Fractal Optimization Through a Procedural Lens

1. Fractals and How to Measure Them

A fractal is a shape whose detail does not run out when you zoom in. Its dimension says how fast the detail grows, and two tools measure it.

1.1 How Long Is a Coastline?

Richardson measured coastlines with a pair of dividers and found that the answer depends on the dividers. Set them to 100 km and walk the coast of Britain: so many steps. Set them to 10 km and the walk goes into bays the long steps cut across, and the coast is longer. It keeps getting longer as the ruler shrinks, with no sign of stopping [1]. Mandelbrot read Richardson's table in 1967 and saw that the way the length grows is the thing to measure [2].

The Koch curve does it exactly [3]. Replace a segment by four segments a third as long, and repeat on every segment. Each round multiplies the length by 4/3, forever. Below, walk it with a ruler of your own. Halve the ruler and the measured length grows by the same factor every time; on the log plot the points fall on a line, and the slope of that line is the dimension minus one. Between the powers of three the measure wobbles, which real coastlines also do; the fit uses the powers.

1.2 The Sierpinski Carpet

Cut a square into nine, remove the middle one, repeat on the eight that remain. Each round keeps 8 copies at a third of the size, so D = log 8 / log 3 = 1.893: more than a line, less than a plane. The Mandelbrot set is the same idea with a different rule. Mirrorbrot [4] folds it, the 16K edition [5] pushes the zoom, and the Nebulabrot [6] draws the escape paths instead of the boundary.

1.3 The Menger Sponge

The same cut in three dimensions keeps 20 cubes of 27, D = log 20 / log 3 = 2.727. Drag to rotate. It is also the first, smallest example of what self-similarity buys. The sponge is drawn as copies of one cube, so the work grows with the 20 cubes kept, not the 27 cells of a grid; at level 3 that is 8,000 cubes against 19,683 cells, and the gap widens by 27/20 every level. Chapter 6 is the large version of this. The N-Dimensional Viewer [7] slices fractals in more dimensions, and the tesseract [8] shows what one slice of a 4D object loses.

1.4 Measuring: Difference of Gaussians

Two tools carry every number in this paper: one finds the edges, the other counts boxes. Blur the image twice with different widths and subtract. What survives is edges, the Marr–Hildreth operator [9]. Here the two blurs are box blurs, a cheap stand-in for the Gaussians. Upload your own image, or use the sample: the carpet from 1.2 at level 4.

σ₁ 1.0 σ₂ 2.0

Count boxesD = –

1.5 Measuring: Box Counting

Cover the edges with boxes of side 2, 4, 8, 16, 32 and 64 pixels, count the boxes that touch an edge, and fit log count against log 1/size. The slope is D. The carpet's answer is known, 1.893; the counter reads 1.81. The gap is the method. The smallest boxes see pixel-wide edges, dimension 1; the largest see the whole image, dimension 2; only the middle of the range sees the carpet. A dimension is never read off one number. It is read off a range of scales, and the range has to be stated.

1.6 The Space of Fractals

Every fractal made by a formula is one point in a space of fractals: change a number and the shape moves to a neighbouring point. That space has a geometry of its own. The Mandelbrot set is its most famous chart: each point of it stands for a whole Julia set, and near certain points it looks exactly like the Julia set it stands for. In three and four dimensions there is no such theorem, but there is an instrument. Below is the Fractal Engine, inside a Mandelbox. Hold still on a wall and the engine finds the number that moves the surface most there; hold Shift as well and the view becomes the map of two of those numbers around your point, that point's own Mandelbrot set. The companion paper Metafractal: The Space of Fractals Is Fractal (paper 0.1) follows this space from the Mandelbrot set to the engine's maps, walks and species.

2. Brownian Motion: The Casebook

Mandelbrot's Fractal Geometry of Nature is a casebook: a chapter per phenomenon. Most of the phenomena are one machine, the random walk, with one dial turned.

This chapter is that casebook, live [10]. Each section turns one dial and names the dimension it produces.

2.1 The Random Walk

Each step moves up to 5 pixels in a random direction. Stretch time by a factor c and the path stretches by √c, so no zoom looks different from any other: the walk is self-similar by construction. Its trail has dimension 2 in the plane. Measure it: a short walk reads lower and climbs toward 2 as it fills the square. Diffusion Fluid [11] is the same walk run as a field.

2.2 Fractional Brownian Motion

Mandelbrot and Van Ness put a dial on the walk, the Hurst exponent H [12]. Above ½ the steps remember each other and the trace is smooth and persistent; below ½ they contradict each other and it is rough; at ½ it is Brownian. The graph has dimension 2 − H. Hurst found the dial in eight centuries of Nile flood records, at H ≈ 0.7 [13]. The seed below is the same at every H, so the slider changes only the roughness.

2.3 Brownian Landscapes and Coastlines

The same construction on a grid gives Mandelbrot's Brownian landscapes, relief of dimension 3 − H [14]. Flood it to the mean height and the shore is a coastline of dimension 2 − H: section 1.1 again, grown instead of drawn. Every terrain in every game is this machine.

2.4 Lévy Flights

Keep the walk and drop the typical step. Draw the step lengths from a power law and the walk becomes a Lévy flight: clusters of clusters of visits, which Mandelbrot called Lévy dust and used for the distribution of galaxies. With exponent α below 2 the dust has dimension α. Above 2 a typical step returns and the walk is Brownian again.

2.5 Diffusion-Limited Aggregation

Release a Brownian walker far from a seed, let it stick when it touches the cluster, repeat. Witten and Sander: the aggregate has dimension 1.71 [15]. The button gives three numbers. Mass against radius reads 1.63 to 1.70 on this small lattice. Box counting reads about 1.5, because at two cells a branch is a line. Box counting capped at a quarter of the cluster radius is the lesson Petri [16] paid for: on its grid, boxes near the cluster's own size dragged the slope toward 2.

2.6 The Spectrum of a Rough World

A rough world leaves a signature you can measure on any photograph. Break a picture into layers from the broadest shapes to the finest grain and ask how loud each layer is. That is its power spectrum. For a rough world the answer is a straight line on log scales: every octave finer is quieter by the same factor. Field found power falling as 1/f^β with β close to 2 [17]; Ruderman and Bialek measured 1.81 in woodland [18]; Torralba and Oliva fitted twelve thousand scenes at 1.8 to 2.4 depending on scene and direction [19].

The photograph below is the iris behind eye 26 of the Iris Engine (Lizapopova143, Wikimedia Commons, CC BY 4.0). Its texture reads about 2.9 and the whole iris 2.7, steeper than the scene average because one object at one distance is smoother than a scene. The carpet reads 2.35. The landscape from 2.3 should read 2H + 2 and midpoint displacement comes in a little under. White noise reads 0. Every one of them is a straight line, and that is the point. Now move the noise slider: noise is equally loud at every layer, the flat blue line, and every layer under it is gone. Keep this picture; chapter 5 is built on it.

3. Growth: Worlds That Build Themselves

Random walks are not the only machine that makes a rough world. Trees grow one, competing for light, and rivers carve one, competing for ground. Both are searches, and both leave fractals behind.

3.1 Trees and Forests: Synthetic Silviculture

A tree is a branching fractal, but nobody draws it. It grows. Buds compete for light, the well-lit branches get more of the plant's vigour, the shaded ones are shed, and the crown ends up filling the light it can reach. Pałubicki and colleagues grew single trees from those rules [20], with the vigour split between leader and side branch taken from Borchert and Honda's measurements of a tropical tree [21]. Makowski, Pałubicki and colleagues then grew whole forests and called it synthetic silviculture [22]: modules of branches grow by the light they receive, every plant shades its neighbours, grown trees seed young ones around them, and plants die of age or of shade.

Nothing in the rules says forest. Pioneers that need full light come first; shade-tolerant trees germinate under them and take over; the stand thins as it grows. Foresters measured that thinning long before computers: in a crowded stand, mean plant mass rises as density to the power −3/2 [23]. Makowski's full model reproduces that law. The reduced one here does not, cleanly: dense even-aged plantations thin at slopes between −2.3 and −3.1, steeper than the foresters' −3/2, and the mixed forest loops in the second chart as pioneer cohorts boom and bust every fifty years or so. It drops the paper's branch prototypes and climate, and adds three rules of its own; the gap is the price of that. A tree is a shape that is also a search, which is where chapter 4 starts.

Each species in that forest is a handful of numbers: how strongly the leader dominates its side branches (λ), how determinate the branching is (D), which way the branches lean, how long a tree lives, how much shade it can take. Some trees differ by more than numbers. Hallé and Oldeman sorted the world's trees into some two dozen architectural models by how trunk and branches are built [24]; eight of them grow here, from the layered conifer (Massart's model) to the palm (Corner's) and the candelabra (Leeuwenberg's), with reiteration, a strong branch turning into a new trunk, and sprouting after damage. The windows below grow one plant of each form alone, in open light. Every window is cut to the shape of its plant.

3.2 A Landscape Formed by Rivers

The forest needs ground, and the ground is not drawn either. It starts as the Brownian landscape of 2.3, raised by tectonic uplift, and is carved for five million simulated years by the law geomorphologists use for rivers: the ground lowers in proportion to its slope and, roughly, the square root of the area draining through it [25][26]. Hillsides creep, slopes steeper than their angle of repose slide, and hard rock beds erode slower than soft ones, so cliffs and benches form where they alternate [27]. Lakes fill to the height of their spill point [28]. Rivers take their width from the water they carry, and on low slopes they meander and cut off oxbow lakes [29]. Lower the sea and the region becomes coast, islands and, now and then, a strait.

The result can be checked, because river networks on Earth obey tight laws. A river's length grows with its basin's area to a power between 0.57 and 0.60, which is Hack's law; each order of stream gathers three to five tributaries of the order below, which is Horton's [30]. Measured on three generated regions:

Two misses, stated. The hypsometric integral is low because these regions are mostly foothills, and the coastline is a little rougher than Britain's west coast. The other numbers sit inside Earth's ranges or at their edge.

Temperature falls 6.5 °C per kilometre of height, and rain falls on the windward slopes and spares the lee [31]. The region is 10 km across and its summits stand about 1,600 m above the sea, so the forest's bands are now real ones, without the earlier exaggeration. With the sea at 7 °C, broadleaf and conifer mix below about 300 m, conifers run to about 600 m, birch and shrubs above them, and the treeline sits at 700 m, with grass and shrubs to the summits. Warm the climate and the bands climb. Soil grows from the rock and is stripped from steep ground [32]. The rock types keep their signatures: strata on sandstone cliffs, columns in basalt, clints and grikes in limestone, rounded granite boulders, scree under every cliff [33].

Click anywhere on the region and the camera flies down to a plot there, where the forest of 3.1 grows from that spot's soil, climate and neighbours. Trees die at about one percent a year, faster in shade, and in storms. A storm uproots some trees, leaving a root plate, a pit and a mound that outlast the tree by centuries, and snaps others, leaving snags [34][35]. Fallen logs stay where they fell. They decay through five classes at a rate that rises two and a half times with every ten degrees of warmth [36], grow moss, sink into mounds and end as humus, and shade-tolerant seedlings root on the softest of them, as they do on nurse logs. One plot after three centuries: 950 living plants, 81 snags and 984 logs, nearly half of them in the last decay class, 89 pits, and 118 seedlings rooted on logs. Of the 2,012 trees that died there, 85 % died of age, 11 % in storms and 4 % of shade.

What is still chosen rather than derived: the pattern of uplift and the erodibility that sets the relief, the geological map, the rates of storms, the thresholds between materials, and every colour. They are listed in the code.

4. Walks That Fit

The world is rough, and a model has to fit it. Every fitter I know is a walk on a rough landscape.

4.1 The Ising Model and Annealing

Spins on a 100 × 100 grid, each up or down, each wanting to match its four neighbours. The Metropolis rule [37]: a flip that lowers the energy is always taken, one that raises it by ΔE is taken with probability exp(−ΔE/T). Onsager solved this grid exactly [38]: the critical temperature is 2 / ln(1 + √2) = 2.269. The slider starts at 2.3. Below it the domains freeze, above it there is only noise, and at it there are domains of every size: the fractal case. Beggs and Plenz measured the same power law in the avalanches of living cortex [39]. Press Anneal: start hot, cool slowly [40], and the domains grow instead of freezing where they stand.

4.2 A Walker on a Rough Landscape

Take the trace from 2.2 as a landscape and drop a Metropolis walker on it: down is free, up costs exp(−ΔE/T). Cold, it sits in the first dip it finds. Hot, it wanders and settles nowhere. Anneal, and it finds the deep one more often than not. This is all of optimisation on a fractal. The roughness is at every scale, so there is no scale at which a greedy step is safe.

4.3 The Fitters

The Iris Engine [41] fits a procedural iris to a photograph: a search over the shape genes, a material inversion per cell, two thousand Gaussian splats on the relief by gradient descent, then the layer model. On the four isolated eyes the match score reads 90.35, 88.12, 91.40 and 88.32, and the shipped files reproduce those renders within one level on every load. That walk has no noise in it. The splat scenes [42] will be trained with MCMC splatting [43], which is stochastic gradient Langevin dynamics [44]: gradient plus Brownian noise, the walker from 4.2 on millions of parameters. Petri's think tank runs CMA-ES [45], Gaussian mutations on a genome. Evolution is the original [46].

5. Why Image Generators Draw the Big Shapes First

An image generator starts from pure noise and removes it a little at a time. Watch one work and the large shapes come first, the details last. This chapter proves why, in four steps, and each step is a figure.

5.1 Is the Noise in Stable Diffusion Brownian?

For the models most people have used, yes. Stable Diffusion 1.x, 2.x and XL add noise in a thousand small Gaussian steps to a compressed 64 × 64 × 4 picture [47][48]. Those steps are the discrete form of an Ornstein–Uhlenbeck process, Brownian motion with a gentle pull toward zero [49][50]. Stable Diffusion 3 and FLUX do not walk. They slide along a straight line from the picture to one draw of noise [51]. What every family shares is the state at each noise level: the picture plus Gaussian noise of a known width σ. The proof below is written in σ, so it covers all of them.

5.2 The Four Steps, Live

The figure is a world whose spectrum you choose, the exponent β. On the left the world, in the middle what the generator sees at noise level σ, on the right the best possible guess of the world from what it sees. The chart is steps 1 to 3 at once: the falling line is the world, the flat line is the noise, and the shaded band is what the best guess can still recover. Press Generate and σ falls the way a generator lowers it. At β = 2 the picture arrives continents first. Set β to 0, a world as loud at every scale as the noise itself, and press Generate again: there is no order any more, every scale arrives at once.

5.3 The Theorem

5.4 From the Theorem to Real Pictures

Photographs are not Gaussian fields: they have edges and objects. Three things carry the theorem over. The spectrum of photographs is a power law with β near 2, measured, not proved (section 2.6). Trained denoisers act like adaptive versions of this filter, keeping coefficients louder than the noise in a basis fitted to the image [55]. And the generated order is observed: Rissanen and colleagues reported that diffusion models generate coarse to fine [56], and Dieleman named it spectral autoregression [57].

Where the theorem stands among them. Rissanen gave the additive noise floor, Dieleman the phrase, Falck and colleagues the signal-to-noise picture per frequency [58], and Hunt, Kamb and Ganguli wrote the power-law Wiener filter and its rescaling [59]. None of them states the order of generation as a theorem. The contribution here is that statement with the schedule corollary, and figures that let anyone check it.

Two limits keep it honest. Falck's group built a forward process that noises every frequency equally, and it trains as well as the usual one. So the coarse-to-fine order is a consequence of Gaussian noise meeting a power-law world, not a condition for diffusion to work at all. And production schedules do shift with resolution: SD3 shifts by a factor of 3 at 1024², FLUX by the number of tokens. That fits the theorem rather than breaking it. Changing the pixel count moves every frequency in pixel units, and the theorem's statement is in units of the picture.

5.5 The Carpet From Noise

The last figure leaves the Gaussian world. The data are 1,200 points on the carpet, shown faintly; 1,200 points start as noise. Each step runs the reverse-time equation one level down the schedule, with the exact score of the noised data, computed from the points instead of learned by a network. Watch the order in which the carpet returns: the big hole first, then the eight, then the sixty-four. That order is the theorem, on a world that is a fractal instead of a Gaussian field.

6. What the Fractal Toolset Buys: Speed

Self-similarity is also a shortcut. Where a problem has the same structure at every scale, the work can be done once per scale instead of once per pair.

Below are a few thousand bodies scattered as Lévy dust, clusters of clusters. Each pulls on every other, so the direct sum is one interaction per pair: N². Barnes and Hut group the far bodies into boxes of a quadtree and let one box stand for all of its contents when it looks small from where you stand [60]. The opening angle θ says how small is small. The count falls to about N log N. At 4,000 bodies and θ = 0.5 the tree does 41 times less work for an error of 0.9 %; at 32,000 bodies, 235 times less for 2 %. The bigger the problem, the bigger the gain. Push θ past 1 and the error climbs to 5 and then 10 %: the shortcut only holds while a far box really does look small.

The same trade runs through all fast numerical methods. Each buys its speed with one regularity across scales, and each fails where that regularity fails.

None of it makes an arbitrary shape faster. Recursion is cheap only where the structure is really there. For neural networks the open question is the landscape. Sohl-Dickstein found that the boundary between training that converges and training that diverges is fractal in the plane of learning rates, with box dimensions from 1.17 to 1.98 [67]. Stochastic gradient descent behaves, under four stated assumptions, like the walker of 4.2 at a temperature set by learning rate over batch size [68]. Whether that landscape has a regularity a fast method can use is the subject of paper 2.

7. Consequences

Every fit is the same act: a slow walk of a model toward a rough world.

The context an organism adapts to is not a problem with a solution. It is a fitness landscape [46], and the landscape is rugged at every scale: a niche as intricate at the scale of a biome as at the edge of one leaf. Evolution is not a climb. It is a population diffusing across that surface, the walk from 2.1 with selection as its drift. The butterfly's wing and the perceptron's learned weights [69] are two snapshots of the same walk: the slow, painful, and finally beautiful convergence of a model to a world.

Every engine on this site is that act. An iris ID, a Dish ID and a flame preset are compact descriptions, and each engine regenerates a world from one. What none of them has yet is a sense of which descriptions are likely, learned from the fitted cases, and a noisy walk to sample them: diffusion over the description instead of the pixels, with the engine in the loop. The honest gap is size. The fitted set is four eyes. The step this paper implies is the database of fitted eyes the Iris Engine was planned around, and a score learned over it. Then an eye from zero is the carpet from noise.

8. The Series

Brownian motion makes a rough world. A cellular automaton computes one. This paper opens a series, and its last section is a pointer to each of the others.

8.1 Towards the Computational Basis

The next paper, Cellular Automata as the Computational Basis [70], takes local rules as the subject. Conway's rule [71]: a live cell survives with two or three live neighbours and a dead one is born with three. Langton's ant [72]: on white turn right, on black turn left, flip the cell, step; ten thousand steps of mess, then a highway.

Flame Strains [74] burns any 2D automaton as a strain inside one volumetric flame. Petri runs ten growth kernels and 93 organisms as data rows, every dish replayable bit for bit. Its first measured result opens the next paper: Tero and colleagues let Physarum connect 36 food sources laid out as the cities around Tokyo [75], and Petri's adaptive Physarum on Tero's own geometry, 21 replicates, bands fixed before the run, lands inside the published spread.

Slime Mold [76], Ember Engine [77], Mycelium [78] and Frost [79] are the single-rule engines before Petri.

8.2 The Other Papers

• 0.1Metafractal: The Space of Fractals Is FractalOctober 2026

  Section 1.6 made a paper: the Mandelbrot set as the map of the Julia sets, Tan Lei's similarity, and the Fractal Engine's maps, metric, walks and species as the instrument for the spaces no theorem reaches.

• 0.2Radiance Fields: Elements and MorphsOctober 2026

  The engine's object, taken apart: density, emission, view-dependent colour, the volume integral, and the representations (distance fields, voxels, Gaussians, networks), with the morphs between them.

• 1Cellular Automata as the Computational Basisin preparation

  Local rules compute a rough world, measured against known answers: Flame Strains, Petri, the Tero fixture, the think tank.

• 2Fractal Optimization of Tensor Landscapesin preparation

  Chapter 6 made precise: where the landscapes of neural networks are fractal, and whether that structure can be turned into speed.

• 3Diffusion Over the Whole Documentin preparation

  Chapter 5 applied to text. Today's text generators write left to right, a word or a few words at a time. Could a model draw a whole document coarse to fine, the outline first? Text has scale structure, but the diffusion models of text measured so far settle content words first and punctuation last. The paper builds the test.

• 4Lambda Cognition & the TIH Frameworkpublished December 2025

  The substrate: sparse interaction instead of dense matrices.

Glossary

Dimension (fractal)

How fast the detail of a shape grows as you zoom in. A line is 1, a plane is 2; a coastline is in between. Measured as the slope of count against scale on log axes.

Self-similar

Looks the same at every zoom, exactly (the carpet) or statistically (a Brownian walk). Self-affine: the same after stretching the two axes by different amounts, like the trace in 2.2.

Brownian motion

A walk of independent random steps. The simplest machine that makes a rough world, and the noise of the older image generators.

Hurst exponent H

The roughness dial of a walk. ½ is Brownian; above ½ the steps remember each other and the trace is smoother; below ½ it is rougher.

Power spectrum

How loud each scale of a picture or a signal is, from the broadest shapes to the finest grain. A rough world has a spectrum that is a straight falling line on log axes.

Octave

A doubling of frequency, or a halving of scale. From continents to countries is one octave if countries are half the size.

Noise level σ

The width of the Gaussian noise on a picture at one moment of a diffusion model. Generation lowers it from large to nearly zero along a schedule.

Score

The direction in which a noisy picture becomes more likely, ∇ log p. A diffusion network is trained to output it; by Tweedie's formula it is the best guess of the clean picture, minus the noisy one, over σ².

Annealing

Searching hot, then cooling slowly, so a walker can leave shallow dips early and settle into a deep one late.

References

1. Richardson, L. F. The problem of contiguity: an appendix to Statistics of Deadly Quarrels. General Systems Yearbook 6 (1961) 139–187.
2. Mandelbrot, B. B. How long is the coast of Britain? Statistical self-similarity and fractional dimension. Science 156 (1967) 636–638.
3. von Koch, H. Sur une courbe continue sans tangente, obtenue par une construction géométrique élémentaire. Arkiv för Matematik, Astronomi och Fysik 1 (1904) 681–704.
4. Mirrorbrot — Mirror-Fold Deep Zoom. iori.me, 2026.
5. Mirrorbrot — 16K Deep Edition. iori.me, 2026.
6. Image-GS Nebulabrot. iori.me, 2026.
7. N-Dimensional Viewer. iori.me, 2026.
8. Binary-Pure Tesseract — Interactive 4D Projection. iori.me, 2026.
9. Marr, D., Hildreth, E. Theory of edge detection. Proceedings of the Royal Society B 207 (1980) 187–217.
10. Mandelbrot, B. B. The Fractal Geometry of Nature. W. H. Freeman, 1982.
11. Diffusion Fluid. iori.me, 2026.
12. Mandelbrot, B. B., Van Ness, J. W. Fractional Brownian motions, fractional noises and applications. SIAM Review 10 (1968) 422–437.
13. Hurst, H. E. Long-term storage capacity of reservoirs. Transactions of the American Society of Civil Engineers 116 (1951) 770–799.
14. Fournier, A., Fussell, D., Carpenter, L. Computer rendering of stochastic models. Communications of the ACM 25 (1982) 371–384.
15. Witten, T. A., Sander, L. M. Diffusion-limited aggregation, a kinetic critical phenomenon. Physical Review Letters 47 (1981) 1400–1403.
16. Petri — Slime Mold and Colony Growth Engine. iori.me, 2026.
17. Field, D. J. Relations between the statistics of natural images and the response properties of cortical cells. Journal of the Optical Society of America A 4(12) (1987) 2379–2394. doi:10.1364/JOSAA.4.002379
18. Ruderman, D. L., Bialek, W. Statistics of natural images: scaling in the woods. Physical Review Letters 73 (1994) 814–817. doi:10.1103/PhysRevLett.73.814
19. Torralba, A., Oliva, A. Statistics of natural image categories. Network: Computation in Neural Systems 14(3) (2003) 391–412.
20. Pałubicki, W., Horel, K., Longay, S., Runions, A., Lane, B., Měch, R., Prusinkiewicz, P. Self-organizing tree models for image synthesis. ACM Transactions on Graphics 28(3) (2009) 58. doi:10.1145/1531326.1531364
21. Borchert, R., Honda, H. Control of development in the bifurcating branch system of Tabebuia rosea. Botanical Gazette 145(2) (1984) 184–195. doi:10.1086/337445
22. Makowski, M., Hädrich, T., Scheffczyk, J., Michels, D. L., Pirk, S., Pałubicki, W. Synthetic silviculture: multi-scale modeling of plant ecosystems. ACM Transactions on Graphics 38(4) (2019) 131. doi:10.1145/3306346.3323039
23. Yoda, K., Kira, T., Ogawa, H., Hozumi, K. Self-thinning in overcrowded pure stands under cultivated and natural conditions. Journal of Biology, Osaka City University 14 (1963) 107–129.
24. Hallé, F., Oldeman, R. A. A., Tomlinson, P. B. Tropical Trees and Forests: An Architectural Analysis. Springer, 1978.
25. Cordonnier, G., Braun, J., Cani, M.-P., Beneš, B., Galin, É., Peytavie, A., Guérin, É. Large scale terrain generation from tectonic uplift and fluvial erosion. Computer Graphics Forum 35(2) (2016) 165–175. doi:10.1111/cgf.12820
26. Braun, J., Willett, S. D. A very efficient O(n), implicit and parallel method to solve the stream power equation governing fluvial incision and landscape evolution. Geomorphology 180–181 (2013) 170–179. doi:10.1016/j.geomorph.2012.10.008
27. Cordonnier, G., Cani, M.-P., Beneš, B., Braun, J., Galin, É. Sculpting mountains: interactive terrain modeling based on subsurface geology. IEEE Transactions on Visualization and Computer Graphics 24(5) (2018) 1756–1769.
28. Barnes, R., Lehman, C., Mulla, D. Priority-flood: an optimal depression-filling and watershed-labeling algorithm for digital elevation models. Computers & Geosciences 62 (2014) 117–127. doi:10.1016/j.cageo.2013.04.024
29. Paris, A., Guérin, É., Collon, P., Galin, É. Authoring and simulating meandering rivers. ACM Transactions on Graphics 42(6) (2023). doi:10.1145/3618350
30. Dodds, P. S., Rothman, D. H. Unified view of scaling laws for river networks. Physical Review E 59(5) (1999) 4865–4877. doi:10.1103/PhysRevE.59.4865
31. Smith, R. B., Barstad, I. A linear theory of orographic precipitation. Journal of the Atmospheric Sciences 61 (2004) 1377–1391.
32. Heimsath, A. M., Dietrich, W. E., Nishiizumi, K., Finkel, R. C. The soil production function and landscape equilibrium. Nature 388 (1997) 358–361. doi:10.1038/41056
33. Peytavie, A., Galin, É., Grosjean, J., Mérillou, S. Arches: a framework for modeling complex terrains. Computer Graphics Forum 28(2) (2009) 457–467. doi:10.1111/j.1467-8659.2009.01385.x
34. Cordonnier, G., Galin, É., Gain, J., Beneš, B., Guérin, É., Peytavie, A., Cani, M.-P. Authoring landscapes by combining ecosystem and terrain erosion simulation. ACM Transactions on Graphics 36(4) (2017) 134. doi:10.1145/3072959.3073667
35. Peytavie, A., Gain, J., Guérin, É., Argudo, O., Galin, É. DeadWood: including disturbance and decay in the depiction of digital nature. ACM Transactions on Graphics 43(2) (2024) 21. doi:10.1145/3641816
36. Mackensen, J., Bauhus, J., Webber, E. Decomposition rates of coarse woody debris: a review with particular emphasis on Australian tree species. Australian Journal of Botany 51 (2003) 27–37.
37. Metropolis, N., Rosenbluth, A. W., Rosenbluth, M. N., Teller, A. H., Teller, E. Equation of state calculations by fast computing machines. Journal of Chemical Physics 21 (1953) 1087–1092.
38. Onsager, L. Crystal statistics. I. A two-dimensional model with an order-disorder transition. Physical Review 65 (1944) 117–149.
39. Beggs, J. M., Plenz, D. Neuronal avalanches in neocortical circuits. Journal of Neuroscience 23 (2003) 11167–11177. Lecture: Brain Criticality — Optimizing Neural Computations (video).
40. Kirkpatrick, S., Gelatt, C. D., Vecchi, M. P. Optimization by simulated annealing. Science 220 (1983) 671–680.
41. Iris Engine — Photoreal Procedural Iris. iori.me, 2026.
42. iori — Interactive 3D Splat Showcase. iori.me, 2026.
43. Kheradmand, S. et al. 3D Gaussian splatting as Markov chain Monte Carlo. NeurIPS 2024.
44. Welling, M., Teh, Y. W. Bayesian learning via stochastic gradient Langevin dynamics. ICML 2011.
45. Hansen, N., Ostermeier, A. Completely derandomized self-adaptation in evolution strategies. Evolutionary Computation 9 (2001) 159–195.
46. Kauffman, S. A. The Origins of Order: Self-Organization and Selection in Evolution. Oxford University Press, 1993.
47. Rombach, R., Blattmann, A., Lorenz, D., Esser, P., Ommer, B. High-resolution image synthesis with latent diffusion models. CVPR 2022. arXiv:2112.10752
48. Podell, D. et al. SDXL: improving latent diffusion models for high-resolution image synthesis. 2023. arXiv:2307.01952
49. Ho, J., Jain, A., Abbeel, P. Denoising diffusion probabilistic models. NeurIPS 2020. arXiv:2006.11239
50. Song, Y., Sohl-Dickstein, J., Kingma, D. P., Kumar, A., Ermon, S., Poole, B. Score-based generative modeling through stochastic differential equations. ICLR 2021. arXiv:2011.13456
51. Esser, P. et al. Scaling rectified flow transformers for high-resolution image synthesis. ICML 2024. arXiv:2403.03206
52. Efron, B. Tweedie's formula and selection bias. Journal of the American Statistical Association 106(496) (2011) 1602–1614.
53. Wiener, N. Extrapolation, Interpolation, and Smoothing of Stationary Time Series. MIT Press, 1949.
54. Anderson, B. D. O. Reverse-time diffusion equation models. Stochastic Processes and their Applications 12(3) (1982) 313–326. doi:10.1016/0304-4149(82)90051-5
55. Kadkhodaie, Z., Guth, F., Simoncelli, E. P., Mallat, S. Generalization in diffusion models arises from geometry-adaptive harmonic representations. ICLR 2024. arXiv:2310.02557
56. Rissanen, S., Heinonen, M., Solin, A. Generative modelling with inverse heat dissipation. ICLR 2023. arXiv:2206.13397
57. Dieleman, S. Diffusion is spectral autoregression. sander.ai, September 2024.
58. Falck, F. et al. A Fourier space perspective on diffusion models. 2025. arXiv:2505.11278
59. Hunt, Kamb, Ganguli. An exact information theory of generalization phase transitions in Bayesian diffusion models. Preprint, July 2026. arXiv:2607.08041
60. Barnes, J., Hut, P. A hierarchical O(N log N) force-calculation algorithm. Nature 324 (1986) 446–449.
61. Cooley, J. W., Tukey, J. W. An algorithm for the machine calculation of complex Fourier series. Mathematics of Computation 19 (1965) 297–301.
62. Brandt, A. Multi-level adaptive solutions to boundary-value problems. Mathematics of Computation 31(138) (1977) 333–390.
63. Greengard, L., Rokhlin, V. A fast algorithm for particle simulations. Journal of Computational Physics 73(2) (1987) 325–348.
64. Mallat, S. A theory for multiresolution signal decomposition: the wavelet representation. IEEE Transactions on Pattern Analysis and Machine Intelligence 11(7) (1989) 674–693.
65. Hackbusch, W. A sparse matrix arithmetic based on H-matrices. Part I. Computing 62(2) (1999) 89–108.
66. Dao, T. et al. Monarch: expressive structured matrices for efficient and accurate training. ICML 2022. arXiv:2204.00595
67. Sohl-Dickstein, J. The boundary of neural network trainability is fractal. 2024. arXiv:2402.06184
68. Mandt, S., Hoffman, M. D., Blei, D. M. Stochastic gradient descent as approximate Bayesian inference. Journal of Machine Learning Research 18(134) (2017) 1–35.
69. Lambda Cognition & The TIH Framework. iori.me, December 2025.
70. Cellular Automata as the Computational Basis. iori.me, in preparation.
71. Gardner, M. The fantastic combinations of John Conway's new solitaire game "life". Scientific American 223 (1970) 120–123.
72. Langton, C. G. Studying artificial life with cellular automata. Physica D 22 (1986) 120–149.
73. Game of Life — ASCII Edition. iori.me, 2026.
74. Flame Strains — Automata Burning in a Volumetric Flame. iori.me, 2026.
75. Tero, A. et al. Rules for biologically inspired adaptive network design. Science 327 (2010) 439–442.
76. Procedural Physarum Simulation. iori.me, 2026.
77. Ember Engine — Cellular Automata Fire. iori.me, 2026.
78. Mycelium Engine — Cellular Automata. iori.me, 2026.
79. Frost — Generative Crystal Engine. iori.me, 2026.