# Voronoi Diagrams of the Ulam Prime Spiral

The table below shows the first prime with an n-gonal voronoi cell, for n from 3 to 10. There were no 11-gonal voronoi cells for any prime less than 10000. If you aren't sure what all this means, check out the explanation on the main page on this topic.

p"Centre"#VerticesVertex ListAreaPerimeter
2(1,0)5
 ( 32 , 12 ) , ( 32 , - 32 ) , ( 16 , - 56 ) , ( - 14 , 0 ) , ( 0 , 12 )
 6124
 72 + 43 �5
5(-1,1)6
 ( - 2 , 1 ) , ( - 1 , 0 ) , ( - 14 , 0 ) , ( 0 , 12 ) , ( 0 , 73 ) , ( - 12 , 52 )
 15748
 3112 + 52 �2 + 14 �5 + 16 �10
23(0,-2)4
 ( 16 , - 56 ) , ( - 32 , - 52 ) , ( - 1 , - 4 ) , ( 32 , - 32 )
 256
 256 �2 + 23 �5 + 12 �10
61(0,4)7
 ( 0 , 5 ) , ( - 32 , 72 ) , ( - 12 , 52 ) , ( 0 , 73 ) , ( 12 , 52 ) , ( 1 , 3 ) , ( 1 , 5 )
 5512
 3 + 3 �2 + 13 �10
67(-4,2)8
 ( - 5 , 2 ) , ( - 5 , 1 ) , ( - 92 , 12 ) , ( - 4 , 13 ) , ( - 72 , 12 ) , ( - 3 , 1 ) , ( - 3 , 2 ) , ( - 4 , 3 )
 236
 2 + 3 �2 + 13 �10
313(3,9)3
 ( 1 , 8 ) , ( 4 , 11 ) , ( 4 , 7 )
 6
 4 + 3 �2 + �10
2837(27,1)9
 ( 28 , - 1 ) , ( 26 , - 1 ) , ( 743 , - 13 ) , ( 1225 , 1 ) , ( 492 , 32 ) , ( 532 , 72 ) , ( 28 , 195 ) , ( 32111 , 3411 ) , ( 1174 , 114 )
 3647220
 2 + 2 �2 + 23 �5 + 54 �10 + 97132 �26 + 1355 �34
8741(47,45)10
 ( 46 , 44 ) , ( 46 , 47 ) , ( 52611 , 52711 ) , ( 972 , 952 ) , ( 992 , 932 ) , ( 4058 , 3578 ) , ( 86217 , 74917 ) , ( 50 , 43 ) , ( 48 , 43 ) , ( 932 , 872 )
 256931496
 5 + 259136 �2 + 1011 �5 + 12 �10 + 617 �13 + 4588 �34