A Library of Fully Diagonalized Latin k-cubes

Squares, cubes and tesseracts in which every axis line and every diagonal is a rainbow

A fully diagonalized Latin $k$-cube of order $n$ in dimension $k$ is an array of $n^k$ symbols in which every line carries all $n$ symbols exactly once. The lines are the axis lines — fix all but one coordinate and let the last run — together with every unbroken diagonal, meaning every line on which some set of coordinates runs forward, some set runs backward, and the rest stay fixed. At $k = 2$ that is a Latin square whose two main diagonals are also rainbow. For larger $k$, the number of diagonals depends on the order of the cube. You can also think of a fully diagonalized Latin $k$-cube as a Latin $k$-cube whose $(k-1)$-dimensional hyperplane slices are all fully diagonalized Latin $(k-1)$-cubes, though you need to either include slanted $(k-1)$-dimensional hyperplanes or directly stipulate the $2^{k-1}$ $k$-space diagonals.

One consequence is immediate: the $2^k$ corner cells are pairwise joined by diagonals, so they all carry different symbols, and therefore $n \in \lbrace 0, 1\rbrace$ or $n \ge 2^k$. In 1972, Walter Taylor asked whether this is the only obstruction. We now believe that it is not, and in particular that a $9 \times 9 \times 9$ does not exist. Classifying in general when they do and do not exist is a work in progress.

References

1. Walter Taylor, “On the coloration of cubes”, Discrete Mathematics 2 (1972) 187–190.
2. J. Arkin, V. E. Hoggatt Jr. and E. G. Straus, “Systems of magic Latin $k$-cubes”, Canadian Journal of Mathematics 28 (1976) 1153–1161.
3. Ervin Gergely, “A simple method for constructing doubly diagonalized Latin squares”, Journal of Combinatorial Theory Series A 16 (1974) 266–272.
4. A. Hedayat, “A complete solution to the existence and nonexistence of Knut Vik designs and orthogonal Knut Vik designs”, Journal of Combinatorial Theory Series A 22 (1977) 331–337.