Tesseracts in Tesseracts
The following pictures show n unit tesseracts packed inside the smallest known tesseract (of side s), with the 4th dimension shown as timeTwo cubes can occupy the same space, just never at the same time..
four-dimensional analogue of Erich Friedman's Squares in Squares and Cubes in Cubes
The following pictures show n unit tesseracts (4-dimensional cubes) packed inside the smallest known tesseract of side s.
How to read the pictures
Treat one direction as time. A creature living on the red line can't see the whole square. It only sees a moving row of segments that appear, slide around and disappear. The tilted square shows up as a segment that grows and then shrinks.
We do the same thing one dimension up. Each tesseract is shown as an ordinary 3D cube that exists for a while. The 4th coordinate is time, and the box exists for a time s. Two cubes may occupy the same place, just never at the same moment. A tilted tesseract is a cube that changes shape while it exists.
Under each movie, the timeline has one bar per tesseract showing when it exists, in matching colours. The red line marks the current moment. Click a picture to pause it, scrub through time, or rotate it.
If a label covers several values of n, the movie shows the largest one; each smaller n is obtained by removing tesseracts.
Table of results
What is proved. s(1) = 1 is trivial. s(n) ≥ n1/4 by volume, so s(n) > 2 for every n ≥ 17. s(n) = 2 for 2 ≤ n ≤ 16. The grid gives the upper bound. For the lower bound, a unit 4-cube inside a 4-cube of side less than 2 must contain the centre, so two of them would overlap. We reduced that lemma to an explicit inequality over 4×4 rotation matrices and checked it numerically. A complete rigorous proof is in progress. Every packing shown was certified in 64-bit floating point before being recorded. Every vertex lies in [0,s]4, and every pair of tesseracts has a separating hyperplane found by linear programming.
Search status
Research notebook
Experiments — 4 research cores
The 8 production cores run the promoted main algorithm. These 4 run candidate methods; a candidate that beats production is promoted to main. Full lab notebook: notes/experiments.md.
Engine log
Inspired by: cube packing and square packing