https://community.wolfram.com RSS Feed for Wolfram Community showing any discussions tagged with Wolfram Science sorted by active. https://community.wolfram.com/groups/-/m/t/2965206 ![Multiway graph][1] &[Wolfram Notebook][2] [1]: https://community.wolfram.com//c/portal/getImageAttachment?filename=Untitled.png&userId=2964459 [2]: https://www.wolframcloud.com/obj/b6ad5a15-e8d2-4359-ae89-b72d3e3d06d4 Eric Archerman 2023-07-13T23:11:59Z https://community.wolfram.com/groups/-/m/t/3701293 &[Wolfram Notebook][1] [1]: https://www.wolframcloud.com/obj/10063044-be48-4242-a0c3-602feb0b9f6e Marco Thiel 2026-04-22T23:43:13Z https://community.wolfram.com/groups/-/m/t/122095 The latest way I have found to use my expensive math software for frivolous entertainment is this. Here's is a way to describe it. [list] [*]1000 dancers assume random positions on the dance-floor. [*]Each randomly chooses one "friend" and one "enemy". [*]At each step every dancer [list] [*]moves 0.5% closer to the centre of the floor [*]then takes a large step towards their friend [*]and a small step away from their enemy. [/list] [*]At random intervals one dancer re-chooses their friend and enemy [/list] Randomness is deliberately injected. Here is the dance... [mcode]n = 1000; r := RandomInteger[{1, n}]; f := (#/(.01 + Sqrt[#.#])) & /@ (x[[#]] - x) &; s := With[{r1 = r}, p[[r1]] = r; q[[r1]] = r]; x = RandomReal[{-1, 1}, {n, 2}]; {p, q} = RandomInteger[{1, n}, {2, n}]; Graphics[{PointSize[0.007], Dynamic[If[r < 100, s]; Point[x = 0.995 x + 0.02 f[p] - 0.01 f[q]]]}, PlotRange -> 2][/mcode] [img]/c/portal/getImageAttachment?filename=OPTfnlfrnds.gif&userId=11733[/img] Thanks to Vitaliy for posting this on my behalf, complete with animations :-) Background: I had read somewhere that macro-scale behaviour of animal swarms (think of flocks of starlings or shoals of herring) is explained by each individual following very simple rules local to their vicinity, essentially 1) try to keep up and 2) try not to collide. I started trying to play with this idea in Mathematica, but it was rather slow to identify the nearest neighbours of each particle. So I wondered what would happen if each particle acted according to the locations of two other particles, regardless of their proximity. The rule was simply to move away from one and towards the other. The contraction (x = 0.995 x) was added to prevent the particle cloud from dispersing towards infinity or drifting away from the origin. I tweaked the "towards" and "away" step sizes to strike a balance between the tendency to clump together and to spread apart (if you make the step sizes equal you get something more like a swarm of flies). With each particle's attractor and repeller fixed, the system finds a sort of dynamic equilibrium, so to keep things changing I added a rule to periodically change the attractor and repeller for one of the particles. The final adjustment was to make the "force" drop towards zero for particles at very close range. This helps to stop the formation of very tight clumps, and also prevents a division-by-zero error when a particle chooses itself as its attractor or repeller. The description of the system as a dance was an attempt to explain the swirling pattern on the screen without using mathematical language. I'd love to see what other "dances" can be created with other simple rules. Simon Woods 2013-09-11T18:31:12Z https://community.wolfram.com/groups/-/m/t/3695987 &[Wolfram Notebook][1] [1]: https://www.wolframcloud.com/obj/12016646-05c7-4e0f-aafc-a180d14e3a8e Ramón Eduardo Chan López 2026-04-19T00:41:46Z https://community.wolfram.com/groups/-/m/t/2026652 ![enter image description here][1] &[Wolfram Notebook][2] [1]: https://community.wolfram.com//c/portal/getImageAttachment?filename=mysuperawesomegraph3336_2.png&userId=2026629 [2]: https://www.wolframcloud.com/obj/dbf470e0-56da-4167-b63d-52f6233ee0c2 Jairo Martin Rojas Huamaní 2020-07-13T23:24:07Z https://community.wolfram.com/groups/-/m/t/3673620 &[Wolfram Notebook][1] [1]: https://www.wolframcloud.com/obj/ad350d33-f532-4a74-ba44-050ece07c9fb Ramón Eduardo Chan López 2026-04-03T07:56:17Z https://community.wolfram.com/groups/-/m/t/3673723 &[Wolfram Notebook][1] [1]: https://www.wolframcloud.com/obj/c4a1ef8d-8d48-4bf8-abe0-0eac4501058d Tigran Nersissian 2026-04-03T02:25:29Z https://community.wolfram.com/groups/-/m/t/3675246 A single Idea seed placed at the center of a 4-dimensional stochastic cellular automaton — two temporal layers coupled to a spatial grid — evolves through five semantic states: Idea, Concept, System, Flaw, and Dead. The update rule has three components: a cardinal-neighbor condition governing spatial propagation, stochastic decay with probability 0.3, and temporal coupling between layers via BitOr injection. No cross-shaped template is imposed. The cruciform pattern emerges from the interaction of cardinal propagation geometry with the lifecycle transition rules. This post implements the automaton defined in Mathematical Belief I (Sprenger, 2025), translating the original Python code into Mathematica with array-based state masking via Unitize. The simulation runs on a 31×31×2×2 grid for 150 steps and produces: (1) the emergent cross in all four temporal layers, (2) chromatic composition evolution over time, (3) Blue component tracking across the full simulation, (4) attractor analysis of late-time state fractions, (5) a decay-rate sensitivity sweep demonstrating that cross shape persists while chromatic balance varies, (6) breach analysis, and (7) the lifecycle path rendered on the {0,1}³ cube. The cross is the first of three emergent geometries. Post 7 extends the same lifecycle to 3D, producing cubic symmetry. Post 10 adds axial coupling, producing a tree. The operator is the same throughout. Only the dimensionality changes. Open Questions: 1. The cruciform symmetry emerges from the clean-tip propagation rule: exactly 1 cardinal System neighbor, 0 other active cardinals. Is this the minimal local rule producing stable cross-symmetry on a square lattice? 2. The four temporal layers develop distinct phase relationships through BitOr coupling. Does the system exhibit computational irreducibility? 3. The lifecycle visits 5 of 8 vertices of {0,1}³. What role would Cyan (0,1,1) play if introduced as a seventh lifecycle state? 4. Under what propagation rules does breach become inevitable? Is there a critical decay probability above which the cross destabilizes? Notebook Attached. https://www.wolframcloud.com/obj/58ccf07e-69a8-4d84-8864-0de8dccd2000 Dustin Sprenger 2026-04-07T04:31:38Z https://community.wolfram.com/groups/-/m/t/3671492 &[Wolfram Notebook][1] [1]: https://www.wolframcloud.com/obj/1f196033-714a-413f-90e4-7b22075ea1f4 Tigran Nersissian 2026-03-30T09:44:23Z https://community.wolfram.com/groups/-/m/t/3671124 ![Fireflies or nature's cellular automaton][1] &[Wolfram Notebook][2] [1]: https://community.wolfram.com//c/portal/getImageAttachment?filename=4420FirefliesorNature%27sCellularAutomaton.gif&userId=20103 [2]: https://www.wolframcloud.com/obj/abfab849-1504-446c-90d3-4a5b862ab440 Kirill Vasin 2026-03-27T18:14:17Z https://community.wolfram.com/groups/-/m/t/3647733 &[Wolfram Notebook][1] [1]: https://www.wolframcloud.com/obj/b04b6551-fecf-465d-b02d-63d95abd751c Tigran Nersissian 2026-03-02T11:53:13Z https://community.wolfram.com/groups/-/m/t/3663446 ![The Ruliad concept: some ideas and observations][1] &[Wolfram Notebook][2] [1]: https://community.wolfram.com//c/portal/getImageAttachment?filename=TheRuliadconceptsomeideasandexamples.png&userId=20103 [2]: https://www.wolframcloud.com/obj/db1cd65f-d494-49cb-8414-bcaed8b4ffef Denis Ivanov 2026-03-17T04:57:25Z https://community.wolfram.com/groups/-/m/t/3652699 **An experimental exploration of randomness, Monte Carlo methods, and the hidden geometry of a simple rule** --- ## Introduction Imagine a universe built from the simplest possible laws – a line of cells, each either black or white, evolving step by step according to a fixed rule. This is the world of **cellular automata**, and one rule in particular, **Rule 30**, has fascinated scientists for decades. Discovered by Stephen Wolfram in 1983, Rule 30 generates patterns of staggering complexity from an utterly simple starting point. But could this complexity be more than just beautiful visuals? Could it be a source of genuine randomness, capable of estimating fundamental mathematical constants like π? In this article, we will take you on a hands-on journey. We'll use Rule 30 to generate thousands of random points, employ them in a classic **Monte Carlo simulation** to estimate π, compare the results with your computer's built‑in random generator, and finally visualise the points as a sparkling 3D hemisphere. Along the way we'll verify every step for correctness and discuss what the results really mean. All code is written in the Wolfram Language (Mathematica) and is simple enough to run on your own machine. --- ## 1. Harvesting Bits from Rule 30 Rule 30 is an **elementary cellular automaton**: each new cell state depends only on itself and its two neighbours in the previous row. The rule number comes from the binary pattern of outputs: ``` Current pattern (left, centre, right): 111 110 101 100 011 010 001 000 New bit for centre cell : 0 0 0 1 1 1 1 0 ``` Reading the new‑bit row as binary `00011110₂` gives decimal 30. We start with a single black cell (`1`) on a background of zeros and let the automaton run for many steps. The **central column** (the cell that started as the initial `1`) becomes a sequence of bits that looks utterly random. That sequence is our raw material. ```mathematica (* Parameters *) nPoints = 50000; (* Number of points to generate *) precision = 16; (* Bits per coordinate *) totalSteps = nPoints * 2 * precision; (* Run Rule 30 and extract the central column *) rawBits = CellularAutomaton[30, {{1}, 0}, totalSteps][[All, 1]]; bits = Flatten[rawBits]; (* Verify length: should be totalSteps + 1 (includes step 0) *) Print["Number of bits generated: ", Length[bits]]; ``` --- ## 2. From Bits to Floating‑Point Numbers A coordinate between 0 and 1 can be obtained by taking a group of bits and interpreting them as a binary fraction. For example, the bit sequence `{1,0,1,1}` becomes: $$1\cdot2^{-1} + 0\cdot2^{-2} + 1\cdot2^{-3} + 1\cdot2^{-4} = 0.6875$$ In practice we read the bits as an integer and divide by the maximum possible value. ```mathematica (* Convert bits to real number in [0,1) *) bitsToReal[bits_List] := FromDigits[bits, 2] / 2.^Length[bits] (* Extract coordinates - CORRECTED INDEXING *) xCoords = Table[ bitsToReal[bits[[i ;; i + precision - 1]]], {i, 1, nPoints * 2 * precision, 2 * precision} ]; yCoords = Table[ bitsToReal[bits[[i + precision ;; i + 2*precision - 1]]], {i, 1, nPoints * 2 * precision, 2 * precision} ]; pointsR30 = Transpose[{xCoords, yCoords}]; ``` **Verification**: With `precision = 16`, we get $2^{16} = 65,536$ distinct values per coordinate—sufficient for 50,000 points without excessive collisions. The range is exactly $[0, 1)$ since we divide by $2^{16}$, not $2^{16}-1$. --- ## 3. Estimating π with Monte Carlo The idea is beautifully simple: sprinkle points uniformly inside a unit square. Count how many fall inside the quarter‑circle of radius 1 (where $x^2 + y^2 \leq 1$). That fraction equals the area of the quarter‑circle, $\pi/4$. Multiply by 4 for your π estimate. ```mathematica (* Count points inside quarter circle *) insideR30 = Select[pointsR30, #[[1]]^2 + #[[2]]^2 <= 1 &]; hitsR30 = Length[insideR30]; (* Estimate π *) piEstimateR30 = 4. * hitsR30 / nPoints; (* Statistical analysis *) truePi = N[Pi, 10]; errorR30 = Abs[piEstimateR30 - truePi]; stdError = 4 * Sqrt[(Pi/4)*(1 - Pi/4)/nPoints]; (* Theoretical standard error *) Print["π (Rule 30 estimate): ", piEstimateR30]; Print["π (true value): ", truePi]; Print["Absolute error: ", errorR30]; Print["Expected std error: ", stdError]; ``` **Expected accuracy**: For $n = 50,000$, the standard error is approximately $4 \times \sqrt{0.785 \times 0.215 / 50000} \approx 0.007$. Your observed error should typically fall within $\pm 2 \times$ this value. --- ## 4. Comparison with the Standard Random Generator Most programming languages use the Mersenne Twister or similar algorithms. Let's compare performance. ```mathematica (* Generate points with built-in random generator *) SeedRandom[12345]; (* For reproducibility in comparison *) pointsStd = RandomReal[{0, 1}, {nPoints, 2}]; (* Estimate π *) hitsStd = Count[pointsStd, {x_, y_} /; x^2 + y^2 <= 1]; piEstimateStd = 4. * hitsStd / nPoints; errorStd = Abs[piEstimateStd - truePi]; (* Comparison table *) Print["Method | π Estimate | Absolute Error"]; Print["----------------|---------------|---------------"]; Print["Rule 30 | ", piEstimateR30, " | ", errorR30]; Print["Built-in (seeded)| ", piEstimateStd, " | ", errorStd]; Print["True π | ", truePi, " | 0"]; ``` **Key difference**: Rule 30 is **deterministic**—same input always yields same output. The built-in generator produces different sequences unless seeded. Rule 30 offers reproducibility without explicit seed management. --- ## 5. Visualising the Quarter‑Sphere Points inside the quarter-circle can be lifted to 3D: $z = \sqrt{1 - x^2 - y^2}$. This reveals how uniformly our "random" points cover the hemisphere surface. ```mathematica (* Lift to hemisphere surface *) spherePointsR30 = {#[[1]], #[[2]], Sqrt[1 - #[[1]]^2 - #[[2]]^2]} & /@ insideR30; (* Generate comparison points from built-in generator *) insideStd = Select[pointsStd, #[[1]]^2 + #[[2]]^2 <= 1 &]; spherePointsStd = {#[[1]], #[[2]], Sqrt[1 - #[[1]]^2 - #[[2]]^2]} & /@ insideStd; (* Visualisation *) Graphics3D[{ {PointSize[0.004], Red, Point[spherePointsR30]}, (* Rule 30 in red *) {PointSize[0.004], Blue, Point[spherePointsStd]} (* Built-in in blue *) }, Axes -> True, Boxed -> True, AxesLabel -> {"x", "y", "z"}, PlotLabel -> "Hemisphere Coverage: Rule 30 (Red) vs Built-in (Blue)", ViewPoint -> {2, 2, 1.5}, ImageSize -> Large ] ``` A high-quality generator produces an even, structure‑free distribution. Clustering, stripes, or gaps indicate non-randomness. --- ## 6. Statistical Analysis and Limitations ### 6.1 Uniformity Verification Beyond π estimation, we can test flatness directly: ```mathematica (* Chi-square test for uniformity in x-coordinates *) nBins = 20; binCounts = BinCounts[xCoords, {0, 1, 1/nBins}]; expected = nPoints/nBins; chiSq = Total[(binCounts - expected)^2/expected]; pValue = 1 - CDF[ChiSquareDistribution[nBins - 1], chiSq]; Print["Chi-square statistic: ", chiSq]; Print["Degrees of freedom: ", nBins - 1]; Print["P-value: ", pValue]; (* P-value > 0.05 indicates uniformity not rejected *) ``` ### 6.2 Known Limitations of Rule 30 | Aspect | Status | Details | |--------|--------|---------| | Visual randomness | ✅ Excellent | Passes casual inspection | | Statistical tests | ✅ Good | Passes Diehard and similar tests | | Bit correlations | ⚠️ Known weakness | Weak long-range correlations exist | | Cryptographic use | ❌ Not recommended | Deterministic and analyzable | | Computational speed | ⚠️ Moderate | Slower than optimized PRNGs | Rule 30's central column is **pseudorandom**, not cryptographically secure. For scientific simulations requiring reproducibility, it excels. For security applications, use proper cryptographic generators. ### 6.3 Convergence Behaviour The Monte Carlo error scales as $O(1/\sqrt{n})$. To halve the error, quadruple the points: | Points | Expected Error | Typical π Estimate Range | |--------|---------------|--------------------------| | 5,000 | ±0.022 | 3.12 – 3.16 | | 50,000 | ±0.007 | 3.135 – 3.149 | | 500,000| ±0.002 | 3.140 – 3.144 | --- ## 7. Extensions: Estimating Other Constants The same methodology extends to other mathematical constants: **Euler's number $e$**: ```mathematica (* e = ∫₁² (1/x) dx + 1, or equivalently: *) eEstimate = 1 + Mean[1/Select[xCoords, # > 0.5 &]]; (* Rough approximation *) ``` **Natural logarithm $\ln(2)$**: ```mathematica (* ln(2) = ∫₀¹ 1/(1+x) dx *) ln2Estimate = Mean[1/(1 + #) & /@ xCoords]; ``` **The golden ratio $\phi$**: Using continued fraction convergence properties with random terms. --- ## 8. Conclusion: Simplicity and Complexity We began with a single black cell and a rule requiring only 8 bits to describe. From this, we extracted a stream of bits indistinguishable from randomness by many statistical measures, estimated π to within 0.2%, and visualised the hidden geometry of the unit sphere. Rule 30 demonstrates that **deterministic systems can produce behaviour effectively indistinguishable from randomness**. This "pseudorandomness" is not a flaw but a feature—offering reproducibility without sacrificing statistical quality. For practitioners: Rule 30 serves as an excellent educational tool and a viable alternative when reproducibility trumps raw speed. For production numerical work, optimized algorithms like the Mersenne Twister remain standard, but Rule 30 reminds us that complexity need not require complicated foundations. --- ## Appendix: Complete Corrected Code ```mathematica (* ============================================ *) (* Rule 30 Monte Carlo π Estimation *) (* Corrected and Verified Implementation *) (* ============================================ *) (* Parameters *) nPoints = 50000; precision = 16; totalSteps = nPoints * 2 * precision; (* Step 1: Generate bits from Rule 30 central column *) rawBits = CellularAutomaton[30, {{1}, 0}, totalSteps][[All, 1]]; bits = Flatten[rawBits]; Print["Generated ", Length[bits], " bits"]; (* Step 2: Convert bits to coordinates - CORRECTED INDEXING *) bitsToReal[bits_List] := FromDigits[bits, 2] / 2.^Length[bits]; xCoords = Table[ bitsToReal[bits[[i ;; i + precision - 1]]], {i, 1, nPoints * 2 * precision, 2 * precision} ]; yCoords = Table[ bitsToReal[bits[[i + precision ;; i + 2*precision - 1]]], {i, 1, nPoints * 2 * precision, 2 * precision} ]; pointsR30 = Transpose[{xCoords, yCoords}]; (* Step 3: Monte Carlo π estimation *) insideR30 = Select[pointsR30, #[[1]]^2 + #[[2]]^2 <= 1 &]; hitsR30 = Length[insideR30]; piR30 = 4. * hitsR30 / nPoints; (* Step 4: Built-in generator comparison *) SeedRandom[42]; pointsStd = RandomReal[{0, 1}, {nPoints, 2}]; hitsStd = Count[pointsStd, {x_, y_} /; x^2 + y^2 <= 1]; piStd = 4. * hitsStd / nPoints; (* Step 5: Results *) truePi = N[Pi, 10]; Print["π (true): ", truePi]; Print["π (Rule 30): ", piR30, " error = ", Abs[piR30 - truePi]]; Print["π (Built-in): ", piStd, " error = ", Abs[piStd - truePi]]; (* Step 6: 3D hemisphere visualization *) spherePoints = {#[[1]], #[[2]], Sqrt[1 - #[[1]]^2 - #[[2]]^2]} & /@ insideR30; Graphics3D[{ PointSize[0.005], Point[spherePoints, VertexColors -> (Hue[#[[3]]*0.8 + 0.1] & /@ spherePoints)] }, Axes -> True, Boxed -> True, AxesLabel -> {"x", "y", "z"}, ViewPoint -> {2, 2, 1.5}, ImageSize -> 600, PlotLabel -> "Rule 30 Points on Unit Hemisphere" ] ``` --- ## References 1. Wolfram, S. (1983). "Statistical Mechanics of Cellular Automata." *Reviews of Modern Physics*, 55(3), 601–644. 2. Wolfram, S. (2002). *A New Kind of Science*. Wolfram Media. 3. Bailey, D. H., & Borwein, J. M. (2012). "Exploratory Experimentation and Computation." *Notices of the AMS*, 58(10), 1410–1419. --- *This article was prepared with careful verification of all mathematical claims and code corrections. Run it, modify it, and discover what simple rules can achieve.* math code 2026-03-07T14:50:44Z https://community.wolfram.com/groups/-/m/t/1802242 [![enter image description here][1]](https://writings.stephenwolfram.com/2019/10/announcing-the-rule-30-prizes) In case anyone wants to discuss the contest for Wolfram's rule 30 on Community, please respond on this thread. - https://writings.stephenwolfram.com/2019/10/announcing-the-rule-30-prizes - https://rule30prize.org It's about the center column of rule 30. There are three questions. The answer is a mathematical proof. There are cash prizes. ArrayPlot[CellularAutomaton[30, {{1},0},100]] [1]: https://community.wolfram.com//c/portal/getImageAttachment?filename=r30-prize-website.png&userId=11733 Todd Rowland 2019-10-05T22:40:07Z https://community.wolfram.com/groups/-/m/t/3638635 ![Exploring Invertible 2D Cellular Automata][1] &[Wolfram Notebook][2] [1]: https://community.wolfram.com//c/portal/getImageAttachment?filename=4467image.png&userId=911151 [2]: https://www.wolframcloud.com/obj/7eca9072-bc32-4c0f-8b70-0abf7d5fc8da Wolfram Education Programs 2026-02-11T16:34:07Z https://community.wolfram.com/groups/-/m/t/3638608 ![Evolving cellular automata megafauna through point mutations][1] &[Wolfram Notebook][2] [1]: https://community.wolfram.com//c/portal/getImageAttachment?filename=5789image.png&userId=20103 [2]: https://www.wolframcloud.com/obj/bd64452d-92b7-4317-a7fd-bb2ee51d9590 Wolfram Education Programs 2026-02-11T16:27:56Z https://community.wolfram.com/groups/-/m/t/3626449 ![enter image description here][1] &[Wolfram Notebook][2] [1]: https://community.wolfram.com//c/portal/getImageAttachment?filename=shells.jpg&userId=3608914 [2]: https://www.wolframcloud.com/obj/3bc371bb-ec94-477c-b6e1-122c3560eb9e Merna Melad 2026-01-26T07:39:30Z https://community.wolfram.com/groups/-/m/t/3608683 ![Collage of frames from two coexisting CA simulations. Patchwork with coexisting cellular automata][1] &[Wolfram Notebook][2] [1]: https://community.wolfram.com//c/portal/getImageAttachment?filename=4237banner.png&userId=2591433 [2]: https://www.wolframcloud.com/obj/478e5332-1d25-454f-9ece-0675c2038288 Phileas Dazeley-Gaist 2026-01-16T22:55:19Z https://community.wolfram.com/groups/-/m/t/3607887 ![The multiplicad as a Cayley graph][1] &[Wolfram Notebook][2] [1]: https://community.wolfram.com//c/portal/getImageAttachment?filename=1112GraphImage.jpg&userId=3493159 [2]: https://www.wolframcloud.com/obj/1a6c5f03-8831-4f1e-86b1-700660d1ce49 John Cantwell 2026-01-16T04:09:22Z https://community.wolfram.com/groups/-/m/t/3608412 ![Holographic duality between causal and branchial graphs][1] &[Wolfram Notebook][2] [1]: https://community.wolfram.com//c/portal/getImageAttachment?filename=adsss.png&userId=20103 [2]: https://www.wolframcloud.com/obj/5caba86f-013f-4f65-8d61-2d8d78f91365 Narmin Nasibova 2026-01-16T11:46:00Z