Pythagorean recursion
A turning-square dissection repeats inside its own moving centre.
Toroidal harmonics Readme
Short notes and references for the collection.
Mathematical concepts
- Torus
- A surface with two independent closed directions.
- Torus knots
- Integer winding numbers determine how a path closes.
- Villarceau circles
- Oblique cuts reveal two families of exact circles on a torus.
- Braids and interlacing
- Strands alternate above and below one another.
- Periodic approximants
- Repeating rational approximations let quasicrystalline patterns close around this torus.
- Penrose tiling and quasicrystalline geometry
- Ordered, nonperiodic rhomb patterns; these studies use periodic pentagrid approximants.
- Cairo pentagons
- A pentagonal tiling with fourfold organisation.
- Kagome patterns
- Corner-sharing triangles surround hexagonal openings.
- Chair substitution
- An L-shaped tile divides into four smaller chairs.
- Sierpiński carpets
- Repeatedly retain eight surrounding ninths; inlays keep the central squares as tiles.
- Cantor-like constructions
- Repeated paired divisions create nested strips and gaps.
- Julia sets
- Repeated complex squaring separates bounded orbits from escaping ones.
- Möbius transformations
- Complex fractional-linear maps carry circles and lines to circles and lines.
- Jacobi theta functions
- Balanced ratios of quasi-periodic Fourier series create doubly periodic fields with zeros and poles.
- Poincaré-disk geometry
- Hyperbolic distances place infinitely many equal-size regions inside a finite disk.
- Farey fractions
- Reduced rational numbers organise a hierarchy of scales.
- Ford circles
- A reduced fraction p/q gets a tangent circle of radius 1/(2q²).
- Hamiltonian flows
- A stream function generates area-preserving motion on the periodic coordinate chart.
- Arnold cat-map
- An integer torus map stretches and folds coordinates while preserving area.
- Laplace–Beltrami eigenmodes
- Natural standing-wave shapes computed for this curved torus.
- Saddles, centres and separatrices
- Critical points organise contours; saddle passages change which level-set regions connect.
- Changing level-set connections
- Crossing a saddle value joins or separates contour components.
- Phase vortices
- The phase winds around a zero of a complex field.
- Truchet tilings
- Simple tile orientations join local arcs into long paths.
- Moore curves
- A closed recursive space-filling curve, drawn here at finite depth.
- Pythagorean dissections
- Squares and triangles rearrange within a shared boundary.
- Kirigami
- Cuts let a sheet open into moving apertures and folds.
- Origami
- Creases coordinate the movement of connected panels.
- Guilloché and engine-turned engraving
- Repeated interlacing contours make finely engraved ornament.
- Fourier interference
- Complex waves add; their phases determine the resulting pattern.
- Talbot revival
- Quadratic Fourier phases reconstruct shifted copies at rational fractions of a cycle.
- Dirac belt trick
- A complete 720° turn can untwist continuously while the outer attachment stays fixed.
- Phason flips
- Local vertex rearrangements change tile adjacency in a quasicrystalline approximant.
- Torus covering maps
- Integer maps repeat one global organisation at successively smaller scales.
- Recursion and scale
- A construction repeats within its own smaller parts.
- Quadtrees
- Repeated four-way divisions organise nested square regions.
- Golden ratio
- Unequal parts repeat the same proportion at successive scales.
- Voronoi cells
- Each point belongs to its nearest moving seed.
- Tessellation
- Tiles share boundaries and fill a surface without gaps.
- Rotational symmetry
- Related orientations repeat around a common centre.
Artistic influences
- M. C. Escher
- Shared animal contours, figure–ground exchange and recursive worlds.
- Étienne Jacob / bleuje
- Precisely phased geometric loops and choreographed motion.
- Wavegrower
- Monochrome oscillations and collective waves of simple forms.
- Alex Grey
- Watching eyes, branching anatomy and visionary architecture.
- Allyson Grey
- Nested grids and the interplay of order and disorder.
Reptile outline adapted from Sean Michael Ragan’s Escher lizard tile.
The studies 102
Fractal metamorphoses 8
Quadtree genesis
Four child regions grow, subdivide and retract into their parents.
Sierpiński bloom
Eight squares grow around a receding centre, then repeat the same operation.
Cantor reweaving
Paired strips divide repeatedly on alternating axes.
Eyes of emergence
Nested apertures open into eyes as each generation grows.
Golden mitosis
Golden-section proportions guide a growing hierarchy of unequal regions.
Pentagrid genesis
Recursive partitions grow inside a periodic pentagrid mosaic.
Penrose tiling and quasicrystalline geometry · Periodic approximants · Golden ratio
Rhombic chrysalis
Nested quadrilaterals turn and contract inside each pentagrid rhomb.
Penrose tiling and quasicrystalline geometry · Periodic approximants
Revivals and rearrangements 3
Talbot Cathedral
Rosettes disperse into fractional wave revivals, then reconstruct exactly.
Elliptic Eyes
Theta-function zeros and poles exchange, reversing nested eyes and their surrounding light.
Phason Tide
An upward front flips rhomb neighbours; temporary polygonal tiles keep each transition filled.
Phason flips · Penrose tiling and quasicrystalline geometry · Periodic approximants
Living tessellations 5
Square to honeycomb
Moving seed rows change square cells into hexagons and back.
Cairo to crystal
Four seed families move between pentagonal and square partitions.
Octagon bloom
Square corners retreat, growing diamonds between the resulting octagons.
Pythagorean turn
A turning square and four corner triangles always fill their parent.
Truchet reconnection
Four paths meet at a saddle and reconnect with different neighbours.
Escher transformations 5
Square to swan
Shared square boundaries unfold into an interlocking flock of swans.
Fish and fowl
Fish and swans exchange form through continuously shared contours.
Reptile interlock
Three reptile orientations emerge from a hexagonal mosaic.
Moth mosaic
Diamond cells develop wings, antennae and shared moth outlines.
Scale to wing
Scales become wing veins as interlocking fish turn into moths.
Geometric tessellations 8
Square counterweave
Shared square borders flex around turning frames and optional nested inlays.
Hexagonal cadence
Three coordinated tones move through a hexagonal tessellation.
Triangular fugue
Two triangle orientations exchange tone along closed windings.
Rhombic cubes
Three rhombi suggest ambiguous cubes as their facet tones change.
Octagonal inlay
Octagons and small square interstices share a continuous inlay.
Cairo pentagons
Moving seed orbits produce a breathing Cairo-type pentagonal mosaic.
Herringbone tide
Horizontal and vertical dominoes interlock in a herringbone current.
Kagome counterpoint
Hexagons and triangles exchange tone through their shared neighbourhoods.
Figure and ground 1
Interlocking waves
Shared cuts curl square tiles into complementary waves.
Quasicrystal mosaics 10
Penrose counterpoint
Thick and thin rhombi exchange light in a closed pentagrid approximant.
Penrose tiling and quasicrystalline geometry · Periodic approximants
Pentagrid pathways
Arcs join across rhomb-edge midpoints into long mosaic pathways.
Penrose tiling and quasicrystalline geometry · Periodic approximants · Truchet tilings
Golden partitions
Partitions breathe between complementary golden-section proportions inside rhombi.
Rhombic inheritance
Four-way divisions give rhombi a visible ancestry of smaller tiles.
Quadtrees · Penrose tiling and quasicrystalline geometry · Periodic approximants
Rhombic butterflies
Rhomb halves fold about shared diagonals with contrasting tones.
Golden apertures
Nested golden contours open within raised rhombic chambers.
Pentagrid braid
Paired bands weave across a pentagrid mosaic.
Penrose lanterns
Nested golden frames rise from a periodic pentagrid.
Penrose tiling and quasicrystalline geometry · Periodic approximants
Golden rhomb counterpoint
Rhomb families exchange tone while arcs share a travelling phase.
Penrose tiling and quasicrystalline geometry · Periodic approximants
Quasicrystal shutters
A coherent wave raises and lowers hinged quasicrystalline rhombi.
Penrose tiling and quasicrystalline geometry · Periodic approximants · Origami
Fractal tessellations 7
Quadtree parquet
One quadrant remains intact while its neighbours subdivide.
Chair substitution
Each L-shaped chair contains four smaller chairs.
Triangular inheritance
Central triangles remain as tiles while their three neighbours subdivide.
Sierpiński inlay
Central ninths remain as tiles inside a recursively filled carpet.
Dyadic rooms
Alternating rectangular divisions breathe without opening gaps.
Pinwheel parquet
Quarter-turns reveal the directional ancestry of nested square regions.
Truchet carpet
Quarter-circle paths connect across recursively divided squares.
Cubic mechanisms 1
Recursive cube eclipse
Corner cubes repeat and counterturn at progressively smaller scales.
Folds and solids 6
Rolling reconstruction
Fixed shades stay on cube faces as delayed rotations cross the chamber.
Recursive gimbals
Nested frames turn about differently phased axes.
Sierpiński shutters
A perforated Sierpiński carpet opens into delayed hinged shutters.
Kirigami vault
Cuts and coordinated hinges open a sheet into moving vaults.
Recursive kirigami
Nested pleated tiers unfold in alternating directions.
Iris vault
Six curved blades curl around a changing aperture.
Woven circuits 9
Interlaced currents
Broad ribbon families exchange over–under roles at their crossings.
Villarceau ribbons
Two oblique circle families carry ribbons with contrasting tones.
Cable of cables
Each cable bundle contains three groups of three smaller, materially shaded strands.
Jacquard manifold
Nine-ribbon bundles interlace along closed torus windings.
Braided trefoils
Three-strand cables close along trefoil paths with real depth at crossings.
Coprime weave
Coprime winding numbers organise intersecting woven bundles.
Silk braid
Strands pass successively above and below their neighbours.
Ribbon exchange
Paired ribbons change places in an alternating interlacing rhythm.
Villarceau counterflow
Two families of exact torus circles carry travelling intersections.
Fractal visions 14
Julia tapestry
A changing quadratic Julia family reveals nested escape contours.
Paperfolding damask
Binary cell parity determines the handedness of nested, paperfold-inspired frames.
Recursive witness
Smaller eyes occupy the spaces inside larger eyelids.
Ophanim interferometer
Six eye-bearing sectors rotate around a breathing wheel and periodic lattice.
Cathedral of return
Ribbed columns, eye contours and sinusoidal branches form repeating architecture.
Toral substitution
An integer torus automorphism repeats and stretches nested square worlds.
Sierpiński counterpoint
A ternary carpet carries different light phases at each recursive scale.
Renormalisation tide
Logarithmic scale cycling passes ornament between successive generations.
Truchet substitution
Recursive Truchet paths reconnect across smaller inherited chambers.
Indra’s mirrors
Finite torus coverings repeat linked eyes at nested scales.
Ophanim
Eye-bearing circles turn within a counter-rotating corona.
Farey eyes
Rational positions organise a hierarchy of tangent, eye-bearing circles.
Poincaré eyes
Eye rosettes shrink towards the boundary of a hyperbolic disk.
Neural cathedral
Branching ribs and nested eyes join into a continuous architectural field.
Metamorphoses 3
Quadrature labyrinth
Real and imaginary wave components exchange connected dark and light regions.
Metamorphic tesserae
A changing nodal field joins and separates neighbouring tesserae.
Figure-ground braid
Broad ribbon families exchange figure and ground through a shared field.
Symmetry and exchange 7
Rotating squares
Opposing rotations open and close a linked square lattice.
Hexagonal irises
Six petals turn together around a shared hexagonal aperture.
Orbital quartet
Four linked orbits turn with counter-rotating internal markers.
Four-way exchange
A saddle changes the pairing of four fixed boundary connections.
Saddles, centres and separatrices · Changing level-set connections
Saddle metamorphosis
Contour islands join and separate as their saddle levels change.
Saddles, centres and separatrices · Changing level-set connections
Cellular exchange
Moving seed quartets change the neighbours of their surrounding cells.
Truchet currents
Connected arcs move through a smoothly sheared tiling.
Fields and harmonics 15
Nested separatrices
Integer torus maps repeat changing saddle connections at nested scales.
Saddles, centres and separatrices · Changing level-set connections · Torus covering maps
Nodal jacquard
Curved-torus eigenmodes interweave into a moving nodal fabric.
Vortex lace
Complex-field zeros migrate through lace-like phase contours.
Reversible loom
Area-preserving shears stretch filaments and then undo the deformation.
Toroidal eigenmodes
Computed Laplace–Beltrami modes combine into standing and travelling contours.
Hamiltonian silk
Hamiltonian currents stretch fine strands without compressing coordinate area.
Separatrix engraving
Engraved streamlines gather around centres, saddles and their separating curves.
Phase vortices
Interfering complex waves create zeros surrounded by winding phase rays.
Moore filaments
Distance contours follow a finite-depth, closed Moore curve.
Guilloché
Interfering contours form rosettes and engine-turned ornament.
Guilloché and engine-turned engraving · Fourier interference
Breathing lattice
A harmonic lace repeatedly separates and reconnects.
Harmonic ribbons
Wave contours widen into dark ribbons and narrow into engravings.
Fourier interference · Guilloché and engine-turned engraving
Meridian flow
Meridian strands compress and spread through a closed winding flow.
Pearl waves
Two coherent waves move and swell a constellation of beads.
Feather tide
Veined lenses lean and rise in alternately phased rows.