Intuition check-tiling a 3-sphere with 16 tetrahedra

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In summary, the conversation discusses the concept of tiling a 3-sphere with 16 tetrahedra at 90-degree angles. The individual asking for confirmation is directed to a helpful link on triangulation.
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cephron
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Intuition check--tiling a 3-sphere with 16 tetrahedra

Lately I've been trying to understand/visualize the geometry of a 3-sphere (been hanging out in the cosmology section), and I think I'm getting it, at least to some extent.

Intuitively, I think a 3-sphere can be tiled by 16 tetrahedra whose edges (and faces) meet at 90-degree angles. It seems "obvious" to me, and I can hand-wave an explanation for it, but I can't seem to find direct confirmation online (and, topology/geometry not being my field, I have no idea how to go about proving this in any mathematically sound manner). Can someone confirm that this works, or explain how one could show that it works, or correct me if my intuition has led me horribly astray?

Many thanks!
 
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  • #3


Thank you, Tinyboss! That's exactly what I was looking for. I had been googling with the wrong terminology.
 

1. What is an intuition check-tiling of a 3-sphere with 16 tetrahedra?

An intuition check-tiling is a way of arranging geometric shapes, in this case tetrahedra, to cover a 3-sphere, which is a three-dimensional analog of a sphere. It is called an "intuition" check-tiling because it challenges our intuition about how shapes can fit together in higher dimensions.

2. How is this problem related to mathematics?

This problem falls under the field of mathematics known as topology, which studies the properties of geometric shapes and their relationships. Specifically, this problem involves studying the properties of the 3-sphere and how it can be tiled with tetrahedra.

3. What is the significance of using 16 tetrahedra?

The number 16 is significant because it is the lowest number of tetrahedra known to be able to tile a 3-sphere without any gaps or overlaps. This is a unique and interesting property of the 3-sphere, as it is not possible to tile a sphere in three dimensions with a finite number of identical shapes without gaps or overlaps.

4. How does this problem relate to real-world applications?

While this problem may seem abstract and disconnected from real life, it has important applications in fields such as computer graphics and physics. Tiling methods, such as this one, are used in computer graphics to create 3D models and in physics to study the structure of space-time.

5. Are there any unsolved aspects of this problem?

Yes, there are still unsolved aspects of this problem. While we know that 16 tetrahedra can tile a 3-sphere, it is not yet known if there are any other solutions with a different number of tetrahedra. Additionally, the geometry of these tilings and their relationships to other geometric shapes are still being studied and understood.

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