Monthly Archives: March 2014

On Classification of Concave Polygons By Number of Concavities

Concave triangles do not exist, so concavity does not appear in the examination of polygons by ascending side length until the quadrilateral. A quadrilateral may only have one concavity, as shown in the red figure. Any polygon with exactly one … Continue reading

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A Tessellation Featuring Regular Heptagons

Regular heptagons, of course, can’t tile a plane by themselves. Of all tessellations of the plane which include regular heptagons, I think this is the one which minimizes between-heptagon gap-size (the parts of the plane outside any heptagon). However, I do not … Continue reading

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An Icosahedron Variant Featuring Kite-Stars

This variant of the icosahedron has five kites meeting at each of its twelve vertices, forming what I call the twelve “kite-stars” of this polyhedron. Also, two kites meet at the midpoint of each of the icosahedron’s thirty edges. The … Continue reading

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A Variant of Kepler’s Stella Octangula

Johannes Kepler named the compound of two tetrahedra the “stella octangula,” thus helping make it one of the best-known polyhedral compounds today. This variant uses triakis tetrahedra in place of the Platonic tetrahedra in that compound. The triakis tetrahedron is … Continue reading

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The Compound of Six Dodecahedra

Some polyhedral compounds are well-known, such as the compound of five cubes, while others are less famous. I had never heard of this compound before building one today (virtually, not as a physical model). However, a quick Google-search told me that … Continue reading

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Long, Narrow, Multicolored Hexagons As the Edges of a Rotating, Hollow Rhombic Dodecahedron

Software credit: see http://www.software3d.com/stella.php to try or buy Stella 4d, the software I used to create this image.

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A Close-Packing of Space, Using Three Different Polyhedra

This is like a tessellation, but in three dimensions, rather than two. The pattern can be repeated to fill all of space, using cubes (yellow), truncated octahedra (blue), and great rhombcuboctahedra, also known as truncated cuboctahedra (red). Software credit: see … Continue reading

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