About RobertLovesPi

I go by RobertLovesPi on-line, and am interested in many things, a large portion of which are geometrical. Welcome to my own little slice of the Internet. The viewpoints and opinions expressed on this website are my own. They should not be confused with those of my employer, nor any other organization, nor institution, of any kind.

Night Bloodlines 3d6

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Night Bloodlines 3d6

Saint Contradiction

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saint contradiction

Our Red-Haired Lady of Parabolas, Circles, and Probabilities

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our red-haired lady of parabolas, circles, and probabilities

The Snub Dodecahedron and Related Polyhedra, Including Compounds

Posted on 5 March 2016 by RobertLovesPi

The dual of the snub dodecahedron (above) is called the pentagonal hexacontahedron (below, left). The compound of the two is shown below, at right. (Any of the smaller images here may be enlarged with a click.)

Like all chiral polyhedra, both these polyhedra can form compounds with their own mirror-images, as seen below.

Compound of snub dodeca enantiomorphic pair

Finally, all four polyhedra — two snub dodecahedra, and two pentagonal hexacontahedra — can be combined into a single compound.

This polyhedral manipulation and .gif-making was performed using Stella 4d, a program you can find here.

Three Different Views of a 962-Faced Zonohedron

Posted on 4 March 2016 by RobertLovesPi

This zonohedron contains faces which are regular decagons (12 of them), equilateral octagons (30, all of the same type), equilateral hexagons (380 of them, of 7 types, with one of these 7 types, of which there are 20, being regular), squares (60), and non-square rhombi (480 of 8 types, counting reflections as separate types). With each polygon-type, including the reflections, given a different color, this zonohedron looks like this.

If reflected face-types are not counted as separate types, then the coloring-by-face-type uses four fewer colors, and looks like this:

Another view simply colors faces by numbers of sides, and is shown below. Each of these rotating images was created with Stella 4d, a program you may buy, or try for free, at http://www.software3d.com/Stella.php.

The American Historical Clock of War and Peace

Posted on 28 February 2016 by RobertLovesPi

The yellow years are ones in which the USA was getting into or out of major wars — or both, in the case of the brief Spanish-American War. The red years are war years, and the blue years are years of (relative) peace.

The sectors are each bounded by two radii, and a 1.5° arc. The current year is omitted intentionally because 2016 isn’t over yet, and we don’t know what will happen during the rest of it.

Four Convex Polyhedra with Icosidodecahedral Symmetry

Posted on 28 February 2016 by RobertLovesPi

The smaller images above may be enlarged with a click. All these polyhedra were made using Stella 4d, available here.

Seven Fanciful Designs for Polyhedral Throwing Stars with 7-Fold Dihedral Symmetry

Posted on 27 February 2016 by RobertLovesPi

I made these using Stella 4d, a program available at http://www.software3d.com/Stella.php. Any of these images may be enlarged with a click.

A Hollow Faceting of the Rhombicosidodecahedron, and Its Hollow Dual

Posted on 26 February 2016 by RobertLovesPi

Hollow Faceted Rhombicosidodeca showing hexagonal face

Hollow Faceted Rhombicosidodeca showing trapezoidal face

The images above all show a particular faceting of the rhombicosidodecahedron which, to my surprise, is hollow. It has the vertices of a rhombicosidodecahedron, but two different face-types, as seen in the smaller pictures: yellow hexagons, and red isosceles trapezoids. (To enlarge any image in this post, simply click on it.)

The dual of this polyhedron is even more obviously hollow, as seen below. Its faces, as seen in the still picture, are crossed hexagons — with edges which cross three times per hexagon, no less.

Hollow Faceted Rhombicosidodeca dual showing crossed hexagonal face

The software I used to make these polyhedra, Stella 4d, will return an error message if the user attempts to make a polyhedron which is not mathematically valid. When I’ve made things that look (superficially) like this before, I used “hide selected faces” to produce hollow geometrical figures which were not valid polyhedra, but that isn’t what happened here (I hid nothing), so this has me confused. Stella 4d (software you can buy, or try for free, here) apparently considers these valid polyhedra, but I am at a loss to explain such familiar concepts as volume for such unusual polyhedra, or how such things could even exist — yet here they are. Clarifying comments would be most appreciated.

12-Fold Dihedral Polyhedral Explorations

Posted on 23 February 2016 by RobertLovesPi

Above is a dodecagonal antiprism, augmented by 24 more dodecagonal antiprisms. This was the starting point for making all the polyhedra below, using Stella 4d, software available here. Each of these smaller pictures may be enlarged with a click.