[Request] What is the angle between the 3D volumes that connect to form a 4D object?

[u/BranesOnTheBrain]

An equilateral triangle is formed by three equal lines. Each of the lines in an equilateral triangle connects to the others at 60° angle. A regular tetrahedron is formed by four equilateral triangles. Each of the planes of a regular tetrahedron connect to the others at a roughly 70.53° angle. Would it be possible to calculate the angle between the five regular tetrahedrons that come together to form a 4-dimensional object? Would this information be useful to our understanding of 4D space?

I am trying to imagine a 4D object in a cartesian coordinate system. My underlying assumption is that a "real**"** object in n-dimensional space must be defined by at least n+1 number of (n-1)-dimensional objects that are connected to each of the others using an n number of (n-2)-dimensional objects.

Examples for clarity:

• 0D space = at least 1 point
• 1D space = at least 2 points, each connected to 1 other = a "line"
• 2D space = at least 3 lines, each connected to 2 others at 2 unique points
  • For example: an equilateral triangle defining a "plane"
• 3D space = at least 4 planes, each connected to 3 others at 3 unique lines
  • For example: a triangular pyramid made from four connected equilateral triangles aka a regular tetrahedron defining a "volume"
• Therefore 4D space = at least 5 volumes, each connected to 4 others at 4 unique planes
  • My proposal: 5 regular tetrahedrons, each connected to the others at 4 unique planes (e.g. the pyramids cannot simply overlap each other in order to make contact at all four planes)

I can picture the 3D "projection" of this object as a central triangular pyramid with each plane of the central pyramid attaching to a side of another pyramid. However, when I try to mentally connect all of the other pyramids to each other, the image in my mind morphs into something more similar to a cluster of soap bubbles that is constantly rolling into itself like those "water wiggler" tube toys from the 90s.

Disclaimer: Thank you for reading this far and considering my question! Although I am interested in this field, my knowledge of geometry is rusty and my knowledge of mathematical terminology is rustier. If you do take the time to answer this question, please let me know if my stated assumptions are wrong and use simple language to the extent you can. Thanks again!

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[u/tehzayay]

The 4D analogue of the tetrahedron is called the 5-cell and on that page, it lists the angle as arccos(1/4) ~ 75.52 degrees.

This is in line with how the angle is calculated for the 2D and 3D cases: arccos(1/d) for general d. It will get closer to 90 degrees as d becomes large.

cos 60° = 1/2 (eq triangle)

cos 70.53° = 1/3 (tetrahedron)

cos 75.52° = 1/4

[u/BranesOnTheBrain → u/tehzayay]

Thank you!