Draft September 19, 1994.
3-Stresses in 3-Space and Projections of Polyhedral 3-surfaces:
Reciprocals, Liftings and Parallel Configurations.
Henry Crapo † and Walter Whiteley‡
ABSTRACT. Our previous papers analyzed the basic three way connections among
static self stresses of plane frameworks, plane reciprocal diagrams and spatial oriented
polyhedra. We investigate the analogous patterns in 3-space, for projections of spherical polyhedral 3-surfaces of a fixed combinatorial structure:
(i) linear maps (isomorphisms, under appropriate restrictions) between
(a) the affine space of polyhedral 3-surfaces with a fixed projection in 3-space;
(b) the vector space of 3-stresses of the projected configuration;
(c) the space of reciprocal figures (realizations of the dual combinatorial with
dual faces orthogonal to the original edges, dual edges orthogonal to the
original faces) for the projected configuration;
(ii) and, dually, appropriate maps between
(d) the vector space of parallel configurations of a selected polyhedral 3-surface;
(e) the space of parallel configurations of the projection of this polyhedral 3surface into 3-space; and
(f) the space of configurations in 3-space with a fixed reciprocal.
§1.
INTRODUCTION.
Building on the fundamental insights of James Clerk Maxwell for spherical polyhedra and planar graphs, our first paper [3] presented the equivalence, for plane drawings
of 3-connected planar graphs, of the following three properties (Figure 1.1):
(i) a self stress on the drawing (as a bar framework), non-zero on all edges;
(ii) a reciprocal diagram on the dual planar graph, with dual edges of non-zero length,
perpendicular to the original edges;
(iii) a spatial polyhedron, with distinct plane faces at each edge, projecting orthogonally onto the plane drawing;
† Work supported, in part, by a grant from N.S.E.R.C. (Canada).
‡ Work supported, in part, by grants from N.S.E.R.C. (Canada) and F.C.A.R.
(Quebec).
1
Crapo & Whiteley
Spatial Polyhedron:
liftings
Dual Polyhedron:
parallel drawings
6
b
B
4
(6)
(4)
A
e
C
D
(E)
3
d
1
2
c
2
a
3
1
(5)
5
Reciprocal:
perpendicular dual edges
Framework: self stresses
A
B
Plane Framework:
stresses
Reciprocal:
parallel drawing
Figure 1.1.
(iv) a spatial polyhedron on the dual graph, with edges of non-zero length, projecting
to a reciprocal framework.
In our second paper [4], we extended these constructions to the full vector space
of self stresses, and corresponding spaces of reciprocals, and of spatial liftings of a
plane drawing of a 2-connected planar multigraph. To create these spaces, we included
degenerate situations (Figure 1.2), where some scalars of the stress become zero (Figure
1.2A), or correspondingly, several points coincide (Figures 1.2C,D), or several faces go
coplanar (Figure 1.2B). For these degeneracies, we use a broader definition of a drawing
(or framework) - a ‘2-frame’ - where edges are assigned a direction vector, and the two
vertices are allowed to coincide in the plane.
C
A
D
Spatial Polyhedron:
coplanar faces
B
Plane Framework:
stresses zero on edges
Figure 1.2.
2
Dual Polyhedron:
coincident vertices
Reciprocal:
coincident points
3-space Reciprocals and 2-stresses
These more inclusive definitions gave isomorphisms among various vector spaces
and affine spaces:
(i) the vector space of self stresses on the drawing;
(ii) the affine space of reciprocal diagrams on the dual planar graph, with dual edges
perpendicular to the original edges and with one dual vertex fixed;
(iii) the affine space of parallel configurations for a fixed reciprocal diagram on the
dual planar graph, with one dual vertex fixed;
(iv) the affine space of spatial polyhedra, with one face fixed, projecting orthogonally
onto the plane drawing.
(v) the affine space of parallel configurations for a fixed spatial polyhedron on the
dual graph, with one vertex fixed, projecting to reciprocal drawings.
In this paper we present analogs of these results for projections of ‘spherical polyhedral 3-surfaces’, for 3-dimensional ‘reciprocals’, and for ‘stresses’ in spatial frameworks. There are two distinct generalizations of the plane ‘self stresses’. We can assign
scalars to the edges, creating a 2-stress in 3-space, or assign scalars to faces occupying
a hyperplane (here a plane) creating a 3-stress in 3-space. Although we will consider
both spaces of stresses, the strongest analogy with the plane isomorphisms holds for
the 3-stresses.
The basic idea for the reciprocals in 3-space is due to Rankine [11]. As before, it
involves dual objects orthogonal to the originals: in this case, dual faces orthogonal
to the edges, such that the oriented measure (here oriented area) of the dual face
gives the magnitude and sign of the 2-stress in the original (Section 5); and dual
edges orthogonal to the original faces, so that their size and direction gives scalars
for a 3-stress on the original faces (Section 3). As in [3,4], the ‘normal diagram’,
formed by intersecting the cell normals (through a fixed point) with the 3-space is
reciprocal to the orthogonal projection of the polyhedral 3-surface. We have found no
prior comments in the literature about the connections between such reciprocals and
projections of oriented polyhedral 3-surfaces.
The explicit definitions of the appropriate objects for this study are given in Sections 2 and 3. The basic connections between 3-stresses and reciprocals are investigated
in Section 3. In Section 4 we extend the definitions to objects in 4-space and present
the linear maps among the projections, the liftings, the 3-stresses and reciprocals for
one configuration and the parallel configurations in 3-space and 4-space for the reciprocal configuration. In all cases, we include appropriate degeneracies to give full vector
space (or affine space) isomorphisms.
3
Crapo & Whiteley
In Section 5 we consider the weaker links among projections from 4-space, reciprocals and 2-stresses.
These linear maps (isomorphisms under approriate restrictions) are the bridges
between several areas of current geometric research. They connect spaces which arise
in:
(a) the study of the edge-structures of spatial configurations (2-stresses) [2, 12];
(b) in the study of ‘polyhedral pictures’ (liftings of 4-polytopes drawn in 3-space) [22];
(c) the study of Minkowski decompositions of the convex polytopes (parallel configurations) [13, 20];
(d) the study of the independence of dihedral angles of convex polytopes in 4-space;
(e) the study of area-preserving maps of cell complexes in 3-space (3-stresses and
parallel configurations) [15,16];
(f) the related studies of face numbers for convex polytopes (the g-theorem) and
cohomologies of related toric varieties [7, 9, 17].
We note that, unlike many of the studies related to ‘4-polytopes’, we require no restrictions to convex realizations (or realizability) nor to simplicial (or simple) cell
structures. Our combinatorial assumptions lie at the level of homology and cohomology of the structure of cells, faces, edges and vertices. Our geometric assumptions
are minimal: even allowing vertices to coincide, or faces of a cell to be coplanar etc.,
provided there are well defined ‘directions’ for all edges and ‘normals’ for all faces.
In Section 6, we sketch the natural extensions of these results to combinatorial
homology (n)-spheres in n-space. The patterns established for the plane and for 3spahace are clear enough that the details are left to the reader.
All of the results are presented in terms of Euclidean coordinates. However, the
dimensions of the spaces and the underlying connections between projections and 3stresses are projectively invariant. There are alternate projective forms for much of
this theory (see Remark 7.4).
In presenting these results, we are aware of a vast range of results, connections
and questions which we have omitted. Section 7 outlines a few of these questions.
This paper is not the last chapter of a completed theory, but one step of a growing
adventure in the geometry of polytopes and polyhedral surfaces and the connections
among properties in various dimensions.
Acknowledgments. These results grew out of a long term collaboration with Janos
Baracs, within the Structural Topology Research Group at the Université de Montréal.
Janos introduced us to his conjectured connections between 2-stresses in 3-space and
4
3-space Reciprocals and 2-stresses
projections of 4-polytopes (see Remark 5.8). After rediscovering the roots of Maxwell’s
Theorem for plane stresses and projected polyhedra, we created an early version of
this theory around 1978.
The concept of 3-stress goes back to some conjectures of Gil Kalai and some
pioneering work of Carl Lee [7]. Our background on 3-stresses was developed in joint
work of the second author with Tiong-Seng Tay and Neil White [15,16,17].
§2.
COMBINATORIAL 3-spheres AND 3-FRAMES.
Our basic object of study will be generalizations of 4-polytopes, and the projec-
tions of their vertices, edges and faces to form a related 3-dimensional object. We
begin with the combinatorial structure for these structures.
§2.1.
The combinatorial homology 3-spheres.
In [4], we presented an abstract ‘polyhedral surface’ (Figure 2.1 A) by means of a
set of diamond shaped ‘edge patches’ (Figure 2.1 B). These oriented patches are then
matched up (Figure 2.1 C), along their ‘edges’ (incident pairs of faces and vertices) to
form an overall oriented surface, without boundary.
h
k
j
i
A
B
<h,i;j,k>
C
Figure 2.1.
We want a comparable notation for the basic combinatorial topology (homology)
of a combinatorial homology 3-sphere. This will have ‘edge-face’ patches a, b; c, d,
which are labeled with two vertices a, b (sharing the edge) and two cells c, d (sharing the
face) (Figure 2.2A,B). With these quadruples, we will need appropriate cycle conditions
5
Crapo & Whiteley
etc., to ensure that a face has a cycle of edges, an edge has a cycle of faces, a cell has
a polyhedron of faces, vertices and edges, etc.. Effectively, we need the homology of
a 3-sphere. Since duality will be a central part of our analysis, we offer a self-dual
presentation of the combinatorial homology.
Definition 2.1.
A combinatorial homology 3-sphere is a finite set of vertices (ab-
stract points) V = {a1 , . . . , am }, a finite set of cells C = {C 1 , . . . , C n } , and a finite
(non empty) set of edge-face patches B such that:
(i) Each edge-face patch is an ordered quadruple of indices h, i; j, k, where h, i refer
to distinct vertices and j, k refer to distinct cells and if h, i; j, k is in B, then
the companion patch −h, i; j, k = i, h; k, j is in B but the opposite patches:
h, i; k, j and i, h; j, k are not in B;
(ii) An oriented edge is a pair (a, b) of vertices in a patch a, b; j, k, forming the set
E;
(iii) For each oriented edge (a, b), all the edge-face patches in B with these vertices in
the order (a, b) form a cycle N (a,b) : a, b; j s , k s 1 ≤ s ≤ t, (t ≥ 3); with k s = j s+1
(iv)
(v)
(vi)
(vii)
(viii)
and k t+1 = j 1 , and all these cells are otherwise distinct;
An oriented face is an ordered pair (c, d) of cells in a patch h, i; c, d, forming the
set F ;
For each oriented face (c, d), all the edge-face patches in B, with these cells in the
order (c, d), form a cycle N(c,d) : hs , is ; c, d 1 ≤ s ≤ t, (t ≥ 3); with is = hs+1
and it = h1 , and all these vertices are otherwise distinct;
For each pair of vertices a, b, there is a connecting vertex-edge path: a sequence of
oriented edges in E, . . . , (as , as+1 ) . . ., 0 ≤ r ≤ t, such that a0 = a and at+1 = b;
For each pair of cells c, d, there is a connecting cell-face path: a sequence of
oriented faces . . . , (cs , cs+1 ), . . ., 1 ≤ s ≤ t, such that c0 = c, ct+1 = d;
Each simple vertex-edge cycle N = {(a0 , a1 ) . . . , (ar , ar+1 ), . . . (at , a0 )} (closed
path of oriented edges) is sum of face cycles:
(ar , ar+1 ) =
[
(ah , ah+1 )].
F i NF i
N
(summing in the sense of homology over Z - setting (h, i) + (i, h) = 0)
(ix) Each simple cell-face cycle M = {(c0 , c1 ) . . . , (cr , cr+1 ), . . . (ct , c0 )} (closed path of
oriented faces and cells) is sum of edge cycles:
(cr , cr+1 ) =
[
(ck , ck+1 )].
Ei N E i
M
(summing in the sense of cohomology over Z- setting (j, k) + (k, j) = 0).
6
3-space Reciprocals and 2-stresses
d
d
d
<a,b;c,d>
d
b
b
c
b
d
dd
b
a
c
a
a
c
a
b
d
b
a
c
c
c
a
c
A
C
B
(a,b) edge cycle
D
(c,d) face cycle
Figure 2.2.
Remark 2.2.
If we associate the pair of companion edge-face patches h, i; j, k and
i, h; k, j with a single oriented topological ball with a tetrahedral boundary (Figure
2.2B), the combinatorial homology 3-sphere forms a cell complex. In practice, our
‘patches’ are amalgams of four tetrahedra of the first barycentric subdivision of the
cell complex with the given cells, faces, edges and vertices. In our definition, properties
(i)-(vii) just translate properties of a connected, oriented homology manifold [10] into
our presentation via ‘patches’ within the first barycentric subdivision of the underlying
complex. Figure 2.2C illustrates the cell-face cycle which is the ‘coboundary’ of an
edge, and Figure 2.2D illustrates the vertex-edge cycle which is the ‘boundary’ of a
face.
The cycle condition (viii) says that every 1-cycle is the sum (over Z) of the
boundaries of faces, or that the first homology H1 = 0. The dual condition (ix)
on cell-face cycles says that every 2-cocycle (cycle in the dual - see below) is the sum
of edge coboundaries (boundaries of faces in the dual), so that the second cohomology
H 2 = 0. SInce we assume all of the properties of an homology manifold, Poincaré
duality for homology manifolds says these two conditions are equivalent [10], and we
have an homology 3-sphere. (We note that an homology manifold is automatically
a topological pseudo-manifold.) We just have an unusual presentation of such an
homology 3-sphere. Throughout this paper we will work with this homology in the
combinatorial homology 3-sphere to define cycles etc..
Remark 2.3.
In terms of the partially ordered of ‘faces’ of the underlying cell
complex, we are making simple assumptions. Figure 2.3A illustrates our assumptions
about an edge-face patch, with the two vertices of an edge, and the two cells of a
face. Figure 2.3B shows the cell-face cocycle of all faces (and cells) at an edge - the
coboundary cycle of the edge in cohomology. Figure 2.3C shows the vertex-edge cycle
7
Crapo & Whiteley
of all edges and vertices of a face - the boundary cycle of the face in the homology.
edge-face patch
cells
coboundary
face
edge
boundary
edge
A
boundary of a
face is a cycle.
coboundary of an
edge is a cocycle.
vertex-edge cycle
around a face
C
B
vertices
face
cell-face cocycle
around an edge
Figure 2.3.
Remark 2.4. This class of combinatorial homology 3-spheres is closed under duality,
which interchanges vertices and cells (turns the partially ordered set upside down). If
we switch the pairs in each edge-face patch: b = h, i; j, k goes to b∗ = j, k; h, i,
the set of edge patches B goes to B ∗ . The dual combinatorial homology 3-sphere with
‘vertices’ V ∗ = C, ‘cells’ C ∗ = V , and edge-face patches B ∗ , now satisfies Definition
2.1. This duality also switches edges and faces: E goes to F ∗ and F goes to E ∗ .
Property (i) is self-dual, properties (ii), (iii), (vi), (viii) are dual to (iv), (v), (vii), (ix).
Throughout this paper duality will be used in this combinatorial sense.
|V|=8, |E|=24, |F|=32, |C|=16
a
|V|=16, |E|=32, |F|=24, |C|=8
b'
c'
d'
d
c
a'
b
B
A
Figure 2.4.
Example 2.5. Consider the combinatorial homology 3-sphere illustrated in Figure
2.4A. This is the ‘hypercube’ with |C| = 8 cubic cells: the interior cube, the exterior
cube, and the six connecting ‘cubes’. The reader can verify that there are two cells at
each of the |F | = 24 ‘square’ faces, and three cells at each of the |E| = 32 edges.
8
3-space Reciprocals and 2-stresses
Figure 2.4B illustrates the dual ‘cross polytope’ (the 4-space analog of an octahedron), which has |E| = 24 edges, |F | = 32 triangular faces and |C| = 16 tetrahedral
cells.
§2.2.
3-frames.
We will work with two different realizations of the combinatorial homology 3spheres. First we examine 3-space realizations of the underlying structure of edges and
faces, with positions for the vertices, directions for the edges and normals for the faces.
We call these 3-frames. Then we will look at 4-space realizations, with hyperplanes
for the cells, and points for the vertices: the polyhedral 3-surfaces (Section 4). Of
course, our basic objective is to describe the connections between the two realizations
- the projections of a 4-space realization of the combinatorial homology 3-sphere into
3-space, creating a 3-frame, and the liftings of a 3-frame into 4-space polyhedral 3surfaces.
Definition 2.6. A 3-frame is a combinatorial homology 3-sphere (V, C; B), together
with points p : V → IR3 , non-zero directions d : E → IR3 , and non-zero normals
D : F → IR3 such that
(i) for each companion pair of oriented edges (a, b), (b, a), d(a,b) = −d(b,a) = 0;
(ii) for each companion pair of oriented faces (c, d), (d, c), D(c,d) = −D(d,c) = 0;
(iii) for each edge (a, b), pa −pb = β(a,b) d(a,b) for some scalar β(a,b) . (Note that pa −pb
could be 0.);
(iv) for each edge-face patch a, b; c, d, d(a,b) · D(c,d) = 0.
As a convention, we write the 3-frame as H = ((V, C; B); p, d, D) and write
the joints as pi . We write the three coordinates of pi as (p1i , p2i , p3i ), of d(a,b) as
(c,d)
(c,d)
(c,d)
(d1(a,b) , d2(a,b) , d3(a,b) ) and the three coordinates of D(c,d) as (D1 , D2 , D3 ). When
we wish to use affine coordinates of the point, we write p̃i = (p1i , p2i , p3i , 1), but write
(c,d)
(c,d)
(c,d)
d̃(a,b) = (d1(a,b) , d2(a,b) , d3(a,b) , 0) and D̃(c,d) = (D1 , D2 , D3 , 0), since these are
‘direction vectors’ (differences of affine points).
Remark 2.7. Our conditions guarantee a unique plane P(c,d) for each face (c, d),
with equation D(c,d) · (x, y, z) − D(c,d) · pa = 0 for some vertex a on the face. Working
along an edge (a, b) of the face, with pa − pb = β(a,b) d(a,b) , we have
D(c,d) · (pa − pb ) = D(c,d) · β(a,b) d(a,b) = β(c,d) D(c,d) · d(a,b) = 0.
9
Crapo & Whiteley
Therefore D(c,d) · pa = D(c,d) · pb and the face plane is well defined.
§2.3.
Stresses.
The following are direct generalization of the usual ‘self stresses’ used to study
the statics and kinematics of frameworks [2,3,12]. They are, essentially, a record of
dependencies among the lengths of the bars of a framework.
Definition 2.8.
of scalars ω(a,b)
A 2-stress on a 3-frame H = ((V, C; B); p, d, D) is an assignment
to the edges (a, b) ∈ E such that ω(a,b) = ω(b,a) and for each vertex a:
ω(a,b) d(a,b) = 0.
b|(a,b)∈E
A 2-stress is non-trivial if some ω(a,b) d(a,b) = 0 (equivalently, ω(a,b) = 0).
Since the 2-stresses are simply the solutions to a system of 3|V | homogeneous
linear equations ( b ω(a,b) d(a,b) = 0), in the |E| variables ω(a,b) , they form a vector
space. (The matrix which records the 2-stresses as row dependencies is called the
1-rigidity matrix of the 3-frame [2]. Notice that, if the vertices of the 3-frame are all
distinct, then the rank of this matrix only depends on these vertices. More generally,
it depends only on the directions of the edges. )
There is an alternate generalization of plane self stresses. Here we keep the scalars
on the objects of codimension 1 (edge lines in the plane, face planes in 3-space).
These scalars arise in the study of ‘skeletal rigidity’ and geometric results related
to the g-theorem for convex polytopes [7, 15,16]. These ‘3-stresses’ are, essentially,
dependencies among the areas of these plane faces.
Definition 2.9. A 3-stress on a 3-frame H = ((V, C; B); p, d, D) is an assignment of
scalars ω (c,d) to the faces (c, d) ∈ F such that ω (c,d) = ω (d,c) and for each edge (a, b):
ω (c,d) D(c,d) = 0 (sum over (c, d) ∈ F such that a, b; c, d ∈ B).
(c,d)
A 3-stress is non-trivial if some ω (c,d) D(c,d) = 0 (equivalently, ω (c,d) = 0).
10
3-space Reciprocals and 2-stresses
§2.4.
Reciprocals.
We have a basic geometric ‘duality’ for 3-frames in 3-space. (It is not a polarity
in 3-space, but works with the combinatorial duality of the polyhedral 3-surfaces.
This duality will ‘lift’ to a polarity in 4-space (Section 4). Our motivation for this
construction comes from its connections to 3-stresses (Section 3), to projections of
polyhedral 3-surfaces (Section 4) and to 2-stresses (Section 5).
Definition 2.10. Given a 3-frame H = ((V, C; B); p, d, D), a reciprocal 3-frame
H ∗ = ((V ∗ , C ∗ ; B ∗ ); p∗ , d∗ , D∗ ) is a 3-frame on the dual combinatorial homology 3sphere ((V ∗ , C ∗ ; B ∗ ) such that
(a,b)
(i) for each edge and dual face (a, b), d(a,b) = D∗
;
(ii) for each face and dual edge (c, d), D(c,d) = d∗(c,d) .
Note that V ∗ , C ∗ , B ∗ , d∗ , D∗ are just relabelings of the original information, but the
points p∗c are arbitrary solutions to the equations p∗c − p∗d = α(c,d) d∗(c,d) .
Every 3-frame will have trivial reciprocals: place all dual points at a single
(a,b)
point, making p∗c − p∗d = 0 = 0d∗(c,d) for all dual edges (c, d). Since d∗(c,d) · D∗
=
(c,d)
· d(a,b) = 0, this is a (trivial) 3-frame.
D
This is a true geometric duality: the original 3-frame is a reciprocal of its reciprocal
3-frame. However there will be other reciprocals of the reciprocal (see the parallel
configurations below).
|V*|=6, |E*|=15, |F*|=18, |C*|=9
|V|=9, |E|=18, |F|=15, |C|=6
2
a'
3
8
b
9
1
6
7
c'
5
a
c
4
A
B
C
b'
Figure 2.5.
Example 2.11. Consider the reciprocal pair of 3-frames illustrated in Figure 2.5.
Figure 2.5A shows a 3-frame with |V | = 9 vertices, formed into |C| = 6 triangular
prisms (the three in a horizontal ring, the two filling the ‘hole’ of this ring, and the
11
Crapo & Whiteley
exterior prism). Each of the vertices is attached to four cells, and each edge has a
cycle of three cells at that edge. Figure 2.5C shows a dual 3-frame, with |V | = 6 and
|C| = 9. Since, in the original, every cell shared a face with each other cell, the dual
graph is the complete graph on 6 vertices. All of the dual faces are triangles, and the
dual cells are simplices. Figure 2.5B shows, in projection, the reciprocal geometry and should assist in identifying the triangles and tetrahedra of the dual structure.
Remark 2.12.
The reader may be familiar with a related form of ‘reciprocal’ in
3-space: the Voronoi diagram and Delaunay triangulation for a set of points in 3space. Locally, these patterns have edges of the Voronoid diagram perpendicular to
corresponding faces of the triangulation, and faces perpendicular to the dual edges.
This similarity is not an accident. In fact, Voronoi diagrams are projections of ‘open
convex polytopial caps’ and Delaunay triangulations are projections of a special polar.
The plane analog of this connection in explored in [1].
§2.5.
Parallel 3-frames.
We are particularly interested the relationship between two reciprocals of the same
original 3-frame: H1∗ = ((V ∗ , C ∗ ; B ∗ ); p1∗ , d1∗ , D1∗ ) and H2∗ = ((V ∗ , C ∗ ; B ∗ ); p2∗ , d2∗ , D2∗ ).
By the definition of reciprocals, the directions of the edges and the normals to the faces
will be unchanged:
(a,b)
D1∗
(a,b)
= d(a,b) = D2∗
(c,d)
and d∗1
= d∗2
(c,d) = D
(c,d) .
The only difference will be the locations p∗1 , p∗2 of the points. In other settings, including the study of plane reciprocals, this relationship has been called ‘parallel drawing’
(or parallel redrawing).
Definition 2.13. Given a 3-frame H = ((V, C; B); p, d, D), a parallel 3-frame is
a second 3-frame H = ((V, C; B); p , d, D), on the same polyhedral 3-surface, with
the same directions and face normals. We also say that H = ((V, C; B); p, d, D), and
H = ((V, C; B); p , d, D) are parallel .
Remark 2.14. The parallel 3-frames are the solutions p of a set of homogeneous
linear equations: for each edge (a, b)
pa − pb = α(a,b) d(a,b) .
12
3-space Reciprocals and 2-stresses
The solutions . . . pa , α(a,b) . . . for these equations form a vector space. We record the
values pi , but drop the (uniquely induced) values α(a,b) , forming the isomorphic vector
space of parallel 3-frames.
The sum of two parallel 3-frames in this vector space is related to the usual
Minkowski addition of convex polytopes:
H + H = ((V, C; B); p, d, D) + ((V, C; B); p , d, D) = (V, C; B); p + p , d, D).
As we saw above, a 3-frame has a 3-dimensional subspace of trivial parallel 3frames pi = t for some fixed point t. In our figures, we will only indicate the vector
d(a,b) when the two vertices are not distinct, and only indicate D(c,d) when the vertices
of the face do not span a plane.
A
C
B
Figure 2.6.
We also have dilations pi = α(pi − r) + r, for some fixed center r, and scalar α
(Figure 2.6 B). If all the joints of the original 3-frame lie at one point, or the scalar
α is 0, the dilation will be a trivial 3-frame. These dilations and the translations
(pi = pi − t) (‘dilations towards infinity’) form a subspace generated by the initial
2-frame and the trivial 3-frames. (If the original 3-frame is non-trivial, then this space
has dimension 4.) Figure 2.6C shows a parallel 3-frame which is not in this space.
(With three adjacent vertices fixed, and a change for others, it cannot be a dilation or
translation.)
The scalar multiple of a 3-frame H by α in this vector space is just the dilation,
by α, about the origin (0, 0, 0) as center.
Remark 2.15. The set of all reciprocals of a fixed 3-frame is the space of parallel
configurations for a single reciprocal. In this spirit, the set of all reciprocals is also a
13
Crapo & Whiteley
vector space. If we fix an preselected dual vertex, and study parallel 3-frames with
this constraint, we have an affine subspace of parallel 3-frames.
Remark 2.16.
In the plane, parallel 2-frames were isomorphic to ‘infinitesimal
motions’ [2,4]: assignments of velocities (. . . , vi , . . .) to the points, such that, for each
edge (a, b):
(va − vb ) · d(a,b) = 0.
In all dimensions, there is a standard linear algebra duality between 2-stresses and
such infinitesimal motions.
In 3-space, the connection between infinitesimal motions and parallel 3-frames is
modified [20]. Each parallel 3-frame creates a subspace of infinitesimal motions, but
not all infinitesimal motions correspond to parallel 3-frames. The direct linear algebra
duality between 2-stresses and parallel 3-frames is also lost in 3-space. There is a much
closer connection between 3-stresses and parallel 3-frames in 3-space.
Remark 2.17.
The space of parallel 3-frames depends only on the directions
d(a,b) , D(c,d) . Thus, the space of parallel 3-frames H of a parallel 3-frame H , is
the space of parallel 3-frames of the original H.
Moreover, if two 3-frames H, H on the same combinatorial homology 3-sphere
have parallel directions, d(a,b) = α(a,b) d(a,b) (α(a,b) = 0) for all edges (a, b), and have
parallel normals for all faces, they also have isomorphic spaces of 3-stresses and parallel
3-frames. If they also have the same points, then H and H are equivalent. (The only
differences are the magnitudes of the d(a,b) and D(c,d) .)
The spaces of 2-stresses and 3-stresses depend only on the directions d(a,b) , D(c,d) .
Equivalent 3-frames will have isomorphic spaces of stresses. We will effectively be
working with these equivalence classes of 3-frames throughout this paper.
It is clear that any affine transformation of space, and of the edge directions, with
the inverse affine transformation of the face normals, will leave the spaces of 2-stresses,
3-stresses and parallel 3-frames invariant. However, as we noted in the introduction,
the dimensions of all of these spaces are even projectively invariant [15,16, 20].
§3.
3-Stresses and Reciprocal 3-frames.
If we have a reciprocal pair of 3-frames, we have a corresponding 2-stress and a
corresponding 3-stress. We begin with the simpler connections for 3-stresses.
14
3-space Reciprocals and 2-stresses
§3.1.
3-Stresses and Reciprocals.
Remark 3.1.
Assume we have a 3-stress on a 3-frame H. For each edge (a, b), we
have a vector sum
ω (c,d) D(c,d) = 0 (sum over (c, d) ∈ F such that a, b; c, d ∈ B).
Now all the vectors D(c,d) in this sum are perpendicular to a fixed vector d(a,b) , so
this vector sum can be drawn in 3-space as a closed polygon in a plane with d(a,b) as
the normal. This forms a local ‘reciprocal face’ for the edge. The following theorem
says that these faces can be pieced together around each vertex to form a polyhedral
cell, and pieced together over all vertices to form a reciprocal 3-frame.
Theorem 3.2.
For a spherical 3-frame H = ((V, C; E); p, d, D) the following are
equivalent, for a given a, b; c, d ∈ E:
(i) the 3-frame has a non-trivial 3-stress with ω (c,d) D(c,d) = 0;
(ii) there is a non-trivial reciprocal 3-frame, with p∗c = p∗d ;
(iii) any reciprocal 3-frame has a non-trivial parallel 3-frame, with p∗c = p∗d .
Proof. (i) ⇒ (ii). Assume that there is a non-trivial 3-stress on the 3-frame. We choose
an initial cell c0 and assign it an initial point p∗0 . For other cells c1 , the reciprocal
point is defined by
p∗1 = p∗0 +
ω (c,d) D(c,d)
(c,d)∈M
sum over faces on a cell-face path M from c0 to c1 .
We need to show that this definition is independent of the path. Given any two
paths between M, M c0 and c1 , the sum up one path and back the other becomes a
cell-face cycle M, −M . What we really need is that the sum around a cycle is 0, and
the rest will follow.
By the definition of a 3 stress, we know that the sum over the cell-face cycle N (a,b)
around an individual edge (a, b) is 0 (see the remark above). Moreover, any cell-face
cycle is the homological sum of a finite set of cell-face cycles N (a,b) of a set of edges
E (property (ix)):
M
(cr , cr+1 ) +
n{j,k} ω (j,k) [(j, k) + (k, j)] =
[
(a,b)∈E N (a,b)
{j,k}∈F
15
(cs , cs+1 )]
Crapo & Whiteley
for some integers n{j,k} for the faces. Therefore, replacing the formal faces (j, k) with
the vectors ω (j,k) D(j,k) in this formal sum, with the established property ω (j,k) D(j,k) +
ω (k,j) D(k,j) = 0, produces:
r r+1
r r+1
r r+1
r r+1
ω (c ,c ) D(c ,c ) =
ω (c ,c ) D(c ,c ) +
n{j,k} ω (j,k) [D(j,k) + D(k,j) ]
M
M
=
[
{j,k}∈F
k
ω (c
k+1
,c
)
s
D(c
,cs+1 )
] = 0.
(a,b)∈E N (a,b)
We conclude that the sum around any cell-face cycle is 0. Therefore, for any two paths
M, M from c0 to c1 , with M, −M as a cycle,
ω (c,d) D(c,d) +
ω (c,d) D(c,d) = 0 implies
ω (c,d) D(c,d) =
ω (c,d) D(c,d) .
−M
M
The dual points
M
M
p∗j
are well defined by our construction.
In Remark 3.1 we saw that the cycle around each edge produces a plane polygon,
with the edge direction as normal. By the very definition:
p∗c − p∗d = ω (c,d) D(c,d) = ω (c,d) d∗(c,d)
for each dual edge-face patch c, d; a, b. Thus we have a true reciprocal.
Since p∗c − p∗d = ω (c,d) D(c,d) and D(c,d) = 0, we have p∗c = p∗d if, and only if,
ω (c,d) = 0.
(ii) ⇒ (i). Conversely, given a non-trivial reciprocal, we can reverse this construction.
We solve
p∗c − p∗d = ω (c,d) d∗(c,d) = ω (c,d) D(c,d)
for the scalars ω (c,d) , to recreate the unique corresponding 3-stress in the original 3frame. Since the cell-face path around an edge (a, b) in the original becomes a closed
polygon in the reciprocal, we have the required
ω (c,d) D(c,d) =
p∗c − p∗d = 0 (sum over (c, d) such that a, b; c, d ∈ B).
(c,d)
(c,d)
(ii) ⇔ (iii). This follows immediately from the fact thata parallel 3-frame of a reciprocal
is also a reciprocal.
Because the original 3-frame is the reciprocal of its reciprocal, this proof defines
a pair of reciprocal 3-stressesin the two reciprocal 3-frames. Of course, the correspondence between 3-stresses and reciprocals is not unique: the translation of any
reciprocal (adding a trivial reciprocal) will correspond to the same 3-stress. If we fix
the initial dual point p∗0 at (0, 0, 0), we select an appropriate subspace of reciprocals
and a corresponding subspace of parallel 3-frames. This restriction creates a vector
subspace of the space of all parallel 3-frames.
16
3-space Reciprocals and 2-stresses
Theorem 3.3.
For a given 3-frame H = ((V, C; B); p, d, D), with a designated
initial cell, the following spaces are isomorphic:
(a) the space of 3-stresses on the 3-frame;
(b) the space of reciprocal 3-frames with the initial dual vertex at (0, 0, 0);
(c) the affine space of parallel 3-frames of a fixed reciprocal 3-frame, with the
initial vertex fixed.
Proof. We first observe that the construction of the reciprocal from the 3-stress and the
construction of the 3-stress from the reciprocal are inverse functions. We also observe
that the sum of two 3-stresses passes to the sum of the two reciprocals, and the multiple
of a 3-stress passes to the same multiple of the reciprocal. The constructions are linear
isomorphisms.
§4.
POLYHEDRAL 3-SURFACES: PROJECTIONS, LIFTINGS AND
PARALLEL REDRAWINGS.
In [3,4], we saw that 2-stresses (self stresses) in plane 2-frames, and their plane
reciprocals, were directly related to liftings to polyhedral 2-surfaces (‘polyhedra’) in 3space, for simply connected polyhedra. A similar relationship connects the reciprocals
and 3-stresses on a 3-frame to liftings up to polyhedral 3-surfaces in 4-space.
§4.1.
Polyhedral 3-surfaces.
Definition 4.1. A polyhedral 3-surface, ((V, C; B); q, Q), is a combinatorial homology 3-sphere, (V, C; B), with an assignment q of points: qi = (qi1 , qi2 , qi3 , qi4 ) ∈ IR4 to
the vertices i ∈ V , and an assignment Q of dual points: Qj = (Qj1 , Qj2 , Qj3 , Qj4 ) ∈ IR4
(representing the non-vertical hyperplane Qj1 x + Qj2 y + Qj3 z + w + Qj4 = 0), to the cells
j ∈ C, such that if the vertex a and the cell c share an edge-face patch then qa is on
the hyperplane Qc : Qc1 qa1 + Qc2 qa2 + Qc3 qa3 + qa4 + Qc4 = 0.
As a convention, we write the affine coordinates of the point q̃i = (qi1 , qi2 , qi3 , qi4 , 1)
and the extended coordinates of the plane Q̃j = (Qj1 , Qj2 , Qj3 , 1, Qj4 ), so that the statement that the point lies on the plane becomes: q̃a · Q̃c = 0.
We note that our definitions permit polyhedral 3-surfaces in which all cells and
vertices lie on the same plane, or all cells have the same non-vertical space and the
17
Crapo & Whiteley
vertices are placed arbitrarily, or dually, all vertices are the same point and all cells
pass though this point.
We want to study the projections of these polyhedral 3-surfaces into 3-space as
3-frames. Accordingly, we will need a non-zero, non-vertical ‘direction’ w(a,b) for
each oriented edge (a, b) of the combinatorial homology 3-sphere and a non-zero, nonvertical ‘normal’ W(c,d) for each cell such that:
(i) the (non-zero) direction w(a,b) for the edge (a, b) satisfies: qa − qb = β(a,b) w(a,b)
for some scalar β(a,b) , (and q̃a − q̃b = β(a,b) w̃(a,b) );
(ii) the (non-zero) normal W(c,d) for the face (c, d) satisfies: Qc − Qd = β (c,d) W(c,d)
for some scalar β (c,d) ;
(iii) For each oriented edge w(a,b) = −w(b,a) and for each oriented face W(c,d) =
−W(d,c) ;
(iv) For each edge-face patch a, b; c, d, defining the affine directions to be
(c,d)
W̃(c,d) = (W1
(c,d)
, W2
(c,d)
, W3
(c,d)
, 0, W4
)
1
2
3
4
and w̃(a,b) = (w(a,b)
, w(a,b)
, w(a,b)
, w(a,b)
, 0), we have:
(c,d) 1
qa
W̃(c,d) · q̃a = W1
(c,d) 2
qa
+ W2
(c,d) 3
qa
+ W3
(c,d)
+ W4
= 0,
1
2
3
4
+ Qc2 w(a,b)
+ Qc3 w(a,b)
+ w(a,b)
= 0;
Q̃c · w̃(a,b) = Qc1 w(a,b)
and
(c,d)
W̃(c,d) · w̃(a,b) = W1
(c,d)
1
w(a,b)
+ W2
(c,d)
2
w(a,b)
+ W3
3
w(a,b)
= 0.
These selected directions and normals create the augmented polyhedral 3-surface S =
((V, C; B); q, Q, w, W).
If the cells and vertices of an edge-face patch are distinct points and planes, all
of the equations in (iv) follow directly from the incidence conditions of the polyhedral
3-surface. If the two vertices are distinct points, then the third equation of condition
(iv) follows from the first equation (at the two vertices), and similarly if the two cells
are distinct hyperplanes. However, if the vertices and cells are not distinct, then this
equation is required to ensure that the projection is a 3-frame. Of course, we have
equivalent augmentations (different choices of the scalars β(a,b) for the edges and β (c,d)
for the faces). In general, we will choose the augmentation to project to the directions
of a given spatial 3-frame.
18
3-space Reciprocals and 2-stresses
§4.2.
Projections of polyhedral 3-surfaces.
The vertical projection Π of a Euclidean point qa ∈ IR4 is the point
Π(qa ) = Π(qa1 , qa2 , qa3 , qa4 ) = (qa1 , qa2 , qa3 ).
Similarly the projection of an affine point is
Π̃(q̃a ) = Π̃(qa1 , qa2 , qa3 , qa4 , 1) = (qa1 , qa2 , qa3 , 1).
and the vertical projection of the directions w(a,b) and W(c,d) are the vertical projections of the corresponding 4-vectors (dropping the last entry)
Definition 4.2. A 3-frame H = ((V, C; B); p, d, D) contains the vertical projection
of an augmented polyhedral 3-surface S = ((V, C; B); q, Q, w, W) if the 3-frame has
joints pa = Π(qa ), and, for each edge patch a, b; c, d, d(a,b) = Π(w(a,b) ) and D(c,d) =
Π(W(c,d) ).
For an augmented polyhedral 3-surface S = ((V, C; B); q, Q, w, W), the normal
diagram of S is a spatial 3-frame H = ((C, V ; B ∗ ); p∗ , d∗ , D∗ ) on the dual combinatorial homology 3-sphere (C, V ; B ∗ ) with
(i) the point p∗ c = Π(Qc ) for each face (dual vertex) c; and
(ii) the direction d∗ (c,d) = Π(W(c,d) ), and normals D∗ (a,b) = Π(w(a,b) ) for the dual
edges and the dual faces.
Proposition 4.3.
The vertical projection of an augmented polyhedral 3-surface S
and the normal diagram of S are reciprocal 3-frames.
Proof. With the given definitions, what we need to prove is that the vertical projection
and the normal diagram are 3-frames. That is, we must show that:
(i) the defined d(a,b) is parallel to pa − pb ,
(ii) the defined D(c,d) is parallel to p∗ c − p∗ d ,
(iii) and d(a,b) · D(c,d) = 0.
(i) From our definition of an augmentation: β(a,b) w(a,b) = qa − qb , so
β(a,b) d(a,b) = Π(β(a,b) w(a,b) ) = Π(qa − qb ) = pa − pb .
(ii)
From our definition of an augmentation: β (c,d) W(c,d) = Qc − Qd , so
β (c,d) D(c,d) = Π(β (c,d) W(c,d) ) = Π(Qc − Qd ) = p∗c − p∗d .
19
Crapo & Whiteley
(iii)
For each edge-face patch a, b; c, d, condition (iv) gives:
(c,d)
D(c,d) · d(a,b) = W1
(c,d)
1
w(a,b)
+ W2
(c,d)
2
w(a,b)
+ W3
3
w(a,b)
= W̃(c,d) · w̃(a,b) = 0.
We conclude that the projection is a 3-frame. The normal diagram is, by its construction, a 3-frame on the dual combinatorial homology 3-sphere. Since d∗(c,d) = D(c,d)
and D∗(a,b) = d(a,b) , this is a reciprocal 3-frame.
Given the polyhedral 3-surface projecting to one 3-frame, we can reverse the roles
and find a dual polyhedral 3-surface projecting to the reciprocal, with its cell normals
creating the original 3-frame. In 4-space, this ‘duality’ is expressed as a projective
polarity. Throughout this paper, polarity will refer to the 4-space polar given in the
next definition.
Definition 4.4. The Maxwell polarity in 4-space is the pair of transformations L
and L−1 between points in IR4 and non-vertical hyperplanes in 4-space defined by
L(x1 , x2 , x3 , x4 ) = (x1 , x2 , x3 , x4 ) and L−1 (X1 , X2 , X3 , X4 ) = (X1 , X2 , X3 , X4 ).
Remark 4.5.
The Maxwell polarity is a standard projective polarity in 3-space,
about the paraboloid x2 + y 2 + z 2 + 2w = 0. (This is the equation of the points which
lie on their polar hyperplane.) We recall that a Euclidean point pa = (x1a , x2a , x3a , x4a )
has affine coordinates p̃a = (x1a , x2a , x3a , x4a , 1) and non-vertical cell with coordinates
Qc = (X1c , X2c , X3c , X4c ) has affine coordinates Q̃c = (X1c , X2c , X3c , 1, X4c ). The polarity
effectively switches the last two affine (projective) coordinates. (Projective points at
infinity, (x1 , x2 , x3 , x4 , 0), are polarized to vertical planes (X1 , X2 , X3 , 0, X4 ).)
This (and any) projective polarity takes a point qa on a hyperplane Qc to a
polar hyperplane L(qa ) through the polar point L−1 (Qa ). Thus the polarity takes
a polyhedral 3-surface to a dual polyhedral 3-surface. Since L−1 L(qa ) = qa and
L(L−1 (Qc ) = Qc , the polar of the polar is the original polyhedral 3-surface. For the
augmentation, the Maxwell polarity simply reverses the roles of the directions and
(a,b)
∗
= W(c,d) , W∗
= w(a,b) .
normals w(c,d)
The following is an immediate consequence of the definitions of the polarity, the
projection and the normal diagram.
Proposition 4.6.
For an augmented polyhedral 3-surface
S = ((V, C; B); q, Q, w, W)
20
3-space Reciprocals and 2-stresses
and the augmented Maxwell polar polyhedral 3-surface
S ∗ = ((C, V ; B ∗ ); L(q), L−1 (Q), W, w)
(i) the vertical projection of S is the normal diagram of S ∗ ;
(ii) the normal diagram of S is the vertical projection of S ∗ .
§4.3.
Liftings to polyhedral 3-surfaces.
By Remark 2.7, each face with normal D(c,d) and point pa in a 3-frame has
the equation: D(c,d) · (x, y, z) − D(c,d) · pa = 0, with a consistent ‘fourth coordinate’
−D(c,d) · pa .
Definition 4.7. The face moment of an oriented face (c, d) is the scalar m(c,d) =
−D(c,d) · pa for any vertex a on the face. bigskip
The face moment is defined for the oriented face, since all vertices give the same
scalar. For the oppposite orientation (d, c) have
m(d,c) = −D(d,c) · pa = D(c,d) · pa = −m(d,c) .
These face moments will lead to the differences in height between the dual vertices in
our construction of the polyhedral 3-surface.
Lemma 4.8. For any cell-face cycle N of edge-face patches: (. . . , is , hs ; cs , cs+1 , . . .),
1 ≤ s ≤ r, cr + 1 = c1 , and any 3-stress on the 3-frame, the face moments satisfy the
equation: N ω (c,d) m(c,d) = 0.
Proof. This equation holds for the simple cycle N (a,b) around a single edge (a, b):
N
ω (c,d) m(c,d) =
ω (c,d) D(c,d) · pa = (
ω (c,d) D(c,d) ) · pa = 0 · pa = 0.
N
N
Now our original cycle is the homology sum of a finite set of edge cycles, with the
formal cancellation (c, d) + (d, c) = 0. Since m(c,d) + m(d,c) = 0, and ω (c,d) = ω (d,c) , we
have ω (c,d) m(c,d) + ω (d,c) m(d,c) = 0. With this, we can substitute the moments for the
formal faces in the homology sum, to prove that the moment equation for the original
cycle is the sum of the moment equations for the edge cycles, which is zero as required.
21
Crapo & Whiteley
Theorem 4.9.
For a spherical 3-frame H = ((V, C; B); p, d, D) the following are
equivalent, for some edge patch a, b; c, d:
(i) the 3-frame has a 3-stress with ω(a,b) d(a,b) = 0;
(ii) there is a reciprocal 3-frame H ∗ = ((C, V ; B ∗ ); p∗, D, d) with p∗ c = p∗ d ;
(iii) the 3-frame contains the vertical projection of an augmented polyhedral 3surface S = ((V, C; B); q, Q, w, W) with Qc = Qd ;
(iv) the reciprocal 3-frame is the normal diagram of an augmented polyhedral
3-surface S = ((V, C; B); q, Q, w, W) with Qc = Qd .
Proof. Theorem 3.1 gives (i) ⇔ (ii). Proposition 4.6 gives (iii)⇔(i)&(ii).
(i)&(ii) ⇒ (iii),(iv) We first use one 3-stress ω, in the original, to produce the vertices of a polyhedral 3-surface over the reciprocal of this 3-stress. Choose an initial dual vertex 0 at the origin: p∗0 = P0 = (0, 0, 0) in the reciprocal, and set
q∗0 = Q0 = (P10 , P20 , P30 , 0). For each edge-face patch a, b; c, d make the 3-vector
(ω (c,d) D(c,d) , ω (c,d) m(c,d) ). This will become the vector from q∗c to q∗d . Specifically, for
each dual vertex 1 we take an edge-face path N from 0 to 1, and define:
s s+1
s s+1
s s+1
s s+1
q∗1 = (0, 0, 0, 0) + ( (ω (c ,c ) D(c ,c ) ,
ω (c ,c ) m(c ,c ) )
N
N
To show that this is well defined, we must see that the point q∗1 is independent of the
path chosen. For the first three coordinates, this follows from the construction of the
reciprocal, where we showed that
s s+1
s s+1
p∗1 = p∗0 + ( (ω (c ,c ) D(c ,c ) )
N
is independent of the path. For the final coordinate, any two cell-face paths M, M make a cycle M, −M so , by Lemma 4.8:
0=
M
=
s
ω (c
,cs+1 )
s
m(c
,cs+1 )
+
M
,ct+1 )
t
m(c
,ct+1 )
−M s
ω (c
s+1
,c
)
s
m(c
s+1
,c
)
−
t
ω (c
,ct+1 )
t
m(c
,ct+1 )
.
M
M
We conclude that
t
ω (c
s
ω (c
,cs+1 )
s
m(c
,cs+1 )
=
M
t
ω (c
,ct+1 )
t
m(c
,ct+1 )
as required.
We have constructed the points q∗c over the reciprocal. Do we have the required
hyperplanes for the dual cells (original vertices)? We can switch to the reciprocal,
repeat all of the process for its 3-stress and create points qa over the vertices of the
22
3-space Reciprocals and 2-stresses
original, starting with vertex 0 which is held in the hyperplane Q0 with equation w = 0,
but may be at any location on this hyperplane.
The residual question is: are these two constructions compatible? Do the points
qa lie on the polar planes Qc whenever a, b; c, d ∈ B? For the initial face and vertex
Q̃0 · q̃0 = 0(q01 ) + 0(q02 ) + 0(q03 ) + 1(0) + 0 = 0.
Now consider moving out from this pair, through a common edge-face patch:
a, b; c, d. Assume Q̃c · q̃a = 0, and we move through the face (c, d) to the cell
d:
Q̃d − Q̃c = ω (c,d) (D̃(c,d) , m(c,d) ) = ω (c,d) (D(c,d) , 0, −D(c,d) · pa )
so
Q̃d · q̃a = Q̃c · q̃a + ω (c,d) [D(c,d) , 0, −D(c,d) · pa ] · q̃a = 0 + D(c,d) · pa − D(c,d) · pa = 0.
Similarly, assume that we move along the edge (a, b) to the vertex b:
Q̃c · q̃b = Q̃c · q̃a + ω(a,b) Q̃c · [d(a,b) , −d(a,b) · Pc , 0] = 0 + Pc · d(a,b) − d(a,b) · Pc = 0.
By the connectedness of the combinatorial homology 3-sphere, this checks that the
incidence holds for any vertex and cell.
Remark 4.10. The approach of [19] generalizes directly to our setting, interpretting
the 3-stress vectors ω (c,d) D(c,d) as an assignment of ‘relative angular velocities’ to the
faces, showing how its two cells ‘rotate’ instantaneously up into 4-space, with velocities
given by the heights (4th coordinates) of the vertices. In this interpretation, our cycle
conditions for the 3-stress state that the sum of the relative motions around any cellface cycle is zero, as required for an infinitesimal motion.
With this perspective, it is not difficult to see that the 3-frame is the projection
of a convex 4-polytope if, and only if, the spatial realization is a double dissection of
a bounding convex polyhedron (the ‘2-shadow’), into two 3-balls, with disjoint cells,
and all scalars on the faces interior to this ‘boundary’ polyhedron have one sign, while
those on the boundary polyhedron have the opposite sign. This can be demonstrated
by a direct generalization of the techniques of [19].
In general, the sign of the scalar ω (c,d) on a face indicates how the two hyperplanes
of the cells are bending appart as we lift up into 4-space
As we run through the space of 3-stresses, we create a collection of polyhedral
3-surfaceswith the same projection. These form another affine (or vector) space.
23
Crapo & Whiteley
Definition 4.11. The vertical liftings of a 3-frame H = ((V, C; B); p, d, D) are the
set of all polyhedral 3-surfaces S = ((V, C; B); q , Q , w, W), with H as the vertical
projection.
Lemma 4.12.
The set L(S) of all liftings of a 3-frame H = ((V, C; B); p, d, D)
forms an affine space under the addition:
tS + (1 − t)S = ((V, C; B); tq + (1 − t)q , tQ + (1 − t)Q , tw + (1 − t)w, tW + (1 − t)W).
Proof. For each vertex a:
tqa +(1−t)qa = t(p1a , p2a , p3a , qa4 )+(1−t)((p1a , p2a , p3a , qa4 ) = (p1a , p2a , p3a , tqa4 +(1−t)qa4 ),
and for each cell c:
t(Qc1 , Qc2 ,Qc3 , Qc4 ) + (1 − t)(Q1c , Q2c , Q3c , Q4c )
= (tQc1 + (1 − t)Q1c , tQc2 + (1 − t)Q2c , tQc3 + (1 − t)Q3c , tQc4 + (1 − t)Q4c ).
While these points project correctly to H, it remains to prove that this is a polyhedral
3-surface. For each edge-face patch a, b; c, d:
[tQc1 + (1 − t)Q1c ]p1a + . . . + +tqa4 + (1 − t)qa4 + [tQc4 + (1 − t)Q4c ]
= t[Qc1 p1a + . . . + qa4 + Qc4 ] + (1 − t)[Q1c p1a + . . . + qa4 + Q4c ]
= 0 + 0 = 0.
We have an appropriate polyhedral 3-surface.
We must also verify that we have created an appropriate augmentation for the
new polyhedral 3-surface.
(i) tqa + (1 − t)qa − (tqb + (1 − t)qb ) = t(qa − qb ) + (1 − t)(qa − qb )
= β(a,b) tw(a,b) + β(a,b) (1 − t)w(a,b)
, since β(a,b) is determined in the common
projection.
(ii) tQc + (1 − t)Qc − (tQd + (1 − t)Qd ) = t(Qc − Qd ) + (1 − t)(Q c − Q d )
= β (c,d) tw(c,d) + β (c,d) (1 − t)w (c,d) , since β (c,d) is also determined in the common
projection.
(iii) Clearly, reversing the orientations reverses the augmentations for edges and faces.
24
3-space Reciprocals and 2-stresses
(iv) For each edge-face patch a, b; c, d:
[tW̃(c,d) +(1 − t)W̃ (c,d) ] · q̃a
(c,d)
= [tW1
(c,d) 1
qa
= t[W1
+ (1 − t)W1
(c,d)
(c,d)
]qa1 + . . . + [tW1
+ . . . + (1 − t)[W1
(c,d) 1
qa
+ (1 − t)W4
+ . . . + (1 − t)W4
(c,d)
(c,d)
]
]
= t(0) + (1 − t)(0) = 0;
]
Q̃c · [tw̃(a,b) + (1 − t)w̃(a,b)
1
1
4
4
+ (1 − t)w(a,b)
] + . . . + tw(a,b)
+ (1 − t)w(a,b)
= Qc1 [tw(a,b)
1
4
1
4
= t[Qc1 tw(a,b)
+ . . . + w(a,b)
+ (1 − t)[Qc1 w(a,b)
] + . . . + (1 − t)w(a,b)
]
= t[0] + (1 − t)[0] = 0.
and, because Π(W(c,d) ) = D(c,d) = W (c,d) and Πw(a,b) = d(a,b) = w(a,b)
[tW(c,d) + (1 − t)W (c,d) ] · [tw(a,b) + (1 − t)w(a,b)
]
= [tD(c,d) + (1 − t)D(c,d) ] · [td(a,b) + (1 − t)d(a,b) ]
= D(c,d) · d(a,b) = 0.
Therefore, we have an appropriate augmentation.
Remark 4.13. If we restrict to the liftings with a first cell 0 at Q0 = (0, 0, 0, 0), we
have a smaller affine subspace.
Theorem 4.14.
isomorphic:
(i)
(ii)
(iii)
(iv)
For any 3-frame ((V, C; B); p, d, D), there following spaces are
the vector space of 3-stresses of the 3-frame;
the vector space of reciprocal 3-frames with p∗0 = (0, 0, 0);
the vector space of parallel 3-frames of a given reciprocal, with p0∗ = (0, 0, 0);
the affine space of polyhedral 3-surfaces, with initial cell Q0 = (0, 0, 0, 0),
which project into the 3-frame.
Proof. It is simple to verify that the constructions given in Theorems 3.1 and 4.9
are linear with respect to the defined vector spaces and affine spaces. That they are
non-singular follows from the addenda in these theorems that ω (c,d) = 0 if and only if
p∗ c = p∗ d if and only if p∗ c = p∗ c if and only if Qc = Qd .
Remark 4.15. In [3] we showed how the projections of other non-spherical, oriented
polyhedral surfaces from 3-space produced reciprocals and 2-stresses in the plane.
25
Crapo & Whiteley
However, these connections (embeddings of vector spaces) were not, in general, reversible (see [5] for more details). In a similar vein, we can extend our definition from
a combinatorial homology 3-sphere to an oriented homology 3-manifold, dropping conditions (viii) and (ix). With this change, we have:
Corollary 4.16. For any 3-frame H = ((V, C; B); p, d, D) on an oriented homology
3-manifold, there are the following linear maps:
(i) the affine space of polyhedral 3-surfaces, with Q0 = (0, 0, 0, 0), which project
into the 3-frame injects into the vector space of reciprocal 3-frames with
p∗0 = (0, 0, 0);
(ii) the vector space of reciprocal 3-frames with p∗0 = (0, 0, 0) injects into the
vector space of 3-stresses of the 3-frame.
In general, these injections are not isomorphisms. The condition for the reverse
constructions are given by ‘cycle conditions’: for every cell-face cycle N :
(i) the vector cycle condition N ω (c,d) D(c,d) = 0 (condition for the reciprocal construction);
(ii) the moment cycle condition N ω (c,d) m(c,d) = 0 (condition for the reciprocal to
lift) to a polyhedral 3-surface.
§4.4.
Parallel Configurations for Polyhedral 3-surfaces.
There is one space which is missing from our discussion. The polars of the space of
liftings of the original 3-frame form a space of dual augmented polyhedral 3-surfaces
projecting to the space of reciprocals (parallel 3-frames for a reciprocal). All these
dual polyhedral 3-surfaces have the same normal diagram (the original 3-frame) so
their cells have the same normals.
Definition 4.17.
Given an augmented polyhedral 3-surface
S = ((V, C; B); q, Q, w, W)
(with no vertical cells), a parallel configuration is an augmented polyhedral 3-surface
S = ((V, C; B); q , Q , w, W) with cells parallel to the original polyhedral 3-surface:
(Q1c , Q2c , Q3c , 1) = (Qc1 , Qc1 , Qc1 j, 1).
The fact that the augmentations w, W are unchanged, would automatically force
the cells to the same (or parallel) normals, except in certain degenerate situations.
26
3-space Reciprocals and 2-stresses
The set of parallel configurations forms a vector space:
S + αS =((V, C; B); q, Q, w, W) + α(((V, C; B); q , Q , w, W)
= ((V, C; B); q + αq , Q + αQ , w, W).
Those parallel configurations with an selected initial vertex q0 = (0, 0, 0, 0) form a
subspace. Alternately, the space of parallel configurations with a selected vertex held
fixed is an affine subspace.
For a polyhedral 3-surface S, we have the vertical projection Π(S), and the space
of all liftings of this projection. We call this space the 2-shadow equivalent polyhedral
3-surfaces of S.
Theorem 4.18.
Given an augmented polyhedral 3-surface
S = ((V, C; B); q, Q, w, W)
and a spherical 3-frame H which is the vertical projection then:
(i) the affine subspace of 2-shadow equivalent polyhedral 3-surfaces of the polyhedral 3-surface S, (Q0 held constant), is isomorphic to the space of 3-stresses
of the 3-frame;
(ii) the subspace of parallel configurations for the polyhedral 3-surface S with
q0 = (0, 0, 0, 0) is isomorphic to the subspace of parallel 3-frames for the
3-frame H with p0 = (0, 0, 0).
Proof. We have shown part (i) in Theorem 4.14.
(ii) Clearly any orthogonal projection of a parallel configuration for the polyhedral
3-surface is a parallel 3-frame for the first projection.
Finally, parallel 3-frames for the projection are reciprocals of the common normal diagram for all of the polyhedra. Therefore, by Theorem 4.14, polarized, these
lift to augmented polyhedral 3-surfaces with the reciprocal as the normal diagram.
Since these share the same normals to faces as the original, and their augmentations
project to the same directions, these liftings are parallel configurations for the original
augmented polyhedral 3-surface.
If we consider a general configuration of lines and points in 4-space, with the
natural definition of ‘parallel configuration’, it is clear that a parallel configuration in
4-space always projects to a parallel configuration of the projection. However the converse is false - some spatial ‘parallel configurations’ do not lift to parallel configurations
in 4-space. Theorem 4.18 is specifically restricted to the spherical homology.
27
Crapo & Whiteley
§5.
2-Stresses and Reciprocals.
For this section we require an additional observation about our combinatorial
homology 3-spheres. We assumed that the face-edge patches around a face form a
simple vertex-edge cycle, with fixed cells. This defines a clear ‘boundary’ for the face.
We have also assumed that the patches around an edge form a cell-face cycle. As a
result, we know that:
(x) For each cell c, all the face-edge patches in B with first cell c form a set of faces
which is an homology cycle:
each edge occurs twice, with opposite orientations in the two faces of the cell.
We have the equivalent dual statement.
(xi) For each vertex a all the edge-face patches in B with first vertex a form a cohomology cocycle:
each face occurs twice, with opposite orientations in the two edges at the
vertex.
These properties apply any oriented homology 3-manifold (or oriented pseudomanifold). For the following result, we do not require ‘spherical’ conditions (viii) and
(ix), so we can use the projection and reciprocal normal diagram for any oriented
pseudo-manifold formed as a cell complex.
The reciprocal 3-frame induces a 2-stress on the original 3-frame. We will use the
‘oriented area’ of the reciprocal face to define the scalars of the 2-stress. Specifically,
∗
: a, b; js , ks , 1 ≤ s ≤ t of the oriented dual face (a, b),
from the oriented cycle N(a,b)
we have the plane polygon: . . . , p∗js , p∗js+1 , . . ., which has an oriented area, which can
(a,b)
= ω(a,b) d(a,b) .
be written as a multiple of the normal: A∗
To use these areas as local vectors for a 2-stress at a vertex, we must know that
they sum to zero. This will follow from our assumption that the faces around the dual
cell form an oriented homology cycle. The following theorem is an extended form of a
result of Minkowski for convex polyhedra.
Proposition 5.1.
For any homology 2-cycle G (over Z) built with plane faces in
3-space, the oriented areas AF of the faces satisfy:
AF = 0.
F ∈G
Proof. We can, by ‘barycentric subdivision’ (at a combinatorial and geometric level),
assume that the faces are oriented triangles, with the area of the original face in the
28
3-space Reciprocals and 2-stresses
3-frame equal to the sums of the oriented areas of the triangles in the 3-frame of the
subdivision. This takes an homology cycle of the combinatorial homology 3-sphere to
an homology cycle (a ‘chain’ of triangles whose boundary is zero) of the triangulated
combinatorial homology 3-sphere.
For a triangle p1 , p2 , p3 in 3-space, the area can be calculated as
A(1,2,3) =(p2 − p1 ) × (p3 − p2 ) = p2 × p3 + p1 × p2 − p1 × p3
= p2 × p3 + p1 × p2 + p3 × p1 .
This is a ‘vector version’ of the boundary operator: ∂(1, 2, 3) = (1, 2) + (2, 3) + (3, 1).
The entire boundary sum, over the faces, is assumed zero in the formal sum of homology:
(∂F ) =
F ∈G
n{j,k} [(j, k) + (k, j)] = 0.
{j,k}∈F ∈G
Since we also have pa × pb + pb × pa , we can substitute the vectors pj × pk for the
formal faces (j, k) in this equation to get:
0=
n{j,k} [pj × pk + pk × pj ] =
[
pi × pj ] =
AF .
F ∈G
{j,k}∈F ∈G
∂F
F ∈G
Theorem 5.2. Given a spherical 3-frame H = ((V, C; B); p, d, D) with a reciprocal
3-frame H ∗ = ((V ∗ , C ∗ ; B ∗ ); p∗ , d∗ , D∗ ), the scalars ω(a,b) defined by ω(a,b) d(a,b) =
(a,b)
|A∗ | form a 2-stress on H = ((V, C; E); p, d).
Proof. We have given the scalars of the proposed 2-stress. First we verify that ω(a,b) =
∗
∗
ω(b,a) . N(a,b)
= {. . . , a, b; js , ks , . . .} implies N(b,a)
= {. . . , b, a; ks , js , . . .}, in the
reverse order. Therefore the dual polygon for (b, a) is . . . , p∗js+1 , p∗js , . . .. This has the
opposite oriented area, so, as required:
(b,a)
A∗
(a,b)
= −A∗
= −ω (a,b) d(a,b) = ω (a,b) d(b,a) .
For each vertex a, the faces of the dual oriented cell is an homology cycle (an
homology 2-sphere) G. By Proposition 5.1,
ω(a,b) d(a,b) =
F ∈G
b|(a,b)∈E
29
AF
∗ = 0.
Crapo & Whiteley
a
b
a
d
c
b
d
c
c'
b'
b'
c'
d'
A
d'
a'
a'
C
B
Figure 5.1.
Remark 5.3.
We cannot assert that a ‘nontrivial’ dual face implies a non-zero
coefficient for the 2-stress. A self-intersecting face may have oriented area AF = 0
(Figure 5.1A). We can have an entire reciprocal polyhedron for a vertex, with all faces
of zero oriented area (Figure 5.1B). (For comparison, Figure 5.1C indicates a regular
cube with the same combinatorial structure as Figure 5.1B.) However, if the reciprocal
has a triangle, or a convex face, then at least some of the scalars of the 2-stress will
be non-zero.
Is there an entire reciprocal, with all vertices of edges distinct, and with all faces of
area 0? Such a reciprocal would correspond to the projection of a non-trivial polyhedral
3-surface which nevertheless gives the zero 2-stress in its projection.
Remark 5.4. If a strictly convex polytope in 4-space is projected into a shadow
polyhedron in 3-space, it is clear that all the reciprocal faces for edges interior to this
shadow boundary will be convex polygons, with non-zero areas. All of these edges will
have non-zero coefficients in the induced 2-stress. Moreover, the reciprocal will assign
all edges interior to this shadow boundary the same sign (say −) in the 2-stress. For
edges on the shadow boundary, the reciprocal will the opposite sign (+), and these
will also be non-zero. Such a stress is called a spider web
Corollary 5.5.
For any strictly convex 4-polytope, projected orthogonally into
3-space with no cell vertical, the corresponding 2-stress is a strict spider web, with
tension (−) on all interior edges and compression (+) on all boundary edges in the
projection.
Unfortunately, we cannot assert that all 3-frame spider webs are projections of
convex 4-polytopes (see [1] for the true plane analog).
For a strictly convex 4-polytope, all 2-shadow equivalent configurations with a
‘small’ change in vertices from the original are also strictly convex. This guarantees
30
3-space Reciprocals and 2-stresses
that space of their reciprocals will share the property of having all dual faces of nonzero area and therefore inducing nontrivial 2-stresses.
Corollary 5.6.
For any strictly convex 4-polytope, projected orthogonally into
3-space with no cell vertical, there is an injection from the space of 3-stresses (and
space of reciprocals, one dual point fixed) of the projection into the space of 2-stresses
of the projection.
Finally we note that this injection from the 3-stresses into the 2-stresses will hold
for any 3-frame with one reciprocal which has all faces with non-zero area.
All 2-stresses do not correspond to reciprocals.
a'
b
|V| =6
|E| =15
|F| =18
|C| =9
c'
a
c
b'
Figure 5.2.
Example 5.7. Consider the simplicial combinatorial homology 3-sphere whose graph
is shown in Figure 5.2 (also in Figure 2.5C). This polytope has |V | = 6, |E| = 15.
The faces are all possible triangles, except abc, a b c , so |F | = 20 − 2 = 18. The nine
(oriented) cells are all quadruples with an ordered pair from the cycle a, b, c followed
by a ordered pair from the cycle a b c :
aba b , abb c , abc a , bca b , bcb c , bcc a , caa b , cab c , cac a .
(This satisfies Euler’s formula for a spherical 4-polytope: |V | − |E| + |F | − |C| =
6 − 15 + 18 − 9 = 0.)
The space of liftings is easily calculated: fix one cell (4 vertices) in the 3-space,
and lift the other two, independently - generating a two dimensional space of liftings.
31
Crapo & Whiteley
Therefore the space of reciprocals, with one vertex fixed, (Figure 2.5 and 2.6) has
dimension 2.
The space of 2-stresses is also easy to calculate.
As an infinitesimally rigid
framework in 3-space (a complete graph), the space of 2-stresses has dimension [12]:
|E| − (3|V | − 6) = 15 − (3(6) − 6) = 3. We conclude that some 2-stresses do not
correspond to reciprocals.
Remark 5.8.
Our initial investigation of reciprocals in 3-space and projected poly-
hedral 3-surfaces was provoked by the conjectures of Janos Baracs, a practicing architectural engineer. Having independently observed Maxwell’s correspondence between
plane stresses and projected polyhedra, Baracs asked:
Are spatial 2-stresses ‘explained’ by projections of polyhedral 3-surfaces?
Theorem 5.2 is a partial answer: projections of polyhedral 3-surfaces create reciprocals
which in turn create 2-stresses. However, many 2-stresses are not directly connected
with projections of polyhedral 3-surfaces. Our search for other connections between
events in 4-space and 2-stresses in 3-space continues.
§6.
EXTENSIONS TO HIGHER DIMENSIONS.
We have outlined a comprehensive set of interconnected vector spaces and affine
spaces for realizations of the cells of a sphere in 3-space and 4-space. In almost every
respect, these results were generalizations of results in [3,4], for spherical polyhedra in
3-space and their plane projections, where we have converted 2-stresses to 3-stresses.
From these two instances, we can the outline a general theory for n-stresses in n-space
and projections from (n + 1)-space.
For the n-stresses in n-space, we need a combinatorial structure extracted from
homology spheres in (n + 1)-space, with cells C = Fn of dimension n, meeting in pairs
at facets of dimension n − 1, a set Fn−1 (our faces). We assume that the cells at a
(n − 2)-face form a cycle (our cell-face cycles of an edge) (Figure 6.1A). Together the
incidences of the cells, facets and (n − 2)-faces define a set of upper patches B. To
make the constructions among the vector spaces work, we assume that all ‘cell-facet’
paths are sums of the cycles for (n − 2)-faces. (That is, the (n − 1)-cohomology H n−1
is 0.) For purposes of duality (reciprocals etc.) we also need the vertices, edges and
faces, presented as lower patches B. The faces generate boundary cycles of edges (and
vertices) (Figure 6.1B). We also need that all vertex-edge cycles are sums of facial
cycles (the first homology, H1 is 0), and dually, that all cell-facet cocycles are sums
32
3-space Reciprocals and 2-stresses
of the cocycles of n − 2-faces (the cohomology H1 is 0). We require some homology
/ cohomology in between, so that the vertices and edges of a facet ‘span’ both the
facet and its n − 2-faces. We will assume that we have an homology n-sphere, and the
patches form our combinatorial homology (n)-spheres (V, F2 , Fn−2 , C; B, B).
lower patch
upper patch
n-faces (cells)
A
cell-face cocycle
around an (n-2)-face
(n-1)-face (facet)
coboundary
(n-2)-face
face
face
boundary
edge
(n-2)-face
coboundary of an
( n-2)-face is a
cocycle.
B
vertices
boundary of face
is a cycle.
vertex-edge cycle
around a face
Figure 6.1.
We also need geometric realizations of the structure, with points for the vertices
and with specified non-zero ‘directions’ for the facets Fn−1 , indicated as normal vectors
D(c,d) in n-space, such that the directions in the cycle of an (n − 2)-face have a
common (n − 2)-space of orthogonals (for the n − 2-face). Dually, we need nonzero directions d(a,b) for the edges such that the cycle of directions of a 2-face lie in
plane and for any edge of a facet D(c,d) · d(a,b) = 0. These will form our n-frames
H = ((V, F2 , Fn−2 , C; B, B); p, d, D).
Definition 6.1.
An n-stress on an n-frame
H = ((V, F2 , Fn−2 , C; B, B); p, d, D)
is an assignment of scalars ω (c,d) to the facets such that ω (c,d) = ω (d,c) and for each
(n − 2)-face f :
ω (c,d) D(c,d) = 0 (sum over (c, d) such that f ; c, d ∈ B).
(c,d)
Definition 6.2. For an n-frame H = ((V, F2 , Fn−2 , C; B, B); p, d, D), a reciprocal
n-frame is an n-frame on the dual homology n-sphere
∗
∗
, C ∗ ; B ∗ , B ) = (C, Fn−2 , F2 , V ; B, B)
(V ∗ , F2∗ , Fn−2
with d∗(c,d) = D(c,d) for each dual edge and D∗
(a,b)
= d(a,b) for each dual facet.
From an n-stress, we can create the vertices and edges of a reciprocal n-frame,
with edge vectors ω (c,d) D(c,d) . (The homological and cohomological assumptions are
essential to this construction.) We then verify that this extends construction extend
to give the facets of an n-frame, with normals d(a,b) .
With the obvious definition of parallel n-frames (same direction vectors for edges
and same normals for facets), we have the general theorem:
33
Crapo & Whiteley
Theorem 6.3.
For an n-frame H = ((V, F2 , Fn−2 , C; B); p, d, D), the following
spaces are isomorphic:
(a) the space of n-stresses on the n-frame;
(b) the space of reciprocal n-frames with initial vertex at 0.
The definitions of augmented polyhedral n-surfaces in (n + 1)-space, their space
of parallel configurations, their projections and the space of liftings of a projection are
natural extensions of our previous definitions. These lead to the theorems:
Proposition 6.4.
For an augmented polyhedral n-surface S the vertical projection
and the face diagram are reciprocal n-frames.
Theorem 6.5.
Given an augmented polyhedral n-surface and an n-frame H which
is the vertical projection then:
(i) the affine subspace of 2-shadow equivalent polyhedral n-surfaces (Q0 held
constant) is isomorphic to the space of n-stresses of the n-frame;
(ii) the subspace of parallel configurations of the polyhedral n-surface with q0 = 0
is isomorphic to the subspace of parallel n-frames of the n-frame with p0 = 0.
We also have the standard definition of a 2-stress (self-stress in the literature of
statics [2,12]) and the result that the space of n-stresses injects into the space of 2stresses. In fact, if we start with the full combinatorial structure of any oriented homology n-manifold, with appropriate ‘directions’ for cells of all dimensions, the reciprocal
defines a k-stress for each 2 ≤ k ≤ n. We note that, for simplicial combinatorial homology n-spheres, the dimensions of these stress spaces are the h numbers defined by
the numbers of faces of various dimensions, provided that no top dimensional simplex
(cell) is degenerate [17].
§7.
CONCLUDING REMARKS.
As we indicated in the introduction, our work has connections with many other
fields of current geometric research. We outline some directions for further research
which flowfrom these results.
Remark 7.1. We have an isomorphism between parallel drawings of polyhedral
3-surfacesand their projections into 3-space. We have a second isomorphism between
parallel drawings of spherical polyhedral surfaces and parallel drawings of their plane
34
3-space Reciprocals and 2-stresses
projections [4]. The parallel redrawings of a projected polyhedral 3-surface induces
simultaneous parallel drawings of the plane projection of each of its spherical polyhedral cells. Do simultaneous parallel drawings of such a projection from 4-space into
the plane lift back to parallel drawings of the projection into 3-space, and then on into
4-space? Yes - under appropriate homological assumptions. This will be presented
elsewhere.
We expect that, with appropriate assumptions, the parallel drawings of an homology n-sphere in n + 1-space are isomorphic to the parallel drawings of its projections
all the way down into the plane. This remains to be verified in appropriate detail.
We do know that the non-trivial parallel drawings of an object in n-space project
down to non-trivial parallel drawings in the plane. Since non-trivial parallel drawings
of a polytope are equivalent to Minkowski decomposibility of a polytope, we have the
following theorem and conjecture.
Theorem 7.2.
A convex polytope in n-space is Minkowski indecomposible if a
general projection to the plane (from a n − 3-flat not containing any line) has only
trivial parallel drawings in the plane (or equivalently, is an infinitesimally rigid plane
framework).
Conjecture 7.3.
A convex polytope in n-space is Minkowski decomposible if a
general projection to the plane (from a n − 3-flat not containing any line) has a nontrivial parallel drawing in the plane (or equivalently, is an infinitesimally flexible plane
framework).
Remark 7.4. Is it possible for a 3-frame not to be the projection of a non-trivial
polyhedral 3-surface (with at least some cells non-coplanar)? In the plane, it is easy to
give an ‘incorrect’ picture of a polyhedron, which is only the projection of polyhedral
2-surface with all faces coplanar (Figure 7.1B). However the situation is dramatically
different in 3-space.
A
correct plane picture
B
Figure 7.1.
35
'incorrect' plane picture
Crapo & Whiteley
Consider a simple combinatorial homology 3-sphere (a combinatorial homology
3-sphere with all edge cycles of size 3, and all vertices of valence 4). Every proper
realization, with all edges having distinct joints and no three edges at a vertex coplanar,
has a non-trivial 3-stress (unique up to scalar multiplication) [26], and all faces with
non-zero scalars. (This is proven using a reciprocal 3-frame.) It is therefore the
projection of a proper polyhedral 3-surface, with the two cells at any face in distinct
hyperplanes.
At the other extreme, every simplicial combinatorial homology 3-sphere has |V |−4
3-stresses (because it has |V | − 4 liftings with one cell fixed). These are all ‘correct’ in
this general sense, though none of the liftings may be convex, even when the spatial
3-frame appears to be convex.
Conjecture 7.5. For any proper 3-frame H on a combinatorial homology 3-sphere,
with no three edges at a vertex with coplanar directions, and distinct joints for each
edge, there is a proper polyhedral 3-surface S in 4-space on the same combinatorial
homology 3-sphere, with distinct hyperplanes for the cells at each edge, lifting the
3-frame.
This would imply that every proper 3-frame (as defined in the conjecture) has a
non-trivial 3-stress, non-zero on every face.
Remark 7.6.
For projections of polyhedral surfaces into the plane, the projective cross-section of the faces (and edges) provides a form of ‘projective reciprocal’
[23]. Such cross-sections contained essentially the same information as our Euclidean
Maxwell reciprocals. Even for the plane, these cross-section reciprocals have not been
fully examined for non-spherical polyhedra. Nor have we analysed how these crosssections present the information on vector space isomorphisms (the space of all ‘compatible cross-sections’, one line fixed, is isomorphic to the space of 2-stresses).
It would be revealing to elaborate a theory of cross-section reciprocals for polyhedral 3-surfaces projected into 3-space. This remains an direction to be explored. It is
possible that these cross-sections would simplify our understanding of the conditions
for lifting projections of polyhedral 3-surfaces from the plane back into 4-space, as
mentioned in Remark 7.1.
Remark 7.7. As we noted in [4], the theory of plane projections, reciprocals and
liftings for spherical polyhedra in 3-space has an analog for projections of the polyhedron onto the 2-sphere in 3-space, from the center of the sphere. Similarly, the
36
3-space Reciprocals and 2-stresses
theory presented here can be transferred to projections of combinatorial homology
3-spheres onto the 3-sphere in 4-space. Reciprocals will involve ‘planes of dual faces’
(hyperplanes though the center of the sphere) perpendicular to ‘lines of edges’ (planes
through the center of the sphere), and these will still induce 3-stresses in this setting.
‘Parallel 3-frames’ in the 3-sphere are redefined as two configurations sharing a reciprocal (since parallel planes do not exist on the 3-sphere). There are details which have
not been verified, but there is every indication the results of this paper will transfer.
These ‘spherical reciprocals’ would be essential tools for the study of dihedral angles
(angles between cells) in 4-polytopes and polyhedral 3-surfaces.
We also anticipate that there should be a further transfer from reciprocals and
3-stresses from the Euclidean 3-space and the 3-sphere into 3-stresses and reciprocals
in hyperbolic 3-space.
Remark 7.8. It is well known that the theory of 2-stresses on the line is the study
of the homology of the underlying graph [21]. Explicitly,
A framework on the line, with points for the vertices and a non-zero direction
for all edges, has a non-trivial 2-stress if, and only if, it contains a polygon
(an homology 1-cycle).
In several senses, the 2-stresses in the plane, and in higher dimensions, create geometric
extensions of this homology [21]. For example maximal independent sets for 2-stresses
in the plane are formed by 3 edge-disjoint trees, at most two at any vertex, no larger
subset covered by two trees.
In the same sense, the theory of 3-stresses in a cell complex in the plane is the
theory of homology (over Z) [15,16, 17]. Explicitly [15],
A plane realization of the faces, edges and vertices of a cell complex, with
non-zero directions for all edges, has a non-trivial 3-stress if, and only if, the
cell complex contains an homology 2-cycle.
In some appropriate sense, the theory of 3-stresses in 3-space is the geometric extension
of this homology of 2-cycles [17]. The exploration of this connection, and of the
geometry captured by these 3-stresses, is a significant area for continuing research.
Remark 7.9. For the 1-skeleton of a spherical polyhedral 2-surface in 3-space, a
2-stress also implies a Cremona reciprocal in which the dual edges, in the sense of
the spherical polyhedron, are parallel to the original edges [6]. We can extend these
Cremona reciprocals for 2-stresses on the 1-skeleton of a spherical polyhedral 2-surface
in n-space, for all n ≥ 2. We simply move the vectors parallel to the original edges
and turn the vector sum at a vertex into a dual (skew) polygon.
37
Crapo & Whiteley
In the plane, these Cremona reciprocals (Maxwell reciprocals rotated 90◦ ) also
corresponded to projected polar polyhedra, using an alternate ‘Cremona polarity’ in
3-space in place of the Maxwell polarity.
In 3-space, these Cremona reciprocals appear to have no connection with ‘liftings’
or polarities. The original framework and the reciprocal need not even have plane
‘faces’ - just the homology of a sphere and points for the vertices. However, the
Cremona reciprocals do have a space of parallel drawings which is isomorphic (with
one dual vertex fixed) to the space of 2-stresses of the original.
We do not know of an analogous construction for combinatorial homology 3spheresin 3-space or higher dimensions. There may be an analog for combinatorial
polyhedral 5-spheres in some dimension, for which the 2-faces are parallel to the dual
2-faces and oriented areas of the dual faces (forming a homological cycle in the dual cell
for an edge) induce the 3-stress on the original (summing to zero around the original
edge). This has not been investigated.
BIBLIOGRAPHY
[1] P. Ash, E. Bolker, H. Crapo and W. Whiteley; Convex polyhedra, Dirichlet
tesselations and spider webs; in Shaping Space: a Polyhedral Approach M.
Senechal and G. Fleck (eds.) Birkhauser, Boston, 1988, 231-250.
[2] R. Connelly and W. Whiteley; Prestress stability and second order rigidity of
tensegrity frameworks; preprint, York University, North York, Ontario., 1992.
[3] H. Crapo and W. Whiteley; Plane stresses and projected polyhedra I: the basic
pattern; Structural Topology 20 (1993), in press.
[4] H. Crapo and W. Whiteley; Spaces of stresses, projections and parallel drawings
for spherical polyhedra; preprint, York University, North York Ontario, 1993.
[5] H. Crapo and W. Whiteley; Projections, stresses, liftings and reciprocals for
toroidal polyhedra; preprint, York University, North York, Ontario, 1993.
[6] L. Cremona; Graphical Statics; English translation, Oxford University Press
London, 1890, of the 1872 edition.
[7] C. Lee; Some recent results on convex polytopes; Contemporary Math. 114 (1990),
3-19.
[8] J.C. Maxwell; On reciprocal diagrams and diagrams of forces; Phil. Mag. Series
4 27 (1864), 250-261.
[9] P. McMullen; On simple polytopes; preprint, Department of Mathematics, University College London, London, England, 1992.
[10] J.R. Munkres; Elements of Algebraic Topology; Addison-Wesley, Reading,
Mass., 1984.
[11] W.M.R. Rankine; Principle of equilibrium in polyhedral frames; Phil. Mag. Series
4 27 (1864), 92.
38
3-space Reciprocals and 2-stresses
[12] B. Roth and W. Whiteley; Tensegrity frameworks; Trans. A.M.S. 177 (1981),
419-446.
[13] G.C. Shephard; Decomposible convex polyhedra; Mathematika 10 (1963), 89-95.
[14] T-S. Tay; Decomposition of polytopes; preprint, National University of Singapore,
Singapore, 1987 as a draft chapter for The Geometry of Rigid Structures,
H. Crapo and W. Whiteley, eds..
[15] T-S. Tay, N. White and W. Whiteley; Skeletal rigidity of simplicial complexes I;
European J. Combinatorics, to appear.
[16] T-S. Tay, N. White and W. Whiteley; Skeletal rigidity of simplicial complexes II;
European J. Combinatorics, to appear.
[17] T-S. Tay, N. White and W. Whiteley; A homological approach to skeletal rigidity;
preprint York University, North York, Ontario, 1993.
[18] N. White and W. Whiteley; The algebraic geometry of motions in bar and body
frameworks; SIAM J. Alg. Disc. Meth. 8 (1987), 1-32.
[19] W. Whiteley; Motions and stresses of projected polyhedra; Structural Topology
7 (1982), 13-38.
[20] W. Whiteley; Parallel redrawings of configurations in 3-space; preprint York
University, North York, Ontario, (1985).
[21] W. Whiteley; The union of matroids and the rigidity of frameworks; SIAM J.
Disc. Appl. Math. 1 (1988), 237-255.
[22] W. Whiteley; Some matroids on hypergraphs, with applications to scene analysis
and geometry; Disc. Comp. Geometry 4 (1988), 75-95.
[23] W. Whiteley; Weaving, sections and projections of polyhedra; Discrete Applied
Math. 32 (1991), 275-294.
[24] W. Whiteley; How to describe a polyhedron; to appear in Journal of Intelligent
and Robotic Systems, preprint 1992.
[25] W. Whiteley; How to construct a cube; preprint, Department of Mathematics
and Statistics, York University, North York, Ontario, 1993.
[26] W. Whiteley; Diagrams of simple polytopes; to appear.
[27] W. Whiteley; Determination of spherical polyhedra; to appear.
39
Crapo & Whiteley
Authors’ Addresses:
Henry Crapo
Walter Whiteley
Bât 9, INRIA,
Dept. of Mathematics and Statistics
B.P. 105
York University
78153 Le Chesnay
4700 Keele St
Cedex France
North York, Ontario
M3J-1P3 Canada
and
École d’architecture
C. R. M.
Université de Montréal
Université de Montréal
C.P. 6128 succursale A
C.P. 6128 succursale A
Montreal, Quebec
Montreal, Quebec
H3C-3J7 Canada
H3C-3J7 Canada
e-mail
[email protected]
[email protected]
40