Here's the hecatonicosihexapentacosiheptacontihexaexon, more commonly called the 7-dimensional Gosset polytope. A polytope is a higher-dimensional generalization of a polyhedron. This is the most symmetrical one in 7 dimensions!
It has 56 corners, but you can only see 55 here, since it's been projected down to a plane, and the orange one in the middle is directly in front of another. If you draw lines between the opposite corners you get 28 equiangular lines: they're all at equal angles to each other. That's the most equiangular lines you can get in 7 dimensions... and also in 8, 9, 10, 11, 12, 13, or 14 dimensions! So, it's very nice collection of lines.
I'm happy because I asked two hard questions on G+ yesterday, and I got good answers to both. First: why are there 28 equiangular lines in 7 dimensions? +Philip Gibbs gave me the crucial clue I needed to get this nice answer. Second: is it worthwhile making a day trip from Urumqi to Turpan if you don't have much time to explore the far west of China? +Dalibor Smid settled that one: yes!
This 7-dimensional polytope was discovered by Thorold Gosset. Gosset was a student at Pembroke College in Cambridge, and then went on to get a law degree in 1895. When he started he had no clients and so - being a very shrewd and practical fellow - he decided to classify all the regular polytopes in higher dimensions. After succeeding in this (which people had already done), he tried to classify the semiregular polytopes, which have regular polytopes as faces and are so symmetrical that every corner looks alike. In 3 dimensions these are well-known and beautiful things, studied already by Archimedes. But Gosset discovered that in 6, 7 and 8 dimensions there are "exceptional" semiregular polytopes that you'd never expect from lower dimensions. There are none of these in any higher dimension.
Gosset couldn't get any mathematicians interested in his work, so he quit this hobby, started practicing law... and lived until 1962.
The polytopes he discovered turned out to be related to symmetry groups called E6, E7 and E8. E8 is the "king" - the biggest and best - and it contains the other two inside it. I have wasted many hours happily studying these symmetry groups and related structures. It's fun, because at first it seems impossible to understand them... but in fact you can.
To mathematicians, the nicest description of the 7-dimensional Gosset polytope is that its corners are the weights of the smallest irreducible representation of E7. But if that jargon means nothing to you, just take the vectors
(3, 3, -1, -1, -1, -1, -1, -1)
and
(-3, -3, 1, 1, 1, 1, 1, 1)
and permute their coordinates in all possible ways. You'll get 56 vectors in 8 dimensions, but they all point at right angles to the vector
(1, 1, 1, 1, 1, 1, 1, 1)
so they live in a 7-dimensional space... and if you project them down to a pathetic little 2-dimensional plane, you get this shape here!
You can take this 7-dimensional polytope and truncate it in various ways to get 127 different polytopes that are all just as symmetrical. Tom Ruen has drawn pictures of ten of them, here:
http://en.wikipedia.org/wiki/List_of_E7_polytopes
For more on this particular one, see:
http://en.wikipedia.org/wiki/3_21_polytope
Coxeter called it the 3_21 polytope, but someone has more whimsically called it the hecatonicosihexapentacosiheptacontihexaexon.
#geometry #4d
It has 56 corners, but you can only see 55 here, since it's been projected down to a plane, and the orange one in the middle is directly in front of another. If you draw lines between the opposite corners you get 28 equiangular lines: they're all at equal angles to each other. That's the most equiangular lines you can get in 7 dimensions... and also in 8, 9, 10, 11, 12, 13, or 14 dimensions! So, it's very nice collection of lines.
I'm happy because I asked two hard questions on G+ yesterday, and I got good answers to both. First: why are there 28 equiangular lines in 7 dimensions? +Philip Gibbs gave me the crucial clue I needed to get this nice answer. Second: is it worthwhile making a day trip from Urumqi to Turpan if you don't have much time to explore the far west of China? +Dalibor Smid settled that one: yes!
This 7-dimensional polytope was discovered by Thorold Gosset. Gosset was a student at Pembroke College in Cambridge, and then went on to get a law degree in 1895. When he started he had no clients and so - being a very shrewd and practical fellow - he decided to classify all the regular polytopes in higher dimensions. After succeeding in this (which people had already done), he tried to classify the semiregular polytopes, which have regular polytopes as faces and are so symmetrical that every corner looks alike. In 3 dimensions these are well-known and beautiful things, studied already by Archimedes. But Gosset discovered that in 6, 7 and 8 dimensions there are "exceptional" semiregular polytopes that you'd never expect from lower dimensions. There are none of these in any higher dimension.
Gosset couldn't get any mathematicians interested in his work, so he quit this hobby, started practicing law... and lived until 1962.
The polytopes he discovered turned out to be related to symmetry groups called E6, E7 and E8. E8 is the "king" - the biggest and best - and it contains the other two inside it. I have wasted many hours happily studying these symmetry groups and related structures. It's fun, because at first it seems impossible to understand them... but in fact you can.
To mathematicians, the nicest description of the 7-dimensional Gosset polytope is that its corners are the weights of the smallest irreducible representation of E7. But if that jargon means nothing to you, just take the vectors
(3, 3, -1, -1, -1, -1, -1, -1)
and
(-3, -3, 1, 1, 1, 1, 1, 1)
and permute their coordinates in all possible ways. You'll get 56 vectors in 8 dimensions, but they all point at right angles to the vector
(1, 1, 1, 1, 1, 1, 1, 1)
so they live in a 7-dimensional space... and if you project them down to a pathetic little 2-dimensional plane, you get this shape here!
You can take this 7-dimensional polytope and truncate it in various ways to get 127 different polytopes that are all just as symmetrical. Tom Ruen has drawn pictures of ten of them, here:
http://en.wikipedia.org/wiki/List_of_E7_polytopes
For more on this particular one, see:
http://en.wikipedia.org/wiki/3_21_polytope
Coxeter called it the 3_21 polytope, but someone has more whimsically called it the hecatonicosihexapentacosiheptacontihexaexon.
#geometry #4d
