# 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.

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.

The Cairo tiling is a tiling of the plane by non-regular pentagons which is dual to the snub square tiling.

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.

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.

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.

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.

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.

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**.

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.

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.

This picture by Greg Egan shows a hypercube with all vertices except the bottom labelled by **duads**, that is, 2-element subsets of a 6-element set. There are 15 duads, while the hypercube has 16 vertices.

This picture shows the Tutte–Coxeter graph. This graph was discovered by the famous graph theorist William Thomas Tutte in 1947, but its remarkable properties were studied further by him and the geometer H. S. M. Coxeter in a pair of papers published in 1958.

This is the Heawood graph. This graph can be drawn on a torus with no edges crossing in such a way that it divides the torus into 7 hexagons, each pair of which shares an edge. In 1890, Percy John Heawood proved that for any map drawn on a torus, it takes at most 7 colors to ensure that no two countries sharing a common boundary have the same color. The Heawood graph proves that the number 7 is optimal.

This picture by Tilman Piesk shows the 14 Dyck words of length 8. A **Dyck word** is a balanced string of left and parentheses. In the picture, a left parenthesis is shown as upward-slanting line segment, and a right parenthesis as a downward-slanting one.

Suppose you have a set with 5 elements. There are 10 ways to choose a 2-element subset. Form a graph with these 10 choices as vertices, and with two vertices connected by an edge precisely when the corresponding subsets are disjoint. You get the graph shown here, called the **Petersen graph**.