May | 2014 | RobertLovesPi.net

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.

Two Polyhedra Which Feature Heptagons

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It’s unusual to encounter heptagons in any survey of polyhedra . . . so I made a couple. I didn’t see any reason to limit myself to regular heptagons, though.

I made these using Stella 4d, which is available at http://www.software3d.com/Stella.php.

A Ball of Shields

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A Ball of Shields

Software credit: I used Stella 4d to make this, and you can find this program at http://www.software3d.com/Stella.php. A free trial download is available.

Octahedral Lattice of Truncated Octahedra, Meeting At Their Square Faces

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Octahedral Lattice of Truncated Octahedra, Meeting At Their Square Faces

The second image here resulted from stellating the first one many times. It can be enlarged with a click.

The software used to create these rotating images, Stella 4d, may be tried for free at http://www.software3d.com/Stella.php.

Twenty Rainbow Asterisks

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Software credit: see http://www.software3d.com/Stella.php.

A Zonohedrified Rhombic Triacontahedron, and Its Dual

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A Zonohedrified Rhombic Triacontahedron and Its Dual

To create this, I started with a rhombic triacontahedron, and then used software called Stella 4d to zonohedrify it, adding zones to the existing faces along x, y, and z axes meeting at the polyhedron’s center.

I find its dual even more interesting:

The software used to create these rotating images may be tried for free at http://www.software3d.com/Stella.php.

An Elongated Stella Octangula

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An Elongated Stella Octangula

The Stella Octangula is another name for the compound of two tetrahedra. I made this elongated version, which uses narrow isosceles triangles in place of the usual equilateral triangles, using Stella 4d — polyhedron-manipulation software you can find at http://www.software3d.com/Stella.php.

Four Hollow Polyhedron-Variants Related to the Rhombic Triacontahedron and the Icosahedron

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Four Hollow Polyhedron-Variants Related to the Rhombic Triacontahedron and the Icosahedron

To make any of these smaller images as large as the first one, simply click on the smaller image of your choice.

These are all members of the same stellation-sequence, just like the two octahedron-variants in the last post.

I made these in just a few minutes with software, written by a friend of mine, called Stella 4d: Polyhedron Navigator. You can check it out for yourself at http://www.software3d.com/Stella.php. A free trial download is available.