A three-dimensional Kakeya set

A Kakeya set contains a unit line segment in every direction, yet can have astonishingly small volume. Here is the three-dimensional analogue of the planar Perron tree, built from 1 × δ × δ tubes (cylinders). The same N² tubes are shown two ways: sharing a base they fan into a bush of volume ≈ 1/3; translated apart in the Perron pattern they overlap into a far smaller tangle — while every tube still points in its own direction. Drag to rotate; use the compress slider to slide between the two.

An original interactive applet, part of tao-web.

tree depth p (N = 2^p)

thickness δ

compress (bush → Perron)

opacity

colour by direction bounding box axes show one tube clean view

Drag to rotate · scroll to zoom · the readout below reports the exact volume of the tube union.

What you are seeing. The directions form a two-parameter patch d = (s, u, −1) with s, u ranging over a grid of N = 2^p values each, so there are N² tubes — the 3D analogue of the fan of directions in the planar Perron tree. Every tube keeps its direction under translation, so the compressed tangle is still a Kakeya set for this patch of directions; the full “every direction in space” set is a union of finitely many rotated copies. The volume is computed exactly (the union’s cross-section at each height is a product set), and you may notice it is smallest at an intermediate compression, around 0.7–0.8, rather than at the full Perron offset.

Companion to the two-dimensional Kakeya needle applet.