A Chiral Polyhedron Made of Kites and Triangles, Along with Its Dual, Made of Triangles and Isosceles Trapezoids

To make this polyhedron using Stella 4d (available here), one starts with the icosahedron, drops the symmetry of the model down from icosahedral to tetrahedral, and then stellates it once. The result is a chiral solid featuring four triangular faces and twelve kites:

chiral polyhedron made of triangles and kites -- found while exploring tetstells of the icosahedron.gif

The dual of this polyhedron, which is also chiral, has four triangular faces, and twelve faces which are isosceles trapezoids. It is a type of faceted dodecahedron — a partial faceting, meaning it is made without using all of the dodecahedron’s vertices.

Faceted Dodeca.gif

The 43rd Stellation of the Snub Dodecahedron, and Related Polyhedra, Part One

If you stellate the snub dodecahedron 43 times, this is the result. The yellow faces are kites, not rhombi.

Stellated Snub Dodeca refl

Like the snub dodecahedron itself, this polyhedron is chiral. Here is the mirror-image of the polyhedron shown above.

Stellated Snub Dodeca 43rd mirror image

Any chiral polyhedron may be combined with its own mirror-image to create a compound.

Compound of enantiomorphic pair x

This is the dual of the snub dodecahedron’s 43rd stellation.

Stellated Snub Dodeca refl chiral dual

This dual is also chiral. Here is its reflection.

43rd stellation snub dodeca dual reflection

Finally, here is the compound of both duals.

Compound of enantiomorphic pair duals

I used Stella 4d: Polyhedron Navigator to create these images. You may try this program for yourself at http://www.software3d.com/Stella.php.

A Chiral Polyhedron with Tetrahedral Symmetry

tetty

The yellow faces of this polyhedron are parallelograms, while the red ones are trapezoids. To demonstrate its chirality, here is the compound of it, and its own mirror-image.

compound-of-enantiomorphic-pair

Both of these “virtual polyhedra” were made using Stella 4d: Polyhedron Navigator, a program available at this website. It has a free trial download available.

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/.]

Standard and Faceted Versions, Side by Side, of Each of the Thirteen Archimedean Solids

These two polyhedra are the truncated tetrahedron on the left, plus at least one faceted version of that same Archimedean solid on the right. As you can see, in each case, the figures have the same set of vertices — but those vertices are connected in a different way in the two solids, giving the polyhedra different faces and edges.

(To see larger images of any picture in this post, simply click on it.)

The next three are the truncated cube, along with two different faceted truncated cubes on the right. The one at the top right was the first one I made — and then, after noticing its chirality, I made the other one, which is the compound of the first faceted truncated cube, plus its mirror-image. Some facetings of non-chiral polyhedra are themselves non-chiral, but, as you can see, chiral facetings of non-chiral polyhedra are also possible.

The next two images show a truncated octahedron, along with a faceted truncated octahedron. As these images show, sometimes faceted polyhedra are also interesting polyhedra compounds, such as this compound of three cuboids. 

The next polyhedra shown are a truncated dodecahedron, and a faceted truncated dodecahedron. Although faceted polyhedra do not have to be absurdly complex, this pair demonstrates that they certainly can be.

Next are the truncated icosahedron, along with one of its many facetings — and with this one (below, on the right) considerably less complex than the faceted polyhedron shown immediately above.

The next two shown are the cuboctahedron, along with one of its facetings, each face of which is a congruent isosceles triangle. This faceted polyhedron is also a compound — of six irregular triangular pyramids, each of a different color.

The next pair are the standard version, and a faceted version, of the rhombcuboctahedron, also known as the rhombicuboctahedron.

The great rhombcuboctahedron, along with one of its numerous possible facetings, comes next. This polyhedron is also called the great rhombicuboctahedron, as well as the truncated cuboctahedron.

The next pair are the snub cube, one of two Archimedean solids which is chiral, and one of its facetings, which “inherited” its chirality from the original.

The icosidodecahedron, and one of its facetings, are next.

The next pair are the original, and one of the faceted versions, of the rhombicosidodecahedron.

The next two are the great rhombicosidodecahedron, and one of its facetings. This polyhedron is also called the truncated icosidodecahedron.

Finally, here are the snub dodecahedron (the second chiral Archimedean solid, and the only other one, other than the snub cube, which possesses chirality), along with one of the many facetings of that solid. This faceting is also chiral, as are all snub dodecahedron (and snub cube) facetings.

Each of these polyhedral images was created using Stella 4d: Polyhedron Navigator, software available at this website.

Eight Chiral Polyhedra with Icosidodecahedral Symmetry

To see a larger version of any rotating model, simply click on it.

Each of these polyhedral images was created using a program called Stella 4d, which is available here.