45:["$","title",null,{"children":"Unwinding the [Ulam Spiral](https://en.wikipedia.org/wiki/Ulam_spiral) | montyxcantsin | Observable"}] 4d:T40a,daena = { //for the record, for those who care to read the source //I found this on paper at the end of november of 2017 //because my wife(daena) was fascinated by the ulam spiral //and I saw a glint in her eye for the first time in a long time //as she had just finished having a terrible year //and another one was coming (thanks embrace) //so this was and is for her! //also I walked away from javascript years ago so be gentle //observable is awesome though and I am sure this can be built //in much better ways! const canvas = document.createElement("canvas"); canvas.width = 1000; var localn = mutable n; canvas.height = localn * 5; const context = canvas.getContext("2d"); const rows = maskPrimeTest(localn); //this is how many rows you want, 75 just looks nice const params = [context, R.length(R.last(rows)), rows]; //stage.addEventListener("click", R.partial(clickStage, R.append(planchet, params))); mapIndexed(R.partial(createRow, params), rows); return canvas; }4e:T608,md`Initially this was discovered on paper by looking for patterns in the primes within a concentric (hexagonal in this case) configuration of numbers. The first row:

The second row:

7	8
9	10
11	12
13	14
15	16
17	18
ON	OFF

The third row:

19	20	21
22	23	24
25	26	27
28	29	30
31	32	33
34	35	36
ON	ON	OFF

The fourth row:

37	38	39	40
41	42	43	44
45	46	47	48
49	50	51	52
53	54	55	56
57	58	59	60
ON	OFF	ON	OFF

`4f:[] 49:{"id":"dc3cfff0bece89a7","slug":"unwinding-the-ulam-spiral","title":"Unwinding the [Ulam Spiral](https://en.wikipedia.org/wiki/Ulam_spiral)","thumbnail":"31b99039c038a07424a33024c5106a17cb86746e00a010bba2bf7a86dd379916","access_level":"public","create_time":"$D2018-03-24T19:46:45.259Z","update_time":"$D2018-05-19T19:58:13.759Z","view_time":null,"first_public_time":"$D2018-03-26T03:15:02.551Z","has_unpublished_edits":null,"fork":false,"stars":6,"starred":false,"owner":{"type":"team","id":"f54f35b12ca6e6f9","login":"montyxcantsin","name":"","avatar_url":"https://avatars-next.observableusercontent.com/b2559328c151508c8fd83e2d85ab4644b35f1e7aa7d66a110ab597d6925e6fe3","tier":"starter_2024","bio":"","followed_by_you":false,"follows_you":false},"version":504,"publish_version":504,"role":null,"authored":false,"authors":[{"type":"individual","id":"d41bfab2835e2f76","login":"montyxcantsin","name":"","avatar_url":"https://avatars-next.observableusercontent.com/b2559328c151508c8fd83e2d85ab4644b35f1e7aa7d66a110ab597d6925e6fe3","tier":null,"bio":"","followed_by_you":false,"follows_you":false}],"members":[{"type":"individual","id":"d41bfab2835e2f76","login":"montyxcantsin","name":"","avatar_url":"https://avatars-next.observableusercontent.com/b2559328c151508c8fd83e2d85ab4644b35f1e7aa7d66a110ab597d6925e6fe3","tier":null,"role":"editor"}],"comments":[],"custom_thumbnail":false,"trashed":false,"lapsed":null,"subscribed":null,"forks":11,"collection":null,"type":"notebook","body":{"titling":"auto","title":"Unwinding the [Ulam Spiral](https://en.wikipedia.org/wiki/Ulam_spiral)","cells":[{"id":0,"value":"md`# Unwinding the [Ulam Spiral](https://en.wikipedia.org/wiki/Ulam_spiral)`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":435,"value":"mutable n = 74","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":499,"value":"md`Feel free to adjust the above but know that for n > 128 the primality testing becomes slower.`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":301,"value":"$4d","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":306,"value":"md`This is created by layering quadrants of the ulam spiral on top of each other and then testing for primality for each of the layered cells. If there is one or more prime terms we color it black. I call this *parallax compression*.` //parallax was my old band too funnily enough ","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":334,"value":"md`Here is a link to a [mathematica version](https://www.dropbox.com/s/d2dfwhxdmzkp4y4/a-pattern-in-the-primes.nb?dl=0) version which is more accurate and contains things like skipping even or odd rows etc.`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":311,"value":"md`note from [Nova Spivack](http://www.novaspivack.com) and [@novaspivack](http://twitter.com/novaspivack):\n>After many conversations with montyxcantsin, who originated the parallax compression algorithm, I analyzed the resulting pattern. On January 18th, 2018, I found that the sequence of cell values matches sequence [OEIS A054521](http://oeis.org/A054521) -- enumerating from the top down, left to right, where 1 = prime (black), 0 = non-prime (red). Also: montyxcantsin figured out that the number of prime (black) cells per row is the sequence [OEIS A000010](http://oeis.org/A000010) (Euler Totient), and the number of red cells per row is the co-totient.\n`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":339,"value":"md`A note on the note: the fit of the GCD triangle sequence is not perfect. This led me to revert to the method used in mathematica version which does do primality testing for all terms per cell. This yields some even more interesting features e.g. the odd and even differences.`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":262,"value":"$4e","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":139,"value":"size = 5","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":102,"value":"createCell = (context, padding, row, state, col) => {\n context.fillStyle = state == 0 ? \"red\" : \"black\";\n context.fillRect(size * (padding + col), row * size, size, size);\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":152,"value":"R = require(\"https://cdn.jsdelivr.net/npm/ramda@0.25/dist/ramda.min.js\")","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":155,"value":"mapIndexed = R.addIndex(R.map)","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":104,"value":"createRow = (stage, maxWidth, rows, cells, row) => {\n var padding = (maxWidth - 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2)))\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":68,"value":"oblongNumber = (n) => n * (n + 1)","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":70,"value":"squaredNumber = (n) => n * n","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":60,"value":"cellToTerms = (n, sides, cellCountFn, x) => {\n //n is row, x is col - historically named\n var bc = cellCountFn(n);\n var start = (bc * sides) + 1;\n var terms = [];\n if (x > 1) {\n start += x - 1;\n }\n for(var i = 0; i < sides; i++) {\n terms.push(start + (i * n));\n }\n return terms;\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":11,"value":"GCD = (x, y) => {\n if (x == 0) {\n return y;\n } else {\n while (y != 0) {\n if (x > y) {\n x -= y;\n } else {\n y -= x;\n }\n }\n return x;\n }\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":13,"value":"T = (n, k) => {\n return (GCD(n, k) == 1) ? 1 : 0;\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":323,"value":"cellToTerms(17, 30, binomialCoEfficient2, 7).map(isPrime)","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"}],"files":"$W4f","stdlib":"1"}} 44:null