Abstract
The surface of a polyhedron is flat at every point of it — the inside of a face is a plane, and an edge unrolls flat like paper. And the total curvature is $4\pi$ all the same. This paper shows in numbers that both of these are true. NO NEW MATHEMATICAL THEOREM AND NO NEW LAW IS CLAIMED.
First, every regular solid gives $4\pi$. A tetrahedron has four defects of $\pi$, a cube eight of $\pi/2$, an octahedron six of $2\pi/3$ and an icosahedron twelve of $\pi/3$ — each totalling $12.566370614359$ (Section 2).
Second, refining the solid does not move the total. Taking the vertices from $12$ to $42$ to $162$ to $642$, the largest single defect falls from $1.047$ to $0.0228$ while the total agrees to the fourteenth digit (Section 3).
Third, this is the core. Building a torus the same way gives exactly $0$ — positive defects of $+12.3236$ and negative ones of $-12.3236$ cancelling completely (Section 5).
Fourth, a negative defect could not be made by denting. Pushing the centre of a face inwards leaves its defect at $\mathbf{+2.384}$ — a dent is only a cone turned inside out, and only a saddle goes negative (Section 7).
Fifth, a notch does produce one. Cutting a corner out of a cube gives three vertices of exactly $-\pi/2$, with $+17.2788$ on the positive side and $-4.7124$ on the negative — and the total is $4\pi$ still (Section 7).
Scope (what is not claimed)
NO NEW MATHEMATICAL THEOREM AND NO NEW LAW IS CLAIMED — Descartes' theorem on the angular defect, Euler's polyhedron formula, the Gauss–Bonnet theorem and the equivalence of the first two are all standard. No theorem is proved — what is done here is to build polyhedra and add. No general theory of discrete differential geometry is offered. Neither the classification of surfaces nor a general treatment of genus appears — only the sphere and the torus occur. Self-intersecting polyhedra are not treated. Relation to earlier papers: Where an angular defect has appeared before, it has always been on the smooth side — Paper 27 separated the curvature of the ground from that of spacetime, and Paper 33 showed the Wigner angle to be the area of a spherical triangle. This paper is the discrete side of the same quantity, where the curvature gathers at a point. In Paper 212 every map had to distort something — here the distortion may be swept entirely onto the vertices and the bill is unchanged. What is added is adding the defects directly for four regular solids, setting the per-vertex value beside the total along a sequence of refinements, giving the ratio of defect to the area a vertex carries, measuring the cancellation on a torus, checking that denting does not make a defect negative, and obtaining three defects of $-\pi/2$ from a notch.
1. Notation and Position
The angular defect at a vertex of a polyhedron is $2\pi$ less all the face angles meeting there. On a flat sheet the angles come to exactly $2\pi$ and the defect is $0$.
Position: Where an angular defect has appeared in earlier papers it has been on the smooth side — Paper 27 asked which manifold's curvature was meant, and Paper 33 showed the Wigner angle to be an area. This is the discrete side, where the curvature gathers at points.
2. The Regular Solids
Check 2.1: compute the angles from the coordinates and add them vertex by vertex.
| Solid | Vertices | Defect at each | Total | Euler characteristic |
|---|---|---|---|---|
| Tetrahedron | $4$ | $\pi$ ($3.141592653590$) | $12.566370614359$ | $2$ |
| Cube | $8$ | $\pi/2$ ($1.570796326795$) | $12.566370614359$ | $2$ |
| Octahedron | $6$ | $2\pi/3$ ($2.094395102393$) | $12.566370614359$ | $2$ |
| Icosahedron | $12$ | $\pi/3$ ($1.047197551197$) | $12.566370614359$ | $2$ |
Observation 2.2 The defect at a vertex is different in all four cases, and every total is $4\pi=12.566370614359$. The number of vertices and the defect at each balance exactly — the shape does not enter.
3. Refining the Solid Does Not Move the Total
Check 3.1: quarter each face of an icosahedron and project onto the unit sphere, repeatedly.
| Vertices | Faces | Total defect | Largest single defect |
|---|---|---|---|
| $12$ | $20$ | $12.56637061435917$ | $1.04719755119660$ |
| $42$ | $80$ | $12.56637061435917$ | $0.30934133337934$ |
| $162$ | $320$ | $12.56637061435916$ | $0.08227110788505$ |
| $642$ | $1280$ | $12.56637061435901$ | $0.02276083999584$ |
Observation 3.2 The defect at a vertex falls from $1.047$ to $0.0228$, a factor of forty-six, and the total agrees to the fourteenth digit. The curvature spreads thinner, and the bill never moves.
4. The Defect Approaches Curvature Times Area
On the unit sphere the curvature is $1$ everywhere, so the defect at a vertex should match the area that vertex carries.
Check 4.1 (at $642$ vertices):
| Vertex | Defect | Area it carries | Ratio |
|---|---|---|---|
| Of degree $5$ | $0.02276083999584$ | $0.02268154637897$ | $1.0034959529$ |
| Of degree $6$ | $0.01889749080768$ | $0.01885512890356$ | $1.0022467046$ |
| Total | $12.566370614359$ | $12.506492733970$ | ── |
Observation 4.2 The ratio tends to $1$, and the defect behaves like an integral of curvature — but note that the area comes to $12.5065$ and falls short of $4\pi$, the polyhedron being inscribed and a little small, while the defect is $4\pi$ exactly from the start. What approximates is the area, and not the total.
5. The Core: A Torus Gives Zero
Check 5.1: build a torus the same way, with $R=2$, $r=1$ and a $24\times16$ mesh.
| Measured | Value |
|---|---|
| Vertices, edges, faces | $384$, $1152$, $768$ |
| Euler characteristic | $\mathbf{0}$ |
| Total defect | $\mathbf{0}$ |
| Positive defects | $+12.3235628615$ |
| Negative defects | $-12.3235628615$ |
Observation 5.2 The outside curves one way and the inside the other, and the two cancel exactly. $4\pi$ on a sphere and $0$ on a torus — the total is $2\pi$ times the Euler characteristic, settled by the number of holes and not by the shape.
6. Cutting a Face Changes Nothing
Check 6.1: cut one face of a cube into four triangles about a new centre.
| Vertices | Edges | Faces | Total defect | |
|---|---|---|---|---|
| The cube | $8$ | $12$ | $6$ | $12.566370614359$ |
| One face cut | $9$ | $16$ | $9$ | $12.566370614359$ |
Observation 6.2 The new vertex has a defect of exactly $0$ — it was placed on a flat face. Vertices, edges and faces all increased, and neither $V-E+F$ nor the total moved — a vertex is a mark for counting and not where the curvature lives.
7. Denting Fails, and a Notch Succeeds
Check 7.1 (the guess that failed): push the new centre vertex of a cube's face inwards.
| Measured | Value |
|---|---|
| Defect at the pushed-in vertex | $\mathbf{+2.38432329146618}$ |
| Smallest defect anywhere on it | $+0.97471550392835$ |
| Total defect | $12.566370614359$ |
Observation 7.2 The guess was wrong. Pushing a face in leaves that vertex a cone still, merely turned inside out — its angles fall short of $2\pi$ as before and the defect stays positive. A defect goes negative only at a saddle, where the angles exceed $2\pi$.
Check 7.3: remove a corner cube from the cube $[0,2]^3$.
| Measured | Value |
|---|---|
| Vertices, edges, faces | $14$, $36$, $24$ (Euler characteristic $2$) |
| Vertices with a negative defect | $\mathbf{3}$, each of $\mathbf{-\pi/2}$ |
| Positive defects | $+17.278759594744$ |
| Negative defects | $-4.712388980385$ |
| Total | $\mathbf{12.566370614359}$ |
Observation 7.4 Three vertices count against the rest, and the total is $4\pi$ all the same. Convexity is not needed at all — what is needed is only that there be no hole.
8. The Separator and the Fence
Observation 8.1 (the separator) What separates the senses of curvature is whether it is read as a value at a point or as a measure.
| Where | Curvature |
|---|---|
| Inside a face | $0$ |
| Along an edge | $0$ (the surface unrolls flat) |
| At a vertex | a point mass equal to the defect |
Observation 8.2 As a value at a point it is zero almost everywhere, and as a measure its total is $4\pi$. The vertices are a set of measure zero, so the integral of a function that vanishes almost everywhere comes to $4\pi$ — no contradiction, but one word standing for two different objects.
Observation 8.3 (the fence) The fence around this surface is flat is whether a point is meant or a region. Outside it the sentence is not even false but empty — the same shape as this sequence is random in Paper 223 and the blackbody peaks here in Paper 222.
Observation 8.4 And what does not move is the real thing. The defect at a vertex shrinks by a factor of forty-six under refinement, the vertices go from $12$ to $642$, and a notch even changes the sign — and the total never moves once. The same shape as Paper 222, where a change of variable broke the values of a density and left the integrals alone.
9. The Census and the Close
What this paper measured:
- Every regular solid gives $4\pi$ (Check 2.1): the per-vertex value differs in all four.
- Refinement leaves the total (Check 3.1): the largest defect goes $1.047\to0.0228$.
- The defect tends to curvature times area (Check 4.1): ratios $1.0035$ and $1.0022$.
- A torus gives $0$ (Check 5.1): $+12.3236$ against $-12.3236$.
- Cutting a face leaves the total (Check 6.1): the new vertex has defect $0$.
- Denting does not go negative (Check 7.1): it stays at $+2.384$.
- A notch gives three defects of $-\pi/2$ (Check 7.3): and the total is $4\pi$.
References
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- Yamagishi, Y. (2026). Every Map Must Distort Something [Paper 212]. Zenodo. 10.5281/zenodo.22112104
- Yamagishi, Y. (2026). There Are Four Independences, and They Say Nothing to One Another [Paper 220]. Zenodo. 10.5281/zenodo.22119838
- Yamagishi, Y. (2026). The Sun Peaks in the Green, in the Orange and in the Infrared [Paper 222]. Zenodo. 10.5281/zenodo.22120199
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Reference
Yamagishi, Y. (2026). Flat Wherever You Touch It, and Four Pi in Total [Paper 224]. YOI: https://www.yamagishiyuuki.com/yoi/224
DOI: 10.5281/zenodo.22120836
ORCID: 0009-0006-8587-3450
PDF (English) · PDF (Japanese) · Date proof of the English PDF (OpenTimestamps .ots) · SHA256SUMS
The record page for this paper (in Japanese, with the PDFs in each language and the date proof): YOI 224. This paper is a preprint and has not been peer reviewed.