Exploring the Kakeya Conjecture: A Codex‑Powered Interactive Lab After Wang Hong’s Fields Medal

Kakeya Conjecture Background

In 1917 the "needle problem" asked for the smallest planar region that allows a fixed‑length needle to rotate continuously through 180°. If the region is required to be convex, Pál proved the optimal shape is an equilateral triangle of height 1. Removing the convexity restriction, Besicovitch constructed sets of arbitrarily small area.

Modern Kakeya (or Besicovitch) sets are defined in ℝⁿ as sets K that contain a unit line segment in every direction. Such sets may have Lebesgue measure zero. The Kakeya conjecture predicts that any Kakeya set must have Hausdorff dimension n (and the same for Minkowski dimension).

Mathematical Results

Wang Hong and Joshua Zahl proved the three‑dimensional case in 2025, showing that both the Hausdorff and Minkowski dimensions of a three‑dimensional Kakeya set equal 3. In 2026 Larry Guth, Wang Hong, and Joshua Zahl published a streamlined proof of the same result. The conjecture remains open for dimensions four and higher, and the stronger Kakeya maximal conjecture is also unresolved.

Interactive Lab Construction

The aim was to let a first‑time reader develop intuition about the conjecture by rotating, dragging, and adjusting parameters without reading formal definitions.

Prompt to Codex

你是一名擅长科学可视化、交互叙事和前端开发的设计工程师

请为我制作一个中文单页互动科普网站

项目主题：挂谷猜想 Kakeya Conjecture

项目目标：让第一次接触挂谷猜想的普通读者先建立直觉，再理解正式定义、三维结论、证明路线与仍未解决的问题

一、事实与资料要求
1. 先核验资料再写页面
2. 区分以下三个问题：1917 年的挂谷针问题、Besicovitch 集合、现代挂谷集合猜想
3. 准确说明：在 R^n 中，挂谷集合对每个方向都包含一条单位线段；这类集合可以拥有零 Lebesgue 测度；挂谷集合猜想预言 Hausdorff 维数为 n；常见 Minkowski 版本也预言完整维数；王虹与 Joshua Zahl 在 2025 年证明三维情形，两种维数都等于 3；2026 年 Larry Guth、王虹与 Joshua Zahl 发布精简证明；四维及以上仍然开放；Kakeya maximal conjecture 仍未解决
4. 准确说明菲尔兹奖：2026 年王虹获得菲尔兹奖，三维挂谷猜想是其代表性成果之一，官方获奖词还覆盖 Fourier restriction、局部光滑、Falconer 距离问题、Furstenberg 集合等贡献
5. 所有重要结论都要能回到官方资料、原始论文或可靠综述
6. 禁止虚构定理、数字、引文、论文结论和证明步骤
7. 页面必须明确提示：有限线段、有限细管、像素覆盖和动画只用于建立直觉，不能充当数学证明

Three.js Experiment Requirements

Use Three.js.

Represent each sampled direction by a thin cylinder or tube of radius δ.

Sample directions uniformly on the unit sphere.

Provide four arrangement modes: central starburst, dispersed, multiscale adhesion, and wood‑grain particles.

Include a sample‑count slider (≈80–900) and a radius δ slider.

Support drag‑rotate, scroll‑zoom, reset view, and pause rotation.

Update the model and status text immediately after any parameter change.

Avoid per‑frame creation of many objects to control performance.

Reduce animation when the page leaves the viewport or when the user prefers reduced motion.

Project Build and Run

Repository: https://github.com/tjxj/kakeya-conjecture-lab

After cloning, execute:

npm install
npm run dev

To generate a deployable single‑file page: npm run build The build output is dist/index.html, which can be served directly.

Reusable Workflow for Scientific Visualizations

Verify factual information against official sources, original papers, or reliable surveys.

Construct an intuitive geometric or physical model that captures the core phenomenon.

Design interactive controls (sliders, buttons, view manipulation) and a visual layout that works on both desktop and mobile.

State explicitly the scientific boundaries—e.g., finite line segments, thin tubes, pixel coverage, and animations are illustrative only and do not constitute a proof.

Publish a responsive single‑page application using a modern bundler (Vite) and host it on a static site platform.