Research · Discrete geometry · Kakeya

Discrete Kakeya: the area of the union of 128 thin triangles

This page presents a piece of research on the discrete Kakeya problem, in its version of Keich’s thin triangles at n = 128: 128 thin triangles whose slopes are imposed, to be placed so that the area of their union is as small as possible. It shows the exact statement, the landscape of the known public values with their sources, then an area value of 0.106765175154124465, checked by three independent verifiers.

1. The problem

The slopes are imposed: a_i = i/128 for i from 0 to 127, and δ = 1/128. For each i, an intercept b_i is free. Thin triangle number i has, for x between 0 and 1, the vertical section [a_i·x + b_i − δ(1 − x), a_i·x + b_i]: a width δ at x = 0, shrinking to a point at x = 1. The goal is to minimise the area of the union of the 128 triangles.

Each triangle has area δ/2 = 1/256; 128 triangles with no overlap at all would cover 1/2. The 128 intercepts move the triangles relative to one another and decide how they overlap. It is the discrete version of a classical problem in geometry: the minimal size of a set that contains a segment in every direction (Kakeya and Besicovitch sets).

2. The public landscape

The table gathers the area values published for this problem at n = 128, with their source and date. The smaller the area, the more favourable it is. It is the map of the problem: it places the known values on an axis and says nothing about how they are obtained.

Arena ranking read on 9 Oct 2026. The first four entries are given; the value of the experiment is not in it.

3. The value of the experiment

The area of the union of the 128 triangles, for the experiment’s set of intercepts, is exactly the fraction:

723835944252385111836037533790024524449633/6779700807941047711428409834818691902996480

that is about 0.106765175154124465. This area is smaller by 3.26 × 10⁻⁵ than that of CrossingPaths, 2nd in the arena’s ranking, and larger by 1.12 × 10⁻⁵ than that of Ghazt, 1st in the ranking (read on 9 Oct 2026).

Three independent verifiers return the same area, checked on 13 Oct 2026 on a copy of the file:

The three values agree to within 10⁻¹⁵; the last two verifiers compute in floating point, the first in fractions.

4. Fingerprint

5. What this question models

The Kakeya problem asks for the smallest area that contains a unit segment in every direction. Its discrete version with n directions has order of magnitude 1/log n: Córdoba gives a lower bound and Keich, with Schoenberg, an upper bound, of that order. Exact values are known only for n ≤ 4 (Station). At n = 128, the published values range from 0.119207 (Keich, 1999) to 0.10675394011230244 (Ghazt, 6 Oct 2026): the landscape of section 2 shows where each one stands.

Source: arXiv:2603.15617, arXiv:2608.23691, arXiv:2511.02864, Keich (1999) and Einstein Arena, read on the dates indicated.

A research programme, an outside test

These results cover domains with nothing to do with each other: the geometry of packings (Packomania, Friedman Packing Center), space mechanics (ESA SpOC 4), number theory (IBM Ponder This) and graphs (FlyWire Codex). In each, the result is entered on a third party’s page, a table, a ranking or a list, and can be rechecked by computation: ESA evaluates each solution with its simulator, Packomania publishes the entries it accepts, IBM publishes a list of names, Friedman keeps a page that credits authors. These results belong to a single research programme: keep only what a certificate proves.

This page applies the same principle to a problem in discrete geometry: an exact area value, three independent checks and the fingerprint of the file. Unlike the other challenge pages, this value is not entered on any third party’s page: it is published here as research.

The principle is public: every hypothesis is tested by computation and kept only if a certificate proves it. Generative AIs serve me as assistants, interchangeable ones: they propose, my AI decides, the certificate settles. The internal workings remain confidential. The results are public.

The certificate changes the nature of proof. A result can be checked without trusting whoever produced it, as a proof can be checked without knowing its author. That is what makes this page possible: you do not have to believe me, you recheck. The result is fixed by its fingerprint, which is published.

All the work runs on accessible hardware: a desktop PC, a Raspberry Pi 5 and a Raspberry Pi Pico, which also serve to check that the computation stays identical from one machine to another; occasionally, a 32-vCPU instance. The programme aims at a certifiable embedded AI, down to modest machines.

A result that is verifiable without trust, fixed and rechecked: these properties open onto the intended uses:

• certifiable embedded AI, down to modest machines;
• steering generative AIs, keeping only what is proven;
• verifiable identity for AI agents;
• audit after an incident: replaying a past decision exactly;
• reproducible science;
• an open map of certified computations: each result with its certificate, so that what is proven is not recomputed.

The research programme sets out this framework. All the results, with their sources and dates, are gathered on the Results page.