A Chiral Tessellation

A Chiral Tessellation

In this chiral tessellation, the blue triangles and green hexagons are regular. The yellow hexagons are “Golden Hexagons,” which are what you get if you reflect a regular pentagon over one of its own diagonals, then unify the two reflections. The pink and purple quadrilaterals are two types of rhombi, and the red hexagons are a third type of equilateral hexagon. All of the edges of all polygons here have the same length.

There are three different types of points of three-fold rotational symmetry repeated here. Two of these types are centered in the middle of blue triangles, while the third is centered in the middle of some of the green hexagons — specifically, the ones surrounded only by alternating red and yellow hexagons.

When I try to generate the mirror-image of this tessellation, it overloads Geometer’s Sketchpad, and crashes the program. However, inverting the colors of the same reflection, in MS-Paint, to make a color-variant, is easy:

weird tessellation

Prismatic Versions of Part of a Tessellation Using Squares and 60-120-60-120 Rhombi

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Prismatic Versions of Part of a Tessellation Using Squares and 60-120-60-120 Rhombi

Above, the apparent size of the tessellation-prism does not change. In the next image, though, I set the controls of Stella 4d (the program I use to make these images, which you can try at http://www.software3d.com/Stella.php) to fit the image tightly, separately, in each still frame, causing a “breathing” effect. (Click on it if you wish to see it enlarged.)

Augmented Triangular Prism 2

Next, I added two red rhombi, one at the top and one at the bottom, stellated the result many times, and made one final .gif image, leaving the “breathing” effect on.

Augmented Triangular Prism 3 many stellations later

 

A Polyhedron Made of Kites and Rhombi

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A Polyhedron Made of Kites and Rhombi

In this polyhedron, the twelve rhombic faces are in the same position as the faces of a rhombic dodecahedron — but narrowed, so as to make room for the kites. As for the kites, there are 24 of them — in eight sets of three. These set-of-threes’ positions correspond to those of the faces of an octahedron.

Software credit: I made this with polyhedron-manipulation software called Stella 4d. If you wish to purchase it, or try the free trial download, here’s the site to visit:

http://www.software3d.com/Stella.php

Zonohedron Featuring 870 Rhombic Faces of 15 Types

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Zonohedron Featuring 870 Rhombic Faces of 15 Types

 

Software credit: visit http://www.software3d.com/stella.php for more information on the program used to make this rotating image. A free trial download is available.

Compound of Three Rhombic Prisms

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Compound of Three Rhombic Prisms

(See http://www.software3d.com/stella.php for free trial download of the software used to make this.)

24 Blue Rhombi

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24 Blue Rhombi

Created with Stella 4d (site to try it: http://www.software3d.com/stella.php).

Blue Rhombi

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Blue Rhombi

This was created with Stella 4d (site to try it: http://www.software3d.com/stella.php).

Also, here is a multi-colored version, made with the same software:

Augmented Rhombic Dodeca

A Polyhedron In Need of a Name

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A Polyhedron In Need of a Name

This is the dual of the polyhedron from the previous post, although without any faces rendered invisible, as was the case with that one. I find it strongly reminiscent of the rhombic triacontahedron, but it has twice as many rhombi (sixty), as well as twenty triangles. Unlike the rhombic triacontahedron, this figure is chiral.

I can’t think of an appropriate name for this, so, if anyone has one, please leave your suggestion in a comment.

Created with Stella 4d (site to try it: http://www.software3d.com/stella.php).

48 Green Rhombi

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48 Green Rhombi

Created with Stella 4d (site to try it: http://www.software3d.com/stella.php).

A Rotating Array of Rhombi

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A Rotating Array of Rhombi

Software credit: http://www.software3d.com/stella.php