Figures

• The figure crate: every figure the site opens on is one binary in src/bin/<name>.rs, drawn with the mrlyfig kit and pressed by scripts/figures.sh into files/figures/<name>-dark.png and <name>-light.png.
• Cargo autodiscovers the binaries; a new figure is one file here and nothing else.
• bash scripts/figures.sh <name> prints one, no name prints all, check and test are the verbs the desk runs.

FIGURES

• One bullet per figure, subject; artist; parameters; palette; the artist is the kit module the figure leans on: grid, hex, iso, field or plot.
• Names are the routes: site-* the doors and the card, research-<page>, paper-<slug>, blog-<slug>, demo-<name>, wiki-<slug>; research-index also opens REFS.

• site-home: the sponge, code 23 at base 2 grown to level 3, in isometric projection with three-tone faces, standing clear on the ground inside an 8 percent margin; iso; number 3, level 3, side 27; top faces fg, left faces blue, right faces dim
• site-demos: twenty-eight tiles in a 7 by 4 board, each a different base-3 plane design at level 3, one tile per live page, ground gutters between; grid; base 3, level 3, side 27 a tile; inks cycling blue, orange, yellow
• site-papers: twelve thin slabs stacked in isometric and offset so each stays legible, every top face carrying a different design's level-2 cells; iso; 12 sheets, base 2, level 2, 8 percent margin; tops fg, sides blue on ground
• site-research: the moire stack of one design sampled at scales 1 to 20, the layers summed into a scalar field and run through a ramp, twenty layers for twenty pages; field; code 495, base 3, scales 1..20, 1024 board; blue-to-yellow on ground
• site-og: the mark, rows 11111 10101 11111 10101 11111, tiled edge to edge as a lattice across the 1200 by 630 card at 15 px a cell, the pattern centred so the crop is symmetric; grid; level 1, 80 by 42 cells; fg on ground
• site-icon: the mark alone at level 1, five cells a side filling 76 percent of a 512 by 512 board, centred on the ground; grid; level 1; fg on ground
• site-wiki: the door to the wiki, one concept a page in prerequisite order: seven base-2 plane designs at level 2 as a graph of rows one, three and three, each parent joined to its children; grid; codes 7, 14, 3, 5, 9, 6, 15, base 2, level 2, side 9, 283 cells; blue on ground, the root yellow, the edges in line
• site-tools: the door to the tools, the first of them a perforator: the base-2 carpet, code 7, at level 4 as one sheet inside an 8 percent margin, its 4096 cells the metal and every void punched through to the ground; grid; code 7, base 2, level 4, side 81; dim on ground
• site-math: the door to /math, where a design is one string: the eight corner bits of {"kind":"bang","dim":2,"code":7} as a row of eight cells, three of them lit, over the level-3 carpet those bits grow, 512 cells centred below; grid; code 7, base 2, level 3, side 27; fg bits, line empties, blue carpet on ground
• research-index: one design grown by the tree's one move: the base-3 carpet, code 495, at levels 1, 2, 3 and 4 as a 2 by 2 board of equal squares, each level drawn to the same size; grid; code 495, base 3, levels 1 to 4; yellow on ground
• research-core: the 16 base-2 dim 2 designs each grown to level 4, laid as a 4x4 census of 16x16 boards in code order; the six canonical representatives yellow, the ten orbit-mates blue; grid; base 2, dim 2, level 4; yellow + blue on ground
• research-bijection: all 256 base-2 dim 3 designs as a 16x16 census, each a 2x4 stamp of its eight corner bits in binary order; the 22 canonical representatives in yellow, the rest in blue; grid; dim 3, 256 codes, 22 orbits; yellow + blue on ground
• research-complexity: the normalised-Laplacian eigenvalue staircase of the level-6 Sierpinski triangle, 729 values in [0,2], the riser at eigenvalue 1 a full third of the stair and drawn yellow, the rest blue; plot; level 6; yellow + blue on ground
• research-cuts: the two central diagonal cuts of bang dim 3, code 126 at level 7 seen down (1,1,1): 4374 points as six Sierpinski gaskets of 729 tiling a hexagon; hex; level 7, heights t=63,64; three up gaskets blue, three down orange on ground
• research-slices: the central diagonal hexagon of the cube at odd side 11, all 726 unit triangles, the carpet's 486 against the net's 240, so the net paints the carpet's 19 hexagram holes; hex; codes 23 and 232, level 1, base 2; blue carpet, orange net
• research-hexagon: the carpet cut stack at odd sides 3 to 55 read on the hexagon's own mesh of side 110 and painted as excess over the background, the hexagram, the quarter-line bands, the fading centre star; hex; carpet, odd side 3..55, mesh 110; blue through panel to yellow
• research-information: the level-4 base-3 carpet, 81 by 81 and 4096 cells, beside its rearrangement at the 9 by 9 split, the same 4096 cells as the rank-one outer product vec(B) vec(C)^T; grid; base 3, code 495, level 4, split 9; blue and orange on ground
• research-weights: the mass fold ln N(a) - delta a, a lattice sawtooth over a flat nonlattice band, stacked on the length fold ln M(R) - alpha ln R ringing at both weights; plot; base 3, level 8, exact masses, R in 3^4..3^8; blue and orange on ground
• research-dimensions: the pole line of the base-3 Cantor design, an exact column at Re 0.630930 spaced 5.719202 in Im, beside the 21 scattered poles of the two-ratio control; plot; m -6..6, box Re [-1,1] Im [-40,40]; column blue, scatter orange
• research-connectivity: two 64x64 boards side by side: left the level-6 gasket, 729 cells in one green piece; right exactly 729 uniform random cells in pink, shattered; grid; base 2, code 7, 729 cells each; green + pink on ground
• research-magic: one word in both orders: the composite tiles carpet(3) (x) net(5) and net(5) (x) carpet(3), each 15x15, equal fill and different shape; grid; two panels, level 1, side 15; left tile blue, right tile orange on ground
• research-walks: codes 127 and 239 at level 4 side by side, same fill 7, each carrying one blind-ant walker's traced path over its own cells; plot; base 3, level 4, one seed per side, cells dim; 127's path blue, 239's orange
• research-method: the whole 3D design space walked: all 22 symmetry classes as 2x2x2 corner sets in isometric projection, laid in rows by popcount 0 to 8; iso; 22 cubes, three face tones; blue three-tone on ground
• research-farey: the Farey stack on the unit square, the grids of every scale n = 1..60 summed so a node a/b lights at brightness floor(60/b); field; N 60, unit square; ramp ground to blue to yellow at the brightest nodes
• research-mobius: the base-4 anti-symmetric pair: the Mobius meters of digit sets {0,1} and {0,2} walked as staircases, exact mirror images across zero inside a +-sqrt(A_F) envelope; plot; base 4, level 16, 65536 steps; blue, orange, envelope dim
• research-pi: the corner window of the grid at N = 100 with every point of gcd 1 lit as a square dot and the 3913 hidden points left dim, 6087 lit of 10000; plot; N = 100, origin at lower left; yellow lit, dim hidden, on ground
• research-coprime: the base-3 carpet at level 4, side 81, its 4096 cells as squares, those with coprime coordinates lit yellow and the rest dim; grid; base 3, code 495, level 4, origin at the lower left corner; yellow + dim on ground
• research-bases: the hexagonal Eisenstein lattice out to radius 30, points visible from the origin (gcd 1) as filled blue dots, points hidden behind a nearer one as small dim dots; plot; hexagonal lattice, radius 30; blue + dim on ground
• research-spectra: the centroid diagonal slice of the odd-coordinate solid at base 5, level 2, drawn triangle by triangle with the two tile kinds separated: hexagon cells yellow, triangle cells blue; hex; code 23, number 5, level 2, base 5; yellow + blue on ground
• research-spin: the level-5 carpet spun about its corner fixed point: the exact circle mean at every radius painted as a bullseye, rings repeating by a factor of 3; field; code 495, base 3, level 5, centre (0,0); blue-to-yellow ramp on ground
• research-crop: the base-3 carpet at level 4, side 81, under the inscribed circle of radius 1/2: 2908 In cells yellow, 204 Cut cells orange, Out cells left as ground; grid; base 3, code 7, level 4, r = 1/2; yellow + orange on ground
• research-sequences: a census board: plane designs 1, 3, 7, 9, 11, 15 down six rows against odd sides 3, 5, 7, 9 across four columns, each tile the design's fill painted cell by cell; grid; base 2, level 1; yellow fills on ground, gutters in line
• research-integers: the integer census as a 100x100 raster of 1..10000 in reading order, a cell dark where no registry row writes it and brightening with the number of rows that do; grid; window 1..10000, 48-term rows; dim to yellow on ground
• paper-coprime-density-above-dimension-one: the level-5 gasket design in a 32x32 grid, corners (0,0),(1,0),(0,1): its 243 cells as squares, the 122 of coordinate gcd 1 yellow, the other 121 dim, the rest ground; grid; base 2, level 5; yellow + dim on ground
• paper-lemma-b-pincer: the exponent line 0 to 1 as one bar: closed arms [0,0.4475978] and [0.6402122,1] dim, the open window between orange, hairlines at 0.447931, 0.5, 0.605303; plot; window 0.1926 wide; orange + dim + line on ground
• paper-menger-pairwise-coprimality: the 27 base-3 digit vectors as three 3x3 slices c=0,1,2: the 7 sponge-deleted cells left as ground in line outlines, the 13 with at most one zero green, the other 7 survivors dim; grid; base 3, 13/20; green + dim + line
• paper-walsh-spectrometer: the diagonal hexagon slice of code 105 at side 7, all 6 side^2 = 294 unit triangles drawn, the inked ones (even-weight macro parity) blue and the paper ones as line-thin outlines on ground; hex; code 105, side 7, level 1, base 2; blue + line
• paper-slice-recurrence-order: thirteen bar pairs, one per dim 2..14: the free order 2 dim + 1 in dim behind, the proved order ceil(dim/2) in blue in front, a baseline rule in line, no ticks; plot; dim 2..14, 5..29 against 1..7; blue + dim
• paper-slice-sign-even-half: certificate depth against even dimension: base 3's quadratic climb from 0 to 89 as an orange dotted curve, bases 5 to 11 flat at depth at most 2 as dim steps below it; plot; even dim 2..40, K_min 0..89; orange + dim
• paper-gasket-ray-machine: the level-5 base-3 gasket: a ray from the origin along every occupied direction, blue and fainter as its mass falls, the heaviest ray (1,3) yellow, the 243 points as fg dots; plot; level 5, mass(1,3) = F(6)-1; blue + yellow + fg
• paper-order-sensitivity-of-kronecker-words: the witness, two 4x4 pictures side by side: code 3 outside code 6 as four isolated blue cells, code 6 outside code 3 as two orange dominoes, four filled cells each, thin line grid; grid; level 2, comp 4 against 2; blue + orange + line
• paper-component-exponent-of-kronecker-words: the 105 two-letter alphabets over designs 1..15 as an upper-triangular 15x15 grid: the 89 that meet the fill ceiling green, the 16 short orange, a yellow pip on the 27 where the constant rule is exact; grid; 105 cells; green + orange + yellow
• paper-moire-correlation-laws: the 31x31 correlation matrix of parity carpets at odd scales 3 to 63, cell = exact r(C_m,C_n): zero left as ground so prime rows and columns run in ground, positive ramped blue to yellow; field; odd 3..63; blue-to-yellow on ground
• paper-divisor-avatars: the first Menger step at side 3 in isometric: the 20 kept cells as solid cubes in three yellow face tones, the 7 cells with two or more even coordinates as dim wire boxes in their holes; iso; side 3, 20 = d(240); yellow three-tone + dim
• paper-sequence-census: the six planar rules that keep the all-even corner, drawn at side 5 as six 5x5 panels in a 3x2 block: codes 1,3,7 blue, codes 9,11,15 indigo, each with its k=2 inner 3x3 frame in line; grid; side 5, k = 3; blue + indigo + line
• paper-sparse-mertens-under-grh: the level-3 digit transform of a missing-digit set, all 343 frequencies of |hat F_3(a/343)| mirrored, the l^1 mass the proof pays; plot; base 7, digit 3 removed, stems at the per-digit root of the mass; blue to yellow on the ground
• blog-launching-mrlyprod-org: three counted blocks of unit squares on one baseline, 28 demos, 12 papers, 20 research pages, each a tight rectangle with a wide gap between; grid; 28 as 7 by 4, 12 as 3 by 4, 20 as 5 by 4; blue, yellow, green on ground
• research-automata: the 256 elementary rules as a 16 by 16 census of single-seed diagrams in rule order, each 32 generations on a 63-cell window; the 14 nonconstant affine rules yellow, the other 16 surjective rules blue, the rest dim; grid; T 32, 256 panels; yellow + blue + dim on ground
• demo-wolfram: rule 110 from one live cell, 256 generations on a 511-cell window, the space-time triangle with time running down and the live cells the only ink; grid; rule 110, T 256; fg on ground
• demo-mrlylife: B3/S23 on the level-2 base-2 carpet mask (side 9, 64 cells, centre already void) run 64 generations from a seeded soup on a 192 torus, live cells blue, the mask stamped top left in yellow; grid; code 7, side 9, level 2, T 64, fixed seed; blue + yellow on ground
• demo-snail: 1 to 300 wound on the square spiral, each cell a base-2 gasket tile of side 2^floor(log2 n), so the winding doubles at every new digit and nests into nine whorls, the primes lit; grid; base 2, code 7, top 300; blue on dim, line path
• demo-modes: the eigenvalue field of the level-5 base-3 carpet mask, 243 by 243 frequencies of |lambda(t)| / 8^5 read at its per-level root and centred on t = 0, the zero frequency at ratio 1 with the large values yellow along the axes; field; code 495, side 3, level 5, 3 px a frequency; ground to blue to yellow
• demo-tube: the level-5 carpet with its inner tube at eps = 21/729 lining every hole, and on the diagonal below the closed profile G(t) over one period with its 0.3662 percent swing exaggerated; plot; code 495, base 3, level 5, eps 7 cells, t in [1/3, 1); fg cells, blue tube, yellow curve on ground
• demo-weights: the level-5 mass field of the base-3 carpet weighted 5,4,3,4,2,3,2,1 over 24, the two heaviest adjacent so the mass drifts to the top left at every scale, read at gamma 0.12; field; code 495, level 5, 243 by 243, 32768 cells; ground to blue to yellow
• demo-shell: the circle of radius 242 crossing the level-5 carpet, its 485 crossed cells yellow where the design keeps them and blue where it drops them, threaded through the 53, 17 and 5 nested boxes of levels 2, 3 and 4; grid; code 7, base 2, side 3, level 5, r 242; yellow + blue, boxes dim at three weights, carpet line
• demo-star: the carpet's hexagonal cut layers at odd sides 1 to 55 stacked on the hexagon of the deepest, the six-armed star of the arms x = y, y = z, z = x lit over a quiet field of the same ink; field; code 23, base 2, 28 layers, band half-width 6 cells, 400 cells at 5 by 5 samples; blue and yellow over line on ground
• paper-first-base-below-a-quarter: every one-missing-digit set of bases 6 to 40 as a dash at its certified l^1 exponent, dim above the quarter line and yellow below, first yellow at base 21, whole column yellow from 34 on; plot; windows 3 to 5, sub-scan 4; yellow + dim, line blue
• demo-echo: the power spectrum of the base-10 digit design missing the digit 9, its meter M_F(x)/x^(alpha/2) read uniformly in log x, standing against the first thirteen zeta ordinates and the design's own pole lattice; plot; base 10, digits 0 to 8, depth 5, 4096 samples, band 4 to 60; yellow the zeta ordinates, pink the pole lattice, blue the spectrum
• paper-design-dirichlet-inverse: the base-3 digit set {0,1} pole comb on Re s = 0.630930 with three censused zeros right of it, one certified point boxed and two open bands, beside the integers' line at Re s = 1 and its empty half-plane; plot; base 3, fill 2, 11 poles below Im 60, rho 0.720790 + 28.605680i, 2 bands; indigo comb, yellow zeros, half-plane in panel
• research-zeta: the zero set of a design zeta beside the integers', both combs of the pole lattice drawn as rings: base 3 {0,1} carrying yellow live teeth off its abscissa and a blue second family, base 2 full set carrying no tooth at all, fg cofactor-only zeros ON its abscissa and the blue critical line; plot; strip Re s in [alpha-1.2, alpha+0.7], Im s to 40, 13 and 10 zeros; blue family, yellow teeth, fg cofactor
• demo-chladni: Larger-than-Life on the level-3 base-2 carpet mask (side 27, 512 cells) under the Bugs fractions, 64 steps by FFT convolution from a seeded density-0.48 soup on a 256 torus, live cells blue, the mask stamped top left in yellow; grid; code 7, side 27, level 3, birth [0.283, 0.375], survive [0.283, 0.483], T 64, density 0.48, fixed seed; blue + yellow on ground
• demo-moire: the four recipes over odd scales 1 to 9, one a quadrant: the heatmap summed, the weave folded to parity, the hive on the hexagonal lattice, the carpet keeping eight base-3 corners of nine; field; limit 9, 5 layers, 420 a panel; blue-to-yellow on ground
• demo-carry: the sixteen reflection-even carry blocks as a 4 by 4 census, base 3 the top two rows and base 5 the lower two, dim 2 to 9 in reading order, each block ceil(dim/2) cells a side ramped by its own peak; grid; bases 3 and 5, dim 2..9, order 1 to 5; yellow and blue over line
• demo-cuts: the sixteen diagonal cuts of code 126 at level 4, heights 15 to 30, each 81 cells drawn at one shared scale so the shape runs triangle to hexagon to triangle while the size never moves; hex; code 126, base 2, level 4, 16 by 81 = 1296 points; two central heights yellow, the rest blue
• demo-race: the race at 1200 blind steps: 300 walkers on code 127 and 300 on code 239, each design 2401 cells of 6561, the two swarms and their rms rings on the diagonal; grid; base 3, number 3, level 4, side 81, fixed seed; 127 blue, 239 orange, cells in line
• demo-radial: the rosette: the base-3 carpet at level 4 laid down six times over one full turn about its centre and averaged, its rotation order 4 opening lcm(6,4) = 12 petals; field; code 495, level 4, side 81, 6 copies, 60 degrees, mean blend; blue-to-yellow on ground
• demo-graphs: the core graph of the level-5 gasket, its 243 filled cells joined face to face into one tree of 242 branches, every node inked by its degree; plot; code 7, base 2, level 5, side 32, 82 tips, 81 paths, 80 junctions; blue branches and paths, yellow tips, pink junctions
• demo-slices: the carpet's middle hexagon at the sixteen odd sides 2k - 1 as a 4 by 4 census, 32736 unit triangles in all: many separate pieces at odd k, one piece pierced by holes at even k; hex; code 23, base 2, level 1, k 1..16, side 1..31; shattered blue, whole orange
• demo-tile: one tile repeated: the base-3 carpet at level 2, 9 by 9 and 64 cells, laid 5 by 5 into a 45 by 45 sheet of 1600 cells, the 360 seam edges where the tiling buries 720 faces stitched on the gutters; grid; code 495, base 3, level 2, 25 copies; blue cells, orange seams
• demo-crop: the level-3 sponge cut by the inscribed octahedron, of its 8000 cells only the ones the shape reaches left standing, drawn in isometric back to front with three face tones; iso; code 23, number 3, level 3, base 2, side 27, octahedron r 1/2, touching; orange three-tone on ground
• demo-plot: the difference triangle of one ledger row, the sponge's fills at odd side 2k - 1: 8 terms in 36 cells, the third differences flat at 24 and every row under them zero, so the depth is the dimension; plot; sequence_dim=3_code=23_measure=fills_axis=side, k 2..9, 8 rows; blue discs, yellow the constant row, dim rings the zeros
• demo-spectrometer: nine nested designs filling their corners in residue-level order, p = 0 to 8, each cut through the middle at side 11 in a 3 by 3 board so the ink fraction reads the corner count p/8 back; hex; codes 0 1 3 7 23 31 63 127 255, side 11, 726 triangles a panel; blue on a line mesh
• demo-tour: the carpet's perimeter: the base-2 carpet at level 4, side 81 and 4096 filled cells, all 3536 of its exposed unit edges stroked so the boundary the count measures reads as one lace; grid; code 7, number 3, base 2, level 4; yellow edges over cells in line
• demo-spin: the pair on the diagonal: the sponge's central hexagonal cut, 486 unit triangles, and the same cut spun infinitely fast into the exact circle mean at every radius, its rings integrating back to the cut's mass; field; code 23, number 3, level 2, base 2, raster 430; blue, then blue-to-yellow
• demo-gaussian: the Gaussian primes of the window at reach 40, every one of the 6561 points classified and only the primes marked as discs, the four-armed snowflake standing on its mirror symmetry; plot; ring gaussian, reach 40, norms to 3200; split blue, inert orange, the four ramified points yellow
• demo-sequences: the fill law of the ledger: every base-2 design of dimensions 1, 2 and 3 read as fills at odd side 2k - 1, 28 rows of 8 terms on log-log axes, each a polynomial of degree dim so the slope is the dimension; plot; designs 3 + 6 + 22, sides 3 to 17; dim one, yellow two, blue three
• demo-ulam: the square spiral at side 201: 40401 numbers wound from the middle with the 4236 primes lit, Euler's 4k^2-2k+41 laid under them as 101 cells, 80 of them prime and the first 21 unbroken; grid; side 201, ring k holds 8k cells; blue primes, orange line
• demo-sponge: the seed and what it grows: code 127's seven corners of eight as a 2 by 2 by 2 block top left, its level-4 solid of 2401 cubes on side 16 bottom right, the pair on the diagonal; iso; code 127, number 2, base 2, levels 1 and 4; green in its three standard tones
• demo-life: Conway's Life from the R-pentomino on the 96 by 96 dead-edged board, every cell tinted by the generation it first came alive over the page's 512-step fate limit and the population still standing at 512 laid over it; grid; B3/S23, mask the side-3 tile with its centre popped, 5 seed cells, T 512; green fading to line, the live cells yellow
• demo-spectra: the integrated density of states of the sponge's diagonal slice, 306 filled triangles in one piece, the normalised-Laplacian staircase on log-log axes with its low 30 points fitted to d_s = 1.2533; plot; code 23, number 3, level 2, base 2, 286 distinct values, window 10 percent; blue staircase, yellow fit, panel window
• demo-morse: the Thue-Morse plane, the parity lift t(i) xor t(j) at side 32 read plus-minus, 512 warm sites against 512 cool, the Kronecker power of the two-by-two tile; grid; side 32, level 5, base 2, 1024 sites; orange plus, blue minus on ground
• demo-universe: the finite gallery: all 22 base-2 cube designs, each grown at side 3 into its 27-cell solid and stamped in isometric at one scale, in code order in rows of four and five; iso; dim 3, 256 codes folding to 22 orbits, level 1; indigo three-tone in a line cage
• demo-zeta: the walk of zeta on the critical line, s = 1/2 + it traced in the plane for t up to 50, the loop passing through the origin once for each of the 10 zeros below it, the unit circle and the axes as hairlines; plot; t 0 to 50, 8000 samples, first zero 14.134725; blue curve, yellow origin, line axes
• demo-tower: the word dim 3, magic [23 9 23], side [3 3 3] built block by block down the diagonal: the first letter alone at side 3, the pair at side 9, the whole word at side 27, 20 then 200 then 4000 cubes; iso; three letters, base 2, dim 3; orange in its three standard tones
• demo-words: the constructor word carpet(3) net(7) void(5): side 105 and 3432 filled cells, its three prefixes stacked so the nesting depth reads as a ramp, 3200 cells at depth one, 3168 at two, 3432 at three; grid; codes 7, 14, 9 at sides 3, 7, 5, base 2; the word blue over dim and line
• demo-volume: the moire solid of code 23: six layers at the odd scales 1 to 11 summed on a 64 grid, the level set where all six agree drawn as a shell of 34416 voxels of 262144; iso; code 23, base 2, level 1, sum, threshold 6; indigo in its three standard tones
• demo-primes: the sieve of Eratosthenes run to the page's top limit 400 as a 20 by 20 board, the 78 primes yellow and the 321 composites tinted by the rank of the least prime that struck them, the unit left as ground; grid; limit 400, 8 striking primes 2 to 19; yellow primes, dim to orange strikes on ground
• demo-integers: the pinned window 1 to 1000 drawn at the four rendered depths 8, 16, 32 and 48 terms, 1000 cells a panel, the orange draining as the ledger deepens; grid; 18066 rows, four tiers, ceiling 1000, 100000 cells a term; missed orange, written once dim, written by many blue
• research-stack: the spun Farey stack, the Gaussian layers z^-1 Z[i] summed on the unit square, a node u/d lit by every layer z that d divides; field; norm bound 60, 46 layers, 880 nodes, brightness the classes of norm <= 60/N(d); ramp ground to blue to yellow
• demo-tourbillon: the spun odd carpet stack, parity carpets at the odd scales 1 to 55, layer k turned by k golden angles into the inscribed disc, each pixel the share of 28 layers inking it; field; N 55, golden, odd set, plain weights, cells, mean, seed 1, raster 1024, 28 layers, centre 14/28; ramp ground to indigo to pink
• demo-spirograph: the carpet as a spirograph wheel: the base-3 carpet at level 1 with a pencil in each of its 8 fills and its 1 void, rolled inside a circle of radius 13 on a wheel of radius 5, 9 distinct curves closing after 5 orbits with 13-fold symmetry; plot; code 495, base 3, side 3, level 1, both pencils, in, R 13, r 5, reach 1.2, 6000 samples; blue fills, orange void, dim ring
• research-spirograph: the carpet rolled inside a circle at R/r = 7/3: the shape its 8 curves wall off between the two floods, shaded, and their 1288 crossing nodes; plot; code 495, base 3, level 1, fill pencils, in, R 7, r 3, reach 0.9, raster 1024; blue curves, orange nodes, dim shape, line ring
• demo-memory: The supergolden rule 23 read as a width-3 window on the binary digits, its accepted words stacked one row a level; grid; dim 1, k 3, code 23, levels 1 to 9, 8 percent frame; yellow with the deepest level in orange.
• research-beneath: The Koch curve as a radix code: base 3 on the Eisenstein integers, digits 0, 1, 2+w, 2 with unit twists 1, 1+w, -w, 1, the 4^6 word points of level 6 joined in word order; plot; ring Z[omega], base 3, class code 147, level 6, 8 percent frame; one ink, blue on the ground
• demo-radix: the norm-7 tile of the place dial, the seven-digit design at the base 3 + w of the hexagonal lattice; plot; base 3 + w of norm 7, the whole canonical residue system as digits, no twist, level 6, 117649 words as dots, margin 8 percent; one ink, blue
• paper-spin-harmonics: design 45 as a 3x3 grid, its nine cells plated dim and its four filled blue, the six exact chords joining their centres, the two of length 2/3 pink and the other four dim; grid; base 3, level 1, code 45; blue + pink + dim on ground
• research-apollonian: the strip packing on one period, the two lines and every circle to curvature 2048, the 323 Ford circles resting on the lower line ramped by denominator b, the root pair the period's own arcs; plot; strip root (0,0,2,2), T 2048, x in [0,1]; blue at b=1 to fg at b=32, the rest and the two lines dim on ground
• demo-apollonian: the strip packing on one period to curvature 2048, its 323 Ford circles resting on the lower line, and the Farey stack at depth 32 hung under that line, one bar standing on one tangency point; plot; grow strip, T 2048, touches, farey 32, x in [0,1]; circles inked by curvature octave over the six, the bars blue, each ticking through the line at its own tangency point, the two lines dim on ground
• research-cobham: Object Y in the base-2 box [0, 128)^2, which holds every point of both gaskets below 3^5: the base-2 gasket x AND y = 0 as faded dim cells, the 243 points of the base-3 gasket blue, the 111 points in both gaskets yellow, 63 of them on the two axes; grid; base 2 meets base 3, m = 5, side 128, origin at the lower left; yellow + blue + dim on ground
• wiki-kronecker-product: the carpet rule stamped into itself, code 7 at levels 1, 2 and 3 as three equal squares in a row, 8, 64 and 512 cells; grid; code 7, base 2, levels 1 to 3, sides 3, 9, 27; the rule yellow, the products blue on ground
• wiki-farey-sequence: the Farey sequence growing, one row per order 1 to 8, every reduced fraction of the row a dot at its place on the unit line, the fractions the row adds in yellow; plot; orders 1..8, 83 dots, 23 new; blue and yellow dots on line rules
• wiki-visible-lattice-points: the corner window at N = 100, the 6087 points of coprime coordinates as full blue cells and every hidden point as a centred dim square of side one over the divisor it shares; grid; N 100, origin lower left; blue + dim on ground
• wiki-euler-mascheroni-constant: the harmonic staircase standing over the log curve for n up to 30, the shaded strip between them closing on gamma; plot; 30 treads, strip 1.000 at the left and 0.593790 at the right against gamma 0.577216; yellow stair, blue curve, blue strip at 28 percent on ground
• wiki-prime-counting-function: the prime count staircase for n up to 200 with a hairline standing on each of its 46 risers, against the smooth guess n over ln n; plot; n 2..200, 46 primes against a guess of 37.7; yellow stair, blue curve, dim hairlines on ground
• wiki-goldbach-conjecture: the comet, the count of prime pairs summing to every even number from 4 to 400 as 199 bars, the 66 multiples of six riding twice as high as the rest; plot; peak 27 at 390, floor 1; yellow sixes, blue the rest over a line baseline
• wiki-mertens-function: the Mobius marks summed and walked to n = 1000 inside the trumpet of plus and minus the square root of n, up to 7 at 586 and down to minus 12 at 665; plot; 1000 steps, M(1000) = 2; blue walk, dim envelope, line axis and zero rule
• wiki-famous-formulas: the eight systems of the hub as eight panels in a 4 by 2 board, one tiny curve each, every panel the normalised gap on log-log axes over 4.5 decades; plot; n 2..2000, 240 rungs a panel; five chasers blue, the prime count and Goldbach yellow, Mertens orange dots
• wiki-prime-numbers: the sieve of Eratosthenes on the first hundred numbers as a 10 by 10 board, the 25 primes yellow and the 74 composites tinted by the rank of the least prime that struck them, the unit left as ground; grid; limit 100, 4 striking primes 2 to 7; yellow primes, dim to orange strikes on ground
• wiki-greatest-common-divisor: every pair (a, b) drawn from 1 to 24 as a 24 by 24 board, the 359 coprime pairs lit and the other cells shaded by the divisor they share, the diagonal the brightest line; grid; side 24, 576 pairs, gcd up to 24; yellow coprime, dim to blue by shared divisor on ground
• wiki-eulers-totient: phi(n) as a bar for every n from 1 to 60 under the hairline ceiling n - 1, the 17 primes the only bars that touch it; plot; n 1..60, peak phi(59) = 58; yellow primes, blue composites, ceiling and foot in line
• wiki-mobius-function: mu(n) for n = 1 to 100 as a 10 by 10 board in reading order, 31 cells at plus one, 30 at minus one and 39 at zero; grid; top 100, three values; yellow plus, blue minus, dim zero on ground
• wiki-riemann-zeta-function: the partial sums of 1/n^s at s = 2, 3 and 4 over the first forty terms, three curves closing on three hairlines at zeta(2), zeta(3) and zeta(4); plot; 40 terms, window 0.94 to 1.72; blue at s 2, orange at s 3, green at s 4, the limits in line
• wiki-gaussian-integers: the Gaussian primes of the window at reach 20 as dots, the split and ramified ones in a four-armed snowflake and the 16 inert ones alone on the two axes; plot; ring gaussian, reach 20, 1681 points; split blue, inert orange on ground
• wiki-wallis-product: the partial products of 4k^2/(4k^2 - 1) as forty bars climbing towards pi/2, the limit a dashed rule no bar reaches; plot; 40 factors, window 1.273 to 1.611; blue bars, dim limit, line axis
• wiki-leibniz-series: the first thirty partial sums of 1 - 1/3 + 1/5 - ... as dots threaded in order, alternately over and under the dashed pi/4; plot; 30 sums, 15 over and 15 under; blue over, orange under, dim limit on line
• wiki-basel-problem: twelve squares of side 1/k standing end to end on one rule, over the first forty partial sums of their areas as bars against the dashed pi^2/6; plot; 12 squares, 40 sums; blue squares, yellow bars, dim limit
• wiki-eulers-number: (1 + 1/n)^n at n = 1 to 60 as one rising curve with a dot a step, flattening under the dashed line at e; plot; 60 stops, window 1.93 to 2.77; blue curve, yellow dots, dim limit
• wiki-wallis-sieve: the plane Wallis sieve at level 3, its 201 punctures as filled squares over the surviving body, one ink a level so the schedule reads as three sizes; grid; word 3, 5, 7, side 105 drawn 840 px, 9 percent margin; body line, holes yellow, orange, blue
• wiki-parity: the four parity classes of a cell, a 6 by 6 grid with every cell painted by the parity of its row and its column, nine cells a class; grid; side 6, 4 classes, gap 0.05; blue, yellow, orange and green on ground
• wiki-sierpinski-carpet: the carpet at level 4, 4096 filled cells on a side of 81, its holes at four sizes punched through to the ground; grid; code 495, base 3, level 4, side 81; blue on ground
• wiki-menger-sponge: the sponge at level 3, 8000 cubes on a side of 27, standing clear inside an 8 percent margin with a carpet on every face; iso; code 23, number 3, base 2, level 3; top faces yellow, left faces orange, right faces dim
• wiki-fractal-dimension: box counting in three panels, a filled square, the carpet at level 3 and a diagonal segment each under a 3 by 3 lattice with the occupied boxes ringed, 9 against 8 against 3; grid; code 495, base 3, level 3, 3 boxes a side; blue shapes, yellow boxes, line lattice on ground
• wiki-cellular-automaton: rule 90 from one live cell, 64 generations on a 127-cell window, time running down, 729 live cells in a triangle of triangles; grid; rule 90, T 63, window 127; yellow on ground
• wiki-euler-characteristic: the count on two shapes, a cube of 8 vertices, 12 edges and 6 faces beside the level-2 carpet with each of its 9 holes ringed, so 8 - 12 + 6 = 2 against 1 - 9 = -8; iso and grid; cube corners, carpet code 7, base 2, level 2, side 9; blue three-tone, yellow vertices, fg and dim edges, orange rings
• wiki-graphs: the level-3 base-2 gasket as a network, its 27 filled cells dots and its 26 face-to-face joins edges, every dot inked by its degree; grid; code 7, base 2, level 3, side 8, degrees 10 ones 9 twos 8 threes; ends yellow, through blue, junctions pink on blue edges
• wiki-transfer-matrix: a three-state machine over its 3 by 3 table of allowed steps, the counts of walks of length 1 to 8 as bars beneath; plot; 3 states, 4 arrows, lengths 1..8, counts 4 to 60; states and bars blue, arrows and lit cells yellow
• wiki-spectral-radius: one arrow multiplied eight times by a positive 2 by 2 matrix, each cut to the same length so only the turning shows, the settled direction a dim ray; plot; matrix [[2,1],[1,3]], start (1, -0.6), 8 arrows, limit the golden slope; blue to yellow arrows on a dim ray and arc
• wiki-graph-laplacian: the sorted Laplacian eigenvalues of a path of 32 vertices and of a cycle of 32 as two staircases on one 0 to 4 scale; plot; 32 vertices, closed form 2 - 2 cos, 32 and 17 distinct values; path blue, cycle orange on dim rules
• wiki-random-walk: one walk traced on a 64 by 64 lattice with the circle of radius the square root of the step count round its start; plot; 1024 steps, side 64, radius 32 cells, first seed that stays on the lattice and lands within 8 percent of the radius; blue path, yellow start, orange end, dim ring on a faint lattice
• wiki-moire: the unit square ruled at scale 5 and at scale 7, the 92 crossings of one ruling with the other as dim dots and the 4 points both rulings mark lit; grid; scales 5 and 7, gcd 1, 92 crossings, 4 coincidences; scale 5 blue, scale 7 orange, coincidences yellow
• wiki-pascals-triangle: 64 rows of the triangle with the odd entries lit and the even ones as faint dots, drawing the gasket; grid; 64 rows, 2080 entries, 729 odd; odd blue on dim
• wiki-thue-morse-sequence: the first 64 letters as a strip over the plane lift t(i) xor t(j) at side 32; grid; 64 letters, side 32, 512 of each sign; zero orange, one blue
• wiki-ford-circles: the circle of radius 1/(2b^2) over every reduced fraction of denominator at most 8 on the unit line, outlined, with a dot at each tangency; grid; order 8, 23 circles; blue to yellow by denominator on a hairline
• wiki-spirograph: one hypotrochoid, a wheel of radius 3 rolling inside a ring of radius 7 with the pen at reach 0.8, seven arms closing after three laps; plot; ring 7, wheel 3, reach 0.8, 4000 samples; curve blue inside a dim ring
• wiki-ulam-spiral: the whole numbers wound on a square spiral with the primes lit, the diagonals standing out; grid; side 101, top 10201, 1252 primes; blue on ground
• wiki-eisenstein-integers: the hexagonal lattice a + b omega out to eight steps with its bonds, the six units and the origin; grid; radius 8, 217 points, 6 units; lattice blue, units yellow, origin orange
• wiki-burnsides-lemma: the 16 two-by-two patterns as stamps in six rows, one class per row under the symmetries of the square; grid; 16 patterns, 6 classes, 48 fixed points; filled blue, empty dim