap=Min@Table[gapTris@@t, {t,Tuples[{free1, free2}]}]);




(*errorGapShifted[L_ /; Mod[L,4]==1, prec_:$MachinePrecision ]:= ((temp1=N[matrixTH[L-1, prec],prec]) ;

K=M[2].M[1].N[Inverse[temp1],prec].(temp2=N[(matrixTHPerm[L-1,prec]),prec]) .M[2].M[3].M[2].N[Inverse[temp2],prec].temp1.M[1];
end=(fromBary[#, startUp]& /@ Transpose[K.K])[[{1,2,3}]];
start=startUp[[{1,2,3}]];
Max[Norm/@(end-start)]);*)


errorGapShifted[L_ /; Mod[L,4]==1, prec_:$MachinePrecision ]:= (temp1=N[matrixTH[L-1, prec],prec] ;
K=M[2].M[1].N[Inverse[temp1],prec].(temp2=N[(matrixTHPerm[L-1,prec]),prec]) .M[2].M[3].M[2].N[Inverse[temp2],prec].temp1.M[1];
end=(fromBary[#, startUp]& /@ Transpose[K.K])[[{1,2,3}]];
(free1=Sort[Sort/@ ( (((fromBary[#, startUp]& /@ Transpose[M[2]]))[[#]]& /@ faces[[{2,3,4}]]))]);
(free2=Sort[Sort/@ (((fromBary[#1,startUp]&)/@Transpose[K.K])[[#]]& /@ faces[[{2,3,4}]])]);
gap=Min@Table[gapTris@@t, {t,Tuples[{free1, free2}]}]);

(*errorGapShifted[L_ /; Mod[L,4]==0, prec_:$MachinePrecision]:= ((temp3=N[matrixTH[L,prec],prec]);
K=M[1].N[Inverse[temp3],prec].(temp4=N[(matrixTHPerm[L,prec]),prec]) .M[2].N[Inverse[temp4],prec].temp3;
end=(fromBary[#, startUp]& /@ Transpose[K.K])[[faces[[2]]]];
start=startUp[[faces[[2]]]];
Max[Norm/@(end-start)]);*)

errorGapShifted[L_ /; Mod[L,4]==0, prec_:$MachinePrecision]:= (
temp3=N[matrixTH[L,prec],prec];
K=M[1].N[Inverse[temp3],prec].(temp4=N[(matrixTHPerm[L,prec]),prec]) .M[2].N[Inverse[temp4],prec].temp3;
end=(fromBary[#, startUp]& /@ Transpose[K.K])[[faces[[2]]]];
(free1=Sort[Sort/@ ( (((fromBary[#, startUp]& /@ Transpose[M[1]]))[[#]]& /@ faces[[{1,2,3}]]))]);
(free2=Sort[Sort/@ (((fromBary[#1,startUp]&)/@Transpose[K.K])[[#]]& /@ faces[[{2,3,1}]])]);
gap=Min@Table[gapTris@@t, {t,Tuples[{free1, free2}]}]
);





matrixTwoLegs[L_ /; Mod[L,4]==2, prec_]:= (temp1=N[matrixTH[L-2, prec],prec]) .M[1].M[2].M[3].M[2].M[1].N[Inverse[temp1],prec];

matrixTwoLegs[L_ /; Mod[L,4]==0, prec_]:= (temp3=N[matrixTH[L,prec],prec]) .M[1].N[Inverse[temp3],prec];

Collision Code from Italian paper GPR.

CollisionQSW::usage = "CollisionQSW[chain] returns False if there is no collision because a separating plane has been found for every pair; if it returns True, it is likely, but not certain, that there is a collision.";

CollisionTwoTets::usage = "CollisionTwoTets returns False if a a separationg plane is found. It returns true if a vertex of B is inside A. Else it returns True, but it is possible there is no collision in that case.";

allGood[L_] := And@@Thread[0<L<1];
inside3DQ[pts_, pt_] := allGood[baryCom@@ Flatten[{pts,pt}]];
PointInsideOneOrderOnly[A_,B_]:= inside3DQ[A, B[[1]]]||
inside3DQ[A, B[[2]]]||inside3DQ[A, B[[3]]]||inside3DQ[A, B[[4]]];



CollisionTwoTets[A_, B_]:= Which[SepCheckQuickSym[A,B]||
SepPlane[A, B,1]||SepPlane[A, B,2]||SepPlane[A, B,3]||SepPlane[A, B,4], False,
PointInsideOneOrderOnly[A,B ], True,
SepPlane[B, A,1]||SepPlane[B, A,2]||SepPlane[B, A,3]||SepPlane[B, A,4], False,
True, True];


CollisionTwoTetsSym[A_, B_]:= !(SepCheckQuickSym[A,B]||
SepPlaneSym[A, B,1]||SepPlaneSym[A, B,2]||SepPlaneSym[A, B,3]||SepPlaneSym[A, B,4]||
SepPlaneSym[B, A,1]||SepPlaneSym[B, A,2]||SepPlaneSym[B, A,3]||SepPlaneSym[B, A,4]);

GetSepPlaneSym[A_, B_]:= Which[
SepPlaneSym[A, B,1],{1,1},SepPlaneSym[A, B,2],{1,2},
SepPlaneSym[A, B,3],{1,3},SepPlaneSym[A, B,4],{1,4},
SepPlaneSym[B, A,1],{2,1},SepPlaneSym[B, A,2],{2,2},SepPlaneSym[B, A,3],{2,3},SepPlaneSym[B, A,4], {2,4}];

BestRotatedEmbeddedError[sFull_, period_]:= Module[{data,SS},(data=Table[{CollisionTwoTets@@(SeqToTets[SS=RotateLeft[sFull,i]][[{1,-1}]]),i, SeqToDistError [SS]}, {i,0,Length[sFull]/period}];
SortBy[Select[data, #[[1]]==False&], Last][[1]])];

BestRotatedEmbeddedErrorSym[sFull_, period_]:= Module[{data,SS},(data=Table[{CollisionTwoTetsSym@@(SeqToTetsSym[SS=RotateLeft[sFull,i]][[{1,-1}]]),i, SeqToDistErrorSym [SS]}, {i,0,Length[sFull]/period}];
SortBy[Select[data, #[[1]]==False&], Last][[1]])]

loop444= {1,2,3,4,2,1,4,2,1,4,3,4,2,4,1,3,2,1,4,1,2,3,1,4,2,4,3,4,1,2,4,1,2,4,3,2,1,2,4,1,3,4,2,3,4,2,3,1,3,4,3,2,1,4,2,3,2,4,1,2,3,4,3,1,3,2,4,3,2,4,3,1,4,2,1,2,3,4,2,1,4,2,1,4,3,4,2,4,1,3,2,1,4,1,2,3,1,4,2,4,3,4,1,2,4,1,2,4,3,2,1,3,1,4,2,1,3,2,1,3,2,4,2,1,2,3,4,1,3,2,3,1,4,3,2,1,2,4,2,3,1,2,3,1,2,4,1,3,1,2,3,4,2,1,4,2,1,4,3,4,2,4,1,3,2,1,4,1,2,3,1,4,2,4,3,4,1,2,4,1,2,4,3,2,1,2,4,1,3,4,2,3,4,2,3,1,3,4,3,2,1,4,2,3,2,4,1,2,3,4,3,1,3,2,4,3,2,4,3,1,4,2,1,2,3,4,2,1,4,2,1,4,3,4,2,4,1,3,2,1,4,1,2,3,1,4,2,4,3,4,1,2,4,1,2,4,3,2,1,3,1,4,2,1,3,2,1,3,2,4,2,1,2,3,4,1,3,2,3,1,4,3,2,1,2,4,2,3,1,2,3,1,2,4,1,3,1,2,3,4,2,1,4,2,1,4,3,4,2,4,1,3,2,1,4,1,2,3,1,4,2,4,3,4,1,2,4,1,2,4,3,2,1,2,4,1,3,4,2,3,4,2,3,1,3,4,3,2,1,4,2,3,2,4,1,2,3,4,3,1,3,2,4,3,2,4,3,1,4,2,1,2,3,4,2,1,4,2,1,4,3,4,2,4,1,3,2,1,4,1,2,3,1,4,2,4,3,4,1,2,4,1,2,4,3,2,1,3,1,4,2,1,3,2,1,3,2,4,2,1,2,3,4,1,3,2,3,1,4,3,2,1,2,4,2,3,1,2,3,1,2,4,1,3};

loop540={1,2,3,4,1,2,3,4,3,4,1,3,2,3,4,1,2,1,3,4,1,2,4,2,1,4,3,1,2,1,4,3,2,3,1,4,3,4,3,2,1,4,3,2,1,2,3,4,1,2,3,4,1,4,1,2,4,3,4,1,2,3,2,4,1,2,3,1,3,2,1,4,2,3,2,1,4,3,4,2,1,4,1,4,3,2,1,4,3,2,1,2,3,4,1,2,3,4,3,4,1,3,2,3,4,1,2,1,3,4,1,2,4,2,1,4,3,1,2,1,4,3,2,3,1,4,3,4,3,2,1,4,3,2,1,4,1,2,3,4,1,2,3,2,3,4,2,1,2,3,4,1,4,2,3,4,1,3,1,4,3,2,4,1,4,3,2,1,2,4,3,2,3,2,1,4,3,2,1,4,1,2,3,4,1,2,3,4,3,4,1,3,2,3,4,1,2,1,3,4,1,2,4,2,1,4,3,1,2,1,4,3,2,3,1,4,3,4,3,2,1,4,3,2,1,2,3,4,1,2,3,4,1,4,1,2,4,3,4,1,2,3,2,4,1,2,3,1,3,2,1,4,2,3,2,1,4,3,4,2,1,4,1,4,3,2,1,4,3,2,1,2,3,4,1,2,3,4,3,4,1,3,2,3,4,1,2,1,3,4,1,2,4,2,1,4,3,1,2,1,4,3,2,3,1,4,3,4,3,2,1,4,3,2,1,4,1,2,3,4,1,2,3,2,3,4,2,1,2,3,4,1,4,2,3,4,1,3,1,4,3,2,4,1,4,3,2,1,2,4,3,2,3,2,1,4,3,2,1,4,1,2,3,4,1,2,3,4,3,4,1,3,2,3,4,1,2,1,3,4,1,2,4,2,1,4,3,1,2,1,4,3,2,3,1,4,3,4,3,2,1,4,3,2,1,2,3,4,1,2,3,4,1,4,1,2,4,3,4,1,2,3,2,4,1,2,3,1,3,2,1,4,2,3,2,1,4,3,4,2,1,4,1,4,3,2,1,4,3,2,1,2,3,4,1,2,3,4,3,4,1,3,2,3,4,1,2,1,3,4,1,2,4,2,1,4,3,1,2,1,4,3,2,3,1,4,3,4,3,2,1,4,3,2,1,4,1,2,3,4,1,2,3,2,3,4,2,1,2,3,4,1,4,2,3,4,1,3,1,4,3,2,4,1,4,3,2,1,2,4,3,2,3,2,1,4,3,2,1,4};

loop140 = double[seqFromCode[{1,2,3,2,4,1,2,3,4,2,4,3,1,4,3,4,1}, 2, {3,4,2,1}]];
loop124=double[seqFromCode[IntegerDigits@123424312431213, 4, {3,4,2,1}]];

loop116=double[seqFromCode[IntegerDigits@12323132432314, 3, {2,3,4,1}]];

loop92=double[seqFromCode[IntegerDigits@12423142131, 4, {2,3,4,1}]];

loop84=double[seqFromCode[IntegerDigits@1234231243, 2, {4,3,1,2}]];

loop60=double[seqFromCode[IntegerDigits@1234212, 3, {2,3,4,1}]];
loop68 =double[seqFromCode[IntegerDigits@12313143, 1, {2,3,4,1}]]; 

loop36 =double[seqFromCode[IntegerDigits@1231,4, {2,4,1,3}]]; 
loop174A =triple[seqFromCode[IntegerDigits@12342321342132,4, {2,3,1,4}]]; 
loop48 = Table[seqFromCode[IntegerDigits@1,2, {4,3,3,1}], {8}]//Flatten

loop164={1,2,3,2,1,4,1,2,3,2,3,4,1,4,3,2,4,1,2,4,3,4,2,1,4,2,3,4,1,4,3,2,3,2,1,4,1,2,3,2,1,2,4,1,4,2,3,2,4,1,4,1,3,2,3,1,4,3,2,4,3,1,3,4,2,3,4,1,3,2,3,1,4,1,4,2,3,2,4,1,4,2,1,2,3,2,1,4,1,2,3,2,3,4,1,4,3,2,4,1,2,4,3,4,2,1,4,2,3,4,1,4,3,2,3,2,1,4,1,2,3,2,1,2,4,1,4,2,3,2,4,1,4,1,3,2,3,1,4,3,2,4,3,1,3,4,2,3,4,1,3,2,3,1,4,1,4,2,3,2,4,1,4,2};

loop172={1,2,3,1,3,4,3,4,1,2,4,2,1,3,2,1,4,2,1,3,2,1,2,3,1,2,4,1,2,3,1,2,4,2,1,4,3,4,3,1,3,2,1,2,4,1,2,1,3,1,3,2,4,3,4,2,1,4,2,3,4,2,1,4,2,4,1,2,4,3,2,4,1,2,4,3,4,2,3,1,3,1,2,1,4,2,1,2,3,1,3,4,3,4,1,2,4,2,1,3,2,1,4,2,1,3,2,1,2,3,1,2,4,1,2,3,1,2,4,2,1,4,3,4,3,1,3,2,1,2,4,1,2,1,3,1,3,2,4,3,4,2,1,4,2,3,4,2,1,4,2,4,1,2,4,3,2,4,1,2,4,3,4,2,3,1,3,1,2,1,4,2}
(* pretty figure 8 *)


loop108=IntegerDigits[123424341242131242143424321241343132434212434231343142123424341242131242143424321241343132434212434231343142];

loop132=IntegerDigits[123423141232134323431232141324321342142313424321242123424313241243123423141232134323431232141324321342142313424321242123424313241243];
loop148=IntegerDigits[1234231314124321234321234214131324321314214343231241314241314213234341241312342313141243212343212342141313243213142143432312413142413142132343412413 ];



loop174={1,2,3,4,2,3,4,2,3,2,1,4,2,3,4,3,2,4,1,2,3,2,4,3,2,4,3,2,1,2,4,3,1,4,3,1,4,3,4,2,1,4,3,1,3,4,1,2,4,3,4,1,3,4,1,3,4,2,1,2,3,4,2,3,4,2,3,2,1,4,2,3,4,3,2,4,1,2,3,2,4,3,2,4,3,2,1,2,4,3,1,4,3,1,4,3,4,2,1,4,3,1,3,4,1,2,4,3,4,1,3,4,1,3,4,2,1,2,3,4,2,3,4,2,3,2,1,4,2,3,4,3,2,4,1,2,3,2,4,3,2,4,3,2,1,2,4,3,1,4,3,1,4,3,4,2,1,4,3,1,3,4,1,2,4,3,4,1,3,4,1,3,4,2};

loop174Rot=RotateLeft[loop174,11];

perms = Permutations[{1,2,3,4}];permsMoving1=perms1=Select[perms, #[[1]]≠1&];
permsOrder2=Select[perms1, #[[#[[{1,2,3,4}]]]]=={1,2,3,4}&];
permsOrder3=Select[perms1, #[[#[[#[[{1,2,3,4}]]]]]]=={1,2,3,4}&];
permsOrder4=Select[Complement[perms1,permsOrder2], #[[#[[#[[#[[{1,2,3,4}]]]]]]]]=={1,2,3,4}&];

CollisionQSW[chain_] :=  Module[{},
Do[If[CollisionTwoTets[chain[[i]],chain[[j]]], Return[True, Module]] , {i,Length[chain]-2}, {j,Length@ chain, i+2,-1}];
False];

(* CollisionQSWSym[chain_] :=  Module[{},
Do[If[CollisionTwoTetsSym[chain[[i]],chain[[j]]], Return[True, Module]] , {i,1, Length[chain]-2}, {j,Length[ chain], i+2,-1}];
False];  *) (* not sure this works...  *) 

cross=Compile[{a,b,c,d,e,f}, {-c e+b f,c d-a f,-b d+a e}];

SepCheckQuick[TetA_,TetB_]:= Norm[Mean[TetA] -Mean[TetB]]> 1.224744871391589  (* Sqrt[3/2] *) ;

SepCheckQuick[TetA_,TetB_]:= Norm[Total[TetA -TetB]]>4  1.224744871391589  (* Sqrt[3/2]  or  2 Sqrt[6] with the 4 *) ;

SepCheckQuickSym[TetA_,TetB_]:= Norm[Mean[TetA]-Mean[TetB]]≥(Sqrt[3/2]) ;

SepPlane[TetA_,TetB_,i_]:=Module[{ff=faces[[i]],vert,normalV,cen=Mean[TetA]},
vert=TetA[[ff[[1]]]];
normalV=((TetA[[ff[[2]]]]-vert))\[Cross]((TetA[[ff[[3]]]]-vert));
(*normalV= cross@@Flatten[{TetA[[ff[[2]]]]-vert,TetA[[ff[[3]]]]-vert}];*) 
If[(cen-vert).normalV≥0, normalV=-normalV];
Chop[ normalV.(TetB[[1]]-vert)]≥0&&
Chop[ normalV.(TetB[[2]]-vert)]≥0&&
Chop[ normalV.(TetB[[3]]-vert)]≥0&&
Chop[ normalV.(TetB[[4]]-vert)]≥0];

SepPlaneSym[TetA_,TetB_,i_]:=Module[{ff=faces[[i]],vert,normalV,cen=Mean[TetA]},
vert=TetA[[ff[[1]]]];
normalV=((TetA[[ff[[2]]]]-vert))\[Cross]((TetA[[ff[[3]]]]-vert)); 
If[(cen-vert).normalV≥0, normalV=-normalV];
normalV.(TetB[[1]]-vert)>0&&normalV.(TetB[[2]]-vert)>0&&
normalV.(TetB[[3]]-vert)>0&&normalV.(TetB[[4]]-vert)>0]

PermMatrixFromPerm[p_] := IdentityMatrix[Length[p]][[p]];
seqFromCode[S_, j_, p_] := Join[S, {j}, Reverse[S], p[[S]], {p[[j]]}, Reverse[p[[S]]]];

SteinhausWords[n_, avoid_:{}] :=Quiet[Flatten[Table@@(
Join[{x/@Range[n]},  {{x[1], Complement[{1,2,3,4}, Flatten[List[avoid]]]}}, Table[{x[i], Delete[{1,2,3,4}, x[i-1]]},{i,2,n}]]),n-1]];

LSequences[L_, init_]:= Join[init, #]& /@ SteinhausWords[L- Length[init], {Last@init}];

AllColumnsHaveClosePair[M1_, tol_]:=With[{M=Transpose[Abs[M1]]},
And@@Table[(Abs[M[[j,1]]-M[[j,2]]]<tol ||
Abs[M[[j,1]]-M[[j,3]]]<tol ||
Abs[M[[j,1]]-M[[j,4]]]<tol ||
Abs[M[[j,2]]-M[[j,3]]]<tol ||
Abs[M[[j,2]]-M[[j,4]]]<tol ||
Abs[M[[j,3]]-M[[j,4]]]<tol ) ,{j,4}]]

OneRowNice[M1_, tol_] := With[{M=Abs[M1]},
Max[Abs[Sort[M[[1]]] -{0,0,0,1}]]<tol ||
Max[Abs[Sort[M[[2]]] -{0,0,0,1}]]<tol ||
Max[Abs[Sort[M[[3]]] -{0,0,0,1}]]<tol ||
Max[Abs[Sort[M[[4]]] -{0,0,0,1}]]<tol ]

BestRotatedSequence[S_] := (ans=SortBy[Table[{i,SeqToDistError[RotateLeft[
If[per==2, double, triple][S],i]]}, {i,Length[S]}], Last][[1]];
{ans[[2]], RotateLeft[S, ans[[1]]], ans[[1]]})

SeqToDistErrorOld[ss_] :=(mats=SeqToMatrixList[ss]//N;
Do[{Table[next=(fromBary[mats[[i]].#1,startV]&)/@IdentityMatrix[4];{Polygon[next[[f]]]},{f,faces}]},{i,Length[mats]}];
dist[startV, next]);

SeqToDistErrorSym[ss_] :=With[{pair=SeqToTetsSym[ss][[{1,-1}]]},
use=Take[SortBy[Range[4], N[Norm[Nearest[pair[[1]], pair[[-1,#]]][[1]]-pair[[-1,#]]],20]&],3];
Max[N[Table[Norm[Nearest[pair[[1]], pair[[-1,i]]][[1]]-pair[[-1,i]]], {i,use}],20]]];

SeqToDistError[ss_] :=With[{pair=SeqToTets[ss][[{1,-1}]]},
use=Take[SortBy[Range[4], N[Norm[Nearest[pair[[1]], pair[[-1,#]]][[1]]-pair[[-1,#]]]]&],3];
Max[N[Table[Norm[Nearest[pair[[1]], pair[[-1,i]]][[1]]-pair[[-1,i]]], {i,use}]]]];

dist[tet1_, tet2_] := Min[Table[Max[Norm/@ ((tet1-tet2[[pp]]))], {pp,perms}]];

ExponentFromSequence[seq_] :=Round[2π/Arg[Select[Eigenvalues[SeqToMatrixList[seq][[-1]]], Im[#]≠0&,1][[1]]]];

triple[z_] := Flatten[{z,z,z}];
double[z_] := Flatten[{z,z}];
fixDC[s_] := DeleteCases[s, {___, b_, b_, ___}];

baryTri[{{x1_,y1_,z1_},{x2_,y2_,z2_},{x3_,y3_,z3_}}, {X_, Y_, Z_} ] := 
{((x3 y2 Z-x2 y3 Z-x3 Y z2+X y3 z2+x2 Y z3-X y2 z3)/(x3 y2 z1-x2 y3 z1-x3 y1 z2+x1 y3 z2+x2 y1 z3-x1 y2 z3)),((x3 y1 Z-x1 y3 Z-x3 Y z1+X y3 z1+x1 Y z3-X y1 z3)/(-x3 y2 z1+x2 y3 z1+x3 y1 z2-x1 y3 z2-x2 y1 z3+x1 y2 z3)),((x2 y1 Z-x1 y2 Z-x2 Y z1+X y2 z1+x1 Y z2-X y1 z2)/(x3 y2 z1-x2 y3 z1-x3 y1 z2+x1 y3 z2+x2 y1 z3-x1 y2 z3))};



bary[{{x1_,y1_,z1_},{x2_,y2_,z2_},{x3_,y3_,z3_},{x4_,y4_,z4_}}, {X_, Y_, Z_} ] := {(-x2 y3 Z+x2 y4 Z+X y3 z2-X y4 z2+x2 Y z3-X y2 z3+X y4 z3-x2 y4 z3+x4 (-y2 Z+y3 Z+Y z2-y3 z2-Y z3+y2 z3)+(-x2 Y+X y2-X y3+x2 y3) z4+x3 (-y4 Z-Y z2+y4 z2+y2 (Z-z4)+Y z4))/(-x2 y3 z1+x2 y4 z1+x1 y3 z2-x1 y4 z2+x2 y1 z3-x1 y2 z3+x1 y4 z3-x2 y4 z3+x4 (-y2 z1+y3 z1+y1 z2-y3 z2-y1 z3+y2 z3)+(-x2 y1+x1 y2-x1 y3+x2 y3) z4+x3 (-y4 z1-y1 z2+y4 z2+y2 (z1-z4)+y1 z4)),(-x1 y3 Z+x1 y4 Z+X y3 z1-X y4 z1+x1 Y z3-X y1 z3+X y4 z3-x1 y4 z3+x4 (-y1 Z+y3 Z+Y z1-y3 z1-Y z3+y1 z3)+(-x1 Y+X y1-X y3+x1 y3) z4+x3 (-y4 Z-Y z1+y4 z1+y1 (Z-z4)+Y z4))/(x2 y3 z1-x2 y4 z1-x1 y3 z2+x1 y4 z2-x2 y1 z3+x1 y2 z3-x1 y4 z3+x2 y4 z3+x4 (y2 z1-y3 z1-y1 z2+y3 z2+y1 z3-y2 z3)+x2 y1 z4-x1 y2 z4+x1 y3 z4-x2 y3 z4+x3 (y4 z1+y1 z2-y4 z2-y1 z4+y2 (-z1+z4))),(-x1 y2 Z+x1 y4 Z+X y2 z1-X y4 z1+x1 Y z2-X y1 z2+X y4 z2-x1 y4 z2+x4 (-y1 Z+y2 Z+Y z1-y2 z1-Y z2+y1 z2)+(-x1 Y+X y1-X y2+x1 y2) z4+x2 (-y4 Z-Y z1+y4 z1+y1 (Z-z4)+Y z4))/(-x2 y3 z1+x2 y4 z1+x1 y3 z2-x1 y4 z2+x2 y1 z3-x1 y2 z3+x1 y4 z3-x2 y4 z3+x4 (-y2 z1+y3 z1+y1 z2-y3 z2-y1 z3+y2 z3)+(-x2 y1+x1 y2-x1 y3+x2 y3) z4+x3 (-y4 z1-y1 z2+y4 z2+y2 (z1-z4)+y1 z4)),(-x1 y2 Z+x1 y3 Z+X y2 z1-X y3 z1+x1 Y z2-X y1 z2+X y3 z2-x1 y3 z2+x3 (-y1 Z+y2 Z+Y z1-y2 z1-Y z2+y1 z2)+(-x1 Y+X y1-X y2+x1 y2) z3+x2 (-y3 Z-Y z1+y3 z1+y1 (Z-z3)+Y z3))/(x2 y3 z1-x2 y4 z1-x1 y3 z2+x1 y4 z2-x2 y1 z3+x1 y2 z3-x1 y4 z3+x2 y4 z3+x4 (y2 z1-y3 z1-y1 z2+y3 z2+y1 z3-y2 z3)+x2 y1 z4-x1 y2 z4+x1 y3 z4-x2 y3 z4+x3 (y4 z1+y1 z2-y4 z2-y1 z4+y2 (-z1+z4)))};

baryCom = Compile[{x1,y1,z1,x2,y2,z2,x3,y3,z3,x4,y4,z4, X, Y, Z}, {(-x2 y3 Z+x2 y4 Z+X y3 z2-X y4 z2+x2 Y z3-X y2 z3+X y4 z3-x2 y4 z3+x4 (-y2 Z+y3 Z+Y z2-y3 z2-Y z3+y2 z3)+(-x2 Y+X y2-X y3+x2 y3) z4+x3 (-y4 Z-Y z2+y4 z2+y2 (Z-z4)+Y z4))/(-x2 y3 z1+x2 y4 z1+x1 y3 z2-x1 y4 z2+x2 y1 z3-x1 y2 z3+x1 y4 z3-x2 y4 z3+x4 (-y2 z1+y3 z1+y1 z2-y3 z2-y1 z3+y2 z3)+(-x2 y1+x1 y2-x1 y3+x2 y3) z4+x3 (-y4 z1-y1 z2+y4 z2+y2 (z1-z4)+y1 z4)),(-x1 y3 Z+x1 y4 Z+X y3 z1-X y4 z1+x1 Y z3-X y1 z3+X y4 z3-x1 y4 z3+x4 (-y1 Z+y3 Z+Y z1-y3 z1-Y z3+y1 z3)+(-x1 Y+X y1-X y3+x1 y3) z4+x3 (-y4 Z-Y z1+y4 z1+y1 (Z-z4)+Y z4))/(x2 y3 z1-x2 y4 z1-x1 y3 z2+x1 y4 z2-x2 y1 z3+x1 y2 z3-x1 y4 z3+x2 y4 z3+x4 (y2 z1-y3 z1-y1 z2+y3 z2+y1 z3-y2 z3)+x2 y1 z4-x1 y2 z4+x1 y3 z4-x2 y3 z4+x3 (y4 z1+y1 z2-y4 z2-y1 z4+y2 (-z1+z4))),(-x1 y2 Z+x1 y4 Z+X y2 z1-X y4 z1+x1 Y z2-X y1 z2+X y4 z2-x1 y4 z2+x4 (-y1 Z+y2 Z+Y z1-y2 z1-Y z2+y1 z2)+(-x1 Y+X y1-X y2+x1 y2) z4+x2 (-y4 Z-Y z1+y4 z1+y1 (Z-z4)+Y z4))/(-x2 y3 z1+x2 y4 z1+x1 y3 z2-x1 y4 z2+x2 y1 z3-x1 y2 z3+x1 y4 z3-x2 y4 z3+x4 (-y2 z1+y3 z1+y1 z2-y3 z2-y1 z3+y2 z3)+(-x2 y1+x1 y2-x1 y3+x2 y3) z4+x3 (-y4 z1-y1 z2+y4 z2+y2 (z1-z4)+y1 z4)),(-x1 y2 Z+x1 y3 Z+X y2 z1-X y3 z1+x1 Y z2-X y1 z2+X y3 z2-x1 y3 z2+x3 (-y1 Z+y2 Z+Y z1-y2 z1-Y z2+y1 z2)+(-x1 Y+X y1-X y2+x1 y2) z3+x2 (-y3 Z-Y z1+y3 z1+y1 (Z-z3)+Y z3))/(x2 y3 z1-x2 y4 z1-x1 y3 z2+x1 y4 z2-x2 y1 z3+x1 y2 z3-x1 y4 z3+x2 y4 z3+x4 (y2 z1-y3 z1-y1 z2+y3 z2+y1 z3-y2 z3)+x2 y1 z4-x1 y2 z4+x1 y3 z4-x2 y3 z4+x3 (y4 z1+y1 z2-y4 z2-y1 z4+y2 (-z1+z4)))}];


fromBary[bary_, pts4_] := Sum[bary[[i]]*pts4[[i]], {i,4}];

(* insideTetQ[verts_, pt_] := bary[verts, pt]*) 

allGood[L_] := And@@Thread[0<L<1];
inside3DQ[pts_, pt_] := allGood[bary[pts,pt]]
inside3DQ[tet={{0,0,0},{1,0,0}, {1/2, Sqrt[3]/2,0}, {1/2, 1/3, Sqrt[3]/4}}, {.5,.1,.1}]

intersect3D[tri_,{P_,Q_}] := FindInstance [allGood[baryTri[tri, (1-t)P + t Q]] &&0<t<1,t ]

intersect[tet1_, tet2_] := (Or@@(inside3DQ[tet1, #] & /@ tet2))&&(Or@@(inside3DQ[tet2, #] & /@ tet1));

snakeQ[L_] := And@@(ff=Flatten[Table[!intersect[L[[i]], L[[j]]], {i, 1, Length[L]-2}, {j, i+2, Length[L]}],1])

allGood[L_] := And@@Thread[0<L<1];

planeEqn[tri_] :=Module[{a,b,c,d},  (vals={a,b,c,d}/.First@Solve[{c==1,
(a x+b y+c z==d)/. Thread[{x,y,z}->tri[[1]]],(a x+b y+c z==d)/. Thread[{x,y,z}->tri[[2]]],(a x+b y+c z==d)/. Thread[{x,y,z}->tri[[3]]]}];
Most[vals].{x,y,z}==vals[[-1]])];

intersectTime[{{x1_, y1_,z1_}, {x2_, y2_,z2_}, {x3_, y3_,z3_}}, {P_, Q_}]:= 
Module[{t},sss=Solve[
planeEqn[{{x1, y1,z1}, {x2, y2,z2}, {x3, y3,z3}}] /. (Thread[{x,y,z}-> (t)P +(1- t) Q]),t, Reals];
If[sss=={}, {}, t/. First@(sss)]];

lineTriangleStrikeQ[{{x1_, y1_,z1_}, {x2_, y2_,z2_}, {x3_, y3_,z3_}}, {P_, Q_}] := Module[{}, 
t = intersectTime[{{x1, y1,z1}, {x2, y2,z2}, {x3, y3,z3}}, {P, Q}];
mPrint[t];
(zzz=(t=!={})) &&0<t<1 &&(aaa=allGood[baryTri[{{x1, y1,z1}, {x2, y2,z2}, {x3, y3,z3}},t P + (1-t) Q]])];

facesOfTet[tet_] :={ tet[[{1,2,3}]], tet[[{1,2,4}]], tet[[{2,3,4}]], tet[[{1,3,4}]]};
edgesOfTet[tet_] :={ tet[[{1,2}]], tet[[{2,4}]], tet[[{2,3}]], tet[[{1,3}]], tet[[{1,4}]], tet[[{3,4}]]};

lineTetStrikeQ[tet_,{P_, Q_}]:=Or@@((tttLTS=lineTriangleStrikeQ[#, {P,Q}]& /@ facesOfTet[tet]));

intersect[tet1_,tet2_]:=(ooo=Or@@ (ttt= Table[lineTetStrikeQ[tet1,ee], {ee,  edgesOfTet[tet2]}]);mPrint[ttt]; ooo) ||(Or@@  Table[lineTetStrikeQ[tet2,ee], {ee,  edgesOfTet[tet1]}]) 

snakeQ[L_] := And@@(ff=Flatten[Table[mPrint[i,j];!intersect[L[[i]], L[[j]]], {i, 1, Length[L]-2}, {j, i+2, Length[L]}],1])

rr=Append[Range[4], "—"];
faces = {{2,3,4},{4,3,1},{1,2,4},{3,2,1}}; 
startV=N[{{0,0,(Sqrt[(2/3)])-(1/(2 (Sqrt[6])))},{-(1/(2 (Sqrt[3]))),-(1/2),-(1/(2 (Sqrt[6])))},{-(1/(2 (Sqrt[3]))),(1/2),-(1/(2 (Sqrt[6])))},{(1/(Sqrt[3])),0,-(1/(2 (Sqrt[6])))}}];


startVSym={{0,0,(Sqrt[(2/3)])-(1/(2 (Sqrt[6])))},{-(1/(2 (Sqrt[3]))),-(1/2),-(1/(2 (Sqrt[6])))},{-(1/(2 (Sqrt[3]))),(1/2),-(1/(2 (Sqrt[6])))},{(1/(Sqrt[3])),0,-(1/(2 (Sqrt[6])))}};

fromBary[bary_, pts4_] :=bary. pts4;

error[mat_]:=Max[Flatten[Abs[IdentityMatrix[4]-mat]]];
SeqToMatrixList[seq_]:=FoldList[Dot,M[seq[[1]]], N[M /@Rest@ seq]];
SeqToMatrixListSym[seq_]:=FoldList[Dot,M[seq[[1]]], M /@Rest@ seq];

Clear[SeqToMatrixLast];
SeqToMatrixLast[seq_]:=Dot @@ (N[M /@seq]);

SeqToMatrixLast[seq_]:=Fold[Dot,N@M[seq[[1]]], N[M /@Rest@ seq]];

SeqToMatrixLastSym[seq_]:=Fold[Dot,M[seq[[1]]], M /@Rest@ seq];

MatricesToTetsFaces[mats_] := Table[qqq=fromBary[mats[[i]] . #,startV]& /@ IdentityMatrix@4;Table[qqq[[f]], {f, faces}], {i,Length[mats]}]


MatricesToTets[mats_] := Table[fromBary[mats[[i]].#,startV]& /@ IdentityMatrix@4, {i,Length[mats]}];


MatricesToTetsSym[mats_] := Table[fromBary[mats[[i]].#,startVSym]& /@ IdentityMatrix@4, {i,Length[mats]}];

SeqToTets[seq_] := MatricesToTets[SeqToMatrixList[seq]];
SeqToTetsSym[seq_] := MatricesToTetsSym[SeqToMatrixListSym[seq]];
polygons[seq_]:=Polygon/@MatricesToTetsFaces[SeqToMatrixList[seq]];




dist[tet1_, tet2_] := Min[Table[Max[Norm/@ ((tet1-tet2[[pp]]))], {pp,perms}]];

{M[1], M[2], M[3], M[4]} ={{{-1,0,0,0},{(2/3),1,0,0},{(2/3),0,1,0},{(2/3),0,0,1}},{{1,(2/3),0,0},{0,-1,0,0},{0,(2/3),1,0},{0,(2/3),0,1}},{{1,0,(2/3),0},{0,1,(2/3),0},{0,0,-1,0},{0,0,(2/3),1}},{{1,0,0,(2/3)},{0,1,0,(2/3)},{0,0,1,(2/3)},{0,0,0,-1}}};

M3[k_] := 3 M[k];
Mx[k_] :=  M[k]/. 2/3 :> x;
nn=11;
cols=Lighter/@ Flatten[Table[{Red,Green, Blue}, {500}]/. {Red ->Table[Red, {nn}],Green ->Table[Green,{nn}],Blue -> Table[Blue,{nn}]}];

normal[L_] :=Module[{p1, p2, p3},{p1, p2, p3} = L[[{1,2,3}]]; Cross[(p2-p1), p3-p1]/Norm[Cross[(p2-p1), p3-p1]]]

collisionQ::usage = "collisionQ tests whether two tetrahedra collide";

collisionQ[tets1_, ϵ_]:= Module[{},

TetVerticesRaw=Flatten[tets1,1];
TetVertices=N@Transpose[TetVerticesRaw];
TetTriangles = Transpose[Flatten[Table[t[[f]], {t, tets1}, {f, faces}],1]]/. x_?VectorQ :> Position[TetVerticesRaw,x,{1},1][[1,1]];

DimT=Dimensions[TetTriangles];
NumTets=DimT[[2]]/4;
CollisionWithThisTet=0*Range[NumTets];(*CHANGED*)CollisionAnyPair=0;(*Initialize*)Range4=Range[4];(*{1,2,3,4}*)Do[(*beginning of Do ITetA loop*)Ind1=TetTriangles[[All,4*ITetA]];
Ind2=TetTriangles[[All,4*ITetA-1]];
TetAVertexIndices=Union[Ind1,Ind2];(*all vertex indices for TetA*)Va=TetVertices[[All,TetAVertexIndices]];(*4 vertices of TetA*)Do[(*beginning of Do ITetB loop*)Ind1=TetTriangles[[All,4*ITetB]];
Ind2=TetTriangles[[All,4*ITetB-1]];
TetBVertexIndices=Union[Ind1,Ind2];(*all vertex indices for TetB*)Vb=TetVertices[[All,TetBVertexIndices]];(*4 vertices of TetB*)(*----------------------------*)FaceASeparates={1,1,1,1};(*initialize*)FaceBSeparates={1,1,1,1};(*initialize*)(*----------------------------*)(*Check if all vertices of TetB are on one side of face of TetA*)Do[(*beginning of Do Ia loop*)NotIa=Delete[Range4,Ia];(*ind of verts in face opposite vert(Ia)*)VertexI=Va[[All,Ia]];
FaceCenter=(1/3)*Va[[All,NotIa]].{1,1,1};(*CHANGED mult order*)Edge12=Va[[All,NotIa[[1]]]]-Va[[All,NotIa[[2]]]];
Edge13=Va[[All,NotIa[[1]]]]-Va[[All,NotIa[[3]]]];
NormalIaTmp=Cross[Edge12,Edge13];(*normal vector to face of TetA*)SignNormal=NormalIaTmp.(VertexI-FaceCenter);
NormalIa=If[SignNormal>0,-NormalIaTmp,NormalIaTmp];
Do[(*Beginning of Kb loop giving vertex(Kb) of TetB*)(*test if vertex(Kb) of TetB on inward side of face(Ia) of TetA*)ProjectionAB=(Vb[[All,Kb]]-Va[[All,NotIa[[1]]]]).NormalIa;
tmp=If[ProjectionAB<-ϵ,0,1];
FaceASeparates[[Ia]]=Min[FaceASeparates[[Ia]],tmp];,{Kb,Range4}];(*end of Do Kb loop*),{Ia,Range4}];(*end of Do Ia loop*)(*----------------------------*)(*Check if all vertices of TetA are on one side of face of TetB*)Do[(*beginning of Do Ib loop*)NotIb=Delete[Range4,Ib];(*ind of verts in face opposite vert(Ib)*)VertexI=Vb[[All,Ib]];
FaceCenter=(1/3)*Vb[[All,NotIb]].{1,1,1};(*CHANGED mult order*)Edge12=Vb[[All,NotIb[[1]]]]-Vb[[All,NotIb[[2]]]];
Edge13=Vb[[All,NotIb[[1]]]]-Vb[[All,NotIb[[3]]]];
NormalIbTmp=Cross[Edge12,Edge13];(*normal vector to face of TetB*)SignNormal=NormalIbTmp.(VertexI-FaceCenter);
NormalIb=If[SignNormal>0,-NormalIbTmp,NormalIbTmp];
Do[(*Beginning of Ka loop giving vertex(Ka) of TetA*)(*test if vertex(Ka) of TetB on inward side of face(Ib) of TetB*)ProjectionAB=(Va[[All,Ka]]-Vb[[All,NotIb[[1]]]]).NormalIb;
tmp=If[ProjectionAB<-ϵ,0,1];
FaceBSeparates[[Ib]]=Min[FaceBSeparates[[Ib]],tmp];,{Ka,Range4}];(*end of Do Ka loop*),{Ib,Range4}];(*end of Do Ib loop*)CollisionThisPair=If[Max[FaceASeparates,FaceBSeparates]>1/2,0,1];
tmp=If[CollisionThisPair>1/2,ITetB,0];(*CHANGED>0 to>1/2*)CollisionWithThisTet[[ITetA]]=Max[CollisionWithThisTet[[ITetA]],tmp];(*CHANGED*)CollisionAnyPair=Max[CollisionAnyPair,CollisionThisPair];,{ITetB,Range[ITetA+1,NumTets]}];(*end of Do ITetB loop*),{ITetA,Range[NumTets-1]}];(*end of Do ITetA loop*)CollisionAnyPair===1]

collisionQ::usage = "collisionQ tests whether two tetrahedra collide";

collisionQ[tets1_, ϵ_]:= (

TetVerticesRaw=Flatten[tets1,1];
TetVertices=N@Transpose[TetVerticesRaw];
TetTriangles = Transpose[Flatten[Table[t[[f]], {t, tets1}, {f, faces}],1]]/. x_?VectorQ :> Position[TetVerticesRaw,x,{1},1][[1,1]];

DimT=Dimensions[TetTriangles];
NumTets=DimT[[2]]/4;
CollisionWithThisTet=0*Range[NumTets];(*CHANGED*)CollisionAnyPair=0;(*Initialize*)Range4=Range[4];(*{1,2,3,4}*)Do[(*beginning of Do ITetA loop*)Ind1=TetTriangles[[All,4*ITetA]];
Ind2=TetTriangles[[All,4*ITetA-1]];
TetAVertexIndices=Union[Ind1,Ind2];(*all vertex indices for TetA*)Va=TetVertices[[All,TetAVertexIndices]];(*4 vertices of TetA*)Do[(*beginning of Do ITetB loop*)Ind1=TetTriangles[[All,4*ITetB]];
Ind2=TetTriangles[[All,4*ITetB-1]];
TetBVertexIndices=Union[Ind1,Ind2];(*all vertex indices for TetB*)Vb=TetVertices[[All,TetBVertexIndices]];(*4 vertices of TetB*)(*----------------------------*)FaceASeparates={1,1,1,1};(*initialize*)FaceBSeparates={1,1,1,1};(*initialize*)(*----------------------------*)(*Check if all vertices of TetB are on one side of face of TetA*)Do[(*beginning of Do Ia loop*)NotIa=Delete[Range4,Ia];(*ind of verts in face opposite vert(Ia)*)VertexI=Va[[All,Ia]];
FaceCenter=(1/3)*Va[[All,NotIa]].{1,1,1};(*CHANGED mult order*)Edge12=Va[[All,NotIa[[1]]]]-Va[[All,NotIa[[2]]]];
Edge13=Va[[All,NotIa[[1]]]]-Va[[All,NotIa[[3]]]];
NormalIaTmp=Cross[Edge12,Edge13];(*normal vector to face of TetA*)SignNormal=NormalIaTmp.(VertexI-FaceCenter);
NormalIa=If[SignNormal>0,-NormalIaTmp,NormalIaTmp];
Do[(*Beginning of Kb loop giving vertex(Kb) of TetB*)(*test if vertex(Kb) of TetB on inward side of face(Ia) of TetA*)ProjectionAB=(Vb[[All,Kb]]-Va[[All,NotIa[[1]]]]).NormalIa;
tmp=If[ProjectionAB<-ϵ,0,1];
FaceASeparates[[Ia]]=Min[FaceASeparates[[Ia]],tmp];,{Kb,Range4}];(*end of Do Kb loop*),{Ia,Range4}];(*end of Do Ia loop*)(*----------------------------*)(*Check if all vertices of TetA are on one side of face of TetB*)Do[(*beginning of Do Ib loop*)NotIb=Delete[Range4,Ib];(*ind of verts in face opposite vert(Ib)*)VertexI=Vb[[All,Ib]];
FaceCenter=(1/3)*Vb[[All,NotIb]].{1,1,1};(*CHANGED mult order*)Edge12=Vb[[All,NotIb[[1]]]]-Vb[[All,NotIb[[2]]]];
Edge13=Vb[[All,NotIb[[1]]]]-Vb[[All,NotIb[[3]]]];
NormalIbTmp=Cross[Edge12,Edge13];(*normal vector to face of TetB*)SignNormal=NormalIbTmp.(VertexI-FaceCenter);
NormalIb=If[SignNormal>0,-NormalIbTmp,NormalIbTmp];
Do[(*Beginning of Ka loop giving vertex(Ka) of TetA*)(*test if vertex(Ka) of TetB on inward side of face(Ib) of TetB*)ProjectionAB=(Va[[All,Ka]]-Vb[[All,NotIb[[1]]]]).NormalIb;
tmp=If[ProjectionAB<-ϵ,0,1];
FaceBSeparates[[Ib]]=Min[FaceBSeparates[[Ib]],tmp];,{Ka,Range4}];(*end of Do Ka loop*),{Ib,Range4}];(*end of Do Ib loop*)CollisionThisPair=If[Max[FaceASeparates,FaceBSeparates]>1/2,0,1];
tmp=If[CollisionThisPair>1/2,ITetB,0];(*CHANGED>0 to>1/2*)CollisionWithThisTet[[ITetA]]=Max[CollisionWithThisTet[[ITetA]],tmp];(*CHANGED*)CollisionAnyPair=Max[CollisionAnyPair,CollisionThisPair];,{ITetB,Range[ITetA+1,NumTets]}];(*end of Do ITetB loop*),{ITetA,Range[NumTets-1]}];(*end of Do ITetA loop*)CollisionAnyPair===1)

TH[L_]:= Table[{r Cos[n θ],r Sin[n θ],n h},{n,0,L+2}];
THPoint[L_]:={r Cos[L θ],r Sin[L θ],L h};

keyRule =  -4 I+2 (Sqrt[5])+3 I u+3 (Sqrt[5]) u+6 I (u^2)|-4 I+2 (Sqrt[5])+(3 I +3 (Sqrt[5]) )u+6 I (u^2)->((κ u I)/24);

zVal=(1/3) (-2+I(Sqrt[5]) );
 zRule = z -> zVal;
uRule = u -> (zVal^L);
rationalize[expr_]:=expr/. {(1/(Complex[0,b_]+(Sqrt[5]) a_)):>((Complex[0,b]-(Sqrt[5]) a)/(-b-5 (a^2))), (((Complex[0,b_]+(Sqrt[5]) a_))^(j_Integer/; j<0)):>((((Complex[0,b]-(Sqrt[5]) a))^(-j))/(((-(b^2)-5 (a^2)))^(-j)))}
rationalize1[expr_]:=expr/. {(1/(Complex[0,b_]+(Sqrt[5]) a_)):>((Complex[0,b]-(Sqrt[5]) a)/(-b-5 (a^2))), (((Complex[0,b_]+(Sqrt[5]) a_))^(j_Integer/; j<0)):>((((Complex[0,b]-(Sqrt[5]) a))^(-j))/(((-(b^2)-5 (a^2)))^(-j)))}/.( Complex[0, a_Integer]+jj_Integer (Sqrt[5])):> (-I a + jj (Sqrt[-5]))/I/. (Sqrt[5])->- I σ ;;

KSymδ={{(1/3125)(1265-1380 L+138 (Sqrt[5]) Sin[δ]+240 (-14+9 L) ((Sin[δ])^2)-216 (Sqrt[5]) ((Sin[δ])^3)+1440 ((Sin[δ])^4)+6 (1-2 ((Sin[(δ/2)])^2)) (310+230 L-23 (Sqrt[5]) Sin[δ]-120 (-7+L) ((Sin[δ])^2)+12 (Sqrt[5]) ((Sin[δ])^3))),(1/3125)6 (-115-230 L+138 (Sqrt[5]) Sin[δ]+(195+360 L) ((Sin[δ])^2)-216 (Sqrt[5]) ((Sin[δ])^3)+180 ((Sin[δ])^4)+(1-2 ((Sin[(δ/2)])^2)) (115+230 L-138 (Sqrt[5]) Sin[δ]-60 (-5+2 L) ((Sin[δ])^2)+72 (Sqrt[5]) ((Sin[δ])^3))),-(1/3125)6 (10+10 L+9 (Sqrt[5]) Sin[δ]) (23-36 ((Sin[δ])^2)+(1-2 ((Sin[(δ/2)])^2)) (-23+12 ((Sin[δ])^2))),-(1/3125)6 (345+230 L-138 (Sqrt[5]) Sin[δ]-15 (35+24 L) ((Sin[δ])^2)+216 (Sqrt[5]) ((Sin[δ])^3)+180 ((Sin[δ])^4)+(1-2 ((Sin[(δ/2)])^2)) (-345-230 L+138 (Sqrt[5]) Sin[δ]+60 (9+2 L) ((Sin[δ])^2)-72 (Sqrt[5]) ((Sin[δ])^3)))},{-(1/3125)2 (-90-470 L+3 (Sqrt[5]) (89-80 L) Sin[δ]+(-620+840 L) ((Sin[δ])^2)+12 (Sqrt[5]) (-57+10 L) ((Sin[δ])^3)+900 ((Sin[δ])^4)+(1-2 ((Sin[(δ/2)])^2)) (90+470 L+3 (Sqrt[5]) (-89+80 L) Sin[δ]+(2040-480 L) ((Sin[δ])^2)+288 (Sqrt[5]) ((Sin[δ])^3))),(1/3125)(3595+940 L+12 (Sqrt[5]) (-27+40 L) Sin[δ]-30 (65+56 L) ((Sin[δ])^2)-48 (Sqrt[5]) (-26+5 L) ((Sin[δ])^3)-720 ((Sin[δ])^4)+(1-2 ((Sin[(δ/2)])^2)) (-470-940 L+12 (Sqrt[5]) (27-40 L) Sin[δ]+120 (7+8 L) ((Sin[δ])^2)-936 (Sqrt[5]) ((Sin[δ])^3))),-(1/3125)2 (-470-470 L-3 (Sqrt[5]) (221+80 L) Sin[δ]+120 (-2+7 L) ((Sin[δ])^2)+12 (Sqrt[5]) (73+10 L) ((Sin[δ])^3)+540 ((Sin[δ])^4)+(1-2 ((Sin[(δ/2)])^2)) (470+470 L+3 (Sqrt[5]) (221+80 L) Sin[δ]+(600-480 L) ((Sin[δ])^2)-432 (Sqrt[5]) ((Sin[δ])^3))),(1/3125)2 (705+470 L+6 (Sqrt[5]) (13+40 L) Sin[δ]-15 (143+56 L) ((Sin[δ])^2)-24 (Sqrt[5]) (-6+5 L) ((Sin[δ])^3)+1080 ((Sin[δ])^4)-(1-2 ((Sin[(δ/2)])^2)) (705+470 L+6 (Sqrt[5]) (13+40 L) Sin[δ]-60 (33+8 L) ((Sin[δ])^2)+108 (Sqrt[5]) ((Sin[δ])^3)))},{(1/3125)(620+460 L+6 (Sqrt[5]) (139-160 L) Sin[δ]-80 (-20+9 L) ((Sin[δ])^2)+24 (Sqrt[5]) (-97+20 L) ((Sin[δ])^3)-720 ((Sin[δ])^4)+(1-2 ((Sin[(δ/2)])^2)) (-620-460 L+6 (Sqrt[5]) (-139+160 L) Sin[δ]+240 (-9+L) ((Sin[δ])^2)+936 (Sqrt[5]) ((Sin[δ])^3))),(1/3125)2 (115+230 L-6 (Sqrt[5]) (63+80 L) Sin[δ]+(1245-360 L) ((Sin[δ])^2)+24 (Sqrt[5]) (-1+10 L) ((Sin[δ])^3)-900 ((Sin[δ])^4)+(1-2 ((Sin[(δ/2)])^2)) (-115-230 L+6 (Sqrt[5]) (63+80 L) Sin[δ]+60 (-29+2 L) ((Sin[δ])^2)+288 (Sqrt[5]) ((Sin[δ])^3))),(1/3125)(3585+460 L-460 (1+L) (1-2 ((Sin[(δ/2)])^2))-6 (Sqrt[5]) (91+160 L) Sin[δ]+6 (Sqrt[5]) (91+160 L) (1-2 ((Sin[(δ/2)])^2)) Sin[δ]-720 (7+L) ((Sin[δ])^2)+240 (19+L) (1-2 ((Sin[(δ/2)])^2)) ((Sin[δ])^2)+24 (Sqrt[5]) (-7+20 L) ((Sin[δ])^3)+216 (Sqrt[5]) (1-2 ((Sin[(δ/2)])^2)) ((Sin[δ])^3)+2160 ((Sin[δ])^4)),(1/3125)2 (345+230 L-6 (Sqrt[5]) (143+80 L) Sin[δ]+(915-360 L) ((Sin[δ])^2)+24 (Sqrt[5]) (39+10 L) ((Sin[δ])^3)-540 ((Sin[δ])^4)+(1-2 ((Sin[(δ/2)])^2)) (-345-230 L+6 (Sqrt[5]) (143+80 L) Sin[δ]+60 (-15+2 L) ((Sin[δ])^2)-432 (Sqrt[5]) ((Sin[δ])^3)))},{(1/3125)(1060-20 L+6 (Sqrt[5]) (-73+80 L) Sin[δ]+40 (13+6 L) ((Sin[δ])^2)+24 (Sqrt[5]) (49-10 L) ((Sin[δ])^3)+1080 ((Sin[δ])^4)-2 (1-2 ((Sin[(δ/2)])^2)) (530-10 L+3 (Sqrt[5]) (-73+80 L) Sin[δ]+120 (-5+2 L) ((Sin[δ])^2)+216 (Sqrt[5]) ((Sin[δ])^3))),-(1/3125)2 (5+10 L-6 (Sqrt[5]) (21+40 L) Sin[δ]+(855-120 L) ((Sin[δ])^2)+24 (Sqrt[5]) (-2+5 L) ((Sin[δ])^3)-720 ((Sin[δ])^4)+(1-2 ((Sin[(δ/2)])^2)) (-5-10 L+6 (Sqrt[5]) (21+40 L) Sin[δ]+60 (-7+4 L) ((Sin[δ])^2)+36 (Sqrt[5]) ((Sin[δ])^3))),-(1/3125)2 (10+10 L-3 (Sqrt[5]) (77+80 L) Sin[δ]-120 (10+L) ((Sin[δ])^2)+12 (Sqrt[5]) (1+10 L) ((Sin[δ])^3)+540 ((Sin[δ])^4)+(1-2 ((Sin[(δ/2)])^2)) (-10-10 L+3 (Sqrt[5]) (77+80 L) Sin[δ]+120 (11+2 L) ((Sin[δ])^2)+216 (Sqrt[5]) ((Sin[δ])^3))),(1/3125)(3095-20 L+12 (Sqrt[5]) (61+40 L) Sin[δ]+30 (-23+8 L) ((Sin[δ])^2)-48 (Sqrt[5]) (18+5 L) ((Sin[δ])^3)+(1-2 ((Sin[(δ/2)])^2)) (30+20 L-12 (Sqrt[5]) (61+40 L) Sin[δ]-120 (-9+4 L) ((Sin[δ])^2)+648 (Sqrt[5]) ((Sin[δ])^3)))}};

KSym=KSymδ/.δ-> (L+1)θ;

FinalTetrahedron//Clear;
FinalTetrahedronBad[L1_] :=( Transpose[  KSym/. L->L1   ] ).(THPoint/@ Range[-1,2]);
FinalTetrahedron[L1_] :=( (*Transpose[*)  KSym/. L->L1  (* ]*) ).(THPoint/@ Range[-1,2]);
FinalTetrahedron[L1_]:=Table[fromBary[#[[i]]& /@(KSym/. L->L1), (THPoint/@ Range[-1,2])], {i,4}]

planeToPointGlobalInsideTriangle[pt_, face_] := Module[{ans, a,b,c,x1,y1,z1,x2,y2,z2,x3,y3,z3, xsol, ysol}, 
{a,b,c}=pt;
{{x1,y1,z1},{x2,y2,z2},{x3,y3,z3}}=face;

{xsol, ysol}={-((4 (a (x1-x3)-x1 x3+(x3^2)+b y1-b y3-y1 y3+(y3^2)+c z1-c z3-z1 z3+(z3^2)) ((x2^2)-2 x2 x3+(x3^2)+(y2^2)-2 y2 y3+(y3^2)+(z2^2)-2 z2 z3+(z3^2))+4 (x2 x3-(x3^2)+x1 (-x2+x3)-y1 y2+y1 y3+y2 y3-(y3^2)-z1 z2+z1 z3+z2 z3-(z3^2)) (a (x2-x3)-x2 x3+(x3^2)+b y2-b y3-y2 y3+(y3^2)+c z2-c z3-z2 z3+(z3^2)))/(-4 ((x1^2)-2 x1 x3+(x3^2)+(y1^2)-2 y1 y3+(y3^2)+(z1^2)-2 z1 z3+(z3^2)) ((x2^2)-2 x2 x3+(x3^2)+(y2^2)-2 y2 y3+(y3^2)+(z2^2)-2 z2 z3+(z3^2))+4 (((x1 (x2-x3)-x2 x3+(x3^2)+y1 y2-y1 y3-y2 y3+(y3^2)+z1 z2-z1 z3-z2 z3+(z3^2)))^2))),(-x2 x3 (y1^2)+(x3^2) (y1^2)+x1 x3 y1 y2-(x3^2) y1 y2+x1 x2 y1 y3-2 x1 x3 y1 y3+x2 x3 y1 y3-(x1^2) y2 y3+x1 x3 y2 y3+(x1^2) (y3^2)-x1 x2 (y3^2)-c x1 x2 z1+c x1 x3 z1+c x2 x3 z1-c (x3^2) z1-c y1 y2 z1+c y1 y3 z1+c y2 y3 z1-c (y3^2) z1-x2 x3 (z1^2)+(x3^2) (z1^2)-y2 y3 (z1^2)+(y3^2) (z1^2)+c (x1^2) z2-2 c x1 x3 z2+c (x3^2) z2+c (y1^2) z2-2 c y1 y3 z2+c (y3^2) z2+x1 x3 z1 z2-(x3^2) z1 z2+y1 y3 z1 z2-(y3^2) z1 z2-c (x1^2) z3+c x1 x2 z3+c x1 x3 z3-c x2 x3 z3-c (y1^2) z3+c y1 y2 z3+c y1 y3 z3-c y2 y3 z3+x1 x2 z1 z3-2 x1 x3 z1 z3+x2 x3 z1 z3+y1 y2 z1 z3-2 y1 y3 z1 z3+y2 y3 z1 z3-(x1^2) z2 z3+x1 x3 z2 z3-(y1^2) z2 z3+y1 y3 z2 z3+(x1^2) (z3^2)-x1 x2 (z3^2)+(y1^2) (z3^2)-y1 y2 (z3^2)+b (-(x3^2) y1+(x3^2) y2+x2 x3 (y1-y3)+(x1^2) (y2-y3)+x1 (x2 (-y1+y3)+x3 (y1-2 y2+y3))+y2 (z1^2)-y3 (z1^2)-y1 z1 z2+y3 z1 z2+y1 z1 z3-2 y2 z1 z3+y3 z1 z3+y1 z2 z3-y3 z2 z3-y1 (z3^2)+y2 (z3^2))+a (-x3 ((y1^2)+y2 y3-y1 (y2+y3)+(z1-z2) (z1-z3))+x2 ((y1^2)-2 y1 y3+(y3^2)+(((z1-z3))^2))-x1 (y1 (y2-y3)-y2 y3+(y3^2)+z1 z2-z1 z3-z2 z3+(z3^2))))/((x1^2) (y2^2)-2 (x1^2) y2 y3+(x1^2) (y3^2)+(y2^2) (z1^2)-2 y2 y3 (z1^2)+(y3^2) (z1^2)+(x3^2) ((y1^2)-2 y1 y2+(y2^2)+(((z1-z2))^2))-2 y1 y2 z1 z2+2 y1 y3 z1 z2+2 y2 y3 z1 z2-2 (y3^2) z1 z2+(x1^2) (z2^2)+(y1^2) (z2^2)-2 y1 y3 (z2^2)+(y3^2) (z2^2)+(x2^2) ((y1^2)-2 y1 y3+(y3^2)+(((z1-z3))^2))-2 x1 x3 ((y2^2)-y2 y3+y1 (-y2+y3)-(z1-z2) (z2-z3))+2 y1 y2 z1 z3-2 (y2^2) z1 z3-2 y1 y3 z1 z3+2 y2 y3 z1 z3-2 (x1^2) z2 z3-2 (y1^2) z2 z3+2 y1 y2 z2 z3+2 y1 y3 z2 z3-2 y2 y3 z2 z3+(x1^2) (z3^2)+(y1^2) (z3^2)-2 y1 y2 (z3^2)+(y2^2) (z3^2)-2 x2 (x3 ((y1^2)+y2 y3-y1 (y2+y3)+(z1-z2) (z1-z3))+x1 (y1 (y2-y3)-y2 y3+(y3^2)+z1 z2-z1 z3-z2 z3+(z3^2))))};
If[0≤xsol≤1&&0≤ysol≤1&&xsol+ysol ≤1,√((((c (-x3 y1-x1 y2+x3 y2+x2 (y1-y3)+x1 y3)+a y2 z1-x3 y2 z1-a y3 z1+x2 y3 z1-a y1 z2+x3 y1 z2+a y3 z2-x1 y3 z2+a y1 z3-x2 y1 z3-a y2 z3+x1 y2 z3+b (-x2 z1+x3 z1+x1 z2-x3 z2-x1 z3+x2 z3)))^2)/((x1^2) (y2^2)-2 (x1^2) y2 y3+(x1^2) (y3^2)+(y2^2) (z1^2)-2 y2 y3 (z1^2)+(y3^2) (z1^2)+(x3^2) ((((y1-y2))^2)+(((z1-z2))^2))-2 y1 y2 z1 z2+2 y1 y3 z1 z2+2 y2 y3 z1 z2-2 (y3^2) z1 z2+(x1^2) (z2^2)+(y1^2) (z2^2)-2 y1 y3 (z2^2)+(y3^2) (z2^2)-2 x2 (x3 ((y1-y2) (y1-y3)+(z1-z2) (z1-z3))+x1 ((y1-y3) (y2-y3)+(z1-z3) (z2-z3)))+(x2^2) ((((y1-y3))^2)+(((z1-z3))^2))+2 x1 x3 ((y1-y2) (y2-y3)+(z1-z2) (z2-z3))-2 ((x1^2) z2+(y1-y2) (-y2 z1+y3 z1+y1 z2-y3 z2)) z3+((x1^2)+(((y1-y2))^2)) (z3^2))),$Failed]]

minDistPtToSegment[pt_, seg_]:= Module[{a,b,c,x1,y1,z1,x2,y2,z2}, 
{a,b,c}=pt;
{{x1,y1,z1},{x2,y2,z2}}=seg;

xsol=((a-x2) (x1-x2)+(b-y2) (y1-y2)+(c-z2) (z1-z2))/((((x1-x2))^2)+(((y1-y2))^2)+(((z1-z2))^2));
If[0≤xsol≤1, 
Sqrt[sns=(((-a+(x1 (a x1-a x2-x1 x2+(x2^2)+b y1-b y2-y1 y2+(y2^2)+c z1-c z2-z1 z2+(z2^2)))/((x1^2)-2 x1 x2+(x2^2)+(y1^2)-2 y1 y2+(y2^2)+(z1^2)-2 z1 z2+(z2^2))+x2 (1-(a x1-a x2-x1 x2+(x2^2)+b y1-b y2-y1 y2+(y2^2)+c z1-c z2-z1 z2+(z2^2))/((x1^2)-2 x1 x2+(x2^2)+(y1^2)-2 y1 y2+(y2^2)+(z1^2)-2 z1 z2+(z2^2)))))^2)+(((-b+(y1 (a x1-a x2-x1 x2+(x2^2)+b y1-b y2-y1 y2+(y2^2)+c z1-c z2-z1 z2+(z2^2)))/((x1^2)-2 x1 x2+(x2^2)+(y1^2)-2 y1 y2+(y2^2)+(z1^2)-2 z1 z2+(z2^2))+y2 (1-(a x1-a x2-x1 x2+(x2^2)+b y1-b y2-y1 y2+(y2^2)+c z1-c z2-z1 z2+(z2^2))/((x1^2)-2 x1 x2+(x2^2)+(y1^2)-2 y1 y2+(y2^2)+(z1^2)-2 z1 z2+(z2^2)))))^2)+(((-c+(z1 (a x1-a x2-x1 x2+(x2^2)+b y1-b y2-y1 y2+(y2^2)+c z1-c z2-z1 z2+(z2^2)))/((x1^2)-2 x1 x2+(x2^2)+(y1^2)-2 y1 y2+(y2^2)+(z1^2)-2 z1 z2+(z2^2))+z2 (1-(a x1-a x2-x1 x2+(x2^2)+b y1-b y2-y1 y2+(y2^2)+c z1-c z2-z1 z2+(z2^2))/((x1^2)-2 x1 x2+(x2^2)+(y1^2)-2 y1 y2+(y2^2)+(z1^2)-2 z1 z2+(z2^2)))))^2)],
Min[{EuclideanDistance[pt, seg[[1]]],EuclideanDistance[pt, seg[[1]]]}]]];

HausdorffDistanceTrue[T1_, T2_] := (f1=(T1[[#1]]&)/@faces;f2=(T2[[#1]]&)/@faces;
m1=Max[Table[
Min[Table[
aa=planeToPointGlobalInsideTriangle[T1[[j]],face= f2[[i]]];  If[aa =!= $Failed ,aa, 
bb=Min[Table[minDistPtToSegment[T1[[j]], face[[Mod[{k, k+1}, 3,1]]]],{k,3}]]],
{i,4}]],{j,4}]];
m2=Max[Table[
Min[Table[
aa=planeToPointGlobalInsideTriangle[T2[[j]], face=f1[[i]]];  If[aa =!= $Failed ,aa, 
bb=Min[Table[minDistPtToSegment[T1[[j]], face[[Mod[{k, k+1}, 3,1]]]],{k,3}]]],
{i,4}]],{j,4}]];
Max[m1, m2]);

HausdorffDistanceDiscrete[T1_,T2_]:=(m1=Max@Table[Min[Table[EuclideanDistance[T1[[i]],T2[[j]]],{j,4}]] ,{i,4}];
m2=Max@Table[Min[Table[EuclideanDistance[T2[[i]],T1[[j]]],{j,4}]] ,{i,4}];
Max[m1,m2]);

Clear[HausdorffGap];
HausdorffGap[L_] := HausdorffDistanceTrue[THPoint/@ {-1,0,1,2},N[ FinalTetrahedron[L]]];
HausdorffGap[L_, prec_] := HausdorffDistanceTrue[THPoint/@ {-1,0,1,2},N[ FinalTetrahedron[L], prec]];

HausdorffGapOldBad[L_, prec_] := HausdorffDistanceTrue[THPoint/@ {-1,0,1,2},N[ FinalTetrahedronBad[L], prec]];
LLLSolve[{a_,b_,c_},w_] := (A={{a,1,0,0},{b,0,1,0},{-c,0,0,w}};
B=LatticeReduce[A];
B1=Select[B, #[[-1]]≠0&];
SortBy[B1, Abs[Quotient[#[[-1]],w]]&][[1]])

rotationAxis[L_] := {(3 (Sqrt[3])+3 (Sqrt[3]) L+4 (Sqrt[5]) y+(Sqrt[5]) y Cos[L ArcCos[-(2/3)]]-20 y Sin[L ArcCos[-(2/3)]])/(-10+20 Cos[L ArcCos[-(2/3)]]+(Sqrt[5]) Sin[L ArcCos[-(2/3)]]),y, pl1/. x-> (3 (Sqrt[3])+3 (Sqrt[3]) L+4 (Sqrt[5]) y+(Sqrt[5]) y Cos[L ArcCos[-(2/3)]]-20 y Sin[L ArcCos[-(2/3)]])/(-10+20 Cos[L ArcCos[-(2/3)]]+(Sqrt[5]) Sin[L ArcCos[-(2/3)]])};

(* paper has much better formula for this now *)

uniqueQ[set_, poly_] := Select[set,Sort[#[[1]]]==Sort[poly[[1]]]&]=={poly};
color[p_,c_] := p/. Polygon[z_List, c1___] :> Polygon[z, VertexColors->{c,c,c}];
Chain3DNotClean[L_] :=Graphics3D[polygonsUp[RotateLeft[double@specFormBetter[L],L]]];

Options[Print3DChainQuad]= {PivotColor->White,EdgeStyle->None, ShowFiller->True,FillerOnlyQ->False, Background->Black, FillColor->White, ShiftChain->True, Colors-> {RGBColor[1,.4,.4], RGBColor[.3,1,.3],RGBColor[.3,.3,1], Gray, White }};
Print3DChainQuad[L_, opts___] := (
{shiftQ,esty,back,fillQ,fillOnlyQ, fillcol, cols, pivcol}={ShiftChain,EdgeStyle,Background,ShowFiller,FillerOnlyQ, FillColor, Colors, PivotColor}/. {opts} /. Options[Print3DChainQuad];
pp=Flatten[(pp0=N[polygonsUp[RotateLeft[Quadrahelix[L],If[shiftQ,L,0]]]]/. 
Polygon[z_] :> Polygon/@ z),1] 
(* /. {
{X,x_, y_, z_?NumericQ} /;0<Norm[{x,y,z}-a]<.6 :> (mPrint@z;a),
{X,x_, y_, z_?NumericQ} /;0<Norm[{x,y,z}-b]<.6 :> b,
{X,x_, y_, z_?NumericQ} /;0<Norm[{x,y,z}-c]<.6 :> c}*);
pp = pp /. x_?NumericQ :> Round[10^6 x];
     ppClean=Select[pp, uniqueQ[pp,#]&];
ppClean = ppClean/.  x_?NumericQ :>x/800000.;
col=Append[cols,fillcol] ;
 im=Graphics3D[pp=ppClean;
{
If[fillOnlyQ,{},{
EdgeForm[GrayLevel[.2]],EdgeForm[esty],

{
color[pp[[1;;2L+1]],col[[1]]],
color[pp[[2L+2;;2L+3]],pivcol],
color[pp[[2L+4;;4L+3]],col[[2]]],
color[pp[[4L+4;;6L+3]],col[[3]]],
color[pp[[6L+4;;6L+5]],pivcol],
color[pp[[6L+6;;8L+6]],col[[4]]]
},

j = Mod[L+1,4,1];
inds= j /.{1->{1,1},3->{2,2}, 4-> {3,3}, 2->({2,2};{1,1})};
}],

threeRedFaces=First/@ pp[[{1,2,3}]];threeBlackFaces=First/@ pp[[{-3,-2,-1}]];
mPrint[choices=First@First[SortBy[Table[{t,Norm[Mean[t[[1]]]-Mean[t[[2]]]]},{t,Tuples[{threeRedFaces,threeBlackFaces}]}],Last]]];


(*{{a,b,c}=pp[[inds[[1]],1]];
{a1,b1,c1}=Nearest[pp[[16L+10-2;;16L+10]][[inds[[2]],1]],#][[1]]& /@ {a,b,c};
{a1,b1,c1}=pp[[16L+10-2;;16L+10]][[inds[[2]],1]];
{Red, AbsolutePointSize[10], Red,Point[choices[[1]]],Blue, Point[{a1,b1,c1}; choices[[2]]]};
EdgeForm[],
If[fillQ,zz={
EdgeForm[],
color[#,col[[9]]]&/@ Polygon/@ {{a,b,b1}, {a1, b1, a}},
color[#,col[[9]]]&/@ Polygon/@ {{b,c,c1}, {b1, c1, b}},
color[#,col[[9]]]&/@ Polygon/@ {{c,a,a1}, {c1, a1, c}}},{}]}*)

},
Boxed->False,Axes->!True, AxesLabel->{x,y,"z"},ImageSize->400, PlotRange->All, 
ViewPoint-> vp[L], ViewVertical->vv[L],
PlotRangePadding->None, Background->back, Lighting->"Neutral"]);

Clear[vp,vv];
{vp[_], vv[_]}= {{-1,-2.3,2}, {0 0,1}};
{vp[1], vv[1]}={{-1.1321594953000143`,2.598371469441727`,1.5022252108100338`},{-0.09057875782923884`,-1.4810518897136136`,-0.08257754590413302`}};
{vp[2], vv[2]}={{-2.7233051265310286`,-1.3263860159173968`,-1.0556085091493885`},{-0.2693694005446902`,-0.8379800010236252`,-0.5052016401118591`}};
{vp[3], vv[3]}= {{2.3720275909586555`,1.5498457423151089`,1.5038162396910748`},{0.9834376653129644`,0.03563114687045805`,-0.17770981913355688`}};
{vp[4], vv[4]}={{2.747755021873639`,-0.34500755417750245`,-1.6188922531560468`},{1.4558994856197323`,-0.027887826102026555`,-0.003036588746728503`}};
{vp[5], vv[5]}={{-0.8470694943595101`,-0.699128038337188`,-3.0139165976742497`},{-0.6143738991006478`,-0.6099116349418946`,-0.7482741046064467`}};
{vp[6], vv[6]}={{2.876264615359726`,-1.0392679590580234`,1.4481104832522036`},{0.5485449466481384`,0.46739648637445863`,0.8779234229199029`}};
{vp[7], vv[7]} = {{-2.993845608929946`,-0.9018820134701585`,-0.7165872617273455`},{-1.4547501491926975`,0.1589756191563722`,-0.5779333130297644`}};
{vp[8],vv[8]}= {{1.035195480311868`,1.5922207847959193`,-2.5851892174472937`},{0.2460705412074689`,1.0375529703323612`,-0.4017692013939892`}};
{vp[9],vv[9]}= {{1.8725053361894686`,1.9169691683887577`,1.7632223267043998`},{0.061806650485925256`,1.192815233957048`,0.42948583051278477`}};
{vp[10], vv[10]}={{-2.409829785228355`,-0.7165686181602945`,-1.9922976237736434`},{-0.31291545413664784`,-0.7532786299348597`,-0.761161673663979`}};
{vp[13], vv[13]}={{-1.3516097738102983`,0.2829901293039268`,-2.8953527602102342`},{-0.29692353780498437`,-0.34490136703018015`,1.4789445465588271`}};

{vp[11], vv[11]}={{2.180149198677573`,2.35271551578505`,-0.04097771699129434`},{-0.16892889605697986`,-1.437032555053567`,-0.6687278582419959`}};

{vp[12], vv[12]}={{1.9213468303738854`,0.22978687380969645`,-2.5584417816391665`},{0.11420792957852856`,-0.08628262227062182`,0.9897029139673639`}};

{vp[25], vv[25]}={{0.7802285087015227`,0.32958073103844093`,3.093965096108481`},{-0.5877585403943673`,-0.11259378208821114`,1.5823921509273404`}};
{vp[29], vv[29]}={{-1.0265414135243562`,2.329262049324853`,-1.9521145027621714`},{-0.3015382427312685`,-0.6951292888871488`,0.6525871282072497`}};

{vp[30], vv[30]}={{-1.739755400419509`,-0.12511424239011484`,2.6921362471209176`},{-0.19677676510472206`,0.06118829354751308`,-0.9939405598183287`}};

{vp[70], vv[70]}={{-1.7850236478169377`,-0.3401899910491482`,-2.6434752404220268`},{1.247625233799548`,-0.5094680168949465`,0.7001126391794809`}};

{vp[40], vv[40]}={{-2.202330675501434`,2.331990722704213`,0.0394824640426283`},{0.8720815532139217`,-0.2884715698720987`,0.3952947228581743`}};

{vp[182], vv[182]}={{-1.9304721931811195`,-0.6906575134512402`,-2.467036544210041`},{0.8803613637544427`,-0.25387869679135494`,0.4006363394949852`}};

Lval=10; im=Graphics3D[{pp=polygonsUp[QHNew[Lval]];
{Red,pp[[1;;Lval]]},
{GrayLevel[.8],pp[[Lval+1;;3Lval+2;; 2Lval +1]]},
{Green,pp[[Lval+2;;2Lval+1]]},
{Lighter@Lighter@Blue,pp[[2Lval+2;;3Lval+1]]},
{Lighter@Yellow,pp[[3Lval+3;;4Lval+2]]}}, Lighting->"Neutral",Background->Black, Boxed->False]

3D Color Printed Shapeways Model