RobertLovesPi.net

Polyhedra, tessellations, and more.

Tessellation Featuring Regular Octagons and Pentagons, As Well As Two Types of Non-Convex, Equilateral Polygons

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Tessellation Featuring Regular Octagons and Pentagons, As Well As Two Types of Non-Convex, Equilateral Polygons

Variations On Octagonal Prisms

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Variations On Octagonal Prisms

I made these with Stella 4d, a program you can find at http://www.software3d.com/Stella.php.

Octagonal Mandala II

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Octagonal Mandala

If the colors are inverted, but the background remains black, this is what it turns into:

A Polyhedron Featuring Sixty Octagons and Sixty Triangles

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A Polyhedron Featuring Sixty Octagons and Sixty Triangles

If someone had asked me if it were possible to form a symmetric polyhedra out of irregular triangles and octagons, using exactly sixty of one type each, I would have guessed that it were not possible. Why does it work here? Part of the reason is that each triangle borders three octagons, and each octagon borders three triangles — a necessary, but not sufficient, condition. This is a partial truncation of an isomorph of the pentagonal hexacontahedron, the dual of the snub dodecahedron. As such, no surprise — it’s chiral.

This was made while stumbling about in the wilderness of the infinite number of possible polyhedra using Stella 4d: Polyhedron Navigator. You can get it here: http://www.software3d.com/Stella.php.

A Pulsating Compound of Three Octangular Dipyramids

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Pulsating Compound of Three Octangular Dipyramids

Software credit: see http://www.software3d.com/Stella.php for the software used (Stella 4d) to make this image. A free trial download is available.

Tessellation of the Plane with Regular Hexagons, Squares, and Tetraconcave, Equilateral Octagons, #1

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Tessellation of the Plane with Regular Hexagons, Squares, and Tetraconcave, Equilateral Octagons

If you’d like to see the second version of this tessellation, made over a decade later, simply click right here.

Tessellation Using {8/3} Star Octagons and Squares

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The tessellation of the plane which uses regular convex octagons and squares is well-known. This related tessellation, however, is not. I didn’t know it existed until I stumbled across it . . . although I very much doubt I am the first person to do so.