I suspect that this could be continued outward indefinitely, as a radial and aperiodic tessellation, using only the four polygons you see, repeatedly, here. However, I have no proof of this.
Pentagonal Mandala III
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Tag Archives: tessellation
Octagons Can Tile a Plane
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This tessellation is made entirely of octagons. Half of them are regular, while the other half are equilateral and tetraconcave.
A Tessellation Using Tetraconcave Octagons, Convex Hexagons, and Concave Heptagons
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A Rotating Rhombic Dodecahedron with Rotating Tessellations on Its Faces
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Several recent posts here have been of tessellations I have made using Geometer’s Sketchpad and MS-Paint. To create this rotating polyhedron, I selected one of these tessellations, and projected it onto each face of a rhombic dodecahedron, using another program called Stella 4d. Unlike in the last, similar post, though, I set these tessellation-images to keep their orientation, from the point of view of a stationary observer watching the entire polyhedron rotate, from a distance. Since the polyhedron itself is rotating, this creates a rotation-effect for the tessellation-image on each face.
You can try Stella 4d for yourself, right here, for free: http://www.software3d.com.stella.php.
Rotating Rhombic Triacontahedron with Tessellated Faces
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The last several posts here have been of tessellations I have made using Geometer’s Sketchpad and MS-Paint. To create this rotating polyhedron, I selected one of these tessellations (the one in the last post), and projected it onto each face of a rhombic triacontahedron, using another program called Stella 4d. You can try Stella 4d for yourself, right here, for free: http://www.software3d.com.stella.php.
Tessellation Using Regular Hexacontakaihexagons, Regular Dodecagons, and Two Different (and Unusual) Concave, Equilateral Polygons
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Hexacontakaihexagons have 36 sides, and dodecagons, of course, have twelve. When a regular hexacontakaihexagon is surrounded by twelve regular dodecagons, in the manner shown here, adjacent dodecagons almost, but not quite, meet at vertices. The gaps between these near-concurrent vertices are so small that they cannot be seen in this diagram — a zoom-in would be required, with thinner line segments used for the sides of the regular polygons.
As a result, the yellow and purple concave polygons aren’t what they appear, at first, to be. They look like triconcave hexagons, but this is an illusion. The yellow ones, in sets of two regions that aren’t quite separate, are actually tetraconcave, equilateral dodecagons with a very narrow “waist” separating the two large halves of each of them. As for the purple ones, they appear to occur in groups of four — but each set of four is actually one polygon, with three such narrow “waists” separating four regions of near-equal area. These purple polygons are, therefore, equilateral and hexaconcave icosikaitetragons — that is, what most people would call 24-gons.
Tessellation Using Regular Triacontakaihexagons, Equiangular Hexagons, Isosceles Triangles, and Isosceles Trapezoids
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The equiangular hexagons are very nearly regular, with only tiny deviations — probably not visible here — “from equilateralness.”
Tessellation Using Regular Triacontagons, Isosceles Triangles, Equiangular Triangles, and Isosceles Trapezoids
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Little blurbs about posts on this blog get auto-tweeted on my Twitter, @RobertLovesPi. There’s also an A.I. on Twitter, @Hexagonbot, who retweeted my last two tweets about blog-posts here, but will not be retweeting the tweet about this one.
Why is this? Simple: @Hexagonbot is programmed to retweet any tweet which contains the word “hexagon,” which was in the titles of the last two posts here (also tessellations). This tessellation has no hexagons, though, and so the @Hexagonbot will not find it worthy of attention.
I cannot explain why hexagons get their own bot on Twitter, but other polygons do not have such bots. It’s simply one of the mysteries of the Internet.
Tessellation Using Regular Tetrakaiicosagons, Isosceles Triangles, and Equiangular Hexagons
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Tessellation Using Regular Octadecagons and Triconcave, Equilateral Hexagons
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