A Second Coloring-Scheme for the Chiral Tetrated Dodecahedron

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.

chiral tet dod 2nd color scheme

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.

Tetrated Dodeca

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/.]

The Chiral Tetrated Dodecahedron: A New Near-Miss?

The images above show a new near-miss (to the Johnson solids) candidate I just found using Stella 4d, software you can try here. Like the original tetrated dodecahedron (a recognized near-miss shown at left, below), making this polyhedron relies on splitting the Platonic dodecahedron into four three-pentagon panels, moving them apart, and filling the gaps with triangles. Unlike that polyhedron, though, this new near-miss candidate is chiral, as you can see by comparing the left- and right-handed versions, above. The image at the right, below, is the compound of these two enantiomers.

Next are shown nets for both the left- and right-handed versions of the chiral tetrated dodecahedron (on the right, top and bottom), along with the dual of this newly-discovered polyhedron (on the left). Like the rest of the images in this post, any of them may be enlarged with a click.

A key consideration when it is decided if the chiral tetrated dodecahedron will be accepted by the community of polyhedral enthusiasts as a near-miss (almost a Johnson solid), or will be relegated to the less-strict set of “near-near-misses,” will be measures of deviancy from regularity.The pentagons and green triangles are regular, with the same edge length. The blue and yellow triangles are isosceles, with their bases located where blue meets yellow. These bases are each ~9.8% longer than the other edges of the chiral tetrated dodecahedron. By comparison, the longer edges of the original tetrated dodecahedron, where one yellow isosceles triangle meets another, are ~7.0% longer than the other edges of that polyhedron. Also, in the original, the vertex angle of these isosceles triangles measures ~64.7°, while the corresponding figure is ~66.6° for the chiral tetrated dodecahedron.

[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/.]

The Tetrated Dodecahedron

Image

The Tetrated Dodecahedron

The tetrated dodecahedron is a near-miss Johnson solid. It was first discovered in 2002 by Alex Doskey. I then independently rediscovered it in 2003, and named it, not learning of Doskey’s original discovery for several years after that.

It has 28 faces: twelve regular pentagons, arranged in four panels of three pentagons each; four equilateral triangles (shown in blue); and six pairs of isosceles triangles (shown in yellow). All edges of the tetrated dodecahedron have the same length, except for the shared bases of these isosceles triangles, which are approximately 1.07 times as long as the other edges. This polyhedron has tetrahedral symmetry.

Tetrated Dodecanet

(All images here were produced using Stella 4d, which you may try for free, after downloading the trial version from this website: www.software3d.com/Stella.php.)