Is the great icosahedron regular?
Is the great icosahedron regular?
The great icosahedron is a Kepler-Poinsot polyhedron and so it is supposed to be regular. Regular means one of the most symmetrical polyhedra with all faces congruent. It has three types of triangular faces. Two types are scalene and mirror images but the third is isosceles, not the same size or shape and thus not congruent. How can it be regular? I feel I am missing something - somebody put me out of my misery.
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robertw
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Re: Is the great icosahedron regular?
You're describing the "facelets" that are visible from outside, but these are all parts of larger equilateral triangles. It's got 20 equilateral faces overall (icosa- means 20), but they are all intersecting each other. Here's a picture with one of the complete faces highlighted:
Re: Is the great icosahedron regular?
Thank you Robert. So the problem was not with the meaning of "congruent" or "regular" but "faces". Is it standard amongst mathematicians to use "face" to include planes that may have invisible interior parts?
It looks like the nets generated by Stella are always facelets rather than faces. Is it possible to print nets that are faces, ie including interior planes? My reason for asking is that there may be times when assembling faces instead of facelets is an easier model to build.
Nomenclature seems difficult. For example where you say "face angle" I would say "mitre angle" and where you say "mitre angle" I would say "bevel angle". But I am looking at it from the point of building models out of solid materials with more than negligible thickness.
It looks like the nets generated by Stella are always facelets rather than faces. Is it possible to print nets that are faces, ie including interior planes? My reason for asking is that there may be times when assembling faces instead of facelets is an easier model to build.
Nomenclature seems difficult. For example where you say "face angle" I would say "mitre angle" and where you say "mitre angle" I would say "bevel angle". But I am looking at it from the point of building models out of solid materials with more than negligible thickness.
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robertw
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Re: Is the great icosahedron regular?
It's certainly standard among people working with polyhedra, to allow intersecting faces. There are also times when they try to avoid it though, such as with the Stewart toroids.
Nets are always made of the externally visible parts. Using the full faces would make it impossible to buid the model, since in our physical world you can't make the faces intersect, so I'm not sure what you mean by that.
Stella4D also has unfolded nets for 4D polytopes, and in this case the nets consist of the original cells making up the shape, even if they intersect. This is because a 4D model can't physically be folded up anyway, and it's purpose is more for understanding how the cells fit together (and it's mostly of interest for convex polytopes anyway).
Not sure about terminology for woodwork, since it's not my area, but this was based on feedback from people who do. But as long as you can figure out what means what, that's what matters.
Nets are always made of the externally visible parts. Using the full faces would make it impossible to buid the model, since in our physical world you can't make the faces intersect, so I'm not sure what you mean by that.
Stella4D also has unfolded nets for 4D polytopes, and in this case the nets consist of the original cells making up the shape, even if they intersect. This is because a 4D model can't physically be folded up anyway, and it's purpose is more for understanding how the cells fit together (and it's mostly of interest for convex polytopes anyway).
Not sure about terminology for woodwork, since it's not my area, but this was based on feedback from people who do. But as long as you can figure out what means what, that's what matters.