Tag Archives: polyhedra

A Faceted Version of a Truncation of the Icosahedron

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A Faceted Version of a Truncation of the Icosahedron

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

Seven Variants of the Truncated Dodecahedron

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Several Variants of the Truncated Dodecahedron

The smaller pics may be enlarged with a click.

UnbvxcgnamedDual of Convedgex hullConvex hullUnnamed DualDual of Conhdgsavex hullanother compoundStefgasdllated Dual of Convex hull

I used Stella 4d to assemble these polyhedra. You may try this program for free at http://www.software3d.com/Stella.php.

A Rhombic Triacontahedron with Tessellated Faces

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A Rhombic Triacontahedron with Tessellated Faces

The faces of this polyhedron all have the same tessellation projected on them, but with five different coloring schemes. These five different tessellations appear in non-moving form in the last post on this blog, and I used Geometer’s Sketchpad and MS-Paint to make them. Stella 4d, another program, was then used to put the whole thing together, and it may be tried for free at http://www.software3d.com/Stella.php.

The Truncated Truncated Icosahedron

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The Truncated Truncated Icosahedron

The icosahedron has twenty triangular faces. Truncate it once, and the triangles become hexagons, with pentagons appearing under the pyramids removed in the truncation. This is the “soccer ball” shape familiar to millions.

If you take this figure and truncate it again, the twenty hexagons become twenty dodecagons, the twelve pentagons each become decagons, and sixty isosceles triangles appear under the pyramids removed by this second truncation.

I made this image using Stella 4d, a program you can find at www.software3d.com/Stella.php. Also, just for fun, here’s a version of it with the colors switched around, and with a slight bounce as it rotates in the other direction.

truncated trunctaed icosahedron

Three Stellations of the Icosidodecahedron

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Three Stellations of the Icosidodecahedron

The icosidodecahedron’s 24th stellation is above, and the 32nd, then the 36th, are below.

Icosidodeca 32nd StellationIcosidodeca 36th Stellation

I made these images using Stella 4d, a program you can find at www.software3d.com/Stella.php.

The Rhombic Dodecahedron’s Third Stellation

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The Rhombic Dodecahedron's Third Stellation

Above is this polyhedral stellation in three colors. If every face is given its own color, though, except for parallel faces, it looks like this (click to enlarge):

RD stellation 3rd colors b

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

Odd Polyhedral Cluster

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Odd Polyhedral Cluster

I stumbled upon this while using Stella 4d to modify existing polyhedra. You may find this program at http://www.software3d.com/Stella.php.

A Gallery of 27 Polyhedra with Cuboctahedral Symmetry

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Some of these polyhedra have “normal” cuboctahedral symmetry, while others have the chiral variant of that symmetry-type — in other words, the same type of symmetry found in the snub cube.

Some Polyhedra with Cuboctahedral Symmetry

CO1Convex hullCO2 Convex hullCconvex hullDual of Convex hullhexagons and squaresStellated Convex hullStellated Convex hulllAugmented Stellated Convex hullCoonvex hullConvex hull of prism-augmented snub cubedual of Coonvex hulllreaugmented dual of Convex hull of prism-augmented snub cubeStellated Polydual of Convex hull of prism-augmented snub cubeanother augmentation of dual of Convex hull of prism-augmented snub cubedual of Coonvex hullFaceted Convex hullStellated Poly2Stellated Poly3Faceted Convex hulllStellated Poly4Dual of Convex hulglDual of Convex hugFaceted Convex hulfgdsgthxlFaceted Convex hulfgdsgthgfsxlFaceted Convex hufdlfgdsgthgfsxl

I used Stella 4d:  Polyhedron Navigator to make these images, and you can find that program at http://www.software3d.com/Stella.php.

Some Recommeded Websites for Polyhedral Enthusiasts

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There are a lot of websites devoted to polyhedra. Here are some of the best.

This one is run by Jonathan Bowers:  http://www.polytope.net/hedrondude/home.htm.

Here’s the portal-page to George Hart’s pages, with links to a LOT of cool stuff he’s made: http://www.georgehart.com/.

I don’t know who runs this one:  http://polyhedra.org/poly/

This one is Robert Webb’s. He’s the person who wrote Stella 4d, the program I use most often on my own blog to create polyhedral images: http://www.software3d.com/Gallery.php. (Also, while we do share a first name, he is not the same person as me, as a few readers of my blog have thought in the past.)

Craig Kaplan has a page of links to other pages of his, right here: http://www.cgl.uwaterloo.ca/~csk/projects/. Of those, my favorites are the sections on John solid near-misses (http://www.cgl.uwaterloo.ca/~csk/projects/nearmisses/), as well as on symmetrohedra:  http://www.cgl.uwaterloo.ca/~csk/projects/symmetrohedra/

Jim McNeill’s polyhedra site is here:  http://www.orchidpalms.com/polyhedra/

Here’s a good one, but I don’t know its creator’s full name: http://colinspics.org/index.htm

Finally, one by Vladimir Bulatov:  http://www.bulatov.org/polyhedra/index.html

This is definitely not a complete list. If you know of other good polyhedron-oriented websites, please leave links to them in a comment on this post.

A Polyhedral Cage Which Includes the Dodecahedron, the Icosahedron, and the Rhombic Triacontahedron

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A Polyhedral Cage Which Includes the Dodecahedron, the Icosahedron, and the Rhombic Triacontahedron

The dodecahedron’s edges pass through the purple squares (edge midpoints) and red hexagons (vertices), and have blue decagons above their pentagonal face-centers. The blue decagons’ centers also mark the vertices of the triangular faces of the icosahedron, each of which has a purple square as a side-midpoint, and a red hexagon over its face-center. The rhombic triacontahedron’s faces have blue decagons at the vertex of each acute angle, and red hexagons at the obtuse angle vertices, with purple squares above the rhombic faces’ centers.

I used Stella 4d to make this image, and you can find that program at http://www.software3d.com/Stella.php.