The largest polygons in this tessellation are the elongated octagons. There are also equilateral triangles, isosceles triangles, kites of two sizes, and tiny regular hexagons.
A Tessellation in Black and White
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A Tessellation Featuring Regular Dodecagons, and Both Equilateral and Isosceles Triangles
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A Euclidean Construction of the Golden Rectangle, Golden Triangle, Golden Gnomon, and Regular Pentagon
In this construction, the points used are shown in alphabetical order. The colored golden rectangle is rectangle BKOF, and the golden triangle (shown in orange) is triangle FPB. There are two golden gnomons, shown in blue: triangle QFP, and triangle PBR. The regular pentagon is BRPQF. Every circle, line, ray, and segment used, even just to bisect segments, is shown — nothing has been hidden. This construction works because the long-edge-to-short-edge ratio of the golden rectangle is the golden ratio — and so is the diagonal-to-side ratio for the regular pentagon.
I used Geometer’s Sketchpad to make this, but everything shown can be done with the traditional Euclidean construction tools: a compass, and an unmarked straightedge.
Tessellation Featuring Regular Hexagons and Regular Pentagons, as Well as Two Different Types of Kites
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Two Tetstells of the Dodecahedron
To “tetstell” a polyhedron (and yes, I just made that word up) is to drop its symmetry from either octahedral or icosahedral down to tetrahedral, and then stellate it. Here’s an example: the second tetstell of the dodecahedron.
Here’s another one: the eighth tetstell of the dodecahedron.
I made these using Stella 4d, which you can try for free at http://www.software3d.com/Stella.php.
Tessellation of Regular Hexagons and Chevrons
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A 200-Faced Polyhedron Featuring Twenty Regular Enneagons
In addition to the regular enneagons, this polyhedron’s faces include sixty irregular hexagons, as well as sixty each of two different types of irregular pentagon. I made it using Stella 4d, which you can try for free at this website.
Three Compounds From the Stellation-Series of the Tetrakis Hexahedron
If you stellate the tetrakis hexahedron once, you get a compound of two pyritohedral dodecahedra.
The 16th stellation is a compound of three elongated octahedra.
Later, when you get to the 65th stellation, the result is a compound of four triangular dipyramids.
I found these compounds, and created these rotating images, using Stella 4d: Polyhedron Navigator. If you wish, you can try this program for free at http://www.software3d.com/Stella.php.
Tessellation Featuring Regular Enneagons, Convex Pentagons, Rhombi, and Darts
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Having Fun With Zome
This isn’t exactly a polyhedron, or even a polyhedral compound, although it does contain several polyhedra in it. There’s a red rhombic triacontahedron in the center, a blue icosidodecahedron just outside that, and a blue dodecahedron closer to the outside. There are also twelve blue-and-yellow pentagonal pyramids, as well as twenty smaller blue-and-red triangular pyramids. That may not be a complete list, although I did try to include them all. I didn’t build it with the goal of making anything in particular — I was just having fun with Zome. In other words, I was playing.
Zome is available at http://www.zometool.com, if you’d like to try playing with it, or giving it as a gift to someone who would appreciate it. The small parts could cause a choking hazard for babies or toddlers, but they will delight and amaze school-age kids, as well as older people (like me) who still enjoy play for the sake of playing, and doing math for the sake of doing math.
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