The Truncated Cube, with Two Variations Featuring Regular Dodecagons

Posted on 10 December 2016 by RobertLovesPi

This is the truncated cube, one of the thirteen Archimedean solids.

trunc-cube

If the truncation-planes are shifted, and increased in number, in just the right way, this variation is produced. Its purple faces are regular dodecagons, and the orange faces are kites — two dozen, in eight sets of three.

dodecagons-and-kites

Applying yet another truncation, of a specific type, produces the next polyhedron. Here, the regular dodecagons are blue, and the red triangles are equilateral. The yellow triangles are isosceles, with a vertex angle of ~41.4 degrees.

vetex-angle-41p4-degrees

All three of these images were produced using Stella 4d, available at this website.

A Chiral Polyhedron with Tetrahedral Symmetry

Posted on 9 December 2016 by RobertLovesPi

The yellow faces of this polyhedron are parallelograms, while the red ones are trapezoids. To demonstrate its chirality, here is the compound of it, and its own mirror-image.

Both of these “virtual polyhedra” were made using Stella 4d: Polyhedron Navigator, a program available at this website. It has a free trial download available.

A Tetrahedrally-Symmetric Polyhedron Featuring Heptagons

Posted on 3 December 2016 by RobertLovesPi

46-faces-12-of-them-heptagons

Created using Stella 4d: Polyhedron Navigator; see this website to try it for yourself!

Bowtie Cubes in a Polyhedral Honeycomb

Posted on 24 November 2016 by RobertLovesPi

unnamed

This polyhedron has been described here as a “bowtie cube.” It is possible to augment its six dodecagonal faces with additional bowtie cubes. Also, the bowtie cube’s hexagonal faces may be augmented by truncated octahedra.

bizarre

These two polyhedra “tessellate” space, together which square pyramidal bifrustrums, meeting in pairs, which fill the blue-and-green “holes” seen above. This last image shows more of the “honeycomb” produced after yet more of these same polyhedra have been added.

bizarrer

This pattern may be expanded into space without limit. I discovered it while playing with Stella 4d, software you may try for free at this website.

Tetrahedrally-Symmetric Creatures with Polyhedral Legs

Posted on 23 November 2016 by RobertLovesPi

Each of these has a tetrahedron hidden from view in the center.

augmented-tetra

augmented-tetra-2

augmented-tetra-3

augmented-tetra-4

These were made using Stella 4d, which you may try for yourself here.

A Crinkled Octahedron

Posted on 22 November 2016 by RobertLovesPi

unnamedmnmn-octahedral

“Crinkled” is merely descriptive; I offer no mathematical definition of the term. This was a polyhedron I stumbled along while doing random-walk polyhedral manipulations with Stella 4d, available at this website.

The Compound of the Rhombic Dodecahedron and Its Own Third Stellation

Image

compound-of-rd-plus-stellation-of-rd

I used Stella 4d to make this polyhedral compound, and this program may be tried for free at this website.

Shield Mandala

Image

shield-mandala

The Dodecahedron, Made from Smaller Polyhedra

Posted on 14 November 2016 by RobertLovesPi

augmented-rhombic-triaconta-dodeca

The polyhedra at the vertices are rhombic triacontahedra, and the yellow edges are elongated rhombic prisms. This was made using Stella 4d, software you may try for free at this website.

An Icosahedron, Constructed from Smaller Polyhedra

Posted on 13 November 2016 by RobertLovesPi

Augmented Rhombic Triaconta.gif

The polyhedra at the vertices are rhombic triacontahedra, and the yellow edges are elongated rhombic prisms. This was made using Stella 4d, software you may try for free at this website.