The smaller images above may be enlarged with a click. All these polyhedra were made using Stella 4d, available here.
Four Convex Polyhedra with Icosidodecahedral Symmetry
Reply
Tag Archives: polyhedra
A Hollow Faceting of the Rhombicosidodecahedron, and Its Hollow Dual
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.
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
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.
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.
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/.]
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.
Five Faceted Polyhedra
Above, on the left, is a faceted cuboctahedron. To its right are a faceted snub dodecahedron (upper right) which is also a ten-part compound, and a faceted truncated cube below that. Any of these images may be enlarged by clicking on it.
Below, the left figure is a faceting of the great rhombcuboctahedron — one which is also a three-part compound of octagonal prisms. To its right is a faceting of the snub dodecahedron which is markedly different in appearance from the snub dodecahedron faceting shown above.
Faceted polyhedra have the same vertices as the polyhedra from which they are derived, but those vertices are connected in different ways, changing the faces and edges.
All of these were made using Stella 4d, a program you may try for yourself, for free, right here.
A Pyritohedral and Pentagon-Faced Polyhedron
Twelve of the faces of this polyhedron are pink, and the other twenty-four are blue. It has no faces which are not pentagons. I made it using Stella 4d: Polyhedron Navigator, which is avialable at http://www.software3d.com/Stella.php.
Polyhedral Peacock
Created using Stella 4d, available at http://www.software3d.com/Stella.php.
Unsquashing the Squashed Meta-Great-Rhombcuboctahedron
I noticed that I could arrange eight great rhombcuboctahedra into a ring, but that ring, rather than being regular, resembled an ellipse.
I then made a ring of four of these elliptical rings.
After that, I added a few more great rhombcuboctahedra to make a meta-rhombcuboctahedron — that is, a great rhombcuboctahedron made of rhombcuboctahedra. However, it’s squashed. (I believe the official term for this is “oblate,” but “squashed” also works, at least for me.)
So now I’m wondering if I can make this more regular. In other words, can I “unsquash” it? I notice that even this squashed metapolyhedron has regular rings on two opposite sides, so I make such a ring, and start anew.
I then make a ring of those . . .
. . . And, with two more ring-additions, I complete the now-unsquashed meta-great-rhombcuboctahedron. Success!
To celebrate my victory, I make one more picture, in “rainbow color mode.”
[All images made using Stella 4d, available here: http://www.software3d.com/Stella.php.]
RobertLovesPi.net 