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 Post subject: Equivalences.Posted: Thu Feb 07, 2013 1:06 pm

Joined: Sun Aug 26, 2012 10:01 am
I don't know what anyone else could reply to this, whether or not this is the right place to post this, or even if its worth posting at all. But I got thinking today, and I realised because an FTI is 'the same' as a CTD, and a CTI is the same as an FTD does that mean cubedude76's face and corner turning icosahedron has the exact same mechanism as a face and corner turning dodecahedron if it were made? It would be a cool shape mod

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 Post subject: Re: Equivalences.Posted: Fri Feb 08, 2013 4:44 pm

Joined: Fri Dec 28, 2012 1:50 pm
Location: Near Las Vegas, NV
The reason this is true is because the dodecahedron is the dual of the icosahedron and vice versa (that means the corners of the dodecahedron have the same symmetry as the faces of the icosahedron). This applies to lots of other examples, for instance the Face-Turning Octahedron is the same as a corner-turning cube because they are dual to each other, or even a face-turning rhombic dodecahedron is the same as an edge-turning cube.
In fact, Ryan's Face and corner-turning icosahedron could also be viewed as either a face-turning icosidodecahedron or a corner-turning rhombic triacontahedron because the corners of the RT and the faces of the Icosidodecahedron have the same symmetry as both the faces and corners of the dodecahedron or icosahedron (that's because some of them have 3 fold symmetry, the icosahedral symmetry, and others have five-fold symmetry, the dodecahedral symmetry).
What's interesting is that there are some instances of self-duality. The tetrahedron, for example, is dual to itself, which means you could perceive a Pyraminx as both face turning and corner-turning.
I really hope this makes sense...

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 Post subject: Re: Equivalences.Posted: Fri Feb 08, 2013 5:00 pm

Joined: Sun Aug 26, 2012 10:01 am
I think I understand that, pretty sure I do

And I've wondered the same thing! Is the pyraminx face turning or corner turning?

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 Post subject: Re: Equivalences.Posted: Fri Feb 08, 2013 5:05 pm

Joined: Fri Dec 28, 2012 1:50 pm
Location: Near Las Vegas, NV
JackRTully wrote:
Is the pyraminx face turning or corner turning?

I would consider it corner-turning more than face turning since most people turn it by the corners during solving. But geometrically speaking, it's both.

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