Stella Octangula

The term "stella octangula" is another name for a "stellated octahedron" such as is shown in Figure 1.

Figure 1: stellated octahedron
Source

I've only made one previous post about stellated polyhedra and that was The Cubohemioctahedron and other Polyhedra back on the 21st July 2019. Here's what Wikipedia had to say about the stellated octahedron:

The stellated octahedron is the only stellation of the octahedron. It is also called the stella octangula (Latin for "eight-pointed star"), a name given to it by Johannes Kepler in 1609, though it was known to earlier geometers. It was depicted in Pacioli's De Divina Proportione, 1509.

It is the simplest of five regular polyhedral compounds, and the only regular compound of two tetrahedra. It is also the least dense of the regular polyhedral compounds, having a density of 2.

It can be seen as a 3D extension of the hexagram: the hexagram is a two-dimensional shape formed from two overlapping equilateral triangles, centrally symmetric to each other, and in the same way the stellated octahedron can be formed from two centrally symmetric overlapping tetrahedra. This can be generalized to any desired amount of higher dimensions; the four-dimensional equivalent construction is the compound of two 5-cells. It can also be seen as one of the stages in the construction of a 3D Koch snowflake, a fractal shape formed by repeated attachment of smaller tetrahedra to each triangular face of a larger figure. The first stage of the construction of the Koch Snowflake is a single central tetrahedron, and the second stage, formed by adding four smaller tetrahedra to the faces of the central tetrahedron, is the stellated octahedron.

Associated with this shape are the stella octangula numbers. These are figurate numbers  of the form \(n(2n^2 − 1) \) and they form OEIS A007588:

0, 1, 14, 51, 124, 245, 426, 679, 1016, 1449, 1990, 2651, 3444, 4381, 5474, 6735, 8176, 9809, 11646, 13699, 15980, 18501, 21274, 24311, 27624, 31225, 35126, 39339, 43876, 48749, 53970, 59551, 65504, 71841, 78574, 85715, 93276, 101269, 109706, 118599, 127960

To quote from Wikipedia again:

There are only two positive square stella octangula numbers. The first is \(1\) and the other is$$9653449 = 3107^2 = (13 × 239)^2$$corresponding to \(n = 1\) and \(n = 169\) respectively. The elliptic curve describing the square stella octangula numbers is:$$m^2=n(2n^2-1)$$Using Geogebra, this curve is shown in Figure 2.

Figure 2 

The stella octangula numbers arise in a parametric family of instances to the crossed ladders problem in which the lengths and heights of the ladders and the height of their crossing point are all integers. In these instances, the ratio between the heights of the two ladders is a stella octangula number.

The shape and these numbers caught my attention because my diurnal age today is 27624 and this number is a member of the OEIS sequence for the case of \(n=24\). These numbers are quite sparse. The previous number was 24311 and the next will be 31225.

Here is a video which shows how to create a stellated octahedron using origami. It was uploaded on the 13th March 2010 but the technique of course is timeless. Figure 3 shows another representation of the shape.

Figure 3: source

Figure 4 shows how the shape can be inscribed in a cube and also illustrates its connection with the hexagon.

Figure 4: source

The site from which Figures 3 and 4 were taken also contains illustrations of nets can be used to construct the models. This approach is easier than the origami method. Instructions for the construction of wire frame models can also be found there.

Figure 5 shows a screenshot from a site that provides a stellated octahedron calculator. Entering a side length of 53 generates a volume of 26318 to the nearest decimal place. However, the \(a\) shown in Figure 5 is a side length whereas the common formula uses edge length \(b\) where \(b=a/2\). The volume of the shape using edge length of \( b \) is \(b^3 \sqrt {2} \).

Figure 5: source

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:$$1, 32, 156, 436, 935, 1716, 2842, 4376, 6381, 8920, 12056, 15852, 20371, 25676, ...$$