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