Figurative And Non-Transcendental Approach to Mathematical Visualization (FANTA to Math Vis)

Atsushi MIYAZAWA, Masanori NAKAYAMA, and Issei FUJISHIRO
Keio University, Japan

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ABSTRACT

It’s believed that the answers to various problems of basic math consisting of elementary calculus and geometry faced by high-school and college students have already been well-established, and there might not be any room for further study or discussion. But actually, pitfalls are found almost everywhere in this area, especially when it comes to graphing functions to understand their behavior in a higher dimensional space that we can neither experience nor access directly. Although topological abstraction plays an exceptionally important role in modern mathematics, it can greatly reduce the amount of information to what is needed for the problem to be just identified and lead to a lot of misconceptions. Rather than such abstraction processes, we advocate that figurative approaches are often considered as prime solution candidates. The term figurative refers to any form of mathematical visualization that retains strong references to the geometry found in the real world. This presentation explains the figuration process for some basic mathematical functions definable in the 2-dimensional complex projective space. In the latter part of this talk, we raise a question that has been neglected thus far: What does the Riemann sphere’s axis stand for? We show that the answer can be obtained only by observing from the inside the sphere by setting the viewpoint of the immersive environment to the origin (0,0), which is always undefined (and off-limits to everyone) in projective geometry.

Challenge the Old School Knowledge and Practices in Mathematics

for the deep understanding of high-dimensional math visualization

It’s widely believed that …

the answers to various elementary calculus and geometry problems have already been WELL-ESTABLISHED,

and there might be NO ROOM for further study or discussion

But actually …

PITFALLs are found everywhere, when it comes to graphing functions to understand their behavior in a higher dimensional space,

which we can neither experience nor access directly

Nobel winner cited curiosity as key to pursuing basic research

"Do not trust what is written in textbooks. You should have a feeling of wanting to find out what is really going on."

THE ASAHI SHIMBUN (October 2, 2018 at 13 : 40 JST)

Agenda

What is mathematical visualization used in schools, and why does it have so many errors?

Graphical representation of complex roots

Common chord of two non-intersecting circles

Poncelet's continuity principle

What does the Riemann sphere's axis stand for?

Recurrence formula for projective spaces

Understanding the big picture of how mathematical functions behave

The fourth form of the projective plane (Math orbuculum)

Graphical Representation of Complex Roots

Graphical representation of the roots of quadratic equations; w=(z^2)-1 (2 real roots), w=(z^2) (1 double* root), w=(z^2)+1 (no roots)

xmin = -2; xmax = 2; ymin = -2; ymax = 2; umin = -3; 
  umax = 5; GraphicsGrid[
   {{Show[ParametricPlot3D[{x, 0, x^2 - 1}, 
        {x, xmin, xmax}, AxesLabel -> {x, y, u}, 
        PlotRange -> All] /. Line[pts_, rest___] :> 
        Tube[pts, 0.1, rest], 
      ParametricPlot3D[{0, y, -y^2 - 1}, 
        {y, ymin, ymax}] /. Line[pts_, rest___] :> 
        Tube[pts, 0.1, rest], Graphics3D[
       {Dashed, Line[{{-1, 0, 0}, {-1, 0, -1}, 
          {1, 0, -1}, {1, 0, 0}}], 
        Line[{{0, -1, 0}, {0, -1, -2}, {0, 1, -2}, 
          {0, 1, 0}}], Red, Sphere[{1, 0, 0}, 0.2], 
        Sphere[{-1, 0, 0}, 0.2]}], ParametricPlot3D[
       {x, 0, 0}, {x, xmin, xmax}], ParametricPlot3D[
       {0, y, 0}, {y, ymin, ymax}], ParametricPlot3D[
       {0, 0, u}, {u, umin, umax}], PlotLabel -> 
       w == z^2 - 1 == 0], 
     Show[ParametricPlot3D[{x, 0, x^2}, 
        {x, xmin, xmax}, AxesLabel -> {x, y, u}, 
        PlotRange -> All] /. Line[pts_, rest___] :> 
        Tube[pts, 0.1, rest], 
      ParametricPlot3D[{0, y, -y^2}, {y, ymin, 
         ymax}] /. Line[pts_, rest___] :> 
        Tube[pts, 0.1, rest], Graphics3D[
       {Red, Sphere[{0, 0, 0}, 0.2]}], 
      ParametricPlot3D[{x, 0, 0}, {x, xmin, xmax}], 
      ParametricPlot3D[{0, y, 0}, {y, ymin, ymax}], 
      ParametricPlot3D[{0, 0, u}, {u, umin, umax}], 
      PlotLabel -> w == z^2 == 0], 
     Show[ParametricPlot3D[{x, 0, x^2 + 1}, 
        {x, xmin, xmax}, AxesLabel -> {x, y, u}, 
        PlotRange -> All] /. Line[pts_, rest___] :> 
        Tube[pts, 0.1, rest], 
      ParametricPlot3D[{0, y, -y^2 + 1}, 
        {y, ymin, ymax}] /. Line[pts_, rest___] :> 
        Tube[pts, 0.1, rest], Graphics3D[
       {Dashed, Line[{{-1, 0, 0}, {-1, 0, 2}, 
          {1, 0, 2}, {1, 0, 0}}], 
        Line[{{0, -1, 0}, {0, -1, 1}, {0, 1, 1}, 
          {0, 1, 0}}], Red, Sphere[{0, 1, 0}, 0.2], 
        Sphere[{0, -1, 0}, 0.2]}], ParametricPlot3D[
       {x, 0, 0}, {x, xmin, xmax}], ParametricPlot3D[
       {0, y, 0}, {y, ymin, ymax}], ParametricPlot3D[
       {0, 0, u}, {u, umin, umax}], PlotLabel -> 
       w == z^2 + 1 == 0]}}, ImageSize -> Full]

*The root of the real branch can be interpreted as doubled with the root of the imaginary branch

Graphical Representation of Complex Roots (cont’d)

"The usual treatment of complex numbers in the high school is mechanical and meaningless, but the Argand diagram can and should be used to represent such numbers as points in a plane."

Howard F. Fehr, “Graphical representation of complex roots”, 18th Yearbook (National Council of Teachers of Mathematics), p. 130, 1945

Common Chord of Two Non-intersecting Circles

Let (Subscript[C, 1]) and (Subscript[C, 2]) be two circles:

(Subscript[C, 1]):(x^2)+(y^2)+2g x+2f y+c=0
	(Subscript[C, 2]):(x^2)+(y^2)+2g'x+2f'y+c'=0

The difference of two circles gives the equation of the common chord:

(Subscript[C, 1])-(Subscript[C, 2]):2(g-g')x+2(f-f')y+(c-c')=0

Common Chord of Two Non-intersecting Circles (cont’d)

Poncelet’s continuity principle* states that if a certain number of solutions are expected, then there will be the same number of solutions in all cases (although some answers may be imaginary)

*Weisstein, Eric W. “Continuity Principle.” From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/ContinuityPrinciple.html

What is Figurative Art (and Figurative Vis), Anyway?

Definitions of inverse hyperbolic functions in terms of logarithms:

Inverse (/Area*) hyperbolic sine	arsinh x=ln (x+(Sqrt[(x^2)+1])),

Inverse hyperbolic cosine		arcosh x=ln (x+(Sqrt[(x^2)-1]))

Inverse hyperbolic tangent		artanh x=(1/2)ln ((1+x)/(1-x))

and so on …

*"However, arc-, followed by the corresponding hyperbolic function is a misnomer, since the prefix arc is the abbreviation for arcus, while the prefix ar stands for area."

Wikipedia contributors, "Inverse hyperbolic functions," Wikipedia, The Free Encyclopedia, https : // en.wikipedia.org/w/index.php?title = Inverse_hyperbolic _functions & oldid = 858909889 (accessed October 17, 2018)

**In the complex domain, trigonometric functions and hyperbolic functions are not brothers or sisters; they are identical!

Constructing the Graph of the Inverse Hyperbolic Cosine Function

in a Cartesian coordinate 4-space,

by looking at the 3-dimensional CROSS SECTION (above) or the SHADOW cast on our 3-space (below)

(* The orthonormal basis of 4-space {vex, vey, veu, vev} projected down into 3-space so that they lie on the edges of a tetrahedron *)vex={-(1/2),-(1/(2 (Sqrt[3]))),-(1/(2 (Sqrt[6])))};vey={(1/2),-(1/(2 (Sqrt[3]))),-(1/(2 (Sqrt[6])))};veu={0,(1/(Sqrt[3])),-(1/(2 (Sqrt[6])))};vev={0,0,((Sqrt[(3/2)])/2)};

(* Complex cosine function in Cartesian form projected into 3-space *)fu[x_,y_]:=Cos[x] Cosh[y];fv[x_,y_]:=-Sin[x] Sinh[y];fComplexCartesian[x_,y_,vex_,vey_,veu_,vev_]:=Module[{mz,mw},mz=Transpose[{vex,vey}];mw=Transpose[{veu,vev}];mz.{x,y}+mw.{fu[x,y],fv[x,y]}]

Show[Graphics3D[{Arrowheads[.02],Black,Thick,Arrow[{xmin vex,(posaxes xmax)vex}],Text[Style["x",FontFamily->"Times",Black,Bold,Italic,24],(poslabels xmax)vex],Black,Thickness[Medium],Arrow[{ymin vey,(posaxes ymax) vey}],Text[Style["iy",FontFamily->"Times",Black,Bold,Italic,24],(poslabels ymax)vey],Blue,Thick,Arrow[{umin veu,(posaxes umax) veu}],Text[Style["u",FontFamily->"Times",Blue,Bold,Italic,24],(poslabels umax)veu],Red,Thickness[Medium],Arrow[{vmin vev,(posaxes vmax) vev}],Text[Style["iv",FontFamily->"Times",Red,Bold,Italic,24],(poslabels vmax)vev],Black,Dotted,Line[{xmin vex+veu,(posaxes xmax)vex+veu}],Line[{xmin vex-veu,(posaxes xmax)vex-veu}]}],ParametricPlot3D[fComplexCartesian[x,0,vex,vey,veu,vev],{x,xmin,xmax},PlotStyle->Directive[GrayLevel[.2],Thick,Dashed]]/.Line[pts_,rest___]:>Tube[pts,0.1,rest],ParametricPlot3D[{fComplexCartesian[0,y,vex,vey,veu,vev],fComplexCartesian[π,y,vex,vey,veu,vev],fComplexCartesian[2π,y,vex,vey,veu,vev],fComplexCartesian[3π,y,vex,vey,veu,vev]},{y,ymin,ymax},PlotStyle->Directive[Darker[Blue],Thick,Dashed]]/.Line[pts_,rest___]:>Tube[pts,0.1,rest],ParametricPlot3D[{fComplexCartesian[(π/2),y,vex,vey,veu,vev],fComplexCartesian[((3π)/2),y,vex,vey,veu,vev],fComplexCartesian[((5π)/2),y,vex,vey,veu,vev]},{y,ymin,ymax},PlotStyle->Directive[Darker[Red],Thick,Dashed]]/.Line[pts_,rest___]:>Tube[pts,0.1,rest],Axes->False,Boxed->False,BoxRatios->Automatic,ImageSize->Full,Lighting->"Neutral",PlotRange->All,ViewPoint->{-2,2,2},ViewVertical->vex]

xmax=((7π)/2);xmin=-(π/2);ymax=2;ymin=-ymax;umax=fu[0,ymax];umin=-umax;vmax=fv[(π/2),ymax];vmin=-vmax;posaxes=1.2;poslabels=1.3;Show[ParametricPlot3D[fComplexCartesian[x,y,vex,vey,veu,vev],{x,xmin,xmax},{y,ymin,ymax},MaxRecursion->5,Mesh->None,PlotPoints->{100,20},PlotStyle->Directive[Opacity[0.6],Specularity[White,5]]],
(* Draws axes x, iy, u, iv, and their labels *)Graphics3D[{Arrowheads[.02],Black,Thick,Arrow[{xmin vex,(posaxes xmax)vex}],Text[Style["x",FontFamily->"Times",Black,Bold,Italic,24],(poslabels xmax)vex],Black,Thickness[Medium],Arrow[{ymin vey,(posaxes ymax) vey}],Text[Style["iy",FontFamily->"Times",Black,Bold,Italic,24],(poslabels ymax)vey],Blue,Thick,Arrow[{umin veu,(posaxes umax) veu}],Text[Style["u",FontFamily->"Times",Blue,Bold,Italic,24],(poslabels umax)veu],Red,Thickness[Medium],Arrow[{vmin vev,(posaxes vmax) vev}],Text[Style["iv",FontFamily->"Times",Red,Bold,Italic,24],(poslabels vmax)vev],Black,Dotted,Line[{xmin vex+veu,(posaxes xmax)vex+veu}],Line[{xmin vex-veu,(posaxes xmax)vex-veu}]}],(* Draws some branches of the function *)ParametricPlot3D[fComplexCartesian[x,0,vex,vey,veu,vev],{x,xmin,xmax},PlotStyle->Directive[GrayLevel[.2],Thick,Dashed]](*/.Line[pts_,rest___]:>Tube[pts,0.1,rest]*),ParametricPlot3D[{fComplexCartesian[0,y,vex,vey,veu,vev],fComplexCartesian[π,y,vex,vey,veu,vev],fComplexCartesian[2π,y,vex,vey,veu,vev],fComplexCartesian[3π,y,vex,vey,veu,vev],fComplexCartesian[(π/2),y,vex,vey,veu,vev],fComplexCartesian[((3π)/2),y,vex,vey,veu,vev],fComplexCartesian[((5π)/2),y,vex,vey,veu,vev]},{y,ymin,ymax},PlotStyle->Directive[GrayLevel[.2],Thick,Dashed]](*/.Line[pts_,rest___]:>Tube[pts,0.1,rest]*),Axes->False,Boxed->False,BoxRatios->Automatic(*{1,1,.4}*),ImageSize->Full,Lighting->"Neutral",PlotRange->All,ViewPoint->{-2,2,2},ViewVertical->vex(*{0,0,1}*)]

Constructing the Graph of the Inverse Hyperbolic Cosine Function

which is traditionally drawn in abstract form with polylines

Riemann surface for the complex logarithm w=ln z

vex={1,0,0};veu={0,1,0};vey={(1/2),(1/2),-(1/(Sqrt[2]))};vev={-(1/2),-(1/2),-(1/(Sqrt[2]))};

fu[x_,y_]:=Exp[x] Cos[y];fv[x_,y_]:=Exp[x] Sin[y];fComplexCartesian[x_,y_,vex_,vey_,veu_,vev_]:=Module[{mz,mw},mz=Transpose[{vex,vey}];mw=Transpose[{veu,vev}];mz.{x,y}+mw.{fu[x,y],fv[x,y]}]

ymax=4π;ymin=-2π;xmax=2;xmin=-4;umax=fu[xmax,0];umin=-umax;vmax=fv[xmax,(π/2)];vmin=-vmax;posaxes=1.2;poslabels=1.3;Show[ParametricPlot3D[fComplexCartesian[x,y,vex,vey,veu,vev],{x,xmin,xmax},{y,ymin,ymax},MaxRecursion->5,Mesh->None,PlotPoints->{20,200},PlotStyle->Directive[Opacity[0.6],Specularity[White,5]]],
Graphics3D[{Arrowheads[.02],Black,Thick,Arrow[{xmin vex,(posaxes xmax)vex}],Text[Style["x",FontFamily->"Times",Black,Bold,Italic,24],(poslabels xmax)vex],Black,Thickness[Medium],Arrow[{ymin vey,(posaxes ymax) vey}],Text[Style["iy",FontFamily->"Times",Black,Bold,Italic,24],(poslabels ymax)vey],Blue,Thick,Arrow[{umin veu,(posaxes umax) veu}],Text[Style["u",FontFamily->"Times",Blue,Bold,Italic,24],(poslabels umax)veu],Red,Thickness[Medium],Arrow[{vmin vev,(posaxes vmax) vev}],Text[Style["iv",FontFamily->"Times",Red,Bold,Italic,24],(poslabels vmax)vev],Black,Dotted,Line[{ymin vey+veu,(posaxes ymax)vey+veu}],Line[{ymin vey-veu,(posaxes ymax)vey-veu}]}],ParametricPlot3D[fComplexCartesian[0,y,vex,vey,veu,vev],{y,ymin,ymax},PlotStyle->Directive[GrayLevel[.2],Thick,Dashed]],ParametricPlot3D[{fComplexCartesian[x,-((3π)/2),vex,vey,veu,vev],fComplexCartesian[x,-π,vex,vey,veu,vev],fComplexCartesian[x,-(π/2),vex,vey,veu,vev],fComplexCartesian[x,0,vex,vey,veu,vev],fComplexCartesian[x,(π/2),vex,vey,veu,vev],fComplexCartesian[x,π,vex,vey,veu,vev],fComplexCartesian[x,((3π)/2),vex,vey,veu,vev],fComplexCartesian[x,2π,vex,vey,veu,vev],fComplexCartesian[x,((5π)/2),vex,vey,veu,vev],fComplexCartesian[x,3π,vex,vey,veu,vev],fComplexCartesian[x,((7π)/2),vex,vey,veu,vev]},{x,xmin,xmax},PlotStyle->Directive[GrayLevel[.2],Thick,Dashed]],AspectRatio->Automatic,Axes->False,Boxed->False,BoxRatios->Automatic,ImageSize->Full,Lighting->"Neutral",PlotRange->All]

Eight Rules of Thumb for Visualization Design

T. Munzner: Visualization Analysis and Design, AK Peters, 2014

No Unjustified 2D*

Eyes Beat Memory

Resolution over Immersion

Overview First, Zoom and Filter, Details on Demand

Responsiveness Is Required

Get It Right in Black and White

Function First, Form Next

*From a mathematical point of view, these rules can be considered as variants and invariants in projective geometry (If so, then can we expand the rules to apply to a higher dimension?)

Analysis: Four Levels for Validation

T. Munzner: Visualization Analysis and Design, AK Peters, 2014

Domain Situation

Task and Data Abstraction

Visual Encoding and Interaction Idiom

Algorithm*

*A detailed procedure that allows a computer to automatically carry out a desired goal (the goal is to efficiently handle the visual encoding and interaction idioms)

One Day the Famous Mathematics Professor Told Us That,

“Since only equations can prove a theorem is right, we don’t mind even though the visuals are incomplete.”

Does that mean the mathematical visualization may be left incomplete?

Yes, We Found It!

What Does the Riemann Sphere’s Axis Stand for?

Recurrence Formula for Projective Spaces

Recurrence relation between projective spaces must be confirmed graphically or using the equations: 
	R (P^0)=(R^0),
	R (P^1)=(R^1)⋃ ((R(P))^0),
	R (P^2)=(R^2)⋃ R (P^1)=(R^2)⋃ (R^1)⋃ (R^0), …
and so on

On the Visualization of Riemann Surfaces

The function w=(z^(1/n)) is a multiple-valued function having n distinct branches

Sveshnikov, A.G., and Tikhonov, A.N.The Theory Of Functions Of A Complex Variable.(1982)

On the Visualization of Riemann Surfaces (cont’d)

Constructing the Riemann surface of the complex surd function around its singular point in terms of polar coordinates

Ohno, Y., and Miyazawa, A. Graphing 	Complex Functions by Means of Computer Programs. Sugaku Seminar, 21 (11), 2-15 (1982)

Complex Surd Function Graphed around its Singular (Branch) Point

in terms of polar coordinates

(* The set of vectors {vex,veu} (or {vey,vev}) forms an orthonormal basis of the 2-dimensional subspace (plane) to avoid distortion occurring in figures on the plane *)vex={1,0,0};vey={(1/2),(1/2),-(1/(Sqrt[2]))};veu={0,1,0};vev={-(1/2),-(1/2),-(1/(Sqrt[2]))};

CSqrtPFProjected[p_,phi_,vex_,vey_,veu_,vev_]:=Module[{mp,mq},mp=Transpose[{vex,vey}];mq=Transpose[{veu,vev}];p mp.{Cos[phi],Sin[phi]}+(Sqrt[p])mq.{Cos[(phi/2)],Sin[(phi/2)]}](* The square root on complex numbers in polar form can be defined as: q Exp[I psi]=Sqrt[p Exp[I phi]]=Sqrt[p] Exp[I phi/2], that is, q=Sqrt[p] and psi=phi/2 *)

pmin=0;pmax=6;phimin=0;phimax=4π;xyzmax=10;pos=1.1;Show[ParametricPlot3D[CSqrtPFProjected[p,phi,vex,vey,veu,vev],{p,pmin,pmax},{phi,phimin,phimax},MaxRecursion->5,Mesh->None,PlotPoints->{20,120},PlotStyle->Directive[Opacity[0.6],Specularity[White,5]]],(* Draws 5 branches of the function. Each corresponds to phi=0, PI, 2PI, 3PI and PI/3, respectively *)ParametricPlot3D[CSqrtPFProjected[p,0,vex,vey,veu,vev],{p,pmin,pmax},PlotStyle->Directive[Black,Thick]](*/.Line[pts_,rest___]:>Tube[pts,0.1,rest]*),ParametricPlot3D[CSqrtPFProjected[p,(π/2),vex,vey,veu,vev],{p,pmin,pmax},PlotStyle->Directive[Black]],ParametricPlot3D[CSqrtPFProjected[p,π,vex,vey,veu,vev],{p,pmin,pmax},PlotStyle->Directive[Blue,Thick]](*/.Line[pts_,rest___]:>Tube[pts,0.1,rest]*),ParametricPlot3D[CSqrtPFProjected[p,((3π)/2),vex,vey,veu,vev],{p,pmin,pmax},PlotStyle->Directive[Black]],ParametricPlot3D[CSqrtPFProjected[p,2π,vex,vey,veu,vev],{p,pmin,pmax},PlotStyle->Directive[Black,Thick]](*/.Line[pts_,rest___]:>Tube[pts,0.1,rest]*),ParametricPlot3D[CSqrtPFProjected[p,((5π)/2),vex,vey,veu,vev],{p,pmin,pmax},PlotStyle->Directive[Black]],ParametricPlot3D[CSqrtPFProjected[p,3π,vex,vey,veu,vev],{p,pmin,pmax},PlotStyle->Directive[Blue,Thick]](*/.Line[pts_,rest___]:>Tube[pts,0.1,rest]*),ParametricPlot3D[CSqrtPFProjected[p,((7π)/2),vex,vey,veu,vev],{p,pmin,pmax},PlotStyle->Directive[Black]],(* Draws axes x, iy, u, iv, p, and their labels *)Graphics3D[{Arrowheads[.02],Black,Thick,Arrow[{-xyzmax vex,xyzmax vex}],Text[Style[x,Large,Black],(pos xyzmax)vex],Thickness[Medium],Arrow[{((-xyzmax)/(6/2)) vey,(xyzmax/(6/2)) vey}],Text[Style[iy,Large,Black],(pos (xyzmax/(6/2)))vey],Blue,Thick,Arrow[{((-xyzmax)/(3/2)) veu,(xyzmax/(3/2)) veu}],Text[Style[u,Large,Blue],(pos (xyzmax/(3/2)))veu],Blue,Thickness[Medium],Arrow[{((-xyzmax)/(3/2)) vev,(xyzmax/(3/2)) vev}],Text[Style[iv,Large,Blue],(pos (xyzmax/(3/2)))vev]}],Axes->False,Boxed->False,BoxRatios->Automatic,ImageSize->Full,Lighting->"Neutral",PlotRange->All,ViewPoint->{-2.4,1.3,20},ViewVertical->{1,2,0}]

dphi=(π/8);Manipulate[GraphicsRow[Table[ParametricPlot3D[{{r Cos[φ],r Sin[φ],Re[(r^(m/2)) Exp[I m φ/2]]},{r Cos[φ],r Sin[φ],Re[-(r^(m/2)) Exp[I m φ/2]]}},{r,0,1},{φ,-dphi,phimax},AxesLabel->{"x","I y"}(*,BoxRatios->{1,1,1}*),Mesh->{8,{{0,Directive[Red,Thick,Dashing[{}]]},{(π/2),Directive[Red,Thick,Dashing[Medium]]},{π,Directive[Red,Thick,Dashing[{}]]},{((3π)/2),Dashing[Medium]},{2π,Dashing[{}]},{((5π)/2),Dashing[Medium]},{3π,Dashing[{}]},{((7π)/2),Dashing[Medium]},{4π,Dashing[{}]},{((9π)/2),Dashing[Medium]},{5π,Dashing[{}]},{((11π)/2),Dashing[Medium]}}},PlotLabel->Re[±(r^(m/2)) Exp[I m φ/2]],PlotPoints->60,PlotRange->1,PlotStyle->Directive[Specularity[White,20],Opacity[0.8]],PlotTheme->"Marketing"],{m,1,3}],Background->LightRed,ImageSize->Full],{{phimax,dphi},0,4π,dphi}]

On the Visualization of Riemann Surfaces (cont’d)

Projection onto the subspace (y,(x^2)-(y^2),2x y) of the graph of the complex function w=(z^2)

Banchoff, Thomas F. Beyond the Third Dimension: Geometry, Computer Graphics, and Higher Dimensions (Scientific American Library Series). New York: W. H.Freeman & Co., Scientific American Library, 1-210 (1990)

M.Trott, "Trott' s Corner : Visualization of Riemann Surfaces IIa : Compositions of Elementary Transcendental Functions," The Mathematica Journal, 7 (4), 2000 pp.465 - 496.

M.Trott, "Trott' s Corner : Visualization of Riemann Surfaces IIb : Compositions of Elementary Transcendental Functions," The Mathematica Journal, 8 (1), 2001 pp.50 - 62.

M.Trott, "Trott' s Corner : Visualization of Riemann Surfaces IIc : Compositions of Elementary Transcendental Functions," The Mathematica Journal, 8 (3), 2002 pp.409 - 432.

M.Trott, "Trott' s Corner : Visualization of Riemann Surfaces IId : Compositions of Elementary Transcendental Functions," The Mathematica Journal, 8 (4), 2002 pp.532 - 562.

M.Trott, "Trott' s Corner : The Return of the Riemann Surface," The Mathematica Journal, 10 (4), 2008 pp.626 - 656.

What Does the Riemann Sphere’s Axis Stand for? (again)

Is it one of the range (u or v) of the complex function u+i v=f (x+i y) ?

No. Since the Riemann sphere is the domain of the function that is projectivized and then compactified, the range (codomain) has no relation in the first place

The solution candidate is the axis of "1," which we use when we define (z, 1) as the homogeneous coordinates of z.

But in that case, it is strange that the axis is one-dimensional (there is no imaginary axis)

Recurrence Formula for Projective Spaces (again)

R (P^1)=(R^1)⋃ (R^0) actually means {(x,1)}⋃ {(1,0)},
R (P^2)=(R^2)⋃ R (P^1)=(R^2)⋃ R (P^1)=(R^2)⋃ (R^1)⋃ (R^0) means {(x,y,1)}⋃ {(x,1,0)}⋃ {(1,0,0)}, and so on

True Colors of the Riemann Sphere

Conclusion and Future Work

We can show that the understanding of 3D geometry will be deepened by the effect of immersive observations based on a figurative and non-transcendental visualization approach, rather than in an abstract way

By further expanding the abovementioned research results, we plan to develop a visualization method applicable to C (P^2) (complex projective plane)

Thank you for listening