De Bruijn torus

STL model of de Bruijn torus (16,32;3,3)2 with 1s as panels and 0s as holes in the mesh – with consistent orientation, every 3×3 matrix appears exactly once (external viewer)​

In combinatorial mathematics, a De Bruijn torus, named after Dutch mathematician Nicolaas Govert de Bruijn, is an array of symbols from an alphabet (often just 0 and 1) that contains every possible matrix of given dimensions m × n exactly once. It is a torus because the edges are considered wraparound for the purpose of finding matrices. Its name comes from the De Bruijn sequence, which can be considered a special case where n = 1 (one dimension).

One of the main open questions regarding De Bruijn tori is whether a De Bruijn torus for a particular alphabet size can be constructed for a given m and n. It is known that these always exist when n = 1, since then we simply get the De Bruijn sequences, which always exist. It is also known that "square" tori exist whenever m = n and even (for the odd case the resulting tori cannot be square).[1][2][3]

The smallest possible binary "square" de Bruijn torus, depicted above right, denoted as (4,4;2,2)2 de Bruijn torus (or simply as B2), contains all 2×2 binary matrices.

  1. ^ Fan, C. T.; Fan, S. M.; Ma, S. L.; Siu, M. K. (1985). "On de Bruijn arrays". Ars Combinatoria A. 19: 205–213.
  2. ^ Chung, F.; Diaconis, P.; Graham, R. (1992). "Universal cycles for combinatorial structures". Discrete Mathematics. 110 (1): 43–59. doi:10.1016/0012-365x(92)90699-g.
  3. ^ Jackson, Brad; Stevens, Brett; Hurlbert, Glenn (Sep 2009). "Research problems on Gray codes and universal cycles". Discrete Mathematics. 309 (17): 5341–5348. doi:10.1016/j.disc.2009.04.002.