# Leech polytope

Leech polytope
Rank24
TypeIsogonal
Notation
Bowers style acronymLeech
Elements
Facets1197362269604214277200 (232 types)
Edges452088000+4629381120
Vertices196560
Measures (short edge length 1)
Edge lengthsShort (452088000): 1
Long (4629381120): ${\displaystyle {\frac {\sqrt {6}}{2}}\approx 1.22474}$
Circumradius${\displaystyle {\frac {\sqrt {3}}{2}}\approx 0.86603}$
Central density1
Related polytopes
ArmyLeech
RegimentLeech
ConjugateLeech
Abstract & topological properties
Euler characteristic0
OrientableYes
Properties
SymmetryCo0, order 8315553613086720000
ConvexYes
NatureTame

The Leech polytope is a convex isogonal 24-polytope defined as the contact polytope of the Leech lattice. It is perhaps easiest to understand in relation to the kissing number problem, which asks about the maximum number of non-overlapping unit n -balls in n -dimensional Euclidean space that all touch a central unit n -ball. The kissing number in 24 dimensions is exactly 196,560, which requires a packing so restrictive that it admits only one possible configuration up to rotation and reflection. When the convex hull is taken of the centers of the outer 24-balls, the Leech polytope results. It has 1197362269604214277200 facets, 5081469120 edges and 196560 vertices.

The Leech polytope is chiral, and has a lower symmetry order than the regular 24-polytopes. For example, the symmetry order of the 24-dimensional simplex is 15511210043330985984000000, which is 1865325 times higher than the symmetry of the Leech polytope.

The ratio between the longest and shortest edges is ${\displaystyle {\frac {\sqrt {6}}{2}}}$ ≈ 1.22474.

## Vertex coordinates

The vertex coordinates of the Leech polytope are exactly the vectors in the Leech lattice that go to the centers of the 196560 hyperspheres that touch the hypersphere centered at the origin.

To obtain the coordinates of these vectors, we need information from the extended binary Golay code. For our purposes, we can treat the code as a mapping between 12-bit binary numbers and 24-bit binary "codewords", or even more simply as a list of those codewords. A codeword can be obtained by multiplying a 12-bit binary number (viewed as a 12-element row vector of binary digits) by the generation matrix of the code, over the integers modulo 2. (This means that 1+1=0. We can achieve this effect by doing the multiplication in the "natural" manner and then taking the remainder of the results modulo 2.)

The generation matrix of the code is:

${\displaystyle {\begin{bmatrix}1&0&0&0&0&0&0&0&0&0&0&0&1&0&0&1&1&1&1&1&0&0&0&1\\0&1&0&0&0&0&0&0&0&0&0&0&0&1&0&0&1&1&1&1&1&0&1&0\\0&0&1&0&0&0&0&0&0&0&0&0&0&0&1&0&0&1&1&1&1&1&0&1\\0&0&0&1&0&0&0&0&0&0&0&0&1&0&0&1&0&0&1&1&1&1&1&0\\0&0&0&0&1&0&0&0&0&0&0&0&1&1&0&0&1&0&0&1&1&1&0&1\\0&0&0&0&0&1&0&0&0&0&0&0&1&1&1&0&0&1&0&0&1&1&1&0\\0&0&0&0&0&0&1&0&0&0&0&0&1&1&1&1&0&0&1&0&0&1&0&1\\0&0&0&0&0&0&0&1&0&0&0&0&1&1&1&1&1&0&0&1&0&0&1&0\\0&0&0&0&0&0&0&0&1&0&0&0&0&1&1&1&1&1&0&0&1&0&0&1\\0&0&0&0&0&0&0&0&0&1&0&0&0&0&1&1&1&1&1&0&0&1&1&0\\0&0&0&0&0&0&0&0&0&0&1&0&0&1&0&1&0&1&0&1&0&1&1&1\\0&0&0&0&0&0&0&0&0&0&0&1&1&0&1&0&1&0&1&0&1&0&1&1\end{bmatrix}}}$
This matrix (or an equivalent one) can be created by taking the adjacency matrix (for which A ij  = 1 if vertices i  and j  are connected by an edge, and 0 if not) of the regular icosahedron, taking the "complement" of it (changing all the 1s to 0s and vice versa), and appending the resulting matrix to the right of a 12×12 identity matrix.

Of the 4096 codewords, one is all 0s, 759 have eight 1s (and are referred to as "octads"), 2576 have twelve 1s, another 759 have sixteen 1s (and are the complements of the octads), and one is all 1s.

With this in mind, the vertex coordinates of the Leech polytope, centered at the origin and with circumradius ${\displaystyle {\frac {\sqrt {3}}{2}}}$, are given by:

• All permutations and sign changes of
${\displaystyle {\frac {\sqrt {6}}{16}}\left(4,\,4,\,0,\,0,\,0,\,0,\,0,\,0,\,0,\,0,\,0,\,0,\,0,\,0,\,0,\,0,\,0,\,0,\,0,\,0,\,0,\,0,\,0,\,0\right)}$
This produces ${\displaystyle {\binom {24}{2}}\times 4=1104}$ vertices.
• All even sign changes of the nonzero coordinates of the following, where the nonzero coordinates are located at the 1s of an octad:
${\displaystyle {\frac {\sqrt {6}}{16}}\left(2,\,2,\,2,\,2,\,2,\,2,\,2,\,2,\,0,\,0,\,0,\,0,\,0,\,0,\,0,\,0,\,0,\,0,\,0,\,0,\,0,\,0,\,0,\,0\right)}$
This produces 128 × 759 = 97152 vertices.
• All permutations of the following, then with individual coordinate's signs changed at all the locations of the 1s of a codeword:
${\displaystyle {\frac {\sqrt {6}}{16}}\left(-3,\,1,\,1,\,1,\,1,\,1,\,1,\,1,\,1,\,1,\,1,\,1,\,1,\,1,\,1,\,1,\,1,\,1,\,1,\,1,\,1,\,1,\,1,\,1\right)}$
This produces 24 × 4096 = 98304 vertices.

## Related polytopes

The Leech polytope contains the vertices and edges of a rectified 24-orthoplex and an expanded 24-simplex.