Voronoi Diagrams of the Ulam Prime Spiral


The table below shows the primes that have 'less round' Voronoi cells than any smaller prime. I conjecture that this sequence is an infinite sequence - that is, there's no 'least round' voronoi cell. If you aren't sure what all this means, check out the explanation on the main page on this topic.













 
p"Centre"#VerticesVertex ListAreaPerimeter(P/2)2/A
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
4.132018689726583
11(2,0)5
(3
2
,1
2
), (3
2
,-3
2
), (3,-2), (3,0), (2,1)
7
2
4+3
2
�2+1
2
�10
4.237705522975003
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
4.821635921280641
31(3,3)4
(2,3), (3,2), (4,2), (4,5)
7
2
4+3�2
4.852937535496735
47(1,-3)4
(-1,-4), (3
2
,-3
2
), (3,-2), (1
2
,-9
2
)
5
5�2+�10
5.236067977499791
181(7,5)4
(7,4), (14
3
,19
3
), (6,7), (9,4)
16
3
2+16
3
�2+2
3
�5
5.706147946283363
229(8,-4)4
(6,-5), (9,-2), (9,-4), (8,-5)
4
4+4�2
5.82842712474619
1669(8,-20)3
(7,-21), (7,-18), (10,-21)
9
2
6+3�2
5.828427124746191
2153(-23,-13)5
(-24,-13), (-24,-15), (-62
3
,-35
3
), (-167
8
,-85
8
), (-22,-11)
143
24
2+16
3
�2+3
8
�10+5
24
�26
5.832955450142133
4603(-12,34)6
(-14,35), (-11,32), (-10,33), (-14,37), (-43
3
,37), (-29
2
,73
2
)
91
12
1
3
+8�2+2
3
�10
6.2375692795404465
5861(-30,-38)4
(-32,-39), (-28,-35), (-83
3
,-110
3
), (-30,-39)
19
3
2+19
3
�2+1
3
�26
6.32302992683966