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For the purpose of game development, I'm trying to figure out what uniform shapes can be used to create a approximated spherical object.

To give an example of a shape that does not fit my criteria, consider the shapes that compose a soccer ball:

enter image description here

Because the soccer ball has 12 regular pentagonal faces (5 sides) and 20 regular hexagonal faces (6 sides), the shapes are not uniform (they have a different number of sides and a different geometric area).

After a few hours of research online (both here on math.stackoverflow.com and on Google / Wikipedia) the only shape I was able to come across that does match my criteria is the "regular dodecahedron" because it is a sphere-like object composed of only pentagonal faces (5 sides) each with an equal geometric area.

How can I find more geometric shapes that fit my criteria, or what terms could I use to research more in this area?

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  • $\begingroup$ Why not the other way around? Use spheres to approximate objects? $\endgroup$ – mathreadler Feb 10 '18 at 21:47
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    $\begingroup$ What's wrong with a regular icosahedron? $\endgroup$ – Aretino Feb 10 '18 at 21:54
  • $\begingroup$ @mathreadler Please let me know if I misunderstood your thinking, but at a high level the reason I want to use basic geometric shapes to approximate spheres is because in computer graphics everything must be built of triangles (a constraint by modern computer graphics hardware) and I am trying to create a spherical object that would e.g. represent planets. So going the other way around doesn't make sense for me. I need basic uniform shapes approximating a sphere. The uniformity property is desired because the player will interact with them (think of e.g. Minecraft where there are blocks). $\endgroup$ – Stephen Gutekanst Feb 10 '18 at 21:57
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    $\begingroup$ @Aretino Thanks! A regular icosahedron is a good one that definitely matches my criteria. But I'm still interested in how I would find other geometric shapes that match this criteria? (an icosahedron looks a little sharp visually because it is composed of triangles, so out of pure personal preference I don't like it / hope to find other shapes that match my criteria) $\endgroup$ – Stephen Gutekanst Feb 10 '18 at 22:00
  • $\begingroup$ There are tricks to circumnavigate said triangles if you look hard enough $\endgroup$ – mathreadler Feb 11 '18 at 6:23
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You may want to look into Catalan solids. These are polyhedra in which every face is identical, and the larger ones make pretty good approximations to spheres. The faces are not regular, however.

Note however that there are only 13 Catalan solids. If you decide that none of them are a good enough approximation to a sphere for your purposes, you will likely have to relax your uniformity requirement.

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