In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, the Leech lattice is an even
unimodular lattice Λ
24 in 24-dimensional
Euclidean space
Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidea ...
, which is one of the best models for the
kissing number problem
In geometry, the kissing number of a mathematical space is defined as the greatest number of non-overlapping unit spheres that can be arranged in that space such that they each touch a common unit sphere. For a given sphere packing (arrangement o ...
. It was discovered by . It may also have been discovered (but not published) by
Ernst Witt in 1940.
Characterization
The Leech lattice Λ
24 is the unique lattice in 24-dimensional
Euclidean space
Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidea ...
, E
24, with the following list of properties:
*It is
unimodular; i.e., it can be generated by the columns of a certain 24×24
matrix
Matrix most commonly refers to:
* ''The Matrix'' (franchise), an American media franchise
** '' The Matrix'', a 1999 science-fiction action film
** "The Matrix", a fictional setting, a virtual reality environment, within ''The Matrix'' (franchi ...
with
determinant
In mathematics, the determinant is a scalar value that is a function of the entries of a square matrix. It characterizes some properties of the matrix and the linear map represented by the matrix. In particular, the determinant is nonzero if a ...
1.
*It is even; i.e., the square of the length of each vector in Λ
24 is an even integer.
*The length of every non-zero vector in Λ
24 is at least 2.
The last condition is equivalent to the condition that unit balls centered at the points of Λ
24 do not overlap. Each is tangent to 196,560 neighbors, and this is known to be the largest number of non-overlapping 24-dimensional unit balls that can
simultaneously touch a single unit ball. This arrangement of 196,560 unit balls centred about another unit ball is so efficient that there is no room to move any of the balls; this configuration, together with its mirror-image, is the ''only'' 24-dimensional arrangement where 196,560 unit balls simultaneously touch another. This property is also true in 1, 2 and 8 dimensions, with 2, 6 and 240 unit balls, respectively, based on the
integer lattice,
hexagonal tiling
In geometry, the hexagonal tiling or hexagonal tessellation is a regular tiling of the Euclidean plane, in which exactly three hexagons meet at each vertex. It has Schläfli symbol of or (as a truncated triangular tiling).
English mathemati ...
and
E8 lattice, respectively.
It has no
root system
In mathematics, a root system is a configuration of vectors in a Euclidean space satisfying certain geometrical properties. The concept is fundamental in the theory of Lie groups and Lie algebras, especially the classification and representatio ...
and in fact is the first
unimodular lattice with no ''roots'' (vectors of norm less than 4), and therefore has a centre density of 1. By multiplying this value by the volume of a unit ball in 24 dimensions,
, one can derive its absolute density.
showed that the Leech lattice is isometric to the set of simple roots (or the
Dynkin diagram
In the Mathematics, mathematical field of Lie theory, a Dynkin diagram, named for Eugene Dynkin, is a type of Graph (discrete mathematics), graph with some edges doubled or tripled (drawn as a double or triple line). Dynkin diagrams arise in the ...
) of the
reflection group of the 26-dimensional even Lorentzian unimodular lattice
II25,1. By comparison, the Dynkin diagrams of II
9,1 and II
17,1 are finite.
Applications
The
binary Golay code
In mathematics and electronics engineering, a binary Golay code is a type of linear error-correcting code used in digital communications. The binary Golay code, along with the ternary Golay code, has a particularly deep and interesting connection ...
, independently developed in 1949, is an application in
coding theory
Coding theory is the study of the properties of codes and their respective fitness for specific applications. Codes are used for data compression, cryptography, error detection and correction, data transmission and data storage. Codes are studied ...
. More specifically, it is an error-correcting code capable of correcting up to three errors in each 24-bit word, and detecting up to seven. It was used to communicate with the
Voyager probes, as it is much more compact than the previously-used
Hadamard code.
Quantizers, or
analog-to-digital converter
In electronics, an analog-to-digital converter (ADC, A/D, or A-to-D) is a system that converts an analog signal, such as a sound picked up by a microphone or light entering a digital camera, into a digital signal. An ADC may also provide ...
s, can use lattices to minimise the average
root-mean-square
In mathematics and its applications, the root mean square of a set of numbers x_i (abbreviated as RMS, or rms and denoted in formulas as either x_\mathrm or \mathrm_x) is defined as the square root of the mean square (the arithmetic mean of the ...
error. Most quantizers are based on the one-dimensional
integer lattice, but using multi-dimensional lattices reduces the RMS error. The Leech lattice is a good solution to this problem, as the
Voronoi cell
In mathematics, a Voronoi diagram is a partition of a plane into regions close to each of a given set of objects. In the simplest case, these objects are just finitely many points in the plane (called seeds, sites, or generators). For each seed ...
s have a low
second moment.
The
vertex algebra
In mathematics, a vertex operator algebra (VOA) is an algebraic structure that plays an important role in two-dimensional conformal field theory and string theory. In addition to physical applications, vertex operator algebras have proven usef ...
of the
two-dimensional conformal field theory
A two-dimensional conformal field theory is a quantum field theory on a Euclidean two-dimensional space, that is invariant under local conformal transformations.
In contrast to other types of conformal field theories, two-dimensional conformal ...
describing
bosonic string theory, compactified on the 24-dimensional
quotient
In arithmetic, a quotient (from lat, quotiens 'how many times', pronounced ) is a quantity produced by the division of two numbers. The quotient has widespread use throughout mathematics, and is commonly referred to as the integer part of a ...
torus
In geometry, a torus (plural tori, colloquially donut or doughnut) is a surface of revolution generated by revolving a circle in three-dimensional space about an axis that is coplanar with the circle.
If the axis of revolution does n ...
R
24/Λ
24 and
orbifold
In the mathematical disciplines of topology and geometry, an orbifold (for "orbit-manifold") is a generalization of a manifold. Roughly speaking, an orbifold is a topological space which is locally a finite group quotient of a Euclidean space.
D ...
ed by a two-element reflection group, provides an explicit construction of the
Griess algebra In mathematics, the Griess algebra is a commutative non-associative algebra on a real vector space of dimension 196884 that has the Monster group ''M'' as its automorphism group. It is named after mathematician R. L. Griess, who constructed it in ...
that has the
monster group
In the area of abstract algebra known as group theory, the monster group M (also known as the Fischer–Griess monster, or the friendly giant) is the largest sporadic simple group, having order
24632059761121331719232931414759 ...
as its automorphism group. This
monster vertex algebra
The monster vertex algebra (or moonshine module) is a vertex algebra acted on by the monster group that was constructed by Igor Frenkel, James Lepowsky, and Arne Meurman. R. Borcherds used it to prove the monstrous moonshine conjectures, by a ...
was also used to prove the
monstrous moonshine
In mathematics, monstrous moonshine, or moonshine theory, is the unexpected connection between the monster group ''M'' and modular functions, in particular, the ''j'' function. The term was coined by John Conway and Simon P. Norton in 1979. ...
conjectures.
Constructions
The Leech lattice can be constructed in a variety of ways. As with all lattices, it can be constructed by taking the
integral
In mathematics, an integral assigns numbers to functions in a way that describes displacement, area, volume, and other concepts that arise by combining infinitesimal data. The process of finding integrals is called integration. Along with ...
span of the columns of its
generator matrix
In coding theory, a generator matrix is a matrix whose rows form a basis for a linear code. The codewords are all of the linear combinations of the rows of this matrix, that is, the linear code is the row space of its generator matrix.
Terminolo ...
, a 24×24 matrix with
determinant
In mathematics, the determinant is a scalar value that is a function of the entries of a square matrix. It characterizes some properties of the matrix and the linear map represented by the matrix. In particular, the determinant is nonzero if a ...
1.
Using the binary Golay code
The Leech lattice can be explicitly constructed as the set of vectors of the form 2
−3/2(''a''
1, ''a''
2, ..., ''a''
24) where the ''a''
''i'' are integers such that
:
and for each fixed residue class modulo 4, the 24 bit word, whose 1s correspond to the coordinates ''i'' such that ''a''
''i'' belongs to this residue class, is a word in the
binary Golay code
In mathematics and electronics engineering, a binary Golay code is a type of linear error-correcting code used in digital communications. The binary Golay code, along with the ternary Golay code, has a particularly deep and interesting connection ...
. The Golay code, together with the related Witt design, features in a construction for the 196560 minimal vectors in the Leech lattice.
Leech lattice (L mod 8) can be directly constructed by combination of the 3 following sets,
, (
is a ones vector of size n),
* G - 24-bit Golay code
* B - Binary integer sequence
* C -
Thue-Morse Sequence or integer bit parity sum (that give chirality of the lattice)
24-bit Golay ^12 codes 24-bit integer ^24 codes Parity Leech Lattice ^36 codesG = B = C = L = (4B + C) ⊕ 2G
00000000 00000000 00000000 00000000 00000000 00000000 0 00000000 00000000 00000000
11111111 00000000 00000000 10000000 00000000 00000000 1 22222222 00000000 00000000
11110000 11110000 00000000 01000000 00000000 00000000 1 22220000 22220000 00000000
00001111 11110000 00000000 11000000 00000000 00000000 0 ...
11001100 11001100 00000000 00100000 00000000 00000000 1 51111111 11111111 11111111
00110011 11001100 00000000 10100000 00000000 00000000 0 73333333 11111111 11111111
00111100 00111100 00000000 01100000 00000000 00000000 0 ...
11000011 00111100 00000000 11100000 00000000 00000000 1 15111111 11111111 11111111
10101010 10101010 00000000 00010000 00000000 00000000 1 37333333 11111111 11111111
01010101 10101010 00000000 10010000 00000000 00000000 0 ...
01011010 01011010 00000000 01010000 00000000 00000000 0 44000000 00000000 00000000
10100101 01011010 00000000 11010000 00000000 00000000 1 66222222 00000000 00000000
... ... ... ...
11111111 11111111 11111111 11111111 11111111 11111111 0 66666666 66666666 66666666
Using the Lorentzian lattice II25,1
The Leech lattice can also be constructed as
where ''w'' is the Weyl vector:
:
in the 26-dimensional even Lorentzian
unimodular lattice II25,1. The existence of such an integral vector of Lorentzian norm zero relies on the fact that 1
2 + 2
2 + ... + 24
2 is a
perfect square (in fact 70
2); the
number 24 is the only integer bigger than 1 with this property. This was conjectured by
Édouard Lucas, but the proof came much later, based on
elliptic functions
In the mathematical field of complex analysis, elliptic functions are a special kind of meromorphic functions, that satisfy two periodicity conditions. They are named elliptic functions because they come from elliptic integrals. Originally those ...
.
The vector
in this construction is really the
Weyl vector
In mathematics, the Weyl character formula in representation theory describes the characters of irreducible representations of compact Lie groups in terms of their highest weights. It was proved by . There is a closely related formula for the c ...
of the even sublattice ''D''
24 of the odd unimodular lattice ''I''
25. More generally, if ''L'' is any positive definite unimodular lattice of dimension 25 with at least 4 vectors of norm 1, then the Weyl vector of its norm 2 roots has integral length, and there is a similar construction of the Leech lattice using ''L'' and this Weyl vector.
Based on other lattices
described another 23 constructions for the Leech lattice, each based on a
Niemeier lattice. It can also be constructed by using three copies of the
E8 lattice, in the same way that the binary Golay code can be constructed using three copies of the extended
Hamming code
In computer science and telecommunication, Hamming codes are a family of linear error-correcting codes. Hamming codes can detect one-bit and two-bit errors, or correct one-bit errors without detection of uncorrected errors. By contrast, the sim ...
, H
8. This construction is known as the Turyn construction of the Leech lattice.
As a laminated lattice
Starting with a single point, Λ
0, one can stack copies of the lattice Λ
n to form an (''n'' + 1)-dimensional lattice, Λ
''n''+1, without reducing the minimal distance between points. Λ
1 corresponds to the
integer lattice, Λ
2 is to the
hexagonal lattice
The hexagonal lattice or triangular lattice is one of the five two-dimensional Bravais lattice types. The symmetry category of the lattice is wallpaper group p6m. The primitive translation vectors of the hexagonal lattice form an angle of 120� ...
, and Λ
3 is the
face-centered cubic
In crystallography, the cubic (or isometric) crystal system is a crystal system where the unit cell is in the shape of a cube. This is one of the most common and simplest shapes found in crystals and minerals.
There are three main varieties of ...
packing. showed that the Leech lattice is the unique laminated lattice in 24 dimensions.
As a complex lattice
The Leech lattice is also a 12-dimensional lattice over the
Eisenstein integers
In mathematics, the Eisenstein integers (named after Gotthold Eisenstein), occasionally also known as Eulerian integers (after Leonhard Euler), are the complex numbers of the form
:z = a + b\omega ,
where and are integers and
:\omega = ...
. This is known as the complex Leech lattice, and is isomorphic to the 24-dimensional real Leech lattice. In the complex construction of the Leech lattice, the
binary Golay code
In mathematics and electronics engineering, a binary Golay code is a type of linear error-correcting code used in digital communications. The binary Golay code, along with the ternary Golay code, has a particularly deep and interesting connection ...
is replaced with the
ternary Golay code
In coding theory, the ternary Golay codes are two closely related error-correcting codes.
The code generally known simply as the ternary Golay code is an 1, 6, 53-code, that is, it is a linear code over a ternary alphabet; the relative distan ...
, and the
Mathieu group ''M''24 is replaced with the
Mathieu group ''M''12. The ''E''
6 lattice, ''E''
8 lattice and
Coxeter–Todd lattice also have constructions as complex lattices, over either the Eisenstein or
Gaussian integers.
Using the icosian ring
The Leech lattice can also be constructed using the ring of
icosians. The icosian ring is abstractly isomorphic to the
E8 lattice, three copies of which can be used to construct the Leech lattice using the Turyn construction.
Witt's construction
In 1972 Witt gave the following construction, which he said he found in 1940, on January 28. Suppose that ''H'' is an ''n'' by ''n''
Hadamard matrix
In mathematics, a Hadamard matrix, named after the French mathematician Jacques Hadamard, is a square matrix whose entries are either +1 or −1 and whose rows are mutually orthogonal. In geometric terms, this means that each pair of row ...
, where ''n''=4''ab''. Then the matrix
defines a bilinear form in 2''n'' dimensions, whose kernel has ''n'' dimensions. The quotient by this kernel is a nonsingular bilinear form taking values in (1/2)Z. It has 3 sublattices of index 2 that are integral bilinear forms. Witt obtained the Leech lattice as one of these three sublattices by taking ''a''=2, ''b''=3, and taking ''H'' to be the 24 by 24 matrix (indexed by Z/23Z ∪ ∞) with entries Χ(''m''+''n'') where Χ(∞)=1, Χ(0)=−1, Χ(''n'')=is the quadratic residue symbol mod 23 for nonzero ''n''. This matrix ''H'' is a
Paley matrix In mathematics, the Paley construction is a method for constructing Hadamard matrices using finite fields. The construction was described in 1933 by the English mathematician Raymond Paley.
The Paley construction uses quadratic residues in a ...
with some insignificant sign changes.
Using a Paley matrix
described a construction using a
skew Hadamard matrix of
Paley type.
The
Niemeier lattice with root system
can be made into a module
for the ring of integers of the field
. Multiplying this
Niemeier lattice by a non-principal ideal of the ring of integers gives the Leech lattice.
Using octonions
If ''L'' is the set of
octonion
In mathematics, the octonions are a normed division algebra over the real numbers, a kind of hypercomplex number system. The octonions are usually represented by the capital letter O, using boldface or blackboard bold \mathbb O. Octonions hav ...
s with coordinates on the
lattice, then the Leech lattice is the set of triplets
such that
:
:
:
where
. This construction is due to .
Symmetries
The Leech lattice is highly symmetrical. Its
automorphism group
In mathematics, the automorphism group of an object ''X'' is the group consisting of automorphisms of ''X'' under composition of morphisms. For example, if ''X'' is a finite-dimensional vector space, then the automorphism group of ''X'' is the g ...
is the
Conway group
In the area of modern algebra known as group theory, the Conway groups are the three sporadic simple groups Co1, Co2 and Co3 along with the related finite group Co0 introduced by .
The largest of the Conway groups, Co0, is the group of autom ...
Co
0, which is of order 8 315 553 613 086 720 000. The center of Co
0 has two elements, and the quotient of Co
0 by this center is the Conway group Co
1, a finite simple group. Many other
sporadic groups, such as the remaining Conway groups and
Mathieu groups, can be constructed as the stabilizers of various configurations of vectors in the Leech lattice.
Despite having such a high ''rotational'' symmetry group, the Leech lattice does not possess any hyperplanes of reflection symmetry. In other words, the Leech lattice is
chiral
Chirality is a property of asymmetry important in several branches of science. The word ''chirality'' is derived from the Greek (''kheir''), "hand", a familiar chiral object.
An object or a system is ''chiral'' if it is distinguishable from i ...
. It also has far fewer symmetries than the 24-dimensional hypercube and simplex.
The automorphism group was first described by
John Conway. The 398034000 vectors of norm 8 fall into 8292375 'crosses' of 48 vectors. Each cross contains 24 mutually orthogonal vectors and their negatives, and thus describe the vertices of a 24-dimensional
orthoplex
In geometry, a cross-polytope, hyperoctahedron, orthoplex, or cocube is a regular, convex polytope that exists in ''n''- dimensional Euclidean space. A 2-dimensional cross-polytope is a square, a 3-dimensional cross-polytope is a regular octahed ...
. Each of these crosses can be taken to be the coordinate system of the lattice, and has the same symmetry of the
Golay code, namely 2
12 × , M
24, . Hence the full automorphism group of the Leech lattice has order 8292375 × 4096 × 244823040, or 8 315 553 613 086 720 000.
Geometry
showed that the covering radius of the Leech lattice is
; in other words, if we put a closed ball of this radius around each lattice point, then these just cover Euclidean space. The points at distance at least
from all lattice points are called the ''deep holes'' of the Leech lattice. There are 23 orbits of them under the automorphism group of the Leech lattice, and these orbits correspond to the 23
Niemeier lattices other than the Leech lattice: the set of vertices of deep hole is isometric to the affine Dynkin diagram of the corresponding Niemeier lattice.
The Leech lattice has a density of
. showed that it gives the densest lattice
packing of balls in 24-dimensional space. improved this by showing that it is the densest sphere packing, even among non-lattice packings.
The 196560 minimal vectors are of three different varieties, known as ''shapes'':
*
vectors of shape (4
2,0
22), for all permutations and sign choices;
*
vectors of shape (2
8,0
16), where the '2's correspond to an octad in the Golay code, and there are any even number of minus signs;
*
vectors of shape (∓3,±1
23), where the lower sign is used for the '1's of any codeword of the Golay code, and the '∓3' can appear in any position.
The
ternary Golay code
In coding theory, the ternary Golay codes are two closely related error-correcting codes.
The code generally known simply as the ternary Golay code is an 1, 6, 53-code, that is, it is a linear code over a ternary alphabet; the relative distan ...
,
binary Golay code
In mathematics and electronics engineering, a binary Golay code is a type of linear error-correcting code used in digital communications. The binary Golay code, along with the ternary Golay code, has a particularly deep and interesting connection ...
and Leech lattice give very efficient 24-dimensional
spherical code In geometry and coding theory, a spherical code with parameters (''n'',''N'',''t'') is a set of ''N'' points on the unit hypersphere in ''n'' dimensions for which the dot product of unit vectors from the origin to any two points is less than or e ...
s of 729, 4096 and 196560 points, respectively. Spherical codes are higher-dimensional analogues of
Tammes problem
In geometry, the Tammes problem is a problem in packing a given number of circles on the surface of a sphere such that the minimum distance between circles is maximized. It is named after the Dutch botanist Pieter Merkus Lambertus Tammes (the ...
, which arose as an attempt to explain the distribution of pores on pollen grains. These are distributed as to maximise the minimal angle between them. In two dimensions, the problem is trivial, but in three dimensions and higher it is not. An example of a spherical code in three dimensions is the set of the 12 vertices of the regular icosahedron.
Theta series
One can associate to any (positive-definite) lattice Λ a
theta function
In mathematics, theta functions are special functions of several complex variables. They show up in many topics, including Abelian varieties, moduli spaces, quadratic forms, and solitons. As Grassmann algebras, they appear in quantum field ...
given by
:
The theta function of a lattice is then a
holomorphic function
In mathematics, a holomorphic function is a complex-valued function of one or more complex variables that is complex differentiable in a neighbourhood of each point in a domain in complex coordinate space . The existence of a complex deriv ...
on the
upper half-plane
In mathematics, the upper half-plane, \,\mathcal\,, is the set of points in the Cartesian plane with > 0.
Complex plane
Mathematicians sometimes identify the Cartesian plane with the complex plane, and then the upper half-plane corresponds to ...
. Furthermore, the theta function of an even unimodular lattice of rank ''n'' is actually a
modular form
In mathematics, a modular form is a (complex) analytic function on the upper half-plane satisfying a certain kind of functional equation with respect to the group action of the modular group, and also satisfying a growth condition. The theory o ...
of weight ''n''/2 for the full
modular group
In mathematics, the modular group is the projective special linear group of matrices with integer coefficients and determinant 1. The matrices and are identified. The modular group acts on the upper-half of the complex plane by fractional ...
PSL(2,Z). The theta function of an integral lattice is often written as a power series in
so that the coefficient of ''q''
''n'' gives the number of lattice vectors of squared norm 2''n''. In the Leech lattice, there are 196560 vectors of squared norm 4, 16773120 vectors of squared norm 6, 398034000 vectors of squared norm 8 and so on. The theta series of the Leech lattice is
:
where
is the normalized
Eisenstein series of weight 12,
is the
modular discriminant,
is the
divisor function
In mathematics, and specifically in number theory, a divisor function is an arithmetic function related to the divisors of an integer. When referred to as ''the'' divisor function, it counts the ''number of divisors of an integer'' (includin ...
for exponent 11, and
is the
Ramanujan tau function. It follows that for ''m''≥1 the number of vectors of squared norm 2''m'' is
:
History
Many of the cross-sections of the Leech lattice, including the
Coxeter–Todd lattice and
Barnes–Wall lattice, in 12 and 16 dimensions, were found much earlier than the Leech lattice. discovered a related odd unimodular lattice in 24 dimensions, now called the odd Leech lattice, one of whose two even neighbors is the Leech lattice. The Leech lattice was discovered in 1965 by , by improving some earlier sphere packings he found .
calculated the order of the automorphism group of the Leech lattice, and, working with
John G. Thompson, discovered three new
sporadic groups as a by-product: the
Conway groups, Co
1, Co
2, Co
3. They also showed that four other (then) recently announced sporadic groups, namely,
Higman-Sims,
Suzuki
is a Japanese multinational corporation headquartered in Minami-ku, Hamamatsu, Japan. Suzuki manufactures automobiles, motorcycles, all-terrain vehicles (ATVs), outboard marine engines, wheelchairs and a variety of other small internal co ...
,
McLaughlin, and the
Janko group J
2 could be found inside the Conway groups using the geometry of the Leech lattice. (Ronan, p. 155)
, has a single rather cryptic sentence mentioning that he found more than 10 even unimodular lattices in 24 dimensions without giving further details. stated that he found 9 of these lattices earlier in 1938, and found two more, the
Niemeier lattice with A root system and the Leech lattice (and also the odd Leech lattice), in 1940.
See also
*
Sphere packing
In geometry, a sphere packing is an arrangement of non-overlapping spheres within a containing space. The spheres considered are usually all of identical size, and the space is usually three- dimensional Euclidean space. However, sphere pack ...
*
E8 lattice
References
*
*
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*
*
*
*
*
*
*
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*
*
*
*
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*
External links
Leech lattice (CP4space)*
The Leech Lattice, U. of Illinois at Chicago, Mark Ronan's websitePapers by R. E. Borcherds
{{DEFAULTSORT:Leech Lattice
Quadratic forms
Lattice points
Sporadic groups
Moonshine theory