Full Terms & Conditions of access and use can be found athttps://maa.tandfonline.com/action/journalInformation?journalCode=uamm20

The American Mathematical Monthly

ISSN: 0002-9890 (Print) 1930-0972 (Online) Journal homepage: https://maa.tandfonline.com/loi/uamm20

A New Regular (Compound) Polyhedron (of InfiniteKepler–Poinsot Type)

Dirk Huylebrouck

To cite this article: Dirk Huylebrouck (2017) A New Regular (Compound) Polyhedron (of InfiniteKepler–Poinsot Type), The American Mathematical Monthly, 124:3, 265-268

To link to this article: https://doi.org/10.4169/amer.math.monthly.124.3.265

Published online: 13 Dec 2017.

Submit your article to this journal

Article views: 65

View related articles

View Crossmark data

A New Regular (Compound) Polyhedron(of Infinite Kepler–Poinsot Type)

Dirk Huylebrouck

Abstract. The five Platonic polyhedra combine identical regular polygons such that the spatialangles between them are identical. Kepler and later Poinsot added four more regular polyhedraby allowing star polygons or self-intersecting faces. Petrie and Coxeter added three more usinginfinitely repeating elements. In the present note we combine the Kepler–Poinsot generaliza-tion with the Petrie–Coxeter generalization to obtain a new regular (compound) polyhedron.

1. INTRODUCTION. A regular polyhedron has faces that are congruent regu-lar polygons assembled in the same way around each vertex. Or else, followingCoxeter (see [2], p. 16), faces and vertex figures that are all regular polygons. (Thevertex figure at a vertex V of a polygon is the line segment joining the midpoints of thetwo adjacent sides meeting at V . The vertex figure at a vertex V of a polyhedron is thepolygon whose sides are the vertex figures of the faces containing V .) The five regularPlatonic solids are very well known: the tetrahedron, the octahedron, the cube, thedodecahedron, and the icosahedron. These polyhedra combine regular polygons ofthe same type (equilateral triangles, squares, or regular pentagons) and at each vertexthe spatial angle is identical. The so-called Schafli symbols of these polyhedra speakfor themselves: {3, 3}, {3, 4}, {4, 3}, {5, 3}, and {3, 5} (see Figure 1). The reader willrecognize the latter from the cover of this journal.

Tetrahedron Octahedron Cube Dodecahedron Icosahedron

{3,3} {3,4} {4,3} {5,3} {3,5}

Figure 1. The five Platonic solids.

Four more regular solids, the so-called Kepler–Poinsot polyhedra, can be obtainedwhen star polygons or self-intersecting polygons are allowed. Two of them, the smallstellated dodecahedron and the great stellated dodecahedron, use pentagonal stars orpentagrams, represented by the symbol 5/2. The other two, the great icosahedron andthe great dodecahedron, combine triangles and, respectively, pentagons around a vertexin the same way as a pentagonal star rotates around its center and thus their number isalso symbolized by 5/2 (see, for instance, [4]; see Figure 2).

Petrie and Coxeter extended the concept of a regular polyhedron to infinite poly-hedra, where one building element is repeated to infinity, and again regular polygonsmeet at each vertex with identical angles. They obtained three regular configurations,two using hexagons, and one using squares (see Figure 3).

http://dx.doi.org/10.4169/amer.math.monthly.124.3.265MSC: Primary 00A99

March 2017] NOTES 265

Small stellateddodecahedron

Great stellateddodecahedron

Greaticosahedron

Greatdodecahedron

{5/2, 5} {5/2, 3} {3, 5/2} {5, 5/2}

Figure 2. The four Kepler–Poinsot solids.

Open truncatedtetrahedra

Opencubes

Open truncatedoctahedra

{6, 6} {4, 6} {6, 4}

Figure 3. The three Petrie–Coxeter infinite polyhedra.

The infinite polyhedron {4, 6} can be understood as juxtapositions of cubes of whichtwo opposing faces were removed. The {6, 6} can be understood as juxtapositions oftruncated tetrahedra of which the triangles were removed. Finally, his {6, 4} can beinterpreted based on the truncated octahedron, without its square faces (see Figure 4).

Truncated tetrahedron Truncated octahedron

4 triangles and 4 hexagons 6 squares and 8 hexagons

Figure 4. Polyhedra explaining the elements and construction of two of the Petrie–Coxeter infinite polyhedra.

2. A REGULAR INFINITE KEPLER–POINSOT POLYHEDRON. We considerthe cubohemioctahedron, a nonconvex uniform polyhedron composed of four hexag-onal faces passing through its center, intersecting one another so that only triangularportions of each are visible. It shares its vertex and edge arrangements with the betterknown cuboctahedron. From a cubohemioctahedron, we now remove the square facesto get an open cubohemioctahedron. Of course, since the hexagons all go through the

266 c© THE MATHEMATICAL ASSOCIATION OF AMERICA [Monthly 124

center, we cannot look through the polyhedron as was the case with the building ele-ments of the Petrie–Coxeter polyhedra (see Figure 5).

Cuboctahedron Cubohemioctahedron An opencubohemioctahedron

6 squares and 8 triangles 6 squares and 4 hexagons 4 hexagons

Figure 5. Building elements of an infinite Kepler–Poinsot polyhedron.

If we now place an infinity of the open cubohemioctahedra next to each other as inthe Petrie–Coxeter constructions, we get an infinite polyhedron. It is a regular one andof Kepler–Poinsot type as the faces cross each other. Eight hexagons meet in a vertex,and so its symbol is {6, 8} (see Figure 6). It seems new, as Branko Grunbaum, whoknows the vast literature (see [1], [2], [4], [5]), confirmed: “The compound is interest-ing, and I must say that I never encountered it in the literature” (private email com-munication, 2016-04-20). Perhaps it opens up to a wider family of unusual polyhedra(see [3]).

Figure 6. A regular infinite Kepler–Poinsot polyhedron.

A comparison with the {6, 4} infinite Petrie–Coxeter polyhedron above can help usto understand how this {6, 8} regular, infinite, Kepler–Poinsot polyhedron is built up.Four copies of the {6, 4} can form one {6, 8} polyhedron (see Figure 7). Thus, the {6, 8}infinite polyhedron can be seen as a “compound” infinite polyhedron (an aggregation),composed of four {6, 4} infinite polyhedra. Although this may help to explain the 3Dimage, it doesn’t seem to help in, for instance, getting the dual {8, 6} of the {6, 8}(see below).

Another way to get a better understanding for the shape of this regular, infinite,Kepler–Poinsot polyhedron is to introduce prismatic tunnels (open cubes) between theopen cubohemioctahedra. This turns the {6, 8} into an “Archimedean type” infiniteKepler–Poinsot polyhedron {4, 6, 4, 4, 6, 4}, that is, a polyhedron composed from twotypes of regular polygons, in this case, hexagons and squares (see Figure 8).

One might think about obtaining yet another new regular polyhedron by forming thedual of the {6, 8} as that should be an {8, 6}. “Dual” is used here in the sense that the

March 2017] NOTES 267

Figure 7. The same regular infinite Kepler–Poinsot polyhedron as in Figure 6, seen as a combination of 4Petrie–Coxeter {6, 4} polyhedra.

Figure 8. An infinite Kepler–Poinsot polyhedron “of Archimedean type”: the “regular {6,8}” with open cubetunnels, a {4, 6, 4, 4, 6, 4}.

vertices of one polyhedron correspond to the faces of the other. Thus, the tetrahedronis self-dual, while the octahedron and the cube, the icosahedron and the dodecahe-dron, the small stellated dodecahedron and the great dodecahedron, and, finally, thegreat stellated dodecahedron and the great icosahedron are duals of each other. In theinfinite case, Petrie–Coxeter’s {6, 6} was self-dual, while their {4, 6} and {6, 4} wereduals. However, the dual of the (open) cubohemioctahedron, the hexahemioctacron,has vertices at infinity. This is a rather unusual feature and it makes the constructionof the dual left as an open problem.

REFERENCES

1. J. H. Conway, H. Burgiel, Ch. Goodman-Strauss, The Symmetries of Things. A K Peters, Wellesley, MA,2008. 449 pp.

2. H. S. M. Coxeter, Regular Polytopes. Third ed. Dover, New York, 1973. 368 pp.3. D. Huylebrouck, Euler–Cayley’s formula for “unusual” polyhedra, Proceedings of the Finland

Bridges conference, August 2016, pp. 263–268, http://archive.bridgesmathart.org/2016/

bridges2016-263.html.4. A. K. van der Vegt, Order in Space. VSSD, Delft Academic Press, Delft, 2006. 94 pp.5. A. Wachman, M. Burt, M. Kleinman, Infinite Polyhedra. First ed. Technion Publication Faculty of Archi-

tecture and Town Planning, Haifa, 1974; Second ed. 2005. 102 pp.

Faculty of Architecture, KU Leuven, Hoogstraat 51 9000 Gent Belgiumhuylebrouck@gmail.com

268 c© THE MATHEMATICAL ASSOCIATION OF AMERICA [Monthly 124