Watch the evolution of the Ulam Spiral

In 1963, Stanislaw Ulam was attending a talk, during which he became bored. He began to doodle on some paper, but his doodling soon gained some purpose. He drew a grid, and started labeling the positions in his grid with numbers, starting in the center and spiraling outward.

After a bit of this, he circled the prime numbers in his spiral. He noticed a remarkable pattern. Do you see it?

111	112	113	114	115	116	117	118	119	120	121
110	73	74	75	76	77	78	79	80	81	82
109	72	43	44	45	46	47	48	49	50	83
108	71	42	21	22	23	24	25	26	51	84
107	70	41	20	7	8	9	10	27	52	85
106	69	40	19	6	1	2	11	28	53	86
105	68	39	18	5	4	3	12	29	54	87
104	67	38	17	16	15	14	13	30	55	88
103	66	37	36	35	34	33	32	31	56	89
102	65	64	63	62	61	60	59	58	57	90
101	100	99	98	97	96	95	94	93	92	91

The prime numbers seem to cluster along diagonal lines. This pattern persists remarkably even if the spiral is quite different from this one. See this Wikipedia article for a circular version and this page on Wolfram's site for a hexagonal one.

Actually, there are lots of patterns in this grid. Here is another one:

Ulam was a scientist at the Los Alamos National Laboratory, where he had access to some of the best computers available in the world at that time. When he returned home from the trip during which he had made this discovery, he programmed a computer to draw larger grids, to see if the pattern persisted.

Here is what he saw:

(Watch in the section below.)

Stanislaw Ulam's Prime Number Spiral

Watch the evolution of the Ulam Spiral

Spectra

Ulam's Spiral

111	112	113	114	115	116	117	118	119	120	121
110	73	74	75	76	77	78	79	80	81	82
109	72	43	44	45	46	47	48	49	50	83
108	71	42	21	22	23	24	25	26	51	84
107	70	41	20	7	8	9	10	27	52	85
106	69	40	19	6	1	2	11	28	53	86
105	68	39	18	5	4	3	12	29	54	87
104	67	38	17	16	15	14	13	30	55	88
103	66	37	36	35	34	33	32	31	56	89
102	65	64	63	62	61	60	59	58	57	90
101	100	99	98	97	96	95	94	93	92	91

111	112	113	114	115	116	117	118	119	120	121
110	73	74	75	76	77	78	79	80	81	82
109	72	43	44	45	46	47	48	49	50	83
108	71	42	21	22	23	24	25	26	51	84
107	70	41	20	7	8	9	10	27	52	85
106	69	40	19	6	1	2	11	28	53	86
105	68	39	18	5	4	3	12	29	54	87
104	67	38	17	16	15	14	13	30	55	88
103	66	37	36	35	34	33	32	31	56	89
102	65	64	63	62	61	60	59	58	57	90
101	100	99	98	97	96	95	94	93	92	91

111	112	113	114	115	116	117	118	119	120	121
110	73	74	75	76	77	78	79	80	81	82
109	72	43	44	45	46	47	48	49	50	83
108	71	42	21	22	23	24	25	26	51	84
107	70	41	20	7	8	9	10	27	52	85
106	69	40	19	6	1	2	11	28	53	86
105	68	39	18	5	4	3	12	29	54	87
104	67	38	17	16	15	14	13	30	55	88
103	66	37	36	35	34	33	32	31	56	89
102	65	64	63	62	61	60	59	58	57	90
101	100	99	98	97	96	95	94	93	92	91