The Cubohemioctahedron and other Polyhedra

Figure 1: a cubohemioctahedron (source)

Today I turned 25675 days old and one of the properties of this number is that it is figurate, of the centered cubohemioctahedral variety. Specifically it is a member of OEIS A274973: centered cubohemioctahedral numbers:$$ \text{a(}n \text{)} = 2 \times n^3+9 \times n^2+n+1$$In the case of 25675, the value of \(n\) is 22. However, I had no idea what a cubohemioctahedron was and so I set about finding out.

A cubohemioctahedron is shown in Figure 1 and, where F stands for faces, E for edges and V for vertices, it is characterised by \(F = 10, E = 24\) and \(V = 12\). Naturally this shape adheres to $$ \text{Euler's Formula: } V-E+F=2$$It is an impressive shape with the indented tetrahedra (in yellow), meeting at a central point which acts as the first term (1) of the centered cubohemioctahedral numbers. It is described thus in Wikipedia:

In geometry, the cubohemioctahedron is a nonconvex uniform polyhedron, indexed as U15. Its vertex figure is a crossed quadrilateral. It is given Wythoff symbol 4/3 4 | 3, although that is a double-covering of this figure. A nonconvex polyhedron has intersecting faces which do not represent new edges or faces. In the picture vertices are marked by golden spheres, and edges by silver cylinders. It is a hemipolyhedron with 4 hexagonal faces passing through the model centre. The hexagons intersect each other and so only triangle portions of each are visible.

There are several terms is this definition that require further explanation. Firstly though, Figure 2 shows the cubohemioctahedron in the centre with the relates shapes of cuboctahedron and octahemioctahedron on the left and right:

Figure 2: source

Compare the cubohemioctahedron with the first stellation of the cuboctahedron as shown in Figure 3.

Figure 3: first stellation of the cuboctahedron (source)

In the stellation shown in Figure 3, the square and triangular faces have been stellated. If only the triangular faces were stellated then the cubohemioctahedron would represent the opposite of that. In other words, the tetrahedra would be removed rather than added. I don't want to discuss the Wythoff symbol in this post because I'd rather focus on the beauty and practical construction of these objects rather than the more abstract mathematics. 

What follows are some interesting links and resources relating to polyhedra in this post. This is an area of mathematics that has long interested me but which, for whatever reasons, I've rather neglected.

• How To Fold It: The Mathematics of Linkages, Origami, and Polyhedra. An interesting book by Joseph O'Rourke who has a website related to this book and containing additional material. Cover photo in Figure 4.

Figure 4

• There is a nice PowerPoint presentation titled Polyhedra in Art by George W. Hart. that looks at historical examples of polyhedra in art.

• Welcome to Math Craft World! (Bonus: How to Make Your Own Paper Polyhedra)

• Amazing Origami, a book by Kunihiko Kasahara described as "a complete introduction to the mathematical theory of Origami based on the teachings of Freidrich Froebel (1782-1852) and a step-by-step guide to 33 colourful and fun paper folding projects". Cover photo in Figure 5.

Figure 5

• A Constellation of Original Polyhedra, a book by John Montroll described as "origami expert John Montroll provides simple directions and clearly detailed diagrams for creating amazing polyhedral. Step-by-step instructions show how to create 34 different models". Cover photo in Figure 6.

Figure 6

• Unit Polyhedron Origami, a book by Tomoko Fuse described as "With step-by-step diagrams, detailed instructions and over 70 photographs in vibrant full-color, internationally-renowned origamist and author Tomoko Fuse offers an innovative approach to origami based on assembling separate, multi-dimensional shapes into one structure". Figure 7 shows a page from the book in which a structure is made out of twenty cuboctohedra.

Figure 7

• Space-Filling Polyhedron on Wolfram MathWorld. An excerpt states that:

Figure 8

A space-filling polyhedron, sometimes called a plesiohedron, is a polyhedron which can be used to generate a tessellation of space. Although even Aristotle himself proclaimed in his work On the Heavens that the tetrahedron fills space, it in fact does not. Several space-filling polyhedra are in Figure 8. 

The cube is the only Platonic solid possessing this property. However, a combination of tetrahedra and octahedra do fill space. In addition, octahedra, truncated octahedron, and cubes, combined in the ratio 1:1:3, can also fill space. In 1914, Föppl discovered a space-filling compound of tetrahedra and truncated tetrahedra. 

There are only five space-filling convex polyhedra with regular faces: the triangular prism, hexagonal prism, cube, truncated octahedron. The rhombic dodecahedron and elongated dodecahedron, and trapezo-rhombic dodecahedron appearing in sphere packing are also space-fillers, as is any non-self-intersecting quadrilateral prism. The cube, hexagonal prism, rhombic dodecahedron, elongated dodecahedron, and truncated octahedron are all "primary" parallelohedra.

Well, I've made a start on this project but it's far from finished. Today I turned 25676 days old. This was kind of inevitable because yesterday I was 25675 days old but the point is that the number 25676 is associated with the stellated dodecahedron as shown in Figure 9.

Figure 9: stellated dodecahedron (source)

The dodecahedron has 20 vertices and 12 faces. In the stellated docedahedron, the faces become pentagonal pyramids and the associated 12 vertices make for a total of 32 points where edges meet. 25676 is a member of OEIS A318159: figurate numbers based on the small stellated dodecahedron: $$ \text{a(}n \text{)} = \frac{n \times (21 \times n^2 - 33 \times n + 14)}{2}$$The initial terms are:$$1, 32, 156, 436, 935, 1716, 2842, 4376, 6381, 8920, 12056, 15852, 20371, 25676, ...$$