Tag Archives: geometry

The Great Rhombcuboctahedron As a Building-Block

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The Great Rhombcuboctahedron As a Building-Block

This solid, also known as the great rhombicuboctahedron, and the truncated icosidodecahedron, can be used to build many other things. In addition to the elongated ring of eight above, for example, there’s this octagonal prism.

Augmented Trunc Cubocta2

Augmented Trunc Cubocta 2

Remember the elongated ring at the top of this post? This pic, directly above, is of a ring of four of those rings.

Augmented Trunc Cubocta3

And, yes, that’s a (non-great) rhombcuboctahedron made of great rhombcuboctahedra. Here it is again, with a different color-scheme.

Augmented Trunc Cubocta4

For the last of these constructions, eight more great rhombcuboctahedra are added to the figure in the two posts above, which is also returned to its original color-configuration. These eight new polyhedra have positions which correspond to the corners of a cube.

augmented rhombcuboctahedron made of great rhombcuboctahedra

Manipulating polyhedra in this manner is easy with Stella 4d, the program I used to do all of this. You may buy it, and/or try a free trial version first, at www.software3d.com/Stella.php.

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.

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

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.

15Unnamed

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

Co15nvex hull

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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Some Polyhedra Which Feature Heptagons

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.

sixty heptagons 20 hexagons sixty kites 140 faces total

I made these using Stella 4d, which is available at 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:

zonohedron RTC with xyz zones added has an interesting dual

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.