One needle and very little space
How small can a set be if it contains a unit segment in every direction?
The needle problem
A line segment has no arrow, so directions 180° apart are equivalent. A half-turn therefore covers every unoriented direction. Rotating around a fixed center sweeps a disk, but translations and rotations can make many poses overlap.
Small area, difficult construction
A Kakeya set contains a unit segment in every direction. Surprisingly, such sets can have arbitrarily small positive area, and mathematical constructions of area zero exist. The app uses finitely many direction bins and pixel or voxel grids, so its measurements are approximations, not proofs.
Higher dimensions
In three and higher dimensions, the modern conjecture asks about the dimension a set containing a unit segment in every direction must have. The app’s 3D labs project directions onto a sphere and estimate swept volume with voxels to build intuition.