Tag Archives: Mathematics

Counting to Fifty, Part One: Binary through Decimal

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Counting to Fifty, Part One: Binary Through Decimal

Triskelion of Regular Polygons

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Triskelion of Regular Polygons

Tessellation Using Equilateral Triangles, Isosceles Triangles, Squares, Regular Pentagons, and Equilateral, Non-Convex Octakaitetracontagons

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In this tessellation, regular polygons have been given the brighter colors, while the two non-regular polygons have pastel colors.

Tessellation of the Plane with Regular Hexagons, Squares, and Tetraconcave, Equilateral Octagons, #1

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Tessellation of the Plane with Regular Hexagons, Squares, and Tetraconcave, Equilateral Octagons

If you’d like to see the second version of this tessellation, made over a decade later, simply click right here.

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.

persp edge 2-Tes, 8 -cell, Octachoron, Tesseract

The third one is projected vertex-first.

persp vertex 2-Tes, 8 -cell, Octachoron, Tesseract

Lastly, face-first:

persp face 2-Tes, 8 -cell, Octachoron, Tesseract

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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Pulsating Cuboctahedron, Featuring Enneagrammic Mandalas

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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Ten Enneagrams

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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Tessellation Using {8/3} Star Octagons and Squares

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.

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:

Compound of enantiomorphic pair

Both images were created using Stella 4d, software available at www.software3d.com/Stella.php.