arxiv: 2605.01476 · v1 · submitted 2026-05-02 · 🧮 math.MG · math.CA· math.DS

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How Thick Is the Sierpi\'nski Triangle?

Scott Duke Kominers

Pith reviewed 2026-05-10 16:23 UTC · model grok-4.3

classification 🧮 math.MG math.CAmath.DS

keywords Sierpiński triangleFeng-Wu thicknessconvex hullequilateral triangleself-similarityfractal geometrymetric geometry

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The pith

The Sierpiński triangle's local convex hull always contains an equilateral triangle of side equal to the radius, fixing its thickness at exactly √3/6.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The Sierpiński triangle has zero area yet remains uniformly non-flat at every location and scale. For any point x in the set and any radius r, the convex hull of the points inside the disk of radius r contains an equilateral triangle whose side length equals r. This forces the hull to contain a disk of radius (√3/6)r, and the constant is shown to be the largest possible. The argument relies on the survival of boundary edges through the construction and on self-similarity to reduce all cases to a single scale interval.

Core claim

Let E be the standard Sierpiński triangle of side length 1. For every x in E and every 0 < r ≤ 1, the convex hull of E ∩ B(x,r) contains an equilateral triangle of side length r. Consequently, conv(E ∩ B(x,r)) contains a closed disk of radius (√3/6)r, and this constant is best possible.

What carries the argument

Persistence of boundary edges from the iterative construction in the limit set E, which combines with self-similarity to reduce the thickness verification to the normalized interval 1/2 ≤ r ≤ 1.

If this is right

The Feng-Wu thickness of the Sierpiński triangle equals exactly √3/6.
The set is uniformly non-flat: every local convex hull contains a disk whose radius is a fixed positive fraction of the observation scale.
The bound is achieved at the inradius of an equilateral triangle and cannot be improved.
The property holds uniformly for every point in the set and every scale up to the diameter.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The same edge-persistence argument may extend to other self-similar sets built by iterated removal of open triangles.
Finite-stage approximations of the Sierpiński triangle could be checked numerically to see how quickly their local thickness approaches the limit value √3/6.
The result supplies a concrete lower bound on the two-dimensional content of zero-area sets that might be compared with other notions of thickness or dimension in the plane.

Load-bearing premise

Boundary edges of the construction triangles survive in the completed limit set E.

What would settle it

Locate a point x in E and a radius r ≤ 1 such that conv(E ∩ B(x,r)) contains no equilateral triangle of side length r.

Figures

Figures reproduced from arXiv: 2605.01476 by Scott Duke Kominers.

**Figure 1.** Figure 1: A finite-stage approximation to the standard Sierpiński triangle. The limit set is obtained by repeating the corner-triangle construction indefinitely. Feng–Wu thickness [5] packages the idea just described into a numerical invariant. This invariant was originally introduced to study when arithmetic sums of measure-zero sets become large enough to acquire interior; thus, Feng–Wu thickness measures both a … view at source ↗

**Figure 2.** Figure 2: First step of the construction and the cells T1, T2, T3. In particular, E ⊂ ∆. Moreover, v1, v2, v3 ∈ E, since each vertex remains in every stage Km. This construction also makes the zero-area statement visible: Km is a union of 3 m equilateral triangles of side length 2 −m, so area(Km) = 3 4 m area(∆) −→ 0. Thus E has area 0. 1.2. Thickness of compact sets in R 2 . Throughout the paper, B(x, r) denotes… view at source ↗

**Figure 3.** Figure 3: The case i = 1: any point x ∈ T1 lies inside the corner triangle Q1(r), and the three vertices of Q1(r) lie in B(x, r). 3. A local triangle in the normalized range Recall that Ti = ϕi(∆) for i = 1, 2, 3. Each Ti is an equilateral triangle of side length 1/2. For r ∈ [1/2, 1], define three larger “corner triangles:” Q1(r) = convn v1, v1 + r(v2 − v1), v1 + r(v3 − v1) o , Q2(r) = convn v2, v2 + r(v1 − v2), v2… view at source ↗

**Figure 4.** Figure 4: The rescaling step, illustrated for the lower-left level-2 cell, where ϕw(z) = z/4, so x ′ = ϕ −1 w (x) = 4x and r ′ = 4r. The map ϕ −1 w carries the chosen cell ∆w onto the full normalized triangle ∆, so x ′ has the same relative position in ∆ that x has in ∆w. In general, for |w| = n, applying ϕ −1 w sends x to x ′ = ϕ −1 w (x) and rescales the radius to r ′ = 2n r ∈ [1/2, 1], which is the range covered … view at source ↗

**Figure 5.** Figure 5: The Minkowski sum of two segments contained in E is a parallelogram contained in E + E. local convex-geometric invariant enters general sumset theorems. Applying the theorem with F1 = · · · = Fn = E, d = 2, and c = √ 3/6, we obtain the following. Corollary 6. If n > √ 2 q 1 + √ 3/6 − 1 2 ≈ 77.37, then the n-fold sumset nE = E + · · · + E | {z } n times has nonempty interior. In particular, nE has nonemp… view at source ↗

read the original abstract

Although the Sierpi\'nski triangle has planar area $0$, it is uniformly non-flat: at every point and every scale, its nearby points span a two-dimensional region of comparable size. We prove a sharp version of this statement, showing that the Feng--Wu thickness of $E$ is exactly $\sqrt{3}/6$, the inradius of a unit equilateral triangle. More precisely, if $E$ is the standard Sierpi\'nski triangle of side length $1$ and $B(x,r)$ denotes the closed disk of radius $r$ centered at $x$, then for every $x\in E$ and every $0<r\le 1$, the convex hull of $E\cap B(x,r)$ contains an equilateral triangle of side length $r$. Consequently, $\operatorname{conv}(E\cap B(x,r))$ contains a closed disk of radius $(\sqrt{3}/6)r$; this constant is best possible. The proof is elementary -- boundary edges of all construction triangles survive in the limit set, and self-similarity reduces the problem to the normalized range $1/2\le r\le 1$.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 2 minor

Summary. The manuscript proves that the Feng-Wu thickness of the standard Sierpiński triangle E (side length 1) is exactly √3/6. For every x ∈ E and 0 < r ≤ 1, conv(E ∩ B(x,r)) contains an equilateral triangle of side length r and hence a closed disk of radius (√3/6)r; the constant is shown to be sharp. The argument is elementary, using persistence of boundary edges in the construction and self-similarity to reduce to the normalized range 1/2 ≤ r ≤ 1.

Significance. If the result holds, it supplies a sharp, scale-invariant geometric measure of the uniform two-dimensionality of the Sierpiński gasket at every point and scale, despite zero area. The elementary self-similarity proof, free of fitted parameters and relying only on the standard construction, is a clear strength and makes the claim readily verifiable.

minor comments (2)

The reduction to 1/2 ≤ r ≤ 1 via self-similarity is central; a short explicit statement of the surviving boundary-edge property (with a reference to the standard construction) would make the base of the induction fully transparent.
A single illustrative figure showing the equilateral triangle of side r inside conv(E ∩ B(x,r)) for a non-vertex point would help readers visualize the geometric step.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the careful reading, positive summary, and favorable significance assessment of our manuscript. The recommendation of minor revision is noted; however, the major comments section contains no specific points requiring clarification or correction. We have re-examined the argument and believe the elementary self-similarity proof is complete and the sharpness statement is correctly established.

Circularity Check

0 steps flagged

No circularity: direct geometric proof from construction and self-similarity

full rationale

The derivation relies on an elementary geometric argument: boundary edges of the construction triangles persist in the limit set E, and self-similarity reduces the problem to the normalized interval 1/2 ≤ r ≤ 1. The key statement—that conv(E ∩ B(x,r)) contains an equilateral triangle of side exactly r—is established by direct inspection of the iterative construction at each scale, without any parameter fitting, redefinition of the target constant, or load-bearing self-citation. Sharpness follows from exhibiting explicit points in E where the inradius bound is attained. The proof is self-contained against the standard Sierpiński construction and does not reduce any claimed prediction or uniqueness result to its own inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The claim rests on the standard iterative construction of the Sierpiński triangle and the survival of boundary edges in the limit; no free parameters or new entities are introduced.

axioms (2)

domain assumption The Sierpiński triangle E is the attractor of the standard iterated function system consisting of three contractions by factor 1/2 toward the vertices of an equilateral triangle.
Invoked implicitly when self-similarity is used to reduce scales.
domain assumption Boundary edges of every construction-stage triangle remain in the final set E.
Explicitly cited in the abstract as the key fact enabling the reduction to the base range 1/2 ≤ r ≤ 1.

pith-pipeline@v0.9.0 · 5499 in / 1396 out tokens · 45463 ms · 2026-05-10T16:23:51.708227+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

10 extracted references · 1 canonical work pages · 1 internal anchor

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