CC BY
4
УДК 517.55
Construction of Series of Perfect Lattices by Layer Superposition
Sergey B. Vasiliev Nikolai Tarkhanov*
University of Potsdam Karl-Liebknecht-Str. 24/25, Potsdam, 14476
Germany
Received 08.12.2016, received in revised form 16.01.2017, accepted 27.03.2017 We construct a new series of perfect lattices in n dimensions by the layer superposition method of Delaunay-Barnes.
Keywords: lattice packing and covering, polyhedra and polytopes, regular figures, division of spaces. DOI: 10.17516/1997-1397-2017-10-3-353-361.
Introduction
This paper stems out of our discussions at the beginning of the 1990s of those polytopes in 1" whose parallel displacements allow one to form a close partition of the space without open overlaps. The problem was motivated by constructions of optimal numerical schemes the first author elaborated at that time. We continued the discussions in 2009 and prepared a joint project to the German Research Society (DFG) devoted to construction of perfect lattices in the space. Unfortunately, an accident with the first author broke off our cooperation in 2012. This paper is written by the second author and it actually summarises the results of our joint work.
Construction of lattices by the layer superposition method goes back at least as far as [5,6]. It consists in the following. Let 1" be a lattice in 1" determined by a frame f" = {e1,..., в"} of this space. By a Delaunay polytope of 1" is meant any convex polytope with vertices at lattice points, such that there is a solid ball circumscribed about the polytope whose closure does not contain any other points of the lattice except for vertices of the polytope. Pick a Delaunay polytope P" of I". Let B(P") be a solid ball described around P". Denote by с the centre of B(P") and by r its radius. Set в"+1 := с + h, where h is a vector of 1"+1 orthogonal to в1,...,в" and satisfying \h\ = л/l — r2. The vectors e1,..., в", в"+1 constitute a frame f"+1 of 1"+1. The (n + l) -dimensional lattice determined by the frame f "+1 is said to be constructed by the layer superposition method from I". It is denoted by £(£", P"). Obviously, this procedure goes through only if r < l.
If the lattice I" is perfect, then so is 1(1", P"). Even if I" is not perfect, the lattice 1(1", P") may be perfect. The layer superposition method is used successfully for constructing perfect lattices.
By a series of lattices is usually meant an infinite number of lattices corresponding to all n beginning with some dimension n = n0. As developed in [2], the layer superposition method fails to lead to series of perfect lattices, for it does not explain how to find a Delaunay polytope P"+1 of the lattice 1(1", P") lying in a ball B(P"+1) of radius r < l. The present paper is aimed at finding such a Delaunay polytope, which allows one to construct series of perfect lattices by the layer superposition method.
* tarkhanov@math.uni-potsdam.de © Siberian Federal University. All rights reserved
1. Background
As usual, 1" stands for the n-dimensional Euclidean space whose elements are interpreted as n-columns of real numbers. Choose a system e1,...,e" of n linearly independent vectors e1,... ,en in 1". It forms a (non-necessarily orthogonal) frame f" of 1". We can think of f" as the (n x n) -matrix whose columns just amount to e1,... ,e".
The frame f" determines an n-dimensional lattice I" = l(f") in 1" which consists of all points (or vectors) of the form v = x1e1 + ... + x"e" with x1,... ,x" being integer numbers. This can be written as v = f"x, where x is the n-column with entries x1,... ,x". By the very definition, the origin 0 =: e0 is a point of each lattice.
The length of v is evaluated by
\v\2 =^3 aj,k xJxr
j,k=1
where aj,k = (ej,ek) is the inner product of ej and ek in 1". This associates the quadratic form
q(x) = ((f ")Tf" x,x)= xTAx (1.1)
to f", where A = (f")Tf" is the Gram matrix of the frame vectors. By construction, the quadratic form q(x) is positive definite and it allows one to recover the frame f" up to translation and similarity. In that sense one says that q(x) determines the same lattice 1" = l(q).
The convex hull of the vectors e0,e1,... ,e" is called the basic simplex S" of the lattice I". It actually defines the lattice as the set of all points v e 1" whose barycentric coordinates A0, A1,... ,X" with respect to the simplex S" are integer, i.e. v = X0e0 + X1e1 + X"e" where are integer numbers satisfying X0 + X1 +... + X" = 1. In other words, v = (e0, f")X, where X is the (n +1) -column with entries X0, X1,..., X".
With the simplex S" there is associated the quadratic form
Q(X) = —l0,1X°X1 — ¡0,2X0X2 — ... — l"-1,"X" 1X",
where lj}k = \ej — ek\2. The collection of N = n(n + 1)/2 data lj}k with 0 ^ j < k ^ n determines both S"' and I".
It is known [1] that Q(X) = d? — r2, where r is the radius of the ball circumscribed around S and d is the distance of v and the centre of the ball. Given any two vectors v,w e 1" with barycentric coordinates X and n, respectively, the quadrat of the distance of v and w just amounts to Q(^ — X), see ibid.
There are merely a finite number of vectors in the lattice, for which the square of the length is equal to the minimal value m of the form q(x). The lattices which can be obtained from each other by rescaling are said to be equivalent. Hence, one can assume that m = 1. Among all vectors of the lattice the vectors of length one have the least length except for the vector e0 = 0. Two lattice vectors of length one are called equivalent if they differ only in the sign.
Definition 1.1. The quadratic form q(x) and the lattice l(q) are said to be perfect if the matrix A is uniquely determined by the lattice vectors for which q(x) is minimal.
This just amounts to saying that the matrix A of the form is uniquely determined from the system of equations
q(xi) = 1 (1.2)
for i = 1,..., I, where
xi are
lattice vectors of length one. The lattice 1" is called a limit lattice if on 1" the density of packing of the space by equal disjoint balls with centres at lattice points takes on its local maximum. Each limit lattice is
perfect but the inverse is not true, i.e., a perfect lattice is not necessarily limit. For a thorough treatment we refer the reader to [3-7].
By a Delaunay polytope of the lattice is meant any convex polytope with vertices at lattice points which is inscribed in a solid ball whose closure does not contain any lattice points in the interior but the vertices of the polytope on its boundary (such a ball is called "empty ball"). The family of all Delaunay polytopes forms a normal partitioning (Delaunay partitioning) of the space Rn, i.e., a partitioning with the property that any two Delaunay polytopes of the family meet each other at most at a common face of some dimension.
2. General description of the method
Construction of lattices by the layer superposition method goes back at least as far as [5,6]. We follow [2] in presenting it. Let B(Pn) be a solid ball described around a Delaunay polytope Pn of the lattice l(fn). Write c for the centre of B(Pn) and r for its radius. Set en+1 := c + h, where h is a vector of Rn+1 orthogonal to e1,... ,en and satisfying \h\ = \/l — r2. It is clear that the vectors e1,... ,en,en+1 constitute a frame fn+1 of Rn+1. The (n + l) -dimensional lattice ln+1 = £(fn+1) determined by the frame fn+1 is said to be constructed by the layer superposition method from l(fn). This lattice is also denoted by £(£n,Pn). Obviously, this procedure goes through only if r < l.
If the lattice £(fn) is perfect, then so is £(£n,Pn). If £(fn) is a limit lattice, then £(£n,Pn) might be a limit lattice as well as it may fail to be limit. Even if £(fn) is not perfect, the lattice £(£n, Pn) may still prove to be perfect and even limit. The layer superposition method has been used successfully for constructing perfect lattices.
By a series of lattices is usually meant an infinite number of lattices corresponding to all n beginning with some dimension. Sometimes the very construction yields merely one series of perfect lattices. As described above, the layer superposition method fails to lead to series of perfect lattices, for it does not explain how to find a Delaunay polytope Pn+1 of the lattice £(£n, Pn) lying in a ball B(Pn+1) of radius r < l. The paper is aimed at finding such a Delaunay polytope, which allows one to readily construct series of perfect lattices by the layer superposition method.
The convex hull of the point en+1 = c + h and the vertices of the polytope Pn is an (n + 1) -dimensional pyramid Cn+1 whose base is Pn. The presence of the ball B(Pn) gives an evidence to the existence of a ball B(Cn+1). It remains to show that B(Cn+1) is "empty."
By construction, the lattice £(£n, Pn) is constituted by the n-dimensional layers l(fn)+ien+1, for i = 0, ±l,..., which lie in n-dimensional planes ni, and so we get
£(£n,Pn)= (J (£(fn) + ien+1).
The distance between the planes ni and ni+1 is equal to \h\ = a/1 — r2. The lattice £(fn) just amounts to the layer £(fn) + 0 en+1. The centre of the ball B(Cn+1) belongs to the line l(t) = c + th which satifies l(0) = c, l(1) = en+1, l(i) € ni and l(t) is orthogonal to ni for all t and i = 0, ±1,.... The plane n1 is tangent to B(Cn+1) at the point en+1, hence the ball B(Cn+1) has no common points with layers £(fn) + ien+1, for i ^ 2, and it has the only common point en+1 with the layer £(fn) + en+1 lying on the boundary surface of B(Cn+1). The vertices of the polytope Pn lie in the layer l(fn) + 0 en+1 and on the boundary surface of B(Cn+1). In the interior of B(Cn+1) there is no points of the layer £(fn) + 0 en+1 c n0, for the intersection B(Cn+1) n n0 = B(Pn) is empty. We have thus proved
Lemma 2.1. If there is a point of the lattice £(£",Pn) in the interior of the ball B(Cn+1), then this point belongs to a layer £(fn) + ien+1 with i ^ —1.
Let R be the radius of the ball B(C"+1). It is easy to show that R is related to r by
R2 = 1' . 4 1 — r2
This equality gives rise to the recurrence formula
2 1 1
r0 = r, r^ = - --2
4 1 - rU
for i = 1, 2,.... The sequence {ri} satisfies
r2 < rf+i < 0, 5 and Km r2 = 0, 5, if r2 < 0, 5,
ri =0, 5 for all i = 1, 2,..., if r2 = 0, 5,
(2.1)
and {ri} is not defined for i large enough, if r2 > 0,5.
Suppose that the Delaunay polytope Pn satisfies not only the condition r < 1 but also the stronger condition r2 < 0, 5. Then from (2.1) it follows that R2 < 0, 5 and V1 — r2 > R. The diameter 2R of the ball B(Cn+1) is less than the distance 2 \h\ between the planes П1 and П-1. The ball B(Cn+1) is tangent to the plane П1, hence it has no common points with the planes ni, for i ^ —1. By Lemma 2.1, B(Cn+1) is "empty", i.e. the interior of the ball contains no points of the lattice £(£n,Pn). It follows that the pyramid Cn+1 is a Delaunay polytope Pn+1 of the lattice £n+1 := £(£n,Pn) satisfying not only the condition r < 1 but also r2 < 0, 5. The layer superposition method can therefore be applied infinitely many times. As a result we construct an infinite series of lattices ln+m and their Delaunay polytopes Pn+m, for m = 1, 2,....
Let Pn satisfy r2 = 0,5. Then (2.1) yields R = \h\. The ball B(Cn+1) is tangent to the plane П-1 at the point /( — 1) and it has no common points with the planes П for i < —2, i.e., B(Cn+1) is "empty." If /( — 1) g £(fn) — en+1, then the pyramid Cn+1 is a Delaunay polytope Pn+1 of the lattice £n+1 which satisfies r2 = 0, 5. In the opposite case Cn+1 is a part of the Delaunay polytope Pn+1, the vertices of Pn+1 being those of the pyramid Cn+1 and the point l( — 1). Moreover, the balls B(Pn+1) and B(Cn+1) coincide, their common centre c, as the middle of the segment connecting two lattice points en+1 and l( — 1), is a symmetry centre of the lattice ln+1. Hence, c is a symmetry centre of the Delaunay polytopes Pn+1 and Pn. The length of Delaunay diagonals of these polytopes through the centre c satisfies l2 = 2. It is easy to show that these polytopes are actually cross-polytopes. Recall that by a cross-polytope in Rn is meant the convex hull of n segments with common middle point which do not belong to any hyperplane. The layer superposition method can be recurrently applied infinitely many times to yield a series of lattices and their Delaunay polytopes, the polytopes being either all pyramids or all cross-polytopes.
If Pn satisfied 0, 5 < r2 < 1, then R> \h\. The ball B(Cn+1) intersects the plane П-1 and there may be a point of the lattice £n+1 in the interior of B(Cn+1). Then the pyramid Cn+1 fails to be a Delaunay polytope or a part of a Delaunay polytope of the lattice £n+1, in which case no series of lattices £n+1 and their Delaunay polytopes Pn+1 may occur. Even if one succeeds in constructing a series of lattices, the series proves to be finite, for by (2.1) the radius of the ball B(Pn+m) becomes larger than 1 in a finite number m of steps, and so the method no longer works. In particular, for r2 > 0,75 no series of lattices exists, for it consists of one lattice. We thus arrive at the following result.
Theorem 2.1. Assume that the radius r of the ball circumscribed around the Delaunay polytope Pn of the lattice ln = l(fn) satisfies r2 ^ 0, 5. Then on recurrently applying the layer superposition method one is led to an infinite series of lattices £n+m = £(£n+m-i, Pn+m-i) and their Delaunay polytopes Pn+m. If Pn is a cross-polytope, so are all of Pn+m. Otherwise every Pn+m is a pyramid Cn+m over Pn for all m = 1, 2,.... For r2 > 0, 5 the method does not work to produce infinite series of lattices.
Corollary 2.2. If the lattice ln of Theorem 2.1 is perfect, then the corresponding series of lattices is also perfect.
3. Pyramid structure
The structure of the pyramid Cn+m is simple. All its edges emanating from the vertices en+1,en+2, ■ ■ ■, en+m have length 1. Moreover, Cn+m is a right pyramid over the face Pn which is its base. Let (fn+^jn+^K.-JUm—l) be the Euler-Poincare data of the pyramid C n+m-1. Then C n+m has the Euler-Poincare data (fn+m, f[l+m,..., where
xn+m __xn+m—1 I 1
f0 — f0 + 1,
n+m n+m—1 n+m—1
jj — Jj + Jj—i ,
for j = 1,...,n + m — 2, and f^-1 = 1 + K$m-2t ■
If the Delaunay polytope Pn of the lattice ln is basic or of double, tripled, etc. volume, then each pyramid Cn+m for m = 1,2,... has the same property with respect to the corresponding lattice ln+m.
The quadratic forms q(x) of (1.1) which correspond to the lattice series of Theorem 2.1 are more conveniently constructed in the frame whose elements are the edges of the pyramid Cn+m emanating from the vertex en+1. Such a frame f 'n+m = {e[,..., e'n+m} demonstrates rather strikingly common properties of the lattices of the same series. However, it exists provided that the Delaunay polytope Pn of ln is basic.
We now consider this case in more detail. Let the vertices v1,..., vn+1 of the basic Delaunay polytope Pn are end points of the vectors e^,... ,e!n+1, respectively. The end points of vectors eJn+2,..., e'n+m are the vertices en+2,..., en+m of the pyramid Cn+m. The matrix of the quadratic form q(x) with respect to the frame f'n+m has the entries
-1-jj
1, if j — 1, ■ .. ,n + i,
aj,k = 2> (2 \vj — Vk\2), if j,k < n +1,j = k, aj,k = 0,5 otherwise.
It is convenient to write such a form as difference qn+m(x) = q0n+m(x) — rn+1(x), where the form
q'S+m(x) = (x1)2 + ... + (xn+m)2 + x1x2 + ... + xn+m-1xn+m
determines the first perfect lattice and the matrix of the quadratic form rn+1(x) has zero diagonal entries and entries (\vj — vk\2 — 1)/2 for paarweise different j,k ^ n +1. The form rn+1(x) contains information on the length of edges and diagonals of the polytope Pn connecting the vertices v1,... ,vn+1 (and so rn+1 describes the geometry of Pn).
If Pn fails to be a basic Delaunay polytope of the lattice ln, then the frame f 'n+m determines merely a sublattice of the lattice ln+m. Any frame f ''n+m determining the lattice ln+m and maximally close to the frame f 'n+m contains at least one vector emanating from en+1 to the layer ln+m, which is not any edge of the pyramid Cn+m.
We now evaluate the number of vectors of the lattice I'n+1 which have length one. We restrict our attention to those vectors of length one which emanate from the point en+1 € l(fn) + en+1. The only vectors of length one of the lattice ¿n+1 emanating from en+1 and lying in the layer l(fn) + en+1 are those of the lattice ln. Each vector of length one emanating from en+1 and ending in the layer l(fn) + 0 en+1 amounts to an edge of the pyramid Cn+1 emanating from the vertex en+1. There are no vectors of length one which end in a layer l(fn) + ien+1 with i = —1, —2,..., provided that r2 < 0, 5, for the distance of en+1 € l(fn) + en+1 to the plane ni
is equal to (1 + |i|)|fo|, what is greater than 1. The same is true concerning the vectors of length one which end in a layer l(f") + i e"+1 with i = 2, 3,..., for they differ only by sign from those already considered. It follows that the number of vectors of length one of the lattice ("+1 just amounts to N(l") + f0, where N(l") is the number of vectors of length one of the lattice I" and f0 the number of vertices of the polytope P". Hence the number of vectors of length one of the lattice l"+m is equal to N(l") + f0 m +(m — 1)m/2. Note that f0 > n +1. We have thus proved
Lemma 3.1. The lattice l"+m has
N(l"+m) = N(l") — n(n+ 1) + (f0 — n — 1) m +(n + m)(n + m +1) vectors of length 1.
It is worth pointing out that for constructing a series of pyramids C"+m for m = 1,2,... it suffices to have a Delaunay polytope P" of the lattice l" with the property r2 < 0, 5. It gives rise to a pyramid series which can be constructed independently of the lattice. In this sense the pyramid series is of independent interest. Moreover, P" may appear to be a Delaunay polytope simultaneously for two lattices l'" and l''". For one of them P" may be basic and for the other it may be, e.g., of double volume. Then the pyramid series C"+m will be a series of Delaunay polytopes for two lattice series l'"+m and l''"+m. For the first series l'"+m the pyramids will be basic Delaunay polytopes, for the second series l''"+m the pyramids will be Delaunay polytopes of double volume. This shows that to a series of pyramids there may correspond many series of lattices.
When constructing series of perfect lattices and their Delaunay pyramids, the question on the first pyramid and first lattice series is not treated in a unified way. If the generating Delaunay polytope P" of a perfect lattice l" has no vertex v connected with other its vertices by edges of length one, then the first pyramid of the series is C"+1 and the first lattice of the series is l"+1 = l(l", P"). If P" has several vertices v1,... ,vk, each of them being connected with all other vertices of P" by edges of length one, then the series generating Delaunay polytope is the face P"-k of P" which lies opposite to the vertices v1,... ,vk. The first pyramid of the series is the face C"-k+1 containing the face P"-k.
If P" is the basic Delaunay polytope of a perfect lattice l", then l" can be constructed from some lattice l"-k and its Delaunay polytope P"-k by the layer superposition method over k steps. Under this approach the lattice l"-k need not be perfect. It suffices that in a finite number j < k of steps the lattice l"-k+j would become perfect. Then it will be the first lattice of a series of perfect lattices l"-k+j+m with m = 1,2,.... If P" fails to be a basic Delaunay polytope of a perfect series l", the general approach to the choice of the first lattice of the series still remains the same. However, one should take into account that lattices of dimension less than n may fail to exist.
4. Examples
In [2] one described all Delaunay polytopes of perfect lattices in the space 1" with n < 6. We first consider the case n = 2 in detail.
In 12 there is only the first perfect lattice l20, which is usually given by the first quadratic form q2. All Delaunay polytopes of this lattice are equilateral triangles, one of these (the convex span of frame vectors) is denoted by P2. The edges of the triangle are of length one. The quadrat of the radius of the described circle just amounts to r2 = 1/3. The lattice has three vectors of length one.
Since r2 < 0,5, the triangle P2 generates a series of Delaunay polytopes of perfect lattices P2+m for m = 1,2,..., each P2+m being a regular simplex with edges of length one. Since
one-dimensional lattices are hardly of interest, Pi2 can be thought of as the first polytope of the series.
The lattice 10 and its Delaunay polytope P2 generate a series of perfect lattices in determined by the form q]j. To our best knowledge this quadratic form was first studied by G. F. Voronoy and called the first perfect form. The first perfect lattice in is known to have n(n + 1)/2 vectors of length one, as is also seen from Lemma 3.1.
The regular simplex with edges of length one in R7 is a Delaunay polytope of double volume (denoted S7) of a perfect lattice I'7 in R7. This lattice is given by the frame
f7 = {ei, ...,e6, 1 (—ei — e2 — e3 + e4 + e5 + e6 + e7^,
where e1,... ,e7 are edges of the simplex Sr emanating from one vertex. To the frame f7 there corresponds the quadratic form q7(x) = ql(x) — x1x7 — x2x7 — x3x7. The number of vectors of length one in the lattice is 63. The lattice l7 and its Delaunay polytope S7 generate a series of perfect lattices which are given by the form
q (x) = q^n (x) — x x — x x — x x — ^ (x x + ... + x x )
for n = 8,9,.... The Delaunay polytopes of double volume of lattices of this series are precisely regular simplices of the corresponding dimension. By Lemma 3.1, the number of vectors of length one of an n-dimensional lattice is 35 + n(n + 1)/2.
A Delaunay partitioning of the lattice il8 contains regular simplices of tripled volume with edges of length one. Denote by S8 one of them. The lattice can be determined by the frame f8 = {e1,... ,e7, e8}, where e'8 = 1 (—2e1 — 2e2 + e3 + ... + e8) and e1,... ,e8 are the edges of the simplex S8 emanating from one vertex. To the frame f8 there corresponds the quadratic form q8(x) = q8(x) — x1 x8 — x2x8. The lattice contains 120 vectors of length one. Together with its Delaunay polytope S8 the lattice determines a series of perfect lattices. The regular simplex Sn is the Delaunay polytope of tripled volume of the n -dimensional lattice of this series, and the simplex S8 is the 8 -dimensional face of Sn. The frame of the n-dimensional lattice is fn = {e1,... ,e7, e'8, e9,..., en}, where e9,... ,en are the edges of the simplex Sn emanating from a vertex of S8. To this frame there corresponds the form
n/ \ n/ \ 18 2 8 1 / 8 9, ,8 n\
q (x) = q0 (x) — x x — x x — x + ... + x x )
for n = 9,10,.... By Lemma 3.1 the number of vectors of length one of the n-dimensional lattice is 84 + n(n +1)/2.
On the lattices l7 and l8 the density of packing the spaces R7 and R8, respectively, by equal disjoint balls with centres at lattice points takes on its global maximum.
When the dimension of the space increases, one derives always new series of perfect lattices whose Delaunay polytopes are regular simplices. Along with finding new perfect forms, the construction of series of perfect lattices by the layer superposition method allows one to partially order already existing perfect quadratic forms and reduce notation.
In the space R3 there is only the first perfect lattice whose partitioning consists of regular simplices P3 and octahedra P3. The simplex P3 belongs to the series of polytopes considered above. The octahedron P23 gives rise to a series of n -cross-polytopes with edges of length one which are Delaunay polytopes of perfect lattices, where n > 3.
The lattice 10 and its Delaunay polytope P3 give rise to a series of perfect lattices in given by the second perfect form qr{(x) = qfi(x) — x1x2, where n ^ 4. The lattice ir{ has n2 — n vectors of length one. The n-cross-polytope is a basic Delaunay polytope of this lattice.
The cross-polytope with edges of length one is a Delaunay polytope of double volume of the perfect lattice i'8. The frame f8 of this lattice and the corresponding quadratic form are adduced
above. The convex hull of the frame vectors en,... ,e7 constitutes the 7-dimensional face of the partitioning which is a regular simplex. Along this face the simplex S8 borders on an 8 -cross-polytope which we denote by P8. The lattice 18 and its Delaunay polytope P8 generate a series of perfect lattices. The n-dimensional lattice of this series has an n-cross-polytope Pn as a Delaunay polytope of double volume and it is given by the frame {en,... ,e7, e'8, e9, . . . , en}. The vectors en,... ,e7,e9,... ,en form an (n — 1) -dimensional face of an n-cross-polytope Pn which is a regular simplex. Only the vector e'8 of the frame does not belong to the cross-polytope Pn. The distance of the point e'8 to each of the vertices e9,... ,en of the cross-polytope Pn just amounts to 3/2. Hence, to the frame of the lattice of this series there corresponds the quadratic form
Qo (x) — x x — x x — 2 x (x + ... + x ),
where n > 9. The number of lattice vectors of length one is equal to n2 — n + 64, as is easy to see.
In the space R4 there are two perfect lattices i4 and i4. The Delaunay polytopes of the lattices In and ln are well known. All Delaunay polytopes of these lattices, except for the regular simplex Pn of the lattice in and the n-cross-polytope of the lattice in, have circumscribed ball of radius greater than 0,5, and so they give no rise to any countable series of perfect lattices. Therefore, the first and second perfect lattices are not considered in the sequel.
In addition to the first and second perfect lattices in R5 there is also the third perfect lattice . The partitioning of l2 contains Delaunay polytopes of three types whose representatives are U 5, S5 and M5 of [2]. The first polytope has circumscribed ball of radius exceeding 0,5. The second one is a basic simplex of the lattice (r2 = 11/24) and has no vertex connected with other vertices of the simplex by edges of length one. It gives rise to a series of regular simplices over S5, which we denote by Sn for n > 6. The lattice l2 and simplex S5 generate a series of perfect lattices in, for which the simplices Sn are basic Delaunay polytopes. The vertex of the simplex S6 facing
S5
is connected with other vertices of the simplex by edges of length one. On taking these edges as frame we get a quadratic form q3,(x) = q6(x) — 1/2 (xnx2 + x3x4 + x5x6) which gives the first lattice of this series. Then the entire series of lattices in is determined by the form
q3n(x) = qrn(x) — 2 (x1x2 + x3x4 + x5x6),
where n ^ 6. Several of the first forms and lattices given by them are known, however, they do not bear any common designation let alone name. The number of vectors of length one in the lattice l2 amounts to 15, and the simplex S5 has 6 vertices. It follows that the number of vectors of length one in the lattice in is equal to n(n + 1)/2.
The polytope M5 is a regular pyramid over a polytope V22 which is a repartitioning body, see [2]. The radius of circumscribed ball of M5 is r = л/0, 5. Therefore, V42 gives rise to a series of pyramids Mn, n > 5, which are Delaunay polytopes of perfect lattices. The polytope V22 is a basic Delaunay polytope of a lattice i4 which is not perfect. However, on applying the layer superposition method to the lattice i4 and its Delaunay polytope V22 one obtains a perfect lattice i2l. In this way i4 and V22 generate a series of perfect lattices in and a series of Delaunay polytopes Mn of these lattices, where n > 5. The lattice 62 is the first lattice of the series. The body V22 has 6 vertices and the number of vectors of length one in the lattice i4 is equal to 9. Hence, the number of vectors of length one in the lattice in just amounts to — 1 + 1(n — 4) + n(n + 1)/2 or n(n + 3)/2 — 5, which is due to Lemma 3.1. Any 5 of the 6 edges of the polytope M5 emanating from the vertex facing V22 can be taken as a frame of . To this frame there corresponds the quadratic form q5(x) = q0(x) — 1/2 (xnx2 + x3x4 + x3x5 + x4x5). Then the series ir2 is determined by
qO(x) = qS(x) — 2 (xnx2 + x3x4 + x3x5 + x4x5),
where n > 5. This series of perfect quadratic forms is known, see [8]. Unfortunately, G. F. Voronoy found and called the third perfect form only the first quadratic form of this series, cf. [9]. It seems to be more logical to think of the whole series as the third perfect form by analogy with the first and second perfect forms.
In addition to finding new perfect forms the construction of series of perfect lattices by the layer superposition method allows one to partially regulate already known quadratic forms, thus reducing the number of designations in use.
References
[1] E.P.Baranovskii, Simplices of L-subdivisions of Euclidean spaces, Math. Notes, 10(1971), no. 6, 827-834.
[2] E.P.Baranovskii, Decompositions of Euclidean spaces into L-polytopes of some perfect lattices, Proc. Steklov Inst. Math., 196(1992), 29-51.
[3] S.S.Ryshkov, E.P.Baranovskii, C-Types of n-dimensional lattices and 5-dimensional primitive polytopes (with application to the theory of coverings), Proc. Steklov Inst. Math., 137(1976), 1-140.
[4] Baranovskii, E. P., and Ryshkov, S. S., Perfect lattices as admissible centerings, Soviet Math, 37(1987), no. 4, 1155-1183.
[5] E.S.Barnes, The construction of perfect and extreme forms, Acta Arithm., 5(1959), no. 1, 57-79; no. 2, 205-222.
[6] B.N.Delaunay, On the densest parallelepipedal configuration of balls in the spaces of dimension 3 and 4, Trans. of Phys. Math. Inst., 4(1933), 63-65.
[7] P.M.Gruber, C.G.Lekkerkerker, Geometry of Numbers, second edition, Elsevier Science Publishers B. V., Amsterdam, NL, 1987.
[8] S.B.Vasiliev, On a new series of perfect forms, Mathematics i ee prilozheniya. Univ. Ivanovo, 2004, no. 1, 37-41 (in Russian).
[9] G.F.Voronoy, Nouvelles applications de parametres contenues a la theorie de formes quadratiques - Deuxieme memoire, J. fur reine und angewandte Mathematik, 134(1908), 198-287; 136(1909), 67-178.
Построение серий совершенных решеток посредством наложения слоев
Сергей Б. Васильев Николай Тарханов
Потсдамский университет Карл-Либкнехт штрассе, 24/25, Потсдам, 14476
Германия
Построена новая серия совершенных решеток в n 'размерностях методом наложения слоев Делоне-Барнса.
Ключевые слова: решетчатая упаковка и покрытие, многогранники и политопы, правильные фигуры, деление пространств.
