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
(3

2
,1

2
)(3

2
,-3

2
)(1

6
,-5

6
)(-1

4
,0)(0,1

2
)
61

24
7

2
+4

3
5
5(-1,1)6
(-2,1)(-1,0)(-1

4
,0)(0,1

2
)(0,7

3
)(-1

2
,5

2
)
157

48
31

12
+5

2
2+1

4
5+1

6
10
23(0,-2)4
(1

6
,-5

6
)(-3

2
,-5

2
)(-1,-4)(3

2
,-3

2
)
25

6
25

6
2+2

3
5+1

2
10
61(0,4)7
(0,5)(-3

2
,7

2
)(-1

2
,5

2
)(0,7

3
)(1

2
,5

2
)(1,3)(1,5)
55

12
3+32+1

3
10
67(-4,2)8
(-5,2)(-5,1)(-9

2
,1

2
)(-4,1

3
)(-7

2
,1

2
)(-3,1)(-3,2)(-4,3)
23

6
2+32+1

3
10
313(3,9)3
(1,8)(4,11)(4,7)
6
4+32+10
2837(27,1)9
(28,-1)(26,-1)(74

3
,-1

3
)(122

5
,1)(49

2
,3

2
)(53

2
,7

2
)(28,19

5
)(321

11
,34

11
)(117

4
,11

4
)
3647

220
2+22+2

3
5+5

4
10+97

132
26+13

55
34
8741(47,45)10
(46,44)(46,47)(526

11
,527

11
)(97

2
,95

2
)(99

2
,93

2
)(405

8
,357

8
)(862

17
,749

17
)(50,43)(48,43)(93

2
,87

2
)
25693

1496
5+259

136
2+10

11
5+1

2
10+6

17
13+45

88
34