Two (New?) “Near-Miss” Candidates

Posted on 13 March 2016 by RobertLovesPi

Yesterday, I played for the first time with GeoMag toys, which I recently purchased. I was quite surprised to have what I believe to be a near-miss to the Johnson solids appear before me, one I’ve never seen, within just a few minutes:

Here’s what it looks like, when viewed from two other angles.

SANYO DIGITAL CAMERA

The faces of this three-fold dihedral polyedron are six pentagons, twelve triangles, and nine quadrilaterals. The fact that it has been proven that only 92 Johnson solids exist means that all of these faces cannot be regular. However, the irregularity is so small that I could not detect it in this model.

Next, I used Polydrons to build a net of this near-miss candidate.

What to do next was obvious: remove the “belt” of nine quadrilaterals, creating a net for a second near-miss candidate.

Having constructed this net, I then returned GeoMags to build a 3-d model of this second, “unbelted” near-miss candidate.

SANYO DIGITAL CAMERA

I then wondered if I could make a third such solid by removal of the triangles, all of which appeared to be the lateral faces of pyramids.

SANYO DIGITAL CAMERA

Could I remove them? Yes, and I did so. Did this create a third near-miss candidate? No. The resulting polyhedron, shown immediately above, is non-convex, and therefore cannot be a near-miss. The faces with dihedral angles greater than 180° are the triangle-pairs found where the pyramids were in the previous model.

With the “belted” and “unbelted” polyhedra before this non-convex non-candidate, the next step is to share them with other polyhedra enthusiasts, get their input regarding the question of whether these are genuine near-misses, and see if these polyhedra have already been found, unknown to me, by someone else.

[Update: please see the next two posts for more on these near-miss candidates.]

A Gallery of Cuboctahedral Polyhedra, and Polyhedral Compounds, Some of Them Chiral

Posted on 11 March 2016 by RobertLovesPi

Any of these images may be enlarged with a click.

Compound of Trunc Octa and Trunc Cube in 4colors

Zonohedrified Trunc Octa not zonish based on faces and vertices

314 faces includes hexadecagons and enneagons

compound of four polyhedra featuring kites

Compound of trun octa and trun cube by component

24 pentagons and 6 octagons chiral cuboctahedral and convex

56 triangles chiral and cuboctahedral and convex

Compound of triakis octa and tetrakis cube

86 faces 6 octagons 8 green hexagons 24 each of two other types of hexagons 24 pentagons

They were all created using Stella 4d, available at this website.

Hexastar Octahedron

Posted on 10 March 2016 by RobertLovesPi

hexastar octahedron

I wish I remembered exactly how I made this polyhedron, but I don’t. I found it during a “random walk” polyhedral exploration using Stella 4d: Polyhedron Navigator, software you can buy, or try for free, here.

A Collection of Related Polyhedra Featuring Octagons, Heptagons, Hexagons, and Pentagons

Posted on 9 March 2016 by RobertLovesPi

These polyhedra are all part of the same stellation-series, although it appears they were made with truncation, instead. I found them using Stella 4d, a program you may buy, or try for free, right here: http://www.software3d.com/Stella.php. The smaller images may be enlarged with a click.

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.

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.

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.

augmented 12-antiprism Convex hull color version

augmented 12-antiprism Convex hull dual color version

aug 12 Dual of Convex hull dual compound

A Second Coloring-Scheme for the Chiral Tetrated Dodecahedron

Posted on 20 February 2016 by RobertLovesPi

For detailed information on this newly-discovered polyhedron, which is near (or possibly in) the “fuzzy” border-zone between the “near-misses” (irregularities real, but not visually apparent) and “near-near-misses” (irregularities barely visible, but there they are) to the Johnson solids, please see the post immediately before this one. In this post, I simply want to introduce a new coloring-scheme for the chiral tetrated dodecahedron — one with three colors, rather than the four seen in the last post.

In the image above, the two colors of triangle are used to distinguish equilateral triangles (blue) from merely-isosceles triangles (yellow), with these yellow triangles all occurring in pairs, with their bases (slightly longer than their legs) touching, within each pair. This is the same coloring-scheme used for over a decade in most images of the (original and non-chiral) tetrated dodecahedron, such as the one below.

Both of these images were created using polyhedral-navigation software, Stella 4d, which is available here, both for purchase and as a free trial download.

[Later edit: I have now found out I was not the first person to find what I had thought, earlier today, was an original discovery. What I have simply named the chiral tetrated dodecahedron has been on the Internet, in German, since 2008, or possibly earlier, and may be seen here: http://3doro.de/polyeder/.]