H. S. M. Coxeter’s research while affiliated with University of Toronto and other places

When Bolyai János was forty years old, Philip Beecroft discovered that any tetrad of mutually tangent circles determines a complementary tetrad such that each circle of either tetrad intersects three circles of the other tetrad orthogonally. By careful examination of a new proof of this theorem, one can see that it is absolute in Bolyai’s sense. Beecroft’s double-four of circles is seen to resemble Schläfli’s double-six of lines.

We describe three hexacontahedra in which the faces are rectangles, all equivalent under symmetries of the icosahedral group and having all edges in the mirror planes of the symmetry group. Under the restriction that adjacent faces are not coplanar, these are the only possible polyhedra of this kind.

Much has been written about the discovery by physicists of quasicrystals and the almost simultaneous discovery by geometers of nonperiodic tessellations and honeycombs. A somewhat similar serendipity occurred when crystallographers saw what happens when identical balls of plastic clay or lead shot are shaken together and uniformly compressed, or when the bubbles (all of the same size) in a froth are measured; and almost simultaneously geometers investigated statistical honeycombs. Alternate doses of oil and water in a thin tube may be regarded as a one-dimensional "froth" {∞}, each "bubble" having just 2 neighbours. Analogously, soapsuds sandwiched between parallel glass plates (close together) may be regarded as a two-dimensional froth {6,3}, each bubble having just 6 neighbours. Three-dimensional froth presents a far more difficult problem because there is no regular honeycomb having 4 cells at each vertex. The best available substitute seems to be a "statistical" honeycomb {p, 3,3} where p, instead of being rational, is a real number such as π/ arctan $\sqrt {\frac{1}{2}}$ , somewhere between 5 and 6. ({5,3,3} is the regular 120-cell and {6,3,3} is non-Euclidean.) In such a statistical honeycomb, the number of neighbours for each bubble is 13.4, in good agreement with experiments in which the actual number is 12 or 14 and sometimes 15, but most often 13. Hoping not to be too fanciful, we venture to look for a statistical honeycomb {q,3,3,3} in Euclidean 4-space, q being a real number such as π/ arctan $\sqrt {\frac{3}{5}}$ , somewhere between 4 and 5. ({4,3,3,3} is the 5-cube while {5,3,3,3} is non-Euclidean.) In this case the number of neighbours for one bubble in the 4-dimensional froth is computed to be about 28.

With respect to a sphere having centre O and radius r, two points P and P' are said to be inverse if P' lies on the diameter OP and OP × OP' = r²: The transformation P → P' is called inversion. To be able to say that every point has an inverse, we add to the ordinary Euclidean space an extra point O' called the point at infinity. We thus create inversive space. Since circles and spheres through O invert into lines and planes, lines and planes are simply circles and spheres through the point at infinity. Two tangent spheres have just one common point: their point of contact. If this point is O, the two spheres, having no other common point, invert into two parallel planes. Consider 5 mutually tangent spheres having (2⁵) = 10 distinct points of contact. If O is one of these ten points, we obtain by inversion two parallel planes with three ordinary spheres sandwiched between them. Since these three are congruent and mutually tangent, their centres are the vertices of an equilateral triangle. Analogously, if four congruent spheres are mutually tangent, their centres are the vertices of a regular tetrahedron. A fifth sphere, tangent to all these four, may be either a larger sphere enveloping them or a small one in the middle of the tetrahedral cluster. In this article it will be shown that here are fifteen spheres, each passing through six of the ten points of contact of the five mutually tangent spheres.

. The 45 diagonal triangles of the six-dimensional polytope 2 21 (representing the 45 tritangent planes of the cubic surface) are the vertex figures of 45 cubes { 4,3} inscribed in the seven-dimensional polytope 3 21 , which has 56 vertices. Since 45 x 56 = 8 x 315 , there are altogether 315 such cubes. They are the vertex figures of 315 specimens of the four-dimensional polytope { 3,4,3 } , which has 24 vertices. Since 315 x 240 = 24 x 3150 , there are altogether 3150 { 3,4,3 } 's inscribed in the eight-dimensional polytope 4 21 . They are the vertex figures of 3150 four-dimensional honeycombs { 3,3,4,3 } inscribed in the eight-dimensional honeycomb 5 21 . In other words, each point of the 8 lattice belongs to 3150 inscribed 4 lattices of minimal size. Analogously, in unitary 4 -space there are 3150 regular complex polygons 3 { 4 } 3 inscribed in the Witting polytope 3 { 3 } 3 { 3 } 3 { 3 } 3 .

The numerical distance between two circles in the Euclidean plane is defined to be the number c 2 -a 2 -b 2 2ab, where a and b are their radii while c is the ordinary distance between their centres. An infinite sequence of circles is defined to be loxodromic if every four consecutive members are mutually tangent. D n denotes the numerical distance between the mth and (m+n)th circles (the same for all m). Obviously D -n =D n . Since the numerical distance is -1 when a=b and c=0 so that the two circles coincide, D 0 =-1. Since it is 1 when a+b=c so that the circles are externally tangent, D 1 =D 2 =D 3 =1. Any number of further values of D n can be determined successively by the recurrence equation D m +D m+4 =2(D m+1 +D m+2 +D m+3 )· There is also an explicit formula (4.2) for D n (as a function of the sequential distance n) in terms of binomial coefficients and Fibonacci numbers.

For reciprocation with respect to a sphere Σx2 = c in Euclidean n-space, there is a unitary analogue: Hermitian reciprocation with respect to an antisphere Σūu = c. This is now applied, for the first time, to complex polytopes. When a regular polytope Π has a palindromic Schläfli symbol, it is self-reciprocal in the sense that its reciprocal Π′, with respect to a suitable concentric sphere or antisphere, is congruent to Π. The present article reveals that Π and Π′ usually have together the same vertices as a third polytope Π+ and the same facet-hyperplanes as a fourth polytope Π- (where Π+ and Π- are again regular), so as to form a 'compound', Π+[2Π]Π-. When the geometry is real, Π+ is the convex hull of Π and Π′, while Π- is their common content or 'core'. For instance, when Π is a regular p-gon {p}, the compound is {2p}[2{p}]{2p}. The exceptions are of two kinds. In one, Π+ and Π- are not regular. The actual cases are when Π is an n-simplex {3,3, ... , 3} with n ≥ 4 or the real 4-dimensional 24-cell {3, 4, 3} = 2{3}2{4}2{3}2 or the complex 4-dimensional Witting polytope 3{3}3{3}3{3}3. The other kind of exception arises when the vertices of Π are the poles of its own facet-hyperplanes, so that Π, Π′, Π+ and Π- all coincide. Then Π is said to be strongly self-reciprocal.