The Tetrahedral Icosahedron

Posted on 27 June 2021 by RobertLovesPi

This is an icosahedron because it has twenty faces, and it is of tetrahedral symmetry. There are four regular hexagons and four equilateral triangles among these faces, along with twelve isosceles triangles with a leg:base length ratio of 1.73205:1. I found it while investigating variations to the base/dual compound of the truncated tetrahedron using Stella 4d, which you can try for free at this website. This polyhedron is the dual of the convex hull of the base/dual compound of the truncated tetrahedron and the triakis tetrahedron.

Filling Space With Rhombic Dodecahedra

Posted on 27 June 2021 by RobertLovesPi

This is the cuboctahedron, one of the Archimedean solids. Its dual, shown below, is the rhombic dodecahedron.

The rhombic dodecahedron has a property which sets it apart from most other polyhedra: it can fill space with copies of itself, leaving no gaps. The next stage of such growth is shown below.

The next step is to add more rhombic dodecahedra on each face.

One more set added, and the edge-length of the cluster reaches four rhombic dodecahedra.

This could be continued without limit. As is does, the overall shape of the cluster becomes more and more shaped like a cuboctahedron, which is back where we started. You can easily see this in the convex hull of the last cluster.

All of these rotating images were created using Stella 4d: Polyhedron Navigator, which you can try for free at http://www.software3d.com/Stella.php.

A Concave Polyhedron Made of Kites and Stars

Posted on 16 June 2021 by RobertLovesPi

I made this using Stella 4d, which you can try for free at http://www.software3d.com/Stella.php. It contains, as faces, sixty kites and twelve five-pointed stars.

A Spiral of Fifteen Regular Prisms Which Could Continue Indefinitely

Posted on 12 June 2021 by RobertLovesPi

I made this using Stella 4d, which you can try for free at http://www.software3d.com/Stella.php.

The Rhombic Enneacontahedron, With All Faces Augmented With Pyramids

Posted on 4 June 2021 by RobertLovesPi

I made this using Stella 4d: Polyhedron Navigator, which you can try for free at http://www.software3d.com/Stella.php.

A 242-Faced Polyhedron, Along With Its Dual

Posted on 2 June 2021 by RobertLovesPi

This polyhedron’s faces include twelve decagons, thirty rhombi, and 200 triangles. Its dual is shown below.

This dual has 180 faces: 60 hexagons, and 60 each of two types of kites.

A Symmetrohedron Featuring Squares, Regular Hexagons, and “Bowtie” Pairs of Trapezoids

Posted on 2 June 2021 by RobertLovesPi

This symmetrohedron has eight hexagonal faces, six square faces, and twelve pairs of trapezoids, for a total of 38 faces. I made it by using Stella 4d to modify a truncated octahedron. You may try this program for free at www.software3d.com/Stella.php.

A Chiral Polyhedron with 182 Faces

Posted on 2 June 2021 by RobertLovesPi

I made this using Stella 4d, which you can try for free right here.

An Interesting Polyhedron, with 122 Faces

Posted on 31 May 2021 by RobertLovesPi

This 122-faced solid is the dual of the convex hull of the compound of the rhombicosidodecahedron and its dual, the strombic hexecontahedron. Its regular faces are 20 triangles, 12 pentagons, and 30 squares. It also has 60 faces which are isosceles trapezoids.

I made this using Stella 4d, which you can try for free at http://www.software3d.com/Stella.php.

A Polyhedral Journey, Starting with the Compound of Five Dodecahedra

Posted on 30 May 2021 by RobertLovesPi

This is the compound of five dodecahedra, a shape which is included in the built-in polyhedral library of Stella 4d, a program you can try for yourself, free, right here.

I wanted to see what I could make, starting from this compound. My first modification to it was to create its convex hull, which is shown below.

The next move was to use Stella‘s “Try to Make Faces Regular” function, which produced this:

Next, I augmented this figure’s thirty yellow rhombi with prisms.

I then created the convex hull of this augmented polyhedron.

Next, I used the “Try to Make Faces Regular” function again, producing a solid that looks, to me, like a hybrid of the rhombicosidodecahedron and the rhombic triacontahedron.

This polyhedron has yellow faces that are almost squares. Careful inspection reveals that they are actually isosceles trapezoids. The next thing I did was to augment each of these trapezoids with a tall prism.