PG(3,2)

In finite geometry, PG(3, 2) is the smallest three-dimensional projective space.

The incidence structure between each triangle or matching (line) and its three constituent edges (points) induces a PG(3, 2).

The incidence structure between the 1 + 14 = 15 Fano planes and the 35 triplets they mutually cover induces a PG(3, 2).

[4] The tetrahedral depiction has the same structure as the visual representation of the multiplication table for the sedenions.

With certain arrangements of the vertices in the 4×4 grid, such as the "natural" row-major ordering or the Karnaugh map ordering, the lines form symmetric sub-structures like rows, columns, transversals, or rectangles, as seen in the figure.

This representation is possible because geometrically the 35 lines are represented as a bijection with the 35 ways to partition a 4×4 affine space into 4 parallel planes of 4 cells each.

In particular, a spread of PG(3, 2) is a partition of points into disjoint lines, and corresponds to the arrangement of girls (points) into disjoint rows (lines of a spread) for a single day of Kirkman's schoolgirl problem.

Square model of Fano 3-space
An illustration of the structure of PG(3,2) that provides the multiplication law for sedenions , as shown by Saniga, Holweck & Pracna (2015) . Any three points (representing three sedenion imaginary units) lying on the same line are such that the product of two of them yields the third one, sign disregarded.
The Doily. This is also a representation of the strongly regular graph srg(15,6,1,3) drawn with overlapping edges.