3f:null 40:["$","title",null,{"children":"Primordial Particle System | Martin Milan | Observable"}] 46:T4ef,md` The Primordial Particle System is, as the name implies, a model which has a simple rule defining each particle's motion and produces behavior that may be considered life-like. This system was originally described in a paper that can be viewed [here](https://www.nature.com/articles/srep37969). The basis for the model is fairly simple. It begins with a collection of particles with each particle having both a *ϕ*. Each particle advances in its direction at a constant speed. Its direction, however, changes. The parameters of angle *α* and angle *β* define this change in direction. First, a particle's direction is shifted by *α*. This shift in regards to *α* is same for all particles whereas the *β* shift depends on the other particles around it. Specifically, the particles in a circle of a certain radius *r* centered on the particle currently being considered are the only particles used in the calculations. The magnitude of the shift is *N* times *β* where *N* is the number of particles encompassed by the circle. The direction of the shift is either to the left or right of its current direction depending on which side more particles are situated in the circle. Expressed as a differential equation, this is:`47:T540,md`Here, *dϕ* represents the change in a particle's direction, *R* represents the number of particles on the right side, and *L* represents the number of particles on the left side. It is also assumed that a positive increase in *ϕ* corresponds to the right direction. To further illustrate this point, consider this picture:

In this picture, the green particle is under consideration. Also shown is its circle with a radius of *r* and its direction of *ϕ*. Within the circle there are 8 particles, so *N* equals 8 in this example. Additionally, we see that five of these particles are positioned left of the direction, and three are positioned right of the direction. Therefore, the direction will shift counter clockwise, towards the left side where there are more particles, and the magnitude will be *α* - 8*β*. The sign of the *β* term is negative because *R* - *L* is negative. To see it action, play around with the following demonstration. Run the model by hovering your mouse over the window, and the model can be restarted by double clicking the window. The particles in the model are colored based on the number of particles in their circles, and the particles have trails so that their motion is more visible` 48:T4cc,async function* createSystem(type, n, nb, alpha, beta, radius, speed) { let height = width/1.75; let ctx = DOM.context2d(width, height); let ps = randomizeParticles(n, width, height); if (type == 'point') { ps = makeBlob(n, width/2, height/2) } alpha = toRadians(alpha); beta = toRadians(beta); let paused = true; ctx.canvas.onmouseenter = async function() { paused = false; } ctx.canvas.onmouseleave = function() { paused = true; } ctx.canvas.onclick = function(event) { ps = ps.concat(makeBlob(nb, event.offsetX, event.offsetY)); } ctx.canvas.ondblclick = function() { if (type == 'random') { ps = randomizeParticles(n, width, height); } else { ps = makeBlob(n, width/2, height/2) } } let greatest = 0; drawParticles(ps, ctx, 10); yield ctx.canvas; while (true) { await Promises.delay(msPerFrame); if (!paused) { let x = loopParticles(ps, alpha, beta, radius); if (x > greatest) { greatest = x; } ctx.fillStyle = "hsla(0, 100%, 100%," +trailOpacity+")"; ctx.fillRect(0, 0, width, height); moveParticles(ps, speed, width, height); drawParticles(ps, ctx, x); yield ctx.canvas; } } }49:[] 44:{"id":"8d9336d92e915900","slug":"primordial-particle-system","title":"Primordial Particle System","thumbnail":"c097a513c53cd679b5bafab2fdce6817c804e7d1695b0f1e77a8db8a19f552c1","access_level":"public","create_time":"$D2020-03-28T16:54:29.575Z","update_time":"$D2020-03-28T16:54:29.575Z","view_time":null,"first_public_time":"$D2020-03-31T12:03:14.601Z","has_unpublished_edits":null,"fork":true,"stars":1,"starred":false,"owner":{"type":"team","id":"6d3bbae82acc2d78","login":"martin12333","name":"Martin Milan","avatar_url":"https://avatars-next.observableusercontent.com/c7304c0d899196f1b61b2b1a83be600a6aac516768e86aed9ab1dd095612ca83","tier":"starter_2024","bio":"","following":false},"version":1836,"publish_version":1836,"role":null,"authored":false,"authors":[{"type":"individual","id":"c1cfd494ec60e2b3","login":"martin12333","name":"Martin Milan","avatar_url":"https://avatars-next.observableusercontent.com/c7304c0d899196f1b61b2b1a83be600a6aac516768e86aed9ab1dd095612ca83","tier":null,"bio":"","following":false}],"members":[{"type":"individual","id":"c1cfd494ec60e2b3","login":"martin12333","name":"Martin Milan","avatar_url":"https://avatars-next.observableusercontent.com/c7304c0d899196f1b61b2b1a83be600a6aac516768e86aed9ab1dd095612ca83","tier":null,"role":"editor"}],"comments":[],"custom_thumbnail":false,"trashed":false,"lapsed":null,"subscribed":null,"forks":1,"collection":null,"type":"notebook","body":{"titling":"auto","title":"Primordial Particle System","cells":[{"id":1206,"value":"md `# Primordial Particle System`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":0,"value":"$46","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":791,"value":"tex.block`\n\\frac{d\\phi}{dt}=\\alpha + N\\cdot\\beta\\cdot\\text{sign(R - L)}\n`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":815,"value":"$47","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":1812,"value":"createSystem(pgType, pgN, 0, pgAlpha, pgBeta, pgRadius, pgSpeed)","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":1744,"value":"viewof dd = select({\n title: \"Default Systems\",\n options: Object.keys(defaultSystem),\n value: \"Amoeba\"\n})","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":1480,"value":"viewof pgType = radio({\n title: 'System Type',\n options: [\n { label: 'Randomized', value: 'random' },\n { label: 'Cluster', value: 'point' },\n ],\n value: `${this ? defaultSystem[dd][0] : defaultSystem[\"Amoeba\"][0]}`\n})","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":1462,"value":"viewof pgN = slider({\n min: 1, \n max: 800, \n step: 1, \n value: `${this ? defaultSystem[dd][1] : defaultSystem[\"Amoeba\"][1]}`, \n title: \"Number of Particles\", \n})","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":1465,"value":"viewof pgAlpha = slider({\n min: 0, \n max: 180, \n step: 1, \n value: `${this ? defaultSystem[dd][2] : defaultSystem[\"Amoeba\"][2]}`, \n title: \"Alpha\", \n})","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":1467,"value":"viewof pgBeta = slider({\n min: -60, \n max: 60, \n step: 1, \n value: `${this ? defaultSystem[dd][3] : defaultSystem[\"Amoeba\"][3]}`, \n title: \"Beta\", \n})","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":1472,"value":"viewof pgRadius = slider({\n min: 10, \n max: 200, \n step: 1, \n value: `${this ? defaultSystem[dd][4] : defaultSystem[\"Amoeba\"][4]}`, \n title: \"Radius\", \n})","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":1474,"value":"viewof pgSpeed = slider({\n min: 1, \n max: 12, \n step: 0.1, \n value: `${this ? defaultSystem[dd][5] : defaultSystem[\"Amoeba\"][5]}`, \n title: \"Speed\", \n})","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":1401,"value":"md `## References \n\n[Schmickl, T. et al. How a life-like system emerges from a simple particle motion law. Sci. Rep. 6, 37969; doi: 10.1038/srep37969 (2016).](https://www.nature.com/articles/srep37969)`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":123,"value":"md`## Appendix`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":1460,"value":"import {slider, radio, select} from \"@jashkenas/inputs\"","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":397,"value":"$48","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":125,"value":"function randomizeParticles(n, w, h) {\n let ps = [];\n for (let i = 0; i < n; i++) {\n let p = Array(4);\n p[0] = Math.random() * w; \n p[1] = Math.random() * h; \n p[2] = Math.random() * 2 * Math.PI; \n p[3] = 0;\n ps.push(p);\n }\n return ps;\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":437,"value":"function makeBlob(n, x, y) {\n let ps = [];\n for (let i = 0; i < n; i++) {\n let p = Array(4);\n p[0] = x; \n p[1] = y; \n p[2] = Math.random() * 2 * Math.PI; \n p[3] = 0;\n ps.push(p);\n }\n return ps;\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":139,"value":"function drawParticles(ps, ctx, greatest) {\n for (let i = 0; i < ps.length; i++) {\n ctx.beginPath();\n ctx.arc(ps[i][0], ps[i][1], particleRadius, 0, 2 * Math.PI);\n ctx.fillStyle = \"hsl(\"+ps[i][3]/greatest*310+\", 100%, 30%)\";\n ctx.fill();\n ctx.closePath();\n }\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":171,"value":"function loopParticles(ps, alpha, beta, radius) {\n let greatest = 1;\n for (let i = 0; i < ps.length; i++) {\n let left = 0; let right = 0; let N = 0; ps[i][3] = 0;\n for (let j = 0; j < ps.length; j++) {\n let distance = Math.hypot(ps[i][0] - ps[j][0], ps[i][1] - ps[j][1]);\n if (i != j && distance < radius) {\n N++;\n ps[i][3]++;\n let angle = Math.atan2(Math.sin(ps[i][2]),Math.cos(ps[i][2]))-Math.atan2(ps[j][1]-ps[i][1],ps[j][0]-ps[i][0]);\n if (angle > Math.PI) { angle -= 2 * Math.PI; }\n else if (angle < -Math.PI) { angle += 2 * Math.PI; }\n if (angle > 0) { left++; }\n else { right++; }\n }\n }\n \n if (ps[i][3] > greatest) {\n greatest = ps[i][3];\n }\n \n let phi = alpha + N * beta * Math.sign(right - left);\n ps[i][2] += phi;\n }\n return greatest;\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":189,"value":"function moveParticles(ps, speed, width, height) {\n for (let i = 0; i < ps.length; i++) {\n ps[i][0] += Math.cos(ps[i][2]) * speed;\n ps[i][1] += Math.sin(ps[i][2]) * speed;\n \n ps[i][0] = fixInterval(ps[i][0] + 10, width + 20) - 10;\n ps[i][1] = fixInterval(ps[i][1] + 10, height + 20) - 10;\n }\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":237,"value":"function fixInterval(x, max) {\n if (x > max) {\n return x % max; \n }\n else if (x < 0) { \n return (x % max) + max; \n }\n return x;\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":186,"value":"function toRadians(x) {\n return x * (Math.PI/180);\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":140,"value":"particleRadius = 3.5","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":943,"value":"trailOpacity = 0.3","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":432,"value":"msPerFrame = 10;","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":1748,"value":"defaultSystem = {\n let ds = {};\n ds[\"Amoeba\"] = ['random', 800, 120, -6, 55, 6];\n ds[\"Birth of a Cell\"] = ['random', 800, 47, 1, 100, 10];\n ds[\"Straight Lines\"] = ['random', 100, 0, 45, 50, 5];\n ds[\"Classic\"] = ['random', 450, 180, 17, 50, 6.7];\n ds[\"Concentric Circles\"] = ['point', 400, 80, -4, 55, 6];\n return ds;\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"}],"files":"$W49","stdlib":"1"}} 45:"$44:body"