What are your favorite polyhedra?

For general discussion of polyhedra, not necessarily Stella-specific.
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senkoquartz
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What are your favorite polyhedra?

Post by senkoquartz » Sat Sep 19, 2020 9:44 pm

My #1 is the octahedron (perfect)
Some of my favorites are the great rhombicuboctahedron and great deltoidal icositetrahedron (very deep and very spikey, well proportioned)
The pentagonal hexecontahedron (pretty)
The dual of the final icosahedron stellation (DEEP)
And the compound of 5 great dodecahedra

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robertw
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Re: What are your favorite polyhedra?

Post by robertw » Sun Sep 20, 2020 1:24 pm

Good question!

Some of my favourites:

The Great Icosidodecahedron, which I remembering seeing as an orthographic line drawing in an old book and I remember thinking I'd love to make one but didn't know how. I also remember at the time thinking it was very complex, whereas now it's quite simple compared to others.
https://www.software3d.com/GreatIcosidodeca1.php
Image

The Great Icosicosidodecahedron, for the way the faces intersect each other.
https://www.software3d.com/GIID.php
Image

This Stellation of the Small Stellated Truncated Dodecahedron. Just beautiful.
https://www.software3d.com/StelSSTD.php
Image

The Compound of Icosidodecahedron and its dual the rhombic triacontahedron. Just seems very elegant, but also maybe the way this model turned out with the colours and textures of the paper.
https://www.software3d.com/ID+RTC.php
Image

Compound of 5 Tetrahemihexahedra. I like that this looks so familiar, like the compound of 5 octahedra, but with a chiral twist.
https://www.software3d.com/5THH.php
Image

This compound of 15 Cuboids, which is also a faceted truncated icosidodecahedron. There's an infinite series of 15 cuboids compounds, made by moving the vertices around symmetrically, but this one is particularly stunning.
https://www.software3d.com/15RectTID.php
Image

Two interlocking rings of dodecahedra taken from the 4D 120-cell.
https://www.software3d.com/120Cell.php
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This cross-section of the great icosahedral 120-cell. Probably only because I made a model of it, but it's super spiky and looks great in metallic paper.
https://www.software3d.com/Gofix.php
Image

This Genus-41 Stewart Toroid I discovered. Such a big toroid consisting entirely of regular faces
https://www.software3d.com/WebbToroid.php
Image

I reckon those are among my favourites.

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Ulrich
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Re: What are your favorite polyhedra?

Post by Ulrich » Sun Sep 27, 2020 7:27 am

These are beautiful models and very pretty polyhedra/polytopes!
To me it is difficult to say which one is my favorite polyhedron. It changes all day and every model I finished is my favorite one for some time. In principle, I love all uniform ones. And one of my favorite is the rhombic triacontahedron.
Ulrich

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marcelteun
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Re: What are your favorite polyhedra?

Post by marcelteun » Mon Sep 28, 2020 7:56 am

I never thought about that.

I must say I have to agree with Ulrich.

I do like compounds a lot. Don't know why, but I just do. I finished the compound of 15 cubes one week ago and I am very satisfied with the result. I don't have a picture yet, since my wife will have to find the time to take one in her studio.

Another property I like is just having rotational symmetries, i.e. chiral models. E.g. the compound of 5 Tetrahedra. One personal favourite is
Image
because the first time I failed miserably with that one. Also because it took me a while to find that one: a chiral model with folded regular heptagons with A5 symmetry.

Of the uniform ones I would say: Great Rhombicuboctahedron, for same reasons as Robert mentionas the Great Icosicosidodecahedron.
Image
I remember that that model attracted me the first time I browsed through Wenninger's "Polyhedron Models".

I found the snubs in the back of that book amazing and I was blown away by number 117. I couldn't imagine anyone building such a complex model and I couldn't recognise how the faces were organised. I almost decided to build a model for that one many many years later, but in the end I decided to build a model for the compound of two of those instead. And even though the end result was a bulky model it will always be one of my favourites too.
Image

BTW As always, great models, Robert!

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robertw
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Re: What are your favorite polyhedra?

Post by robertw » Tue Sep 29, 2020 2:22 am

Marcel, it's great that you used slightly reflective paper for that compound of two W117s. The reflections around the centre look great!

And yes I also love the great rhombicuboctahedron. Your one looks very well made. Here's my one.

Image

But also the pseudo great rhombicuboctahedron is really nice. Has the same set of faces, but with a 45 degree twist between the two halves. Vertices are still all the same locally, but don't fit into the overall symmetry the same way, so doesn't count as uniform.

Image

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marcelteun
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Re: What are your favorite polyhedra?

Post by marcelteun » Mon Oct 05, 2020 1:48 pm

robertw wrote:
Tue Sep 29, 2020 2:22 am
Marcel, it's great that you used slightly reflective paper for that compound of two W117s. The reflections around the centre look great!
..
And yes I also love the great rhombicuboctahedron. Your one looks very well made. Here's my one.
Thanks, Robert!
robertw wrote:
Tue Sep 29, 2020 2:22 am
But also the pseudo great rhombicuboctahedron is really nice. Has the same set of faces, but with a 45 degree twist between the two halves. Vertices are still all the same locally, but don't fit into the overall symmetry the same way, so doesn't count as uniform.
Thanks is cool! Didn't know that worked with that one too!

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