Can one "chop" up S^3 into little cubes?

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In summary, the conversation discusses the possibility of dividing a three-sphere into small cubes or hexahedra with equal side lengths and angles. It is suggested that this can be achieved by projecting a grid on the faces of a unit cube and then projecting it onto the three-sphere. However, it is noted that the smaller cubes may not have equal side lengths and angles.
  • #1
Spinnor
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Say I live in a very large, fixed radius, three-sphere. Could a mathematician pick a very large set of points at the right locations in my three-sphere such that when each point was connected to their nearest 6 neighbors with short line segments that locally I would have a nearly rectangular 3 dimensional grid that would also be uniform in the sense that from any point the grid would look the same, lengths of line segments all the same and angles between nearest line segments nearly or exactly 90 degrees?

Would be some kind of translation in-variance?

Would this be in effect "chopping" up a three-sphere into little cubes? Is there a smallest number of cubes that a three-sphere can be chopped into or do we only get cubes when the number of cubes is very large?

Thanks for any help!
 
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  • #2
90 degree corners will probably not work, but if all you want to do is divide ##S^3## into hexahedra, then I think the answer is yes. The answer to the analogous question for ##S^2## (dividing it into quadrilaterals) is yes, as follows:

Consider a unit cube, and a sphere circumscribing it, such that they intersect in the 8 corners of the cube. Now project the edges of the cube up onto the sphere. The sphere is now divided into 6 squares, which are quadrilaterals of equal side lengths and equal angles (which will, in this case, be 120 degrees). If you want to divide the sphere into smaller "squares", then put a grid on the faces of your cube, and project that up as well. But the smaller quadrilaterals will not have equal side lengths and equal angles.

You can imagine a similar process should work for ##S^3##; just start with a hypercube and its circumscribing 3-sphere.
 

What is S^3?

S^3 is a mathematical notation for a three-dimensional sphere, also known as a 3-sphere. It is a common object of study in geometry and topology.

What does it mean to "chop" up S^3?

In mathematics, "chopping" up an object refers to dividing it into smaller, more manageable pieces. In this case, it means dividing the 3-sphere into smaller cubes.

Why would someone want to chop up S^3 into little cubes?

Chopping up S^3 into little cubes can help us better understand its geometric and topological properties. It can also make it easier to perform calculations and analyze its structure.

Is it possible to completely chop up S^3 into little cubes?

Yes, it is possible to chop up S^3 into little cubes. This is known as a cube decomposition of S^3 and is a common exercise in topology and geometry.

What is the significance of being able to chop up S^3 into little cubes?

The ability to chop up S^3 into little cubes has significant implications in various fields of mathematics. It allows us to study and understand the 3-sphere in a more systematic and organized manner, leading to new insights and discoveries.

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