A Symmetrohedron Featuring Regular Heptagons and Hexagons, Along with Irregular Quadrilaterals and Triangles

symmetrohedron

Symmetrohedra are symmetric polyhedra which have regular polygons for most (but not necessarily all) of their faces. I made this particular one using Stella 4d, which you can try for yourself at this website. Here’s the net for this polyhedron, also.

symmetrohedron net

This particular symmetrohedron features 12 faces which are regular heptagons, and 8 faces which are regular hexagons. The irregular faces are 12 isosceles triangles, 24 isosceles trapezoids, and 6 rectangles, for a total of 62 faces. It has pyritohedral symmetry. The most unusual thing about this polyhedron are its 12 heptagonal faces.

A Faceted Truncated Octahedron with Twenty Faces

Faceted Trunc Octa

The twenty faces of this polyhedron are six small blue squares, six interpenetrating, larger red squares, and eight irregular, interpenetrating yellow hexagons. I made it using Stella 4d: Polyhedron Navigator, which you can try for free at http://www.software3d.com/Stella.php. The squares are easy to find, but it can be a challenge to see the yellow hexagons. In the .gif below, all of the yellow faces but one are hidden, which should make it easier to see where the hexagons are positioned, relative to the squares.

 

Complexes of 101 Great Stellated Dodecahedra Each, Shown with Three Different Coloring Schemes

GSD complex 101 parts

GSD complex 101 parts parallel faces same color

GSD complex 101 parts rb

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

 

Complexes of 61 Small Stellated Dodecahedra Each, Shown with Three Different Coloring Schemes

spectral small stellated dodecahedra

spectral small stellated dodecahedra 61

spectral small stellated dodecahedra rb

I made these .gifs using Stella 4d: Polyhedron Navigator, which you can try for free at this website.

A Compound of the Rhombic Triacontahedron and a Truncation of the Icosahedron

Stellated Dual Morph 50.0%

In the compound above, the yellow hexagons are not quite regular, which is why I’m calling the yellow-and-orange polyhedron a truncation of the icosahedron, rather than simply the truncated icosahedron. I stumbled upon it while playing with Stella 4d, which you may try for free at http://www.software3d.com/Stella.php.

Cubes Made of Lux Blox

What is a cube? That’s a simple question, and I thought it had a simple answer . . . until I took on the project of building cubes with Lux Blox. Lux can be bought at this website, but one thing you won’t find there, or in shipments of Lux, are directions. This was a little frustrating at first, but I understand it now: the makers of Lux don’t want directions getting in the way of customers’ creativity.

A cube has six square faces. This is the six-piece Lux model based on that statement.

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This first cube model is interesting, but it is also severely limited. Lux Blox connect at their edges, and all edges in this model are already used, joining one face to another. The model has no openings where more can be attached, and added to it.

Next, I made a cube out of Lux Blox which is open, in the sense that more Lux Blox can be attached to it. It also has an edge length of two.

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Besides the openness of this model to new attachments, it also has another characteristic the smaller cube did not have: it can be stretched. If you take two opposite corners of this model and gently pull them away from each other, here’s what you get:

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Stretching a cube in this manner creates a six-faced rhombic polyhedron known as a parallelopiped.

The third cube model I’ve built of Lux Blox uses Lux Trigons in addition to the normal square-based Lux Blox.

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In this model, the black pieces in the center are the Lux Trigons — twelve of them, occupying the positions of twelve of the twenty faces of an icosahedron. The other eight faces are where the orange triangles (or triangular prisms, if you prefer) are attached. The orange triangles mark the eight corners of a cube. This model has pyritohedral symmetry — the symmetry of a volleyball — as I hope this last picture, a close-up of this third type of cube, helps to illustrate.

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