The Fractal Engine, Tool by Tool

1. One Object: the Orbit

A formula is a list of operations. Applied to a point over and over, it gives the point's orbit, and the orbit is the only thing the engine computes.

Take a point in space, fold it into a box, fold it again through a sphere, scale it by 2.5 and add back where it started. Do it twelve times. Points near the shape stay near it; points away from it fly off. That is the Mandelbox [1], the shape you arrive in. The rule add back where it started is what makes it a Mandelbrot-type set, the same rule as z → z² + c in the plane [2].

The engine follows each orbit together with how it stretches space (its Jacobian), which branch of a fold it took, and how close it came to the origin. Colour, the surface, the glow, the chaos meter and the handles you grab are all statistics of that one trace. So the look is separate from the geometry by construction, and every tool below reads the same object.

1.1 Notation

A formula is written as one line of text. Operations are separated by a dot, numbers follow their operation, and ×12 says how many times the list runs. The arrival formula reads

fold.box 1 · fold.sphere .5 1 · scale 2.5 · +c ×12 bail=100

Twenty-nine operations make up the library: folds through a plane, a box, a box with a limit per axis, a sphere, the absolute value and a sort of the coordinates, a stepped fold (section 5.7); powers (the triplex bulb, the complex power, inversion, a twist); the affine moves (translate, +c, scale, rotate); the 4-D forms (rotate4, sort4, bulb4, quat); the lattice wrap, the Kleinian step and three seeds (section 5.6); and five flame variations for the push engine [3]. A letter word after the count is a schedule: ×12 AAB alternates two op lists, a hybrid. c=x,y,z fixes the constant and gives a Julia set; slice=…, shade=… and drive=… carry the 4-D slice, the colour network and the drivers (sections 5, 6). The line, the JSON behind it, the graph of section 5.5 and the URL are one object: every formula is its own address, and its hash is its gallery id.

2. Three Ways to Draw

The same formula is drawn three ways: forwards from every pixel (pull), backwards from random points (push), and as an image fed back through its own map (loop). P or a double click cycles them.

2.1 Pull: the Cave

Pull marches a ray from the camera through every pixel. At each step it asks the formula how far the nearest surface could be, the distance estimate, and moves that far. Near the surface the steps shrink, and where they fall below a pixel the ray has hit [4] [5]. The estimate comes from the orbit itself: how far the point travelled divided by how much the formula stretched space along the way [6].

A lamp rides just off the camera, so enclosed caves are lit wherever you are; shadows and ambient occlusion appear when the view is still; depth of field focuses on whatever the centre ray hits. Every speed is the distance estimate at the camera: a hall and a crevice feel the same under the hand, and moves that would enter the surface are refused.

Two shaders draw the same formula. An interpreter reads the op list from a buffer and never recompiles, so editing is instant. A JIT writes the formula out as its own shader, every operation inlined and the loop unrolled, compiled in the background once the structure is known; it runs two to twelve times faster and takes over by itself. The two are checked against each other and against a float64 CPU reference on every change. Hover the line under a figure to see which one is drawing.

2.2 Push: the Nebula

Push runs the formula backwards. The inverse of an expanding map is a contracting one, so walking a point through random inverse branches draws the attractor, Barnsley's chaos game [7]. 32,768 walkers take 24 steps a frame; every sample lands as a small Gaussian splat whose shape follows how the inverse steps squeezed space, and the histogram is tone-mapped by log density as flames are [3]. A walk that hits a dead end backs up instead of retrying in place, so thin branches are not swallowed.

Pull and push are one formula in opposite directions. Without a fixed constant, push shows the Julia set of the point you are looking at, so one toggle gives the cave and the nebula from the same seed.

2.3 Loop: the Optical Machine

Point a camera at a monitor showing the camera's own picture and the image feeds back through the optics: the distance is a scale, the roll a rotation, the tone knobs a nonlinearity. Crutchfield showed in 1984 that this is an iterated map acting on images [9], and Dave Blair's Light Herder turned it into an instrument [10]. The loop engine is that machine with the formula as its optics: every frame, each pixel looks up where the formula sends it in the last frame, so the escape-time picture grows out of noise instead of being computed.

The screen is a window on a plane. Its knobs are the monitor's: hue rotation, saturation, gain, contrast about a dark pivot, decay, blur, a seed of sparks and how much of the field is fed in. Gain times decay just under one gives the nested halos of escape time; at or above one the set saturates.

3. Derivatives: Chaos and the Metric

Every orbit is followed together with its derivative, in position and in one parameter. From that one mechanism come the normals, the Lyapunov exponent, and a measure of how far each parameter moves the surface on screen.

The shader carries dual numbers: alongside each coordinate, how it changes when the starting point or one parameter changes [13]. Five columns ride through every orbit: three for the position gradient (the surface normal), one for the selected parameter, one for the Lyapunov exponent. The exponent says how fast nearby orbits part: negative where the dynamics is calm, positive where it is chaotic [14].

L colours the surface by it, about the view's own median: calm surfaces dark and cool, chaotic ones hot and glowing. In the Mandelbox the sphere-fold pits run hot and the box faces stay calm.

The parameter column gives the metric. A sensing pass marches a coarse grid of rays to the surface and measures, for every parameter, how many pixels the surface moves per unit. Raw sensitivities differ sixfold between the Mandelbox's parameters; divided by the metric, one pixel of hand motion moves the surface about one pixel whichever parameter it is. Every tool in section 4 is built on it.

3.1 The Derivatives, Compiled

Nothing here is for the hand: the same pictures arrive faster. Section 2.1's JIT writes the forward orbit out as its own shader; the dual orbit that runs at every hit is compiled beside it, into the same pipeline, and the interpreter draws until it is ready. One op becomes one block: what its forward map and its derivatives share is computed once, and a branch that leaves the derivatives alone holds no code. What no hit reads (the selected parameter's column, the colour) is left out. Chimeras unroll, and drivers are written in with their derivatives. Collage pieces compile only the Lyapunov column, and a light collage loses a little: the three-piece test scene runs 3 to 5 % slower, three compiled orbits behind a switch costing about what one saves.

GPU time per frame at 1440 × 900 on an M3 Pro. The images match the interpreter's to at least 61 dB (SSIM ≥ 0.9999), collages exactly. Push and the loop do not run the duals, so they are unchanged.

4. The Hand: Grab, Lens, Lineage, Breeding

There are no sliders. You edit by holding the image where you want it to change; the engine finds what moves the surface there.

4.1 Grab

Press and hold still for a moment. A sensing pass of 25 rays around the cursor ranks every parameter by how far it moves the surface there, scaled by the parameter's own size so lengths and angles compete fairly. The first is the attention op: its handle appears as its own geometry (the fold plane, the box's edges, the sphere fold's two radii, the scale as nested squares), with the reticle around the held point and the point's real orbit inside it. The ticks on the reticle's ring are the Lyapunov exponent: dense and bright where a change is about to turn chaotic. Drag sideways to edit at the metric's rate. The value appears on a dimension line; it is the only number on screen.

A press that moves more than 5 pixels in the first 280 ms is a look, not a grab, and a quick click does neither. Only a sideways drag edits; up and down do nothing yet. There is no grab in push, which has no surface to sense, and in the loop a hold takes the knobs (section 2.3).

4.2 The Atlas Lens

The Mandelbrot set is the map of the Julia sets: each of its points is a parameter, coloured by what that parameter does. The lens generalises the idea to any two parameters of any formula. Hold Shift in a grab and the view becomes the map of the attention parameter and a second one, coloured by how far (in pixels) the surface would be from the held point under each pair. The key colour sits where the surface would touch the point; a hairline marks every doubling of distance. The map spans exactly what a drag across the screen reaches. Moving travels it, both parameters change, and the field itself updates live in a round inset; letting go closes the lens and the grab carries on from there.

4.3 The Lineage

Every formula you make is a node in a tree: going back and editing starts a branch, so nothing is ever overwritten and yesterday's shape is always one click away, the feature the physical machine cannot have. The structure layer shows it as dots and hairlines, the big dot being now; hover a dot for its thumbnail. Moving between nodes is never a crossfade: within one structure the parameters and the camera move in 350 ms, so every frame in between is a real fractal; across structures the old image flows through the new formula's map in the loop engine. R replays the whole lineage from its root at constant visual speed. Any input stops a replay. In the live engine the lineage is kept in your browser (600 nodes, thumbnails for the newest 160); in these figures it lives only as long as the figure. Every edit makes one node when it ends, and so does a change of mode. A dot can also be dragged onto the field: its formula joins what is on screen as a collage's piece (section 7.3).

4.4 The Breed Ring, the Archive and the Drift

Hold still for longer, 1.2 s, and eight neighbours open around the cursor. Four are near: the formula stepped 35, 60, 90 and 130 pixels across the screen in a random direction. Four come from an archive of variety that grows in the background whenever your hand is off: a MAP-Elites grid of the best formula found for each combination of how much of the view is surface and how chaotic it is [15]. They sit around the cursor in the direction of their features, and a spoke's length is how different the neighbour looks. Release on one and the field walks there at a constant visual speed; the wheel brings farther or nearer neighbours. Picking from a gallery of variations is old [16] [17]; here the distances are measured.

The drift is the same idea without the hand: the camera glides toward the most open way and every unpinned parameter wanders a few pixels about its home value. The engine arrives drifting. Your first press or key turns the drift off for the session, so nothing moves under a working hand; Space, or the drift glyph on touch, turns it back on, and it then starts four seconds after your last input. It waits while a grab, the graph or the Metasurface is open. F in a grab pins that region: the parameters that move it stop wandering, the pin shows in the structure layer, and F on a pin removes it. Pins are how a composition is herded.

5. New Species

Three ways to mix formulas are kept apart. Breeding interpolates the parameters of one structure. A chimera blends two structures inside every iteration, a new dynamics. A collage joins finished shapes (section 7).

5.1 The Fourth Dimension

The orbit lives in four dimensions and the camera in three. A slice, an offset along w and a tilt toward it, places the 3-D world inside the 4-D one. Moving the slice is flying through the fourth dimension; tilting it turns the world toward it. A 3-D formula is a 4-D formula whose ops leave w alone, so the slice changes nothing for it until a 4-D op (rotate4, bulb4, quat, sort4) mixes w in. Quaternion Julia sets are the classic case [20].

5.2 The Chimera

Write two op lists in brackets and both run on the same point in the same iteration; the point is then blended between them. Neither formula's shape survives intact: the blend has dynamics of its own, a new species. The weight can be a constant, or a field over the point, so one species gives way to the other across a wall.

[fold.box 1 · fold.sphere .5 1 · scale 2.5 ‖ abs · sort · scale 3 · + -2 -2 0] axis .5 1.5 1 1 0 0 · +c ×12

5.3 Fields and Tissue

The weight of a chimera is a field, and the field library has eight kinds: a constant, an axis, a sphere, four-dimensional noise [21], Voronoi tissue [22], and three fields read from the fractal itself, its orbit trap, its depth and its running stretch, so a formula can mix itself by its own dynamics. Tissue takes two to seven species: n cells, each one species, with soft walls between them. A field is read either at the orbit point every iteration (new dynamics) or at the world point the orbit started from (a spatial tissue: the shape varies across space).

5.4 The Metasurface

Sixteen families (Mandelbulb, 4-D bulb, quaternion Mandelbrot and Julia, Mandelbox and its −1.5 form, Amazing Surf, Mandalay, a 4-D-rotated Mandelbox, Menger, Sierpiński, KIFS, pseudo-Kleinian with and without pillars, Apollonian, Kleinian) lie on a square as Voronoi cells, each site its family's thumbnail. Drag the crosshair and the formula follows: inside a cell the family is bred along its own two axes; across a wall, sites of one structure blend their parameters and different structures become a chimera [23]. A stroke can be played in time (a walk through the families) or laid in space (species ordered across the world along the stroke). Any formula can become a site of its own. The four lattice families of section 5.6 have no outside to frame, so their sites open with the camera inside a hall.

Your own sites live in this browser, up to thirteen beside the sixteen families; moved families are remembered too, and a corner mark stays bright while the layout differs from the default. A site that is a chimera has a hard wall, drawn as one hairline: crossing it is a cut, unless the chimera is a constant blend. While the surface is open on a touch screen, only the structure layer's key row answers outside it.

5.5 The Graph

The strip in the structure layer is a folded network. Unfolded, it is the formula drawn top to bottom: each op a glyph node, a chimera a fork into its species and a join at its field's glyph, the iteration a repeat zone with its loop up the right side and the count and schedule below. It is the formula itself, not a picture of it: every edit here is an edit of the line of text, and one node in the lineage.

5.6 Seeds: the Families That Never Escape

The formulas so far stretch space, and the distance estimate measures that stretch. A second kind only folds and inverts: at scale 1 an orbit never escapes, so there is nothing to measure and the old estimate fills all of space. These are the sphere lattices, tunnels and packings of the pseudo-Kleinian family [24], Apollonian packings [25] and Jos Leys's Kleinian limit sets [26].

They are drawn with a seed: a shape placed at the end of the iteration. An op does nothing to the point; the estimate reads the distance to the shape where the orbit ended, divided by how much the folds stretched space on the way. The fractal is the seed seen through every fold. There are three, seed.slab, seed.sphere and seed.pillar. A seed is an op like any other, so the grab, the lens, the ring and the drivers all reach its numbers. fold.cell is the box fold with a limit per axis, wrap folds space into a lattice, and kleinian is one step of Leys's algorithm, its two numbers the real and imaginary parts of the group's trace on Maskit's slice.

Two limits. The Apollonian slab's estimate is scaled by .4, which keeps the measured stretch under 1 at the 99th percentile, so the march stays safe. The Kleinian estimate jumps at the wrap's seam, where nearby points land on opposite sides of the lattice: the measured stretch there reaches 11 to 13 where a true distance has 1. An overstep at that seam is possible; none has been seen. In push these families read as star clusters and webs, sparse beside pull. Walking the Metasurface between two halls, a blended camera can land on a surface for a frame or two, never at rest.

5.7 The Mandalay Fold

DarkBeam's stepped fold, adapted by knighty [27]: where the box fold reflects each axis at one wall, fold.mandalay sorts |z| and steps the axes against each other, so faces rise into towers and domes. It stands in the Mandelbox's place in the formula, and it is what Tom Beddard's Aurullia series was built from [28].

fold.mandalay .5 .15 · fold.sphere .5 1 · scale 2.3 · rotate 0 0 10 · +c ×12

5.8 Subgraphs

A run of ops can be named and used in several places, so that an edit to one reaches them all. In the text it is a letter: a( … ) the first time, with its ops, and a( … ) after that with only what is its own.

fold.box 1 · a( fold.sphere .5 1 · scale ^1.8 ) · +c · rotate 0 0 15 · a( 1.4 ) ×8

The two uses are linked: the structure and every number are shared, so moving the sphere fold's radius in either moves both. A number marked with ^ is exposed: it belongs to each use, here 1.8 and 1.4, and a folded use shows it as an ordinary dimension line. Detach turns one use back into plain ops and leaves the others linked. The same link works on a piece of the shading network or the driver lane. Breeding treats a shared number as one gene that moves every use, an exposed number as one gene per use, and never touches the structure.

A definition lives in its formula, so a URL or an export carries it. To use a subgraph in another formula, the shelf keeps saved ones in this browser; placing one copies it in, with no link back.

The limits. The engine's eight slots and 64 ops count every use, because each use is real ops: a subgraph saves typing, not GPU space, and a chimera subgraph used twice takes its species' slots twice. The lens's preview of a shared number moves the first use only; the value applied on release is exact. Ramp stops can't be exposed (a ramp's map lines can). A subgraph can't contain a subgraph, and no link crosses collage pieces. A shelf holds eight entries: an entry can't be removed, and a ninth saved one pushes out the oldest.

5.9 Workflow: Subgraphs, Start to Finish

Follow these in the figure below: the arrival Mandelbox with its graph open and no subgraph yet.

1. Wrap a run. A run is a stretch of ops in one chain, clear of other subgraphs. Choose its two ends; a bracket with a plus appears beside them; the plus wraps the run. It takes a letter, a, and folds: its glyphs shrink between brackets. Desktop Shift-click the first op, Shift-click the last, click the plus.Touch Hold the first op, tap the last with a second finger, tap the plus.
2. Open and fold it. Open, a use is drawn in place in a dashed frame with the rest of the graph in sight. Edit inside: every edit reaches every use. Hold an edge at the frame's inner ends to add an op to the run itself. Desktop Double-click the use. Esc closes it.Touch Double tap. Two-finger tap closes it.
3. Place another use. The new use is linked: it shares the structure and every number. Desktop Hold an edge: the ring lists this formula's subgraphs first. Or Shift-drag a use onto an edge.Touch Hold an edge. Or hold a use, rest a second finger, and drag it onto an edge.
4. Expose a number. Exposing makes a number belong to each use instead of being shared. With a use open, select an op, pull its line off itself and drop it on the dashed frame: a dot appears there. One number, one inner parameter. In the text it gets a ^. Desktop Double-click the use, click the op, drag its line up and onto the frame. Pull it back inside to share it again.Touch The same with a finger.
5. Override it per use. Close the use and select it: the exposed number is an ordinary dimension line, on this use alone. It can be dragged, and bound to a driver with a band. Bindings belong to their use. Desktop Click the folded use, drag its line.Touch Tap, drag.
6. Detach or delete. Detach turns one use back into plain ops, with its numbers as they are, and leaves the others linked. Desktop Hold the use: the ring's detach use. Delete removes the selected use.Touch Hold; three-finger tap removes.
7. Keep it on the shelf. A subgraph lives in its formula. To carry one to another formula, save it to the shelf, in this browser, and place it there. Placing copies it in, with its numbers, and keeps no link. The shelf holds eight, newest first. An entry can't be removed, and a ninth pushes out the oldest. Desktop Hold a use, pick save to shelf. In the other formula, hold an edge and pick from shelf.Touch The same.
8. Do the same in shading and drivers. The same link works on nodes. A use's last node is its output. Opening any use opens the first, and the frame's empty ground takes new members. A folded use has no input dots, so wire its links from outside with it open. A placed use starts free, reading what the first reads, with its output unconnected. From the shelf, a shading or driver subgraph loses its links from outside. Desktop Shift-click nodes, click the bracket's plus, double-click to open. Hold empty frame for another use.Touch Hold a node, tap another with a second finger, tap the plus, double tap to open.

6. Colour and Motion

Colour and motion are networks in the same graph, fed by the orbit's readings: shading nodes for colour, drivers for parameters that vary across space, along the orbit or in time.

6.1 Shading Nodes

Beside the formula's graph sits its shading frame. The orbit's readings flow out of the repeat zone into it: the fold index, the orbit trap, the depth, the Lyapunov exponent, facing, the leading species, the w axis. A colour ramp turns a number into a colour; maths nodes combine numbers (add, multiply, power, sine, bands for terraces); colour mixes combine colours; two outputs take the surface colour and the glow. The ramp has up to sixteen stops, five interpolations, three colour spaces (RGB, HSV, Oklab [29]), an input map (scale, offset, clamp, repeat, mirror, log, invert) and presets: oil, malachite, bone, prism, strata, beetle and abyss, each with the reading it suits (the orbit trap, or facing for the thin films). A new ramp starts as the colour already there, and its presets open beside it, one tap away. Colours are picked on a hue ring around a held stop: angle is hue, distance is saturation. The network is stored in the formula, so the URL, the lineage, breeding and every export reproduce the colour exactly.

6.2 Drivers

A driver makes a parameter a function instead of a number. Four kinds drive: orbit readings, read every iteration (the iteration count, the radius, the trap, the running stretch, the point's coordinates), which change the dynamics; world fields read where the orbit starts (axis, sphere, noise, tissue, the start point), which vary the shape across space; time, which animates it; and other parameters, which link one op's value to another's. Maths and remap nodes sit between. A bound parameter shows its range as a band on its dimension line, with a live tick at the value the centre's orbit gives it.

A driven shape is no longer guaranteed to have a safe distance estimate, so the engine measures how steep the drivers make it and shortens the march's steps by that factor, only for driven formulas.

6.3 Drivers into Colour

The drivers are visible to the shading frame. Any driver can feed a ramp, a maths node or a mix, so one noise can bend the shape and paint it in step. Everything in colour can be bound with bands as formula parameters are: the ramp's scale, offset and hue turn, a stop's position, a maths constant, a mix's amount, and the glow's gain. A clock on the hue turns the whole ramp around the hue ring.

6.4 The Kaliset

Kali's map is a shader artist's favourite: fold by absolute value, divide by the squared length, subtract a constant, and a few iterations give dense, glittering detail [30]. The forums used it for colour and bump on other fractals. Here it is the kali node: a number computed from the point the ray hit, so any species can take it, and it feeds a ramp, a maths node or a mix like any reading. It does not change the shape.

7. Collage

A collage leaves every piece's dynamics alone and decides only where one finished surface meets the next, and what the seam looks like.

7.1 Joins

Pieces are whole formulas, up to eight, each with its own iterations, chimeras, drivers and colour. Their distance estimates are joined by smooth minima [31]: a smooth union, and the same kernels turned around for intersection and subtraction; terraced and bevelled seams, the stairs and chamfer of hg_sdf [32]; and a morph, where a field decides which piece is there. The seam width is a parameter like any other, so drivers can vary it across space. A fractal's estimate has no inside, so a cutter is given a grow: its surface pushed out by a margin, which is what a subtraction carves with.

7.2 Placement and Repetition

Each piece has a rigid placement and a uniform scale, which keep its estimate exact, and may repeat: a grid with counts, a mirror, or a polar array. The repetition is measured in the piece's own units, before its size: a polar radius of 4 on a piece of size 0.25 is a ring of radius 1. Click a surface and the piece that leads there is picked; its affine frame appears in hairlines. Drag an axis to move along it, the ring to turn, the corner to scale, the centre dot to move it freely in the screen's plane. A tap on the centre dot switches the axes between the piece's own and the world's; the world frame is drawn a little dimmer. With a piece picked, its repetition cells are drawn too, the original's solid and the copies' dashed.

7.3 Workflow: A Collage, Start to Finish

Follow these in the figure below, or in the engine itself, which keeps its lineage. The figure starts on a plain Mandelbulb: there is no collage until step 1 makes one.

1. Start one. Four ways, all with the same result: the formula on screen becomes piece 1, the new formula lands on the surface under the pointer (or at the focus), about a quarter of the view across, joined by a smooth union a tenth of its size, and it is picked. Desktop (a) The collage glyph in the row: the Metasurface opens with a plus beside its crosshair; click a site. (b) M, then hold still on a site until the ring around it fills. (c) Tab, then drag a lineage dot onto the field. (d) Open the breed ring and carry a neighbour 60 px past its thumbnail, then release on the field.Touch (a) The collage glyph, then tap a site. (b) The Metasurface glyph, then hold a site until its ring fills. (c) Tap a corner mark, drag a lineage dot out. (d) Hold 1.2 s for the ring, carry a neighbour out.

On the Metasurface	Tap or click	Hold 0.45 s
A site, as usual	goes there: the formula on screen (or the picked piece's) becomes the site's; nothing is added	adds it as a piece, where you pressed
A site, plus at the crosshair (add mode)	adds it as a piece, where the view looks	adds it, where you pressed
Open ground, as usual	moves the crosshair	drops your formula there as a new site
Open ground, add mode	adds the formula at that point of the Metasurface, where the view looks; a drag does too	drops your formula there as a new site

2. Pick a piece. A quick click traces the surface and picks the piece that leads there; its frame appears. Picking again elsewhere moves the pick. With a piece picked, a tap on a Metasurface site changes that piece's formula, not the whole collage's. Desktop Click its surface; click open ground, or Esc, to unpick.Touch Tap its surface; tap open ground to unpick.
3. Move, turn, scale. The frame's three axes are the piece's own. The ring turns about your line of sight. The corner between x and y scales. A dimension line, an arc or the size shows while you drag. Desktop Drag an axis, the ring or the corner. Shift snaps: a tenth of the frame, 15°, two figures.Touch The same, with larger hit areas. Rest a second finger to snap.
4. Switch the axes, or drag freely. The centre dot does two things. A tap flips the axes between the piece's own and the world's, and the world frame draws dimmer so you can tell which you have. Dragging it moves the piece anywhere in the screen's plane through its origin. The choice lasts the session. Desktop Click the dot; drag it.Touch Tap the dot; drag it. Holding it past 0.4 s names the state and does not flip it.
5. Open the collage graph. Pieces side by side, each its own small formula, a join under each neighbouring pair, the output at the bottom. Desktop G, or click the strip in the structure layer. G or Esc closes it.Touch The collage glyph, or tap the strip. Two-finger tap closes it.
6. Change a seam. A join's glyph is the join itself, the kernel's real contour. The line under it is the seam width at its true size on screen. Desktop Hold the join: the four ops (with this kernel) and seven kernels (with this op), each drawn as itself. Drag the line for the width. Click the join for its numbers: width, steps, morph weight.Touch Hold it; drag the line; tap it.
7. Edit one piece's own formula. Click a piece to pick it: its placement and repetition appear as dimension lines below, behind glyphs for move, turn, scale, grow and repeat. Click one of its ops: that piece's formula graph opens on that op, the ordinary editor of section 5.5. Desktop Click the piece, then its op. G or Esc comes back to the collage.Touch Tap, tap. Two-finger tap comes back.
8. Repeat a piece. Hold a piece for the ring: none, grid, mirror, polar. The render shows every copy at once; with the piece picked, the cells are drawn on the field, the original solid and the copies dashed. Spacing is in the piece's own units (section 7.2). Desktop Hold the piece in the collage graph, point, release. Its counts and spacing are lines below.Touch Hold, point, lift.
9. Carve with a cutter. Change a join to subtraction or intersection and the piece after it becomes a cutter. A fractal's estimate has no inside, so a cutter needs a grow to cut with: a new one gets a tenth of its size automatically. The cut surface sits that far out from the fractal's. Desktop Hold the join, pick the op; drag the picked piece's grow line.Touch The same.
10. Give the collage its own colour and drivers. Each piece keeps its shading, blended across seams. The collage's own shading, when set, wins, and it has two more readings: piece and seam. Drivers live in a row above the pieces (start x, y, z, w, the four fields, time, maths, remap), and can bind any seam or placement number with a band. A collage's colour reads the collage's drivers only, never a piece's. Desktop Click the output for the shading frame. Hold the row for the ring of drivers. Drag a driver's dot onto a line or a seam line to bind it. Delete removes the selected driver or band.Touch Tap the output. Hold the row. Drag the dot. Three-finger tap removes.
11. Add and remove pieces. Up to eight. The plus left of the pieces opens the Metasurface in add mode and comes back with the new piece in the graph. It is gone once the collage is full. Desktop The plus. Click a piece, Delete.Touch The plus. Tap a piece, three-finger tap.
12. Save, breed, restore. A collage is a document like a formula: the URL, the lineage, stills and splat files carry it, and every edit above is one lineage node. The breed ring works on it: neighbours move placements, joins and each piece's numbers. Two collages walk into each other when they have the same pieces and joins; otherwise the change is a cut. Desktop ⌘Z and ⇧⌘Z; ⌘S; drop the still back on the engine; hold still 1.2 s for the ring.Touch Two- and three-finger taps; the frame glyph; hold 1.2 s.

Refused along the way: a collage dropped onto a collage, a ninth piece, the lens, and the loop (P skips it). A push of a collage walks each piece's attractor and keeps samples by their share of the join.

8. Keeping It: Exports and State

What is on screen leaves as a 4K still, a video, a cloud of Gaussian splats or a mesh, and every file carries the formula that made it.

The still is 3840 pixels on the long side at 64 samples, rendered by the JIT in 512-pixel tiles that are each exactly their window of the full frame, so the seams do not show. The video walks the lineage from its root to here at constant visual speed, one PNG per frame. The splats are push samples captured with their covariance and written as a 3D Gaussian splatting file [33]. The mesh samples the distance estimate on a 128³ grid around the focus and extracts the surface with surface nets [34], as a binary STL.

Stills and splat files hold the whole state (formula or collage, camera, mode, look, focus and pins): drop one back onto the engine and it opens where it was made. A mesh does not, and neither does the video. The URL holds the formula, camera and mode, so a link is a scene, without the look. The live view pauses while an export runs, and a hairline at the bottom of the structure layer fills. A video's PNG sequence goes to a folder you pick, or to a streamed ZIP where the browser has no folder access.

9. Touch and the Overlay Toggles

Desktop decides what happens; touch adapts the gesture and the layout, never the behaviour, and adds no buttons.

A phone has no keyboard, no Shift, no ⌥, no wheel and no hover. The engine answers with one rule: a second finger is the modifier, everywhere. Resting, it is Shift (the lens opens, the gizmo snaps); a quick tap is one notch of the wheel (the next op, the next knob pair, farther neighbours); a double tap pins. A touch grab aims at the crosshair in the middle of the screen, so nothing hides under the thumb. Three fingers move the 4-D slice. Two- and three-finger taps walk the lineage back and forward.

The menu mark at the top right unfolds a row of hairline glyphs, the same set on every device: the graph, shading, drivers, Metasurface, collage, structure, Lyapunov and the mode, and last a page glyph that opens this paper in a new tab. A glyph is dim while off and bright while on; every key keeps working, the row only shows them.

The mark takes three presses. The first unfolds the row. A second, while the row is out, folds it and opens the site's own menu, whose docs item opens this paper in a new tab. A third closes the menu. In these figures the mark only folds the row. The mode glyph shows the mode and cycles it: pull, push, loop. Inside the graph, the shading and drivers glyphs fold that part away so the network takes the room, and bring it back; with the graph closed they open it. In a collage the shading glyph opens the collage's own shading and the drivers glyph its graph. The collage glyph opens that graph too, and before there is a collage it opens the Metasurface to add the first piece. One drawing at a time: opening one closes the others. Replay and drift have no glyph here; they sit in the structure layer's key row, with Lyapunov and the Metasurface, on a touch screen.

Names on demand. The drawings have no words, so every control can say its name when asked. Rest the mouse still on a node, a port, a line, a ring item or a toggle for less than half a second, or hold a finger still on it, and its name appears in small text beside it: an op by its notation (fold.box), anything else in one to three words (colour ramp, smooth union · quad, orbit trap). On a desktop the key follows where there is one. The label only reads: a hold still does what it always did (the thumbnail, the ring of choices, the grab), and a second finger or any move takes the name away. These are the names this paper uses.

Not yet on touch. One thing a desktop does has no finger: opening a saved still or splat file, which a desktop drops on the engine.

10. Examples: As Published, and One Step Further

Seven fractals people know, each drawn as it was published and then taken one step further with a tool from the chapters above. Left, the original; right, the step.

The sources are the forums and programs where 3-D fractals grew up: Fractal Forums, where the Mandelbulb, the Mandelbox and the kaleidoscopic folds were posted and argued over; Fragmentarium and Mandelbulber, whose open source code holds the formulas exactly [35] [36]; and the artists who made them a medium. Each pair starts from that formula as published.

10.1 The Mandelbulb

Daniel White and Paul Nylander's power-8 bulb of 2009: in spherical coordinates, raise the radius to the eighth power and multiply both angles by eight, then add the starting point [18]. The engine's bulb 8 · +c ×10 is that formula with its derivative carried alongside (section 3). The step: a tissue in which every cell of space is a bulb of its own power, 8 or 5, with soft walls between them (section 5.3). The classic hybrids alternated whole formulas from one iteration to the next; here the species are laid out in space.

10.2 The Mandelbox at −1.5

Tom Lowe's Mandelbox, posted as the "amazing fractal" in February 2010: fold into a box, fold through a sphere, scale, add the starting point [1]. At scale −1.5 the faces turn into castles. The step is a collage (section 7): six small −1.5 boxes in a polar ring around a seventh, joined by smooth unions, a cathedral ring no single formula draws.

10.3 The Kaleidoscopic Folds

Knighty's kaleidoscopic iterated function systems of 2010: fold space across mirror planes, then stretch it about a corner [37]. The Menger sponge is abs, sort, scale 3; the Sierpiński tetrahedron three plane folds and scale 2. The step: a chimera that runs both inside every iteration and averages the point (section 5.2), a species between the sponge and the tetrahedron.

10.4 The Quaternion Julia Set

Alan Norton drew Julia sets of the quaternions in 1982; Hart, Sandin and Kauffman ray traced them with a distance estimate in 1989 [20] [5]. A three-dimensional slice of a four-dimensional set: q → q² + c. The step: drivers (section 6.2) put the constant c on a slow clock, so the slice drifts through the family of quaternion Julia sets on its own.

10.5 Amazing Surf

A Mandelbox variant from the Mandelbulber library (credited there to Kali): the box fold on two axes only, so the shape runs out into sheets and terraced "lands" [36]. It is still one of the most used formulas in Mandelbulber galleries, usually with depth of field [38]. The step: painted in one hue in terraces of the orbit trap (section 6.1), with the rotation inside its iteration on a slow clock (section 6.2), so the lands turn as you watch.

10.6 A Cave in the Manner of Fractal Nature

Julius Horsthuis's films (Foreign Nature, Recombination) made fractals into places: enveloping halls with natural light, rendered in Mandelbulb3D [39]. The engine arrives in such a hall. Painted in one hue, in terraces, with the camera's depth of field, it is the left figure. The step: the same hall with its scale changing through space, a noise driver moving it between 2.35 and 2.65 (section 6.2), so the architecture shifts as you fly through it. Enter and fly (the wheel, or a pinch). A tissue of two Mandelboxes, the obvious step, measured over half a second a frame here: too heavy for a page.

10.7 A Fractal as Gaussian Splats

3D Gaussian splatting represents a scene as millions of small coloured ellipsoids [33], and it is usually fitted to photographs. The engine makes them from the formula: push walks a Julia-type bulb backwards and every sample becomes an ellipsoid shaped by its walk (section 2.2), and the structure layer's dots glyph writes them as a file any splat viewer opens. Left, the bulb as a surface; right, the same set as splats.

11. Planned

What the engine does not do yet, in the same form as the chapters above. Nothing here runs: each section is a place held for a tool that is coming.

Two things are by design, not missing: a collage has no loop, since it has no one map to feed back through; and the real-iPhone checks (three-finger gestures, the corner marks against rounded screens) wait for a first session on the device. The design document beside the engine, fractal-engine/DESIGN.md, records every stretch, its decisions, what was measured and what was left.

11.1 New Capability

11.2 Speed

11.3 Finishing Touches

11.4 Further Out

Reference: Every Key and Gesture

One page to keep open beside the engine. An empty cell means that side has no gesture for it.

The field

Grab, lens, pins

The loop

Lineage and the ring

The fourth dimension

The graph

Subgraphs

Shading and drivers

The Metasurface

Collage

The row and exports

What the Engine Refuses, and Its Limits

• Formulas. Eight op lists (slots) and 64 ops, counting every use of a subgraph. A chimera holds two to seven species, with no chimera inside a chimera.
• Grab. Pull only: push has no surface to sense, and in the loop a hold takes the knobs. Moves that would enter the surface are refused.
• Lens. Op parameters of a formula. Not rotation angles or plane normals, not driver or shading parameters, not collages. Its map is computed when it opens and does not follow an edit made inside it. On a shared subgraph number, the preview moves the first use.
• Breeding. Parameters only, never structure. The archive restarts after a flight, because its features depend on the view.
• Drivers. 32 nodes and 16 bindings; four rebuilt ops, eight such bindings. Four driven ramp stops. kali can't be driven. A driven formula's shader variant takes a few seconds to compile the first time, and it draws with its own values until then.
• Collages. Eight pieces. No collage inside a collage. No loop and no lens. A repetition reads the nearest copy only, so a piece wider than its cell is overstepped at the wall.
• Subgraphs. None inside another, none in a species if it holds a chimera, no link across collage pieces, no ramp stops exposed. Eight on the shelf, none removable, and a ninth pushes out the oldest.
• The Metasurface. 29 sites, 16 of them the families. At most four species blend across a wall. A chimera's wall is a cut unless it is a constant blend.
• Stored here only. The lineage, your sites and the shelf live in this browser. A link carries the formula, camera and mode; a still or a splat file also carries the look, focus and pins.
• The figures in this paper. Store nothing, and their mark only folds the row.

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