xii OVERVIEW
Our treatment of Voronoi’s theory is presented in a way such that it can be
generalized naturally from a lattice theory to a theory for m-periodic point sets.
As lattices can be viewed as linear images of
Zd
in
Rd,
m-periodic sets are linear
images of a standard periodic set
(0.1)
m
i=1
ti +
Zd
with ti
Rd.
In Section 3.2, we introduce a new parameter space
Sd,m
0
to deal with m-periodic
sets in Rd up to isometries. We introduce generalized Ryshkov sets, on which
determinant minimization yields the densest m-periodic sphere packings.
In Section 3.3 we analyze local optima (m-extreme periodic sets) of the packing
density in
Sd,m.
0
We obtain necessary and sufficient conditions. It turns out that
our framework provides a new explanation for a previously by Conway and Sloane
observed phenomenon; namely for the existence of uncountably many 9-dimensional
2-periodic sphere packings (fluid diamond packings) which are as dense as the
densest known lattice sphere packing. We investigate the possibility of improving
the densest lattices (represented as points in
Sd,m)
0
locally, to obtain a periodic
non-lattice set with larger packing density. We show that this is not possible (see
Corollary 3.18). More generally, we show in Theorem 3.17 that a perfect, strongly
eutactic lattice is periodic extreme, that is, it cannot locally be improved to a denser
periodic point set.
In Section 3.4 we give extensions of Voronoi’s algorithm, hereby laying the the-
oretical foundations for systematic, computer assisted searches for dense periodic
sphere packings. We develop a “t-theory” which enables us to find local sphere pack-
ing optima (t-extreme sets) among all t-periodic point sets. These are linear images
of a standard periodic set (0.1) with a fixed translational part t = (t1, . . . , tm). For
the case of a rational matrix t we show in Theorem 3.23, in analogy to the classical
theory, that there exist only finitely many t-extreme sets. These can be enumerated
by a generalized Voronoi algorithm. On top of the t-theory, we extend the theory
of G-perfect and T -perfect lattices (where G is a finite subgroup of the orthogonal
group and T is a linear subspace in the space of quadratic forms).
This allows us to restrict searches for dense t-periodic sphere packings to ones
with specific features, for example with a fixed finite symmetry group. For rational
t and a finite subgroup of the orthogonal group (leaving (0.1) invariant) we show
in Theorem 3.25 that there exist only finitely many local optima among sphere
packings with the corresponding properties. This generalizes works of Berg´ e, Mar-
tinet, Sigrist and others. Our proof relies on a general observation (Lemma 3.26)
regarding group actions on polyhedral subdivisions. The section ends with three
examples of the G-theory, describing new results for Eisenstein, Gaussian and Hur-
witz quaternionic lattices.
Chapter 4: Voronoi’s second reduction theory. In the fourth chapter we
generalize Voronoi’s second reduction theory which is based on Delone polyhedra
and Delone subdivisions of lattices. In Section 4.1 we first give the necessary defini-
tions, some background and we explain Voronoi’s theory on secondary cones (also
called L-type domains). We give a simplified and generalized proof for Voronoi’s
theory in Section 4.2. Theorem 4.7 extends Voronoi’s theory to a “t-theory” for
Delone subdivisions having a standard periodic vertex-set (0.1). For rational t we
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