Mathematics | RobertLovesPi.net

Four Views of a Tesseract, Rotating in Hyperspace

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Four Views of a Tesseract, Rotating in Hyperspace

These tesseract views are all of the perspective projection-type, with the first one, above, being done in cell-first fashion.

The next one is projected edge-first.

The third one is projected vertex-first.

Lastly, face-first:

Although all of these are rotating in the same direction in hyperspace, the different projection-choices make the second and fourth images appear to be rotating in different directions. Why? I’m still trying to figure that out!

These animations were created with Stella 4d, software available at www.software3d.com/Stella.php.

Pulsating Cuboctahedron, Featuring Enneagrammic Mandalas

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The enneagramic mandalas on the square faces of this cuboctahedron are from the last post, with inverted-color, smaller versions of the same image on the triangular faces. These mandalas were created using Geometer’s Sketchpad and MS-Paint. Projecting them onto the faces of the cuboctahedron, and then creating this rotating, pulsating .gif image, however, took a third program: Stella 4d, which you can buy, or try for free, at http://www.software3d.com/Stella.php.

Ten Enneagrams

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These enneagrams are of the {9/3} variety, which means each one is made of three concentric, equilateral triangles. One of these enneagrams is at the center; the other nine surround it.

Tessellation Using {8/3} Star Octagons and Squares

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The tessellation of the plane which uses regular convex octagons and squares is well-known. This related tessellation, however, is not. I didn’t know it existed until I stumbled across it . . . although I very much doubt I am the first person to do so.

The Compund of Three Octahedra

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The Compund of Three Octahedra

This polyhedral compound is part of the built-in library of polyhedra that comes with Stella 4d: Polyhedron Navigator. You can find this software here: http://www.software3d.com/Stella.php.

Four-Color Radial Enneagon-Centered Mandala

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Four-Color Radial Enneagon-Centered Mandala

Thank You, Coca-Cola Inc., for Promoting Archimedean Solid Awareness!

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Thank You, Coca-Cola Inc., for Promoting Archimedean Solid Awareness!

I was delighted to see that cans of Coca-Cola^® now on sale feature a picture of a truncated icosahedron. This is real progress!

Snub Dodecahedron Variant

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Snub Dodecahedron Variant

In this polyhedron, there are twelve pentagons and sixty kites. It can be made by augmenting twenty of the triangles in a snub dodecahedron with short pyramids, but the pyramid-height has to be just right, in order to make those pyramids’ lateral faces coplanar with the non-augmented triangles, which produces the kites.

Since this polyhedron is chiral, a compound can be made by adding it to its own mirror image: