Divide a Sphere into Many Shapes

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To divide a sphere into many shapes, a dodecahedron is a starting point with 12 pentagonal faces, but it may not provide enough divisions. An icosahedron, which has 20 triangular faces, offers a more rounded option. For non-regular solids, the number of faces can be increased significantly, with a myriahedron being a possibility. Geodesic domes are also suggested as a creative approach for achieving multiple divisions. Exploring these geometric shapes can help meet the division requirements for the sphere.
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Hi a bit random this question, but I'm trying to divide a sphere into many equal shapes (round-ness isn't important). So far I've come up with a dodecahedron, 12 pentagons. This isn't enough of a division for my needs. Can anyone give me the names of geometric shapes that have more divisions please. I found an image of a hexadecagon, but this only showed the divisions for a circle, so I could see the spherical division shape. I hope I've made enough sense for someone to understand what I'm after. It's not homework by the way.

Thanks
 
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It's really hard for me to guess your meaning. You're looking for a shape of some kind, one with many sides, I take it. But I don't know what "divisions for a circle" means, nor "spherical division shape".

If your solid needs to be regular, the only step up from a dodecahedron to a "rounder" figure is an icosahedron, a 20-sided figure with triangles for faces. If you want the individual faces to be as "round" as possible, then a dodecahedron is the best regular solid; the others have triangles or squares as faces.

If you don't need the solid to be regular, you can have as many faces as you want. A myriahedron is possible, if desired...
 
Here is a little puzzle from the book 100 Geometric Games by Pierre Berloquin. The side of a small square is one meter long and the side of a larger square one and a half meters long. One vertex of the large square is at the center of the small square. The side of the large square cuts two sides of the small square into one- third parts and two-thirds parts. What is the area where the squares overlap?

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