The Double Icosahedron, and Some of Its “Relatives”

The double icosahedron is simply an icosahedron, augmented on a single face by a second icosahedron. I thought it might be interesting to explore some transformations of this solid, using Stella 4d: Polyhedron Navigator (available here), and I was not disappointed. I used Stella to produce all the images in this post.

Augmented Icosa

It is well-known that the dual of the icosahedron is another Platonic solid, the dodecahedron. Naturally, I wanted to see the double icosahedron’s dual, and here it is — a simple operation for Stella. This dual resembles a dodecahedron in its center, but gets more unusual-looking as one moves further out from its core. 

Augmented Icosa dual

I next examined stellations of the double icosahedron, but did not find any which seemed attractive enough to post, until I saw its sixteenth stellation, which features six kites as faces, in sets of three, on opposite sides of the solid.

Stellated Augmented Icosa 16th.gif

What proved most fruitful was my examination of various zonohedra based on the double icosahedron. Here’s what I found for the zonohedron based on the faces of the double icosahedron: a large number of rhombic faces, with Northern and Southern “hemispheres” separated by an “equator” of hexagonal zonogons.

Zonohedrified Augmented Icosa faces.gif

The next image is the zonohedron based on the edges of the double icosahedron.

Zonohedrified Augmented Icosa edges.gif

The next zonohedron shown is based on the vertices of the double icosahedron.

Zonohedrified Augmented Icosa vertices

All of these zonohedra have 6-fold dihedral symmetry, while the double icosahedron itself has 3-fold dihedral symmetry. The next image shows the zonohedron based on both the vertices and edges of the double icosahedron.

Zonohedrified Augmented Icosa v and e.gif

Zonohedrification based on vertices and faces produces the next zonohedron shown here.

Zonohedrified Augmented Icosa v and f.gif

The next logical step was to create a zonohedron based on the double icosahedron’s edges and faces.

Zonohedrified Augmented Icosa e and f.gif

Finally, here is the zonohedron based on all three characteristics: the vertices, edges, and faces of the double icosahedron.

Zonohedrified Augmented Icosa VEF.gif

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Two Excellent Mathematical Websites

I usually only post my own work here, but today I’m giving a shout-out to the websites of a German friend of mine named Tadeusz E. Doroziński. He made this snub polyhedron with 362 faces, which I’m posting here as a sample of his work. All of its edges are of equal length. Like me, he uses Stella 4d: Polyhedron Navigator frequently (available here), and he used that program to create this polyhedron. 

35297218_10212322040197904_264340303467511808_o

His two geometry-focused and polyhedron-filled websites, https://geometryka.wordpress.com/ and http://www.3doro.de/, contain much more, including some mathematics which flies right over my head, as the saying goes. If you like the image above, or you are a fan of my own blog, I strongly recommend following the links above to check out his work. Every time I visit either of his websites, I always find something amazing.

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The Compound of the Octahedron and the Small Stellated Dodecahedron

compound of the small stellated dodecahedron and the octahedron

I made this rotating virtual model using Stella 4d: Polyhedron Navigator, which you can try for yourself at http://www.software3d.com/Stella.php. This solid is different from most two-part polyhedral compounds because an unusually high fraction of one polyhedron, the yellow octahedron, is hidden inside the compound’s other component.

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A Zoo of Zonohedra

Zonohedra are a subset of polyhedra with all faces in pairs of parallel and congruent zonogons. Zonogons are polygons with sides which occur only as parallel and congruent pairs of line segments. As a consequence of this, the faces of zonohedra must have even numbers of sides.

Considering all the restrictions on zonohedra, it may be surprising that there is so much variety among them. Every polyhedron shown in this post is a zonohedron. The colors are chosen so that all four-sided zonogons have one color, all six-sided zonogons have a second color, and so on.

Octagon-dominated zonohedron.gif

Zonohedrified Cobvjnvex hull.gif

Zonohedrified Conjhvjvvex hull.gif

Zonohedrified Conjhvvex hull.gif

Zonohedrified Connbvj,njkvex hull.gif

Zonohedrified Convb bvvex hull.gif

Zonohedrified Convehckhcx hull.gif

Zonohedrified Convex hull  186 faces.gif

Zonohedrified Convex hull 132 faces

Zonohedrified Convex hull 138 faces.gif

Zonohedrified Convex hull 306 faces colored by number of edges per face.gif

Zonohedrified Convex hull features octadecagons.gif

Zonohedrified Convex hull.gif

Zonohedrified Convex hullygduyd.gif

Zonohedrified Cube 2.gif

Zonohedrified Ochjgta.gif

Zonohedrified Octa 2

Zonohedrified Octa 3.gif

Zonohedrified Octa z.gif

Zonohedrified Octa.gif

Zonohedrified Pmhgcholy.gif

Zonohedrified Polly.gif

Zonohedrified Poly.gif

Zonohedrified Snub Cube.gif

Zonohedrified tet.gif

Zonohedrified Trunc Dodeca featuring octadecagons.gif

Zonohedrified Trunc Dodeca.gif

Zonohedrified Trunc Tetra vef.gif

Zonohedrified Trunc Tetra.gif

Zonohedrified Trunc Tetrahedron.gif

I made all of these using Stella 4d: Polyhedron Navigator. This program may be tried for free at this website.

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Seven Different Facetings of the Truncated Icosahedron

Trunc Icosa.gif

The polyhedron above is the truncated icosahedron, widely known as the pattern for most soccer balls. In the image below, the faces and edges have been hidden, leaving only the vertices.

Trunc Icosa vertices only

To make a faceted version of this polyhedron, these vertices must be connected in novel ways, creating new edges and faces. There are many faceted versions of this polyhedron, of which seven are shown below.

Faceted Trunc Icosa

Faceted Trunc Icosa 8

Faceted Trunc Icosa 7

Faceted Trunc Icosa 5.gif

Faceted Trunc Icosa 4.gif

Faceted Trunc Icosa 3

Faceted Trunc Icosa 2.gif

I used Stella 4d to make these polyhedral images, and you’re invited to try the program for yourself at http://www.software3d.com/Stella.php.

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Four Symmetrohedra

Symmetrohedra are polyhedra with some form of polyhedral symmetry, all faces convex, and many (but not all) faces regular. Here are four I have found using Stella 4d, a polyhedron-manipulation program you can try for yourself at http://www.software3d.com/Stella.php.

octagons and elongates dodecagons.gif

Octagon-dominated zonohedron

regular decagons and triangles, plus elongated octagons.gif
dual of GRID and dual's compound's convex hull 182 faces incl 12 deca 20 hexa 30 squares and 120 triangles

The second of these symmetrohedra is also a zonohedron, and is colored the way I usually color zonohedra, coloring faces simply by number of sides per face. That is why some of the red octagons in that solid are regular, while others are elongated. The other three symmetrohedra are colored by face type, with the modification that the fourth one’s scalene triangles are all given the same color.

These symmetrohedra were all generated using Stella 4d, a program you may try for yourself at http://www.software3d.com/Stella.php.

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Some Ten-Part Polyhedral Compounds

While examining different facetings of the dodecahedron, I stumbled across one which is also a compound of ten elongated octahedra.

Faceted Dodeca and compound of ten elongated octahedra.gif

Here’s what this compound looks like with the edges and vertices hidden:

Faceted Dodeca and compound of ten elongated octahedra without edges and vertices.gif

Next, I’ll put the edges and vertices back, but hide nine of the ten components of the compound. This makes it easier to see the single elongated octahedron which is still shown.

Faceted Dodeca one part of ten with edges and vertices.gif

Here’s what this elongated octahedron looks like with all those vertices and edges hidden from view.

Faceted Dodeca one part of ten.gif

I made all these polyhedral transformations using Stella 4d, a program you can try for yourself at this website. Stella includes a “measurement mode,” and, using that, I was able to determine that the short edge to long edge ratio in these elongated octahedra is 1:sqrt(2).

The next thing I wanted to try was to make the octahedra regular. Stella has a function for that, too, and here’s the result: a compound of ten regular octahedra.

compound of ten regular octahedra.gif

My last step in this polyhedral exploration was to form the dual of this solid. Since the octahedron’s dual is the cube, this dual is a compound of ten cubes.

compound of ten cubes.gif

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