Generic Orthotopes David Richter Department of Mathematics Western Michigan University

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Generic Orthotopes

David Richter?

?Department of Mathematics, Western Michigan University,

Kalamazoo MI 49008 (USA), david.richter@wmich.edu

October 24, 2022

Abstract

This article studies a large, general class of orthogonal polytopes which

we may call generic orthotopes. These objects emerged from a desire to

represent a Coxeter complex by an orthogonal polytope that is particu-

larly nice with respect to traditional topological, structural, or combina-

torial considerations. Generic orthotopes have a pleasant “homogeneity”

property, somewhat like a smoothly bounded compact subset of Euclidean

space. Thus, as soon as we demand that every vertex of an orthogonal

polytope be a ﬂoral arrangement, as deﬁned here, many derivative struc-

tures such as faces and cross-sections are also described by ﬂoral arrange-

ments. We also give formulas for the volume and Euler characteristic of

a generic orthotope using a couple of statistics that are deﬁned naturally

for ﬂoral arrangements.

1 Introduction

Suppose dis a non-negative integer. By an orthogonal polytope we mean a union

of ﬁnitely many axis-aligned boxes in Euclidean space Rd. This article lays a

foundation for a theory of a particular set of orthogonal polytopes which repre-

sents an elementary generalization of the d-dimensional cube to d-dimensional

orthogonal polytopes. We summarize the salient properties of these “generic

orthotopes”:

•Every face of a generic orthotope is a generic orthotope.

•Every orthographic cross-section of a generic orthotope is a generic ortho-

tope.

•The vertex ﬁgure of every vertex a generic orthotope is a simplex.

•The 1-dimensional skeleton of a generic orthotope is a bipartite d-regular

graph.

arXiv:2210.12012v1 [math.CO] 21 Oct 2022

•There are elementary formulas which relate the volume and Euler charac-

teristic of a generic orthotope.

•The structural and combinatorial properties of a generic orthotope remain

intact through small perturbations of their facets.

•We may approximate any compact subset of Euclidean space to any degree

of accuracy with a generic orthotope.

Notice that all but the last of these properties remain valid when “generic

orthotope” is replaced by the word “cube”. We establish all of these properties

in this article. Moreover, by what seems like good fortune, all of these properties

follow in an elementary manner, given an understanding of the local structure

of a generic orthotope.

The local structure of a generic orthotope has a convenient construction

using read-once Boolean functions. Thus, one ﬁnds “ﬂoral arrangements” and

“ﬂoral vertices” at the core of this theory, where a ﬂoral arrangement is deter-

mined by applying a read-once Boolean function to a set of half-spaces possessing

distinct supporting hyperplanes. We may encode a read-once Boolean function

by a series parallel diagram, and this leads to another bit of good fortune: We

may use a topological invariant of these diagrams, namely the number of loops

modulo 2, to obtain an expression for the Euler characteristic of a generic ortho-

tope using only the values of this invariant at its vertices. Our statement of this

formula appears below as Theorem 4.9. If one accepts the thesis that generic or-

thotopes are analogous to smoothly-bounded subsets of Euclidean space, then

one cannot help but recognize the similarity of this formula to the Poincar´e-

Hopf theorem which expresses the Euler characteristic as the sum of indices of

a vector ﬁeld with a ﬁnite number of singularities.

The emergence of generic orthotopes is somewhat convoluted. The original

motivation came from this author’s desire to represent Coxeter complexes by

orthogonal polytopes which are somehow “nice”. In conceiving this problem,

however, it was not clear what “nice” should mean with regards to orthogo-

nal polytopes. This author regards convex polytopes which are simple (having

exactly dedges at every vertex) as particularly “nice”, but it was not a priori

clear what higher structure one might borrow to study orthogonal polytopes.

The present article precisely develops what “nice” should mean for an orthog-

onal polytope, and we pose the general problem for Coxeter complexes in the

concluding section.

1.1 Examples in Low Dimensions

In order to illustrate the main ideas of this article, we consider the cases d= 2

and d= 3.

In two dimensions, we draw a contrast between the polygons which appear

in Figure 1.1; one of these is homeomorphic to a disc, while the other has

“singular points” where the boundary is self-intersecting. Using the terminology

developed here, the former of these is a generic orthotope and the latter is not.

(a) (b)

Figure 1.1: (a) A generic orthogon. (b) Self-intersecting boundary.

One may relate the numbers of corners of the two types that one sees in a generic

orthogon. In the polygon on the left in Figure 1.1, one notices that there are

n1= 9 corners that “point outward” and n3= 5 corners that “point inward”.

The authors of [14, 4] call these “salient” and “reentrant” points, respectively.

The subscripts 1 and 3 here specify the number of quadrants occupied by the

polygon at that type of vertex. An immediate corollary of the 2-dimensional

version of our formula in Theorem 4.9 is that one always has n1−n3= 4 for

every generic orthogon.

As one would expect, the situation when d= 3 is more complicated. In the

terminology developed here, each of the vertices which appear in Figure 1.2 is a

ﬂoral vertex. Superﬁcially, the properties that make these vertices “nice” are (a)

there are easily identiﬁed faces incident to the vertex and (b) the faces incident

to each vertex coincide with the face lattice of a 2-dimensional simplex (i.e. a

triangle). By contrast, if a 3-dimensional orthogonal polytope has a “degenerate

vertex” as one appearing in Figure 1.3, then we do not regard it as a generic

orthotope. One can quickly conceive of other kinds of degenerate points in

3 dimensions, and one imagines that the number of types of degeneracies that

might arise when d≥4 grows quickly, perhaps exponentially. Up to congruence,

the only types of ﬂoral vertices when d= 3 appear in Figure 1.2.

We may relate the numbers of congruence types of vertices which appear in

such a polytope. Thus, suppose Pis a 3-dimensional orthogonal polytope that

such that every vertex appears as one of the four congruence types as depicted

in Figure 1.2. For each i∈ {1,3,5,7}, let nidenote the number of vertices of

these corresponding types, where iindicates the number of octants occupied by

its tangent cone. Then Theorem 4.9 yields, for d= 3,

n1−n3−n5+n7= 8χ(P),

where χ(P) is the (combinatorial) Euler characteristic of P.

We illustrate some of these ideas with an example. Deﬁne an orthogonal

polytope by P=Sv∈Sv+ [0,1]3, where S⊂R3appears in Figure 1.4, and

v+[0,1]3denotes the translation of the unit cube [0,1]3by adding v. One should

imagine Pas an assembly of several stacks of unit cubes resting “skyscraper

style” on a ﬂat surface representing the (x, y)-plane in R3. A view of P“from

Figure 1.2: Three-dimensional ﬂoral vertices.

Figure 1.3: Degenerate vertices in 3 dimensions.

S=









(0,0,0),(1,0,0),(2,0,0),(3,0,0),(4,0,0),(0,1,0),(1,1,0),

(2,1,0),(3,1,0),(0,2,0),(2,2,0),(0,3,0),(1,3,0),(2,3,0),

(1,0,1),(2,0,1),(3,0,1),(4,0,1),(2,1,1),(3,1,1),(2,2,1),

(0,3,1),(1,3,1),(2,3,1),(1,0,2),(2,0,2),(3,0,2),(4,0,2)











Figure 1.4: Generating corners of the example.

Figure 1.5: A 3-dimensional generic orthotope.

above” appears in Figure 1.5. The numbers which appear in the ﬁgure give the

heights of these stacks. In order to see that Pis a generic orthotope, notice that

every vertex of Pis congruent to one vertices appearing in Figure 1.2. Here, we

have n1= 15, n3= 11, n5= 5, n7= 1, and thus

n1−n3−n5+n7= 15 −11 −5 + 1 = 0,

which we expect because Pis homeomorphic to a solid 3-dimensional torus.

1.2 Flowers

Dating to the 1950’s, the Kneser-Poulsen conjecture asserts roughly that the

union of a ﬁnite set of Euclidean balls cannot increase if the distances between

their centers decrease or remain equal. In their work on this problem for general

d, Bezdek and Connelly [5] demonstrated equivalences between various state-

ments which generalize the conjecture to ﬂower weight functions. In their ter-

minology, which can be traced to work by Csik´os [12] and earlier to Gordon and

Meyer [17], a “ﬂower” is a subset of Euclidean space that is obtained by applying

a certain type of Boolean function to a collection of balls. In their description,

Bezdek and Connelly used the phrase “exactly once” to describe the variables

used for the Boolean functions which they employed to deﬁne ﬂowers. This led

the present author to [11], where such functions are described as “read-once

Boolean functions”, terminology which this author employs throughout. Thus,

the underlying construction of a ﬂoral arrangement as deﬁned here is essentially

identical with the construction of ﬂowers.

We can see how to use read-once functions to deﬁne 3-dimensional ﬂoral

vertices. Thus, denote H1={(x, y, z) : x≥0},H2={(x, y, z) : y≥0}, and

H3={(x, y, z) : z≥0}as three half-spaces in R3. Using union ∪and ∩for

union and intersection respectively, we may describe these four conﬁgurations

H1∩H2∩H3,(H1∪H2)∩H3,(H1∩H2)∪H3,and H1∪H2∪H3.

By contrast, neither of the degenerate vertices depicted in Figure 1.3 possesses

such a representation. The graphs which appear in Figure 1.2 are the repre-

sentations of these read-once functions by series-parallel diagrams, where ∩is

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GenericOrthotopesDavidRichter??DepartmentofMathematics,WesternMichiganUniversity,KalamazooMI49008(USA),david.richter@wmich.eduOctober24,2022AbstractThisarticlestudiesalarge,generalclassoforthogonalpolytopeswhichwemaycallgenericorthotopes.TheseobjectsemergedfromadesiretorepresentaCoxetercomplexbyanor...

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