April | 2014 | RobertLovesPi.net

Sunrise 2014

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Sunrise 2014

Drawn with Geometer’s Sketchpad, and based on the numbers four and seven.

A Cube-Based “Bowtie” Symmetrohedron Featuring Six Regular Hexadecagons, Eight Equilateral Triangles, and Two Dozen Each of Two Types of Icosceles Trapezoid

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A Cube-Based Symmetrohedron Featuring Six Regular Hexadecagons, Eight Equilateral Triangles, and Two Types of Icosceles Trapezoids

The two types of trapezoid are shown in blue and green. There are twenty-four blue ones (in eights set of three, surrounding each triangle) and twenty-four green ones (in twelve sets of two, with each set in “bowtie” formation).

This symmetrohedron follows logically from one that was already known, and pictured at http://www.cgl.uwaterloo.ca/~csk/projects/symmetrohedra/, with the name “bowtie cube.” Here’s a rotating version of it.

(Images created with Stella 4d — software you can try yourself at http://www.software3d.com/Stella.php.)

Symmetrohedron Featuring Eighteen Regular Octagons, Eight Equiangular Hexagons, and Twenty-four Isosceles Trapezoids

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Symmetrohedron Featuring Eighteen Regular Octagons, Eight Equiangular Hexagons, and Twenty-four Isosceles Trapezoids

The regular octagons are of the same size, but of two different types, when one considers the pattern of other faces surrounding them. This is why six of them are yellow, and twelve are red.

If the hexagons and isosceles trapezoids were closer to regularity, this would qualify as a near-miss to the Johnson solids, but it falls short on this test. Is is, instead, a “near-near-miss” — and not the first such polyhedron to appear on this blog, either.

(Image created with Stella 4d — software you can try yourself at http://www.software3d.com/Stella.php.)

A Symmetrical Arrangement of Regular Octagons, Triangles, and Squares

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A Symmetrical Arrangement of Regular Octagons, Triangles, and Squares

This contains twelve octagons, six squares, and eight triangles. The “holes” in it keep it from being a true polyhedron, but it is my hope than further study of this arrangement may lead to the discovery of new, interesting, and symmetrical polyhedra.

(Image created with Stella 4d — software you can try yourself at http://www.software3d.com/Stella.php.)

150 Irregular Hexagons, Rotating About a Common Axis

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150 Irregular Hexagons, Rotating About a Common Axis

There are sixty of the irregaular, pentagonal gaps. Also, the hexagons themselves are of three types, two of which are sixty in number, and one of which is thirty in number.

If the gaps are filled, and the color scheme changed to make each of the four polygon-types into its own color-group, this looks, instead, like this (click on it if you wish to see it enlarged). It has 210 faces.

(Images created with Stella 4d — software you can try yourself at http://www.software3d.com/Stella.php.)

A Fifty-Faced, Zonohedrified Form of the Truncated Octahedron

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A Zonohedrified Form of the Truncated Octahedron

This zonohedron has fifty faces:

6 regular octagons
8 regular hexagons
24 squares
12 equilateral octagons, the only irregular polygons needed as faces of this polyhedron

(Image created with Stella 4d — software you can try yourself at http://www.software3d.com/Stella.php.)

The MetaSnubDodecahedron

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The MetaSnubDodecahedron

It’s like the snub dodecahedron’s big brother.

(Image created with Stella 4d — software you can try yourself at http://www.software3d.com/Stella.php.)

A Snub Dodecahedron, Caught In the Act of Attempting To Disguise Itself As an Icosahedron

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A Snub Dodecahedron, Caught In the Act of Attempting To Disguise Itself As an Icosahedron

(Image created with Stella 4d — software you can try yourself at http://www.software3d.com/Stella.php.)

The Hextrated Pentagonal Icositetrahedron

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The Hextrated Pentagonal Icositetrahedron

Years ago, I split a dodecahedron into four panels of pentagons, rotated the pentagon-panels and moved them outward from the center, and did so just the right amount to create gaps that could be filled with triangles. Thus was named the tetrated dodecahedron, which you can read more about here: https://en.wikipedia.org/wiki/Tetrated_dodecahedron

The choice of word “tetrated” was somewhat unfortunate, for tetration already exists in mathematics, as a means of expressing very large numbers, and which I shall not explain here. I didn’t learn this until much later, though, and by that time, it was too late to turn “tetrate” into something else. It had come to mean an operation one does on a polyhedron: break it into four multi-face panels, move them out and rotate them just enough, and fill the resulting gaps with triangles.

As such, “tetrate” can, in the geometrical sense, be modified for differing numbers of panels of multiple faces from a polyhedron. Consider the pentagonal icositetrahedron, the dual of the snub cube. Here, it has been split into six panels, and then each panel moved out from the center and rotated, with triangles filling the gaps. The triangles differ between color-groups slightly, but are close to equilateral, except for the ones shown in green, which simply are equilateral.

(Image created with Stella 4d — software you can try yourself at http://www.software3d.com/Stella.php.)