Eight led to this.

If you think about eight long enough, you will understand.

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# Day: April 13, 2014

## I Started with Eight.

## News Flash: Big Snake on the Loose in Texas

## Circumslices

## A Rotating Rhombic Dodecahedron with Rotating Tessellations On Its Faces

## Rotating Rhombic Triacontahedron with Tessellated Faces

## Tessellation Using Regular Hexacontakaihexagons, Regular Dodecagons, and Two Different (and Unusual) Concave, Equilateral Polygons

## Tessellation Using Regular Triacontakaihexagons, Equiangular Hexagons, Isosceles Triangles, and Isosceles Trapezoids

Eight led to this.

If you think about eight long enough, you will understand.

This just in from Longview, Texas — a five-meter (~16 foot) Burmese Python is on the loose.

If you see it, do not approach.

Retreat and dial 911.

No kidding.

Also: learn the metric system.

Regions between close-packed circles of equal radius resemble triangles, but with 60 degree arcs replacing the sides. As these regions are the only things left of a plane after all such circles are sliced out, and they each are outside all the circles used, I’ve decided to name them “circumslices.” Interestingly, the three interior angles of a circumslice each asymptotically approach zero degrees, as one approaches circumslice-vertices, which are also the points of contact of the circles.

Why did I name these things “circumslices?” Because they needed a name, that’s why!

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.

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.

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.

The equiangular hexagons are very nearly regular, with only tiny deviations — probably not visible here — “from equilateralness.”

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