Hoffman–Singleton Graph
This is the Hoffman–Singleton graph, a remarkably symmetrical graph with 50 vertices and 175 edges. There is a beautiful way to construct the Hoffman–Singleton graph by connecting 5 pentagons to 5 pentagrams.
Cairo Tiling
The Cairo tiling is a tiling of the plane by non-regular pentagons which is dual to the snub square tiling.
Free Modular Lattice on 3 Generators
This is the free modular lattice on 3 generators, as drawn by Jesse McKeown. First discovered by Dedekind in 1900, this structure turns out to have an interesting connection to 8-dimensional Euclidean space.
Kaleidocycle
This image by intocontinuum show how you can take 8 perfectly rigid regular tetrahedra and connect them along their edges to form a ring that you can turn inside-out.
Golay Code
The extended binary Golay code, or Golay code for short, is a way to encode 12 bits of data in a 24-bit word in such a way that any 3-bit error can be corrected, and any 7-bit error can at least be detected. The easiest way to understand this code uses the geometry of the dodecahedron, as shown in this image by Gerard Westendorp.
Newton’s Apsidal Precession Theorem
A little-known result in Newton’s Principia Mathematica is that by adding an inverse cube force to the usual force of gravity, you could change the angular motion of a planet while leaving its radial motion unchanged. This can have strange effects, shown in the animation here.
Balaban 11-Cage
This picture shows part of a graph called the Balaban 11-cage. A (3,11)-graph is a simple graph where every vertex has 3 neighbors and the shortest cycle has length 11. A (3,11)-cage is a (3,11)-graph with the least possible number of vertices. The Balaban 11-cage is the unique (3,11)-cage.
Harries Graph
This is the Harries graph. It is a graph with the minimum number of vertices such that each vertex is connected to 3 others and every cycle has length at least 10. Such graphs are called (3,10)-cages.
Balaban 10-Cage
This is the Balaban 10-cage, the first known (3,10)-cage. An (r,g)-cage is graph where every vertex has r neighbors, the shortest cycle has length at least g, and the number of vertices is maximal given these constraints.
McGee Graph
This is the McGee graph. It is 3-regular graph, meaning that every vertex has 3 neighbors. It also has girth 7, meaning that the shortest cycles have length 7. What makes the McGee graph special is that it has the least number of vertices of any 3-regular graph of girth 7.