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Weary Crescent computer nerd - rest yourself here.
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Blob - "Calling Mathos"
Question: It's (fairly) well known that a four-dimensional "cube" is a tesseract, but what are the terms for other 4D shapes - such as sphere, tetrahedron, dodecahedron etc. I've seen them referred to as hyper-whatever but this seems a bit weak, are there no coined names ?
(09:45, Wed 07-Jan-2004)
rab - "Well-known"
[Blob] I didn't know that, but if I have cause to consider a 4d cube in my
work, I shall insist on the proper terminology. Can you actually have
higher-dimensional dodecahedra etc? Not sure it's topologically
possible, but I'm no mathematician. Certainly a spherical object can
easily be defined in all (even noninteger) dimensions once you have a
notion of length in that space. However I've never seen a term other
than 'hypersphere'. Although, when the number of dimensions is fewer
than 3, I attempt to use the term 'hyposphere'...
(10:02, Wed 07-Jan-2004)
snorgle - "hypospherical"
[rab]Do you usually get slapped at that point?
Wouldn't a 2D sphere be a circle and a 1D sphere a point?
(11:31, Wed 07-Jan-2004)
Raak
[Blob, rab] The tetrahedron generalises to all dimensions, and is
generally known as the n-dimensional simplex. It has n vertexes and n
faces, each of which is an n-1-dimensional simplex. The cube and the
octahedron also generalise to all dimensions, being known as the
hypercube (in 4 dimensions, tesseract), and the
cross-polytope. The icosahedron and dodecahedron don't
generalise. In four dimensions you get a few other regular polyhedra,
such as the one made of 600 tetrahedra.
(11:38, Wed 07-Jan-2004)
matt - "polytopia"
[Raak] I have no idea why, but that post made my day. Thanks.
(12:38, Wed 07-Jan-2004)
rab - "Hypothetical"
[snorgle] Not usually, for whilst I agree with your nomenclature, what on
earth do you call a 2.57-dimensional sphere?
(13:54, Wed 07-Jan-2004)
Blob - "Multimathos"
[rab, Raak] Thanks, that's interesting. I had a feeling that the dodecahedron & isos~ might not translate to four dimensions, but do you know if there are any regular 4D solids that don't spring from a 3D equivalent, it seems unlikely - in which case presumably there are very few regular hyper-polyhedra - Just (1) n-sphere, (2) n-tetrhedron, (3) n-cube and (4) n-octahedron. I'm still rather amazed that there is no specific name for a multidimensional sphere - perhaps a job for the Mornington Dictionary.
[rab] O.k. I'm quite open-minded but how in Blue Blazes to you have a fraction of a dimension ?
(14:29, Wed 07-Jan-2004)
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