*After finishing writing this post, I realized that I may have already written this exact same post a couple weeks ago. I couldn't find it though, so I'm posting this. This one is probably better anyway.*
So I was trying to make a brute force approach to figuring out different combinations of regular polygons that fit around a point. (For example, 3 hexagons, or a triangle and 2 dodecahedrons) So into Mathematica I went.

First I tried this:

FindInstance[**(180 - (360/x)) + (180 - (360/y)) + (180 - (360/z)) + (180 - (360/w)) == 360**, {x, y, z, w}, Integers, 100]

In normal person language, this finds 100, 4 integer combinations that fulfill that equation up there where the 4 integers are x, y, z, and w. The thinking behind this equation is this: 180 - (360/x) is the formula for the interior angle of a regular polygon with x number of sides. So, in fact, this equation only solves for combinations of polygons with a maximum of 4 polygons.

Anyways, this didn't work. If you don't understand this equation, message me up in the comments. I think the main reason why this made Mathematica go crazy was the it overflowed. So I added some minimum and maximums for the number of sides.

FindInstance[(180 - (360/x)) + (180 - (360/y)) + (180 - (360/z)) + (180 - (360/w)) == 360 && x > min && y > min && z > min && w > min && x < max && y < max && z < max && w < max, {x, y, z, w}, Integers, 100]

You don't really need to understand this. All you need to know is that I put restrictions on the number of sides for each polygon so that the computer didn't explode.

After a couple more trivial little Mathematica troubles I finally got it to work. This is what is output:

{{2, 3, 10, 15}, {4, 6, 12, 2}, {15, 10, 3, 2}, {6, 3, 4, 4}, {6, 6, 2,

6}, {3, 8, 2, 24}, {5, 10, 5, 2}, {20, 5, 2, 4}, {3, 10, 2, 15}, {2, 9, 3,

18}, {12, 2, 3, 12}, {12, 6, 2, 4}, {4, 8, 2, 8}, {6, 4, 3, 4}, {5, 20, 4,

2}, {4, 8, 8, 2}, {4, 12, 3, 3}, {4, 20, 2, 5}, {5, 10, 2, 5}, {10, 3, 2,

15}, {12, 4, 3, 3}, {15, 3, 2, 10}, {3, 4, 3, 12}, {4, 12, 2, 6}, {5, 5, 10,

2}, {4, 2, 20, 5}, {10, 2, 15, 3}, {4, 20, 5, 2}, {2, 12, 12, 3}, {15, 2,

3, 10}, {9, 3, 18, 2}, {4, 4, 6, 3}, {4, 2, 12, 6}, {8, 2, 4, 8}, {2, 4, 6,

12}, {3, 3, 12, 4}, {10, 3, 15, 2}, {5, 5, 2, 10}, {12, 2, 6, 4}, {3, 2, 12,

12}, {2, 4, 12, 6}, {2, 8, 3, 24}, {4, 6, 3, 4}, {6, 6, 6, 2}, {3, 2, 18,

9}, {2, 8, 24, 3}, {3, 4, 4, 6}, {2, 20, 5, 4}, {3, 18, 9, 2}, {18, 9, 2,

3}, {5, 2, 10, 5}, {18, 3, 2, 9}, {18, 2, 9, 3}, {12, 4, 2, 6}, {18, 9, 3,

2}, {2, 6, 12, 4}, {12, 3, 4, 3}, {8, 2, 8, 4}, {2, 12, 3, 12}, {6, 3, 3,

6}, {2, 5, 10, 5}, {6, 6, 3, 3}, {2, 5, 4, 20}, {24, 3, 8, 2}, {3, 24, 8,

2}, {2, 15, 10, 3}, {2, 10, 15, 3}, {6, 4, 4, 3}, {3, 6, 3, 6}, {4, 5, 20,

2}, {2, 10, 5, 5}, {2, 12, 4, 6}, {10, 15, 3, 2}, {5, 2, 5, 10}, {6, 2, 6,

6}, {8, 2, 24, 3}, {2, 5, 5, 10}, {3, 6, 6, 3}, {8, 3, 2, 24}, {24, 8, 3,

2}, {8, 24, 3, 2}, {4, 3, 12, 3}, {3, 4, 6, 4}, {8, 8, 4, 2}, {3, 12, 4,

3}, {4, 12, 6, 2}, {9, 2, 18, 3}, {2, 15, 3, 10}, {18, 2, 3, 9}, {12, 2, 4,

6}, {12, 3, 2, 12}, {4, 6, 2, 12}, {24, 3, 2, 8}, {2, 6, 6, 6}, {9, 3, 2,

18}, {6, 2, 12, 4}, {2, 12, 6, 4}, {10, 2, 5, 5}, {2, 4, 20, 5}, {2, 18, 9,

3}}

These are 100 different combinations. For instance, the first combination--{2,3,10,15} is saying a triangle, a dodecahedron, and a 15-gon will fit around a point. The 2 represents a line, which takes up 0 degrees around angle. This is caused by me using 4 variables. In other words, just ignore any 2s.

Question: are there infinite combinations?

It would be nice to organize this data a little bit better. For instance, in this case, order *does *matter, but it shouldn't. I'd also like to think about writing a process like this but for full tilings. It might be a lot more difficult though. Also, making Mathematica output graphics would be nice.