Five-Point “Circle” Paradox: A Radius-Zero Projective Solution

Atsushi MIYAZAWA, Aichi Sangyo University

#WolframTechConf

ABSTRACT

A well‑known Japanese brain‑teaser asks for one circle that passes through five fixed points arranged like the “five‑spot” on a die—an obviously impossible task in Euclidean geometry, so the customary punch line substitutes the kanji character 円 (“circle/yen”) for a true circle. In this paper we show that the puzzle, far from being merely a linguistic joke, admits a rigorous geometric resolution when the ambient space is extended to the complex projective plane  ((ℂℙ)^2). Any five points in general position determine a unique (possibly degenerate) conic; for the five‑spot configuration this conic factors into two real diagonal lines, which are projectively equivalent to the radius‑zero circle defined by (x^2)+(y^2)=0. We analyze the transformation that maps the degenerate conic to the standard circle, discuss the role of isotropic lines, and propose classroom and outreach applications illustrating how everyday puzzles can motivate fundamental concepts in projective and conformal geometry.

The Puzzle

Draw a circle through these five points.

The Puzzle (Cont’d)

Usual “solution”: draw the kanji 円 (“yen / circle”).

Unexpected Claim

There is a mathematically meaningful resolution.

Not Euclidean-concyclic... but complex-concyclic in ((ℂℙ)^2).

Same Family, Different Topology

○ (a circle) and × (two real lines crossing) are not homeomorphic.

‘Quiz O/X Crowd’ by Atsushi Miyazawa, 2025, licensed CC BY 4.0; created with ChatGPT.

Yet they lie in the same pencil of conics; × is a degenerate member.

Complex-Concyclic = Through I,J

In CP2, a “circle” ⇔ a conic through the circular points

I=[1:i:0],J=[1:−i:0].

Replace × by the pair of imaginary lines y=±ix → complex-concyclic.

Five-Spot Mechanics (why the conic splits)

Each diagonal contains three of the five points.

A conic meeting a line in three points contains that line.

⇒ Unique conic = union of the two diagonals .

Conic Tomography (Fehrian Branches*) — I

Work in ((ℝ)^4)≅((ℂ)^2) with coordinates (x,y,u,v) for z=x+iy, w=u+iv.

Intrinsic complex planes:

((Subscript[P, xy])) = the (z)-plane; ((Subscript[P, uv])) = the (w)-plane.

Composite (mixed) planes: ((Subscript[P, xv])),((Subscript[P, yu])),((Subscript[P, xu])),((Subscript[P, yv])).

*Miyazawa, Atsushi . Fehrian Branches or : How I Learned to Reduce y = x^2 + 1 and Love Mathematica, Wolfram Technology Conference 2023 presentation . https : // youtu . be/IPiL0qtfQ1M

Conic Tomography (Fehrian Branches) — II

Complexify the conic, then take hyperplane cuts of ((ℝ)^4).

Read off the real branches in each 3-D slice = Fehrian Branches.

For a point-circle (x^2)+(y^2)=0, Fehrian Branches appear as the imaginary line pair in the mixed planes, and collapse to a point in ((Subscript[P, xy])).

Two Lenses: Figure and Shadow

Figure: rigid rules (no stretching) — motions, scalings, inversions, projections.

Shadow: projective/incidence — horizon line, cross-ratio, group actions.

Two Lenses: Figure and Shadow (Cont’d)

What the Two Lenses Say

Topology: ○ ≄ × (not homeomorphic).

Stone-and-Shadow: both live in one conic pencil, compared via invariants (circular points I,J, coaxal families, cross-ratio).

From Five-Spot to Radius-Zero

Five distinct points impose five linear conditions on conics.

Five-spot ⇒ reducible conic = the two diagonals.

Over ℂ, (x,y)↦(x,iy)⇒y=±ix⇒(x^2)+(y^2)=0.

Definition Recap

Radius-zero circle (point-circle): (x^2)+(y^2)=0.

Complex-concyclic: lying on a conic through I and J.

Möbius invariants for 4 points: cross-ratio is real.

Visualizing I and J

Homogeneous form for circles: a((X^2)+(Y^2))+d X Z+e Y Z+f (Z^2)=0.

Set Z=0: meet (Subscript[ℓ, ∞]) in I=[1:i:0],J=[1:−i:0].

All circles pass through I,J.

Visualizing I and J (Cont’d)

Unification

Lines and circles form a coaxal pencil through I,J.

The five-spot becomes complex-concyclic in ((ℂℙ)^2).

“Parlor puzzle” → projective-conformal case study with explicit invariants.

Classroom & Outreach Notes

Three-point vs five-point distinction (nondegenerate vs degenerate).

Visual demos: Cayley map, Fehrian Branches, circular points.

Possibility of an immersive virtual environment for the visualization of complex and/or infinitely distant territory .

Classroom & Outreach Notes (Cont’d)

Takeaways

Don’t stretch: move, scale, invert, project; track invariants.

The “no-circle” turns into a radius-zero circle.

Five points : not Euclidean - concyclic, but complex - concyclic .

Acknowledgments / References

Poncelet; Coxeter, The Real Projective Plane; standard projective & inversive geometry texts.

Thanks to collaborators and the WTC community. I also thank “ChatGPT (GPT-5 Thinking),” “‎Gemini 2.5 Pro,” and “Wolfram Notebook Assistant + LLM Kit” for early discussions.