28 | May | 2014 | RobertLovesPi.net

The Royal Octahedron, with Half-a-Dozen Crowns

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The Royal Octahedron, with Half-a-Dozen Crowns

Software credit: see http://www.software3d.com/Stella.php to check out Stella 4d, the program I used to make this. A free trial download is available.

Seventeen Truncated Tetrahedra Rotating in Symmetrical Formation

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Seventeen Truncated Tetrahedra in Symmetrical Formation

Software credit: you can try the free trial download of Stella 4d at http://www.software3d.com/Stella.php.

Eight Hexagons, Rotating with Six Tetrastars

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Eight Hexagons, Rotating with Six Tetrastars

Software credit: see http://www.software3d.com/Stella.php for more information on Stella 4d, the program I used to make this. A free trial download is available.

Rippling Tessellation Using Squares, Regular Octagons, and Octaconcave, Equilateral Hexadecagons II

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← Tessellation Using Squares, Regular Octagons, and Octaconcave, Equilateral Hexadecagons Rippling Tessellation Using Squares, Regular Octagons, and Octaconcave, Equilateral Hexadecagons II

Rippling Tessellation Using Squares, Regular Octagons, and Octaconcave, Equilateral Hexadecagons

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Rippling Tessellation Using Squares, Regular Octagons, and Octaconcave, Equilateral Hexadecagons

Tessellation Using Squares, Regular Octagons, and Octaconcave, Equilateral Hexadecagons

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Tessellation Using Squares, Regular Octagons, and Octaconcave, Equilateral Hexadecagons

Five Polyhedron-Inspired, Rotating Images of Many Colors

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Four Polyhedron-Inspired, Rotating Images of Many Colors

The three images below appear smaller than the one above, at first, but can be enlarged with a single click.

These were all created using Stella 4d: Polyhedron Navigator. You can try or buy this software at http://www.software3d.com/Stella.php.

In case you’re wondering, I’m simply calling these polyhedron-inspired images, rather than actual polyhedra, because of the holes in most of them. With the software I use to make these, creating these empty spaces is simply a matter of selecting faces, and then hiding them from view.

For this last one, just for fun, I made the vertices visible — even those where hidden faces touch. The idea was to create a snow-like effect.

Three Polyhedra Which Feature Regular Pentadecagons

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Some Polyhedra Which Feature Regular Pentadecagons

I haven’t encountered many polyhedra which feature regular pentadecagons, and geometry textbooks generally don’t even use that word, calling them “15-gons,” instead. The pentadecagon happens to be one of my favorite polygons, though, and has been ever since I independently figured out, a few years back, how to duplicate the ancient Greeks’ accomplishment of combining the Euclidean constructions for the regular pentagon and equilateral triangle, in order to construct a regular pentadecagon.

The one above also includes regular decagons as faces — but I had to let the pentadecagons intersect each other to get that to work.

This third polyhedron resembles a truncated icosahedron, but with pentadecagons replacing that solid’s twenty hexagons. The pentagons are still in place, with two types of trapezoid and some very thin rectangles needed to fill the gaps.

These images were all created using Stella 4d, software you may try or buy at http://www.software3d.com/Stella.php.