Set of points in S^3, way to show spaced equal or not?

In summary, the conversation discusses the possibility of dividing a three-sphere into cubes using a computer program to check the angles and distances between nearby points. The set of points in S^3 is given by the exponential equation, and it is suggested that using a spin down spinor may yield different results. The question of whether one set of points can be rotated into the other is also raised.
  • #1
Spinnor
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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!
 
Last edited:
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  • #2
If I had defined my set of points using the spin down spinor,

\begin{pmatrix}
0 \\
1\\
\end{pmatrix}

would there be a fundamental difference between the two sets of points? Could one set of points be rotated into the other set?

Thanks for any help!
 

1. How do you define a set of points in S^3?

A set of points in S^3 is a collection of points that are located on a three-dimensional sphere, where each point is defined by its coordinates (x, y, z, w). These points can be represented using spherical coordinates or Cartesian coordinates.

2. What does it mean for points in S^3 to be spaced equally?

In order for points in S^3 to be spaced equally, they must be equidistant from each other on the three-dimensional sphere. This can be visualized by imagining a sphere with evenly spaced points along its surface, similar to the latitude and longitude lines on a globe.

3. How can you show whether points in S^3 are spaced equally or not?

One way to show if points in S^3 are spaced equally is by calculating the distance between each point and comparing it to the distance between neighboring points. If the distances are equal, then the points are spaced equally. This can also be demonstrated by plotting the points on a three-dimensional graph and visually inspecting the spacing between them.

4. What is the significance of studying the spacing of points in S^3?

The study of the spacing of points in S^3 has applications in various fields such as mathematics, physics, and computer science. It can help in understanding the geometry of four-dimensional space, as well as in the development of algorithms for 3D graphics and simulations.

5. Are there any real-world examples of points in S^3 being spaced equally?

Yes, there are several real-world examples where points in S^3 are spaced equally. One example is the distribution of stars in a galaxy, where stars can be considered as points on a three-dimensional sphere. Another example is the arrangement of atoms in a crystal lattice, where the atoms are evenly spaced on a three-dimensional grid.

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