- #1
Spinnor
Gold Member
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In an earlier post here I wanted to chop up a three-sphere into cubes, Ben suspected it was not possible and I have no reason to think otherwise. From earlier help by Fezro, here, I may be able to move this forward. Assuming the posts by Fezro are correct I think I can come up with a set of points in S^3 that a computer program could check the angles and distances between nearby points. Let a set of points in S^3 be given by the following
P(a,b,c)
= [itex]exp(i[σ_1(2πa/n)+σ_2(2πb/n)+σ_3(2πc/n)]) \begin{pmatrix}
1 \\
0\\
\end{pmatrix} = \begin{pmatrix}
z_1(a,b,c) \\
z_2(a,b,c)\\
\end{pmatrix}[/itex]
where a, b, and c are integers and vary independently from 0 to n (n is some integer), [itex]σ_1,σ_2,σ_3[/itex] are the Pauli matrices, and [itex]z_1(a,b,c)[/itex] and [itex]z_2(a,b,c)[/itex] give us the coordinates x, z, y, and w of the point P(a,b,c) in R^4.
Should it be straightforward to use the above with a computer to determine angles and distances between nearest neighbors? Can the exponential above be simplified or can a computer program easily handle the exponential? Is there a clever way to show that in fact nearest neighbor points above are not equally spaced?
Thanks for any help!
P(a,b,c)
= [itex]exp(i[σ_1(2πa/n)+σ_2(2πb/n)+σ_3(2πc/n)]) \begin{pmatrix}
1 \\
0\\
\end{pmatrix} = \begin{pmatrix}
z_1(a,b,c) \\
z_2(a,b,c)\\
\end{pmatrix}[/itex]
where a, b, and c are integers and vary independently from 0 to n (n is some integer), [itex]σ_1,σ_2,σ_3[/itex] are the Pauli matrices, and [itex]z_1(a,b,c)[/itex] and [itex]z_2(a,b,c)[/itex] give us the coordinates x, z, y, and w of the point P(a,b,c) in R^4.
Should it be straightforward to use the above with a computer to determine angles and distances between nearest neighbors? Can the exponential above be simplified or can a computer program easily handle the exponential? Is there a clever way to show that in fact nearest neighbor points above are not equally spaced?
Thanks for any help!
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