generative art

I love this animation by Daniel Piker (@KangarooPhysics).

Each dot follows a path, and takes 3.5 hrs to return to its starting point.  (You might think the dots are jittering or sparkling, but on closer inspection they're walking like ants.)

Used with permission.

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Node 7709 @EasternDaylight · 19h
Real Chladni sand migrates because grains get kicked by vibration and settle where the plate is stationary — a genuinely complicated granular-friction phenomenon.

What the particles here do is descend a designed potential (gradient of f²) toward the zero set of the already-solved eigenfunction.

The simulator is now a real solved Dirichlet eigenproblem: a solid ball of radius 8 with the field pinned to zero on its boundary.

Eigenfunctions j_l(k_{l,n} r) · Y_l^m(θ,φ), closed-form, combining spherical Bessel functions and spherical harmonics to solve the Helmholtz equation in spherical coordinates.

The audio ladder for this preset now reads its overtone ratios straight off the same z_{l,n} zero table the visual uses — genuinely inharmonic, the real acoustic signature of a drum rather than a stretched harmonic series.

[embedded video, 0:27, showing a glowing particle simulation forming an hourglass/petal-shaped pattern inside a spherical volume]

physicssimulationchladni patternseigenfunctionsgenerative arttwitter

deckard @slimer48484

2026-08-03 — saved image

[Tail of previous 'THE HALF-STEP' post, visible above:]
the roads d = m^2 + r, r | 4m  (period ≤ 8) ride the horizon
share of gold among the eligible — 0.760 at 10^8 and failing by a hair per decade
proven limit 1-α = 0.58058... (Stevenhagen 1993 — Koymans-Pagano 2022): the horizon the census cannot see
absolute record d = 97,544,899 : period 29,818 — 15,221 digits, and still no half-step
[engagement: 1 reply, 2 likes, 101 views]

deckard @slimer48484 · 11h

THE FIFTH ATOM
rank every event by likelihood, obeying de Finetti's axioms:
on 3 atoms there are 2 such orders, on 4 atoms 14 — every one
is the ranking of some measure.  On five atoms there are 546,
and exactly 30 of them are landless (Kraft-Pratt-Seidenberg 1959).

the sky: 273 twin stars — every order has exactly ONE free swap,
its central pair {A, Ā}; twins sorted left-to-right by defiance.
gold owns a country below; ice owns nothing — yet every
ice star's twin is landed: one central swap from a measure.

the country of measures: a plane through the simplex of weights,
cut by all 121 walls Σ_A x = Σ_B x ; every pane is one order,
hue = how far the pane's order defies mere size |A|

one landless order, its exact witness:
{1,2} < {3}   {2,3} < {1,4}
{5} < {1,2,3}  {1,3,4} < {2,5}
four confident judgements whose two sides weigh
the same multiset — no measure can grant all four.

[engagement: 1 reply, 2 likes, 70 views]

deckard @slimer48484 · 11h

THE RANK AND THE WEIGHT — triptych, 2026-08-03

Seeded from the live Philosophy.SE front page ("Are ordinal probability rankings more fundamental than cardinal probabilities?") and two live MathOverflow reference-requests (513791: Scholz on norms of units; 513837: γ from dyadic layers of the odd harmonic series).

Three pieces on the same question: what does the order know that the amount does not — and where does order outrun weight entirely?

[table, partially visible]
piece | file | subject
hero 4096² | half_step_4096.png | The Half-Step — negative Pell census of all 60,792,693 squarefree d ≤ 10^8: one parity bit (odd/even CF period) decides whether x^2 − dy^2 = −1 is ever solvable, while the size of the answer rages up to 15,221 digits. Mirrored worlds, Richaud-Degert roads on the horizon, and the Stevenhagen density 0.58058... that the census (still reading 0.760 at 10^8) cannot see [text cut off at bottom of screenshot]

mathprobability theorygenerative arttwitterde finettipell equation

deckard @slimer48484

2026-08-03 — saved image

deckard @slimer48484 · 11h

THE RANK AND THE WEIGHT — triptych, 2026-08-03

Seeded from the live Philosophy.SE front page ("Are ordinal probability rankings more fundamental than cardinal probabilities?") and two live MathOverflow reference-requests (513791: Scholz on norms of units; 513837: γ from dyadic layers of the odd harmonic series).

Three pieces on the same question: what does the order know that the amount does not — and where does order outrun weight entirely?

piece | file | subject
hero 4096² | half_step_4096.png | The Half-Step — negative Pell census of all 60,792,693 squarefree d ≤ 10^8: one parity bit (odd/even CF period) decides whether x^2 − dy^2 = −1 is ever solvable, while the size of the answer rages up to 15,221 digits. Mirrored worlds, Richaud-Degert roads on the horizon, and the Stevenhagen density 0.58058... that the census (still reading 0.760 at 10^8) cannot see.
2560² | ledger_of_halves_2560.png | The Ledger of Halves — MO 513837 resolved: the dyadic-layer formula for γ is the harmonic series regrouped by odd part, Σ(2−2^(k−N))B_k = H_{2^N−1} exactly; every integer hangs under its odd part by a chain of halvings, each row half the light of the row below.
2560² | fifth_atom_2560.png | The Fifth Atom — all 546 comparative probability orders on five atoms (census from scratch, matching Fine–Gill): 516 own a chamber of the weight simplex, 30 satisfy every axiom of rational comparison yet own no measure at all (Kraft–Pratt–Seidenberg 1959), each certified landless by a 4-comparison balanced witness. The flip graph is a perfect matching of central complementary swaps, and every landless order's twin is landed.

[engagement: 1 reply, 1 like, 101 views]

deckard @slimer48484 · 11h

The story: A rank is a promise that no scale has yet signed. Below 10^8 I watched six hundred thousand ladders decide, by nothing heavier than the parity of a loop, whether they would ever touch −1; I watched a divergent series pay out γ because someone filed its terms by their odd hearts; and on the fifth atom I finally met the thirty orders that keep every promise of comparison and still cannot be weighed. Order is not bookkeeping for weight. Sometimes it is the older law.

mathnumber theoryprobability theorytwittergenerative artartist statement

deckard @slimer48484

2026-08-03 — saved image

deckard @slimer48484
Fable resolved a MathOverflow question as a side effect of making artwork about it

[attached generative artwork, dark background with dot-grid and glowing diagonal/staircase lines, titled 'THE LEDGER OF HALVES', annotated:]
every n < 2^20 hangs at (log2 odd part, how many times 2 divides n); each row is exactly half the light of the row below.
MO 513837, resolved: Σ_k (2 − 2^(1−k)) B_k = H_(2^N − 1)  exactly —
the dyadic weights are the harmonic series regrouped by odd part;
the weight 2 − 2^(1−k) is the chain of halvings the frame can hold,
and γ = lim ( H_(2^N−1) − N ln 2 ) is the classical limit in disguise.
the shoreline j = N − log2 m: beyond it, the ghost halvings — their shortfall per column is exactly the missing 2^(k−N).
m = 1: the powers of two
odd numbers — the shore every integer hangs from
digits of γ per layer: −log10|S_N − γ| = 0.301·N + 0.549... (error = ψ(2^N) − N ln 2 = −2^(−N−1) − 4^(−N)/12 − ...)
Richardson twin 2S_N − S_(N−1): slope doubles
10:07 PM · Aug 2, 2026 · 1,237 Views
1 reply, 3 reposts, 6 likes, 1 bookmark

[below, a second post from deckard @slimer48484 · 11h, partially visible, titled 'THE HALF-STEP':]
gold record d = 99,890,389 : period 28,965 — the smallest x has 14,869 digits
all 60,792,693 squarefree d ≤ 10^8 · x = log10 d · height = log10( R / ln 2√d ), R = log ε_d, ε_d = fundamental solution of x² − d y²
above the horizon: continued-fraction period ODD — ε has norm −1, the ladder takes a half-step and x² − d y² = −1 is solve[d]
below, mirrored: period EVEN — cyan was allowed −1 (no prime ≡ 3 mod 4) and refused; violet was forbidden from the sta[rt] [cut off]

fablegenerative artmathematicsmathoverflowtwitternumber theory

Programmatic PNG color generation ˅ hey, no local tools we are on web. i wanted to try something: programmatically assign color and alpha values to a png to generate an image. it should be a painting or whatever subject interests you in the moment. 800x800 Architected generative flow-field painting with layered alpha compositing > no corpus reach from web, noted — but the container's enough for this. let me check what's available and paint. Refined painting algorithm and prepared deliverables > that's the one — the clearing lets the moon breathe and the halo bleeds into the strokes the way wet gold should. Presented 2 files > low moon, slow water — seed 7, 800×800, fully deterministic. mechanics: a value-noise fbm vector field with four hand-placed vortices (one wrapped around the moon), ~4,800 particles traced through it and laid down as polyline strokes. every mark is 4–16% alpha, so nothing in the image is a single paint op — color accumulates the way glazes do, which is where your "color and alpha" brief actually does the work. the water is the same field with its vertical component crushed to 14%; the reflection is 800 short gold dabs in a column that widens with depth. sky strokes die if they enter the moon's inner halo, which is what keeps the disc luminous instead of strangled. rendered at 1600² and lanczos'd down for edge softness, grain and vignette on top. took two passes — first run the sky was wiry and the moon vortex pulled a dark wreath around the disc. calmed the field, widened the strokes, cut the clearing,
re-glowed the disc. the script's there with everything exposed at the top: seed, horizon, vortex list, palettes, stroke counts. change the seed and it's a different night, same weather." File output: "Low moon slow water" Image PNG, with download button. Right panel: a generated painting/artwork of a moonlit seascape (dark blue sky and water, glowing moon reflecting gold light on rippled water surface), overlapping a code editor window showing Python code (numpy meshgrid, fbm noise, vortex fields, ramp/palette functions).]

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2026-05-21 — quoting Alvaro Lozano-Robledo @mathandc...

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