0 ≤ D < 1 — Points & Fractal Dust
Totally disconnected sets: more than a point, less than a continuous line
Isolated Point Set
A point has position, but no length, area, or volume. Even for a scatter of separate points, zooming in never splits a point into smaller pieces—each one remains a single dimensionless dot.
Harmonic Sequence {1/n}
Even an infinite sequence of points 1, 1/2, 1/3, 1/4, … crowding closer and closer toward zero still has Hausdorff dimension 0: every point remains isolated from its neighbors, taking up zero total length.
Ultra-Sparse Cantor Dust
Splitting a bar and keeping only 2 tiny pieces at the far ends (scaled by 1/1000 each step) removes almost everything, leaving a sparse fractal dust of dimension log(2)/log(1000) = 0.1.
1D Lévy Flight Dust (α = 0.2)
A random walk that takes many tiny steps mixed with occasional giant leaps leaves behind tight clusters of landing points separated by wide empty gaps, with fractal dimension 0.2.
Sparse 1/9 Cantor Dust
Removing the middle seven-ninths of each bar and keeping only the 2 outer ninths (r = 1/9) leaves a sparse Cantor dust with exactly half the dimension of the classic Middle-Third Cantor Set.
Generalized 1/9 Devil’s Staircase Singularities
A continuous staircase curve that climbs from 0 to 1 while staying completely flat across the middle seven-ninths of every interval; all of its upward steps are concentrated on a Cantor dust of dimension log₉ 2 ≈ 0.3155.
Nonlinear Cookie-Cutter Repeller (3x² − 1)
Points in [−1, 1] that never escape when repeatedly mapped by f(x) = 3x² − 1 form a non-uniform Cantor dust: because the parabola is steeper near the edges than near the center, outer intervals shrink faster than inner ones.
Kleinian Schottky Limit Set
Repeatedly inverting four mutually external circles into one another via Möbius transformations shrinks disks into a curving necklace of fractal dust inside the complex plane.
Zeros of a Wiener Process
The instants where a 1D Brownian random walk crosses zero form a random, clustering dust on the time axis with exact Hausdorff dimension 1/2.
Gauss–Cantor Continued-Fraction Set (E₂)
Numbers in [0, 1] whose continued-fraction expansions use only the digits 1 and 2 are generated by repeating x ↦ 1/(1+x) and x ↦ 1/(2+x), carving a non-uniform Cantor dust of dimension ≈ 0.5313.
Ford Circles & Bounded-Type Diophantine Spectrum
Placing a tangent circle of radius 1/(2q²) above every simplified fraction p/q packs the line with mutually tangent circles; the irrational gaps that avoid all large circles with digits above 2 share the exact same dimension ≈ 0.5313.
Feigenbaum Attractor
As the growth rate r of the logistic map x ↦ rx(1 − x) approaches the onset of chaos (r∞ ≈ 3.56995), its period-doubling branches split infinitely many times and land on a universal fractal dust of dimension ≈ 0.5380.
Golden-Conjugate Cantor Set
Keeping 2 copies scaled by r = 2^(−φ) ≈ 0.3258 at each step produces a symmetric Cantor dust whose Hausdorff dimension is the exact golden ratio conjugate 1/φ ≈ 0.6180.
Fibonacci Cut-and-Project Quasicrystal
Selecting 2D grid points inside a tilted strip of golden slope 1/φ and projecting them onto the strip’s baseline creates an aperiodic chain of Long and Short intervals whose diffraction spectrum lives on a fractal support of dimension 1/φ.
Middle-Third Cantor Set
Repeatedly removing the open middle third of every interval leaves N = 2 copies at scale r = 1/3: uncountably infinite points, yet zero total length.
Disconnected Fatou Dust Julia Set
When the complex constant c = −0.76 + 0.28i lies outside the Mandelbrot set, the Julia set shatters from a connected loop into a swirling cloud of disconnected Cantor dust.
Asymmetric Golden Cantor Set
Splitting each interval into a left piece of scale r₁ = 1/2 and a right piece of scale r₂ = 1/4 satisfies r₁^D + r₂^D = 1, yielding the golden ratio exponent.
Hofstadter’s Butterfly (Quantum Hall Spectrum)
Electrons moving through a 2D crystal lattice in a magnetic field can only occupy specific energy bands; plotting allowed energies against magnetic flux reveals a self-similar butterfly made of Cantor-set slices.
Minkowski’s Question-Mark Curve ?(x)
Mapping each Farey mediant fraction (p₁+p₂)/(q₁+q₂) to the midpoint between its parents builds a continuous staircase curve whose growth is concentrated on a fractal dust of dimension ≈ 0.7875.
Four-Quarter Cantor Set
Dividing each bar into 4 equal quarters and removing the third quarter leaves 3 copies at scale r = 1/4, producing an asymmetric Cantor dust of dimension log(3)/log(4) ≈ 0.7925.
Smale Horseshoe Strange Saddle
Stretching a square vertically, folding it into a horseshoe, and intersecting its forward vertical strips with backward horizontal strips traps an invariant 2D Cantor dust at the saddle points.
Arnold Tongues & Circle Map Mode-Locking Spectrum
When a rotating oscillator is periodically driven, rational frequency ratios p/q lock into flaring V-shaped resonance tongues; along the critical top line where the tongues first touch, the remaining unlocked frequencies form a universal Cantor dust of dimension ≈ 0.8700.
Zaslavsky Quasicrystal Stochastic Web
A harmonic oscillator periodically kicked at 5-fold rotational resonance (α = 2π/5) weaves a pentagonal quasicrystal skeleton of chaotic filaments and resonant island chains in 2D phase space.
Kempner Missing-Digit Set (No Digit 9)
Omitting a single decimal digit in base 10 keeps N = 9 intervals of scale r = 1/10 at every step: although its total length is zero, its Hausdorff dimension log₁₀ 9 ≈ 0.9542 makes it nearly indistinguishable from a solid 1D line.
1 ≤ D < 2 — Curves, Boundaries & Dendrites
Rough paths, self-similar coastlines, and branching trees that crinkle into the plane
Smith–Volterra–Cantor Set (Fat Cantor Dust)
Removing a shrinking middle slice (length 1/4^k at step k) from every interval leaves a disconnected dust that still keeps half of the original bar’s total length, giving it full Hausdorff dimension 1.
Takagi Blancmange Curve
Adding together a stack of triangle waves where each wave has twice the frequency and half the height of the previous one creates a curve that is crinkled at every scale and has no tangent anywhere, yet still has Hausdorff dimension 1.
Quasi-Fuchsian Bers Slice Limit Loop
Deforming a Fuchsian circle group into complex Kleinian space twists its invariant circle (D = 1) into a nowhere-differentiable Jordan loop ("quasi-circle") separating two hyperbolic domains.
Cauliflower Julia Set (z² + 1/4)
Sitting directly at the parabolic cusp c = 1/4 of the Mandelbrot cardioid, this Julia set has no hyperbolic bulbs—only a billowing, scalloped chain of parabolic petals of dimension ≈ 1.0812.
Rauzy Tribonacci Fractal Boundary
Projecting the Tribonacci substitution (1 ↦ 12, 2 ↦ 13, 3 ↦ 1) onto its complex contracting eigenplane tiles the plane with three self-similar spiral lobes whose fractal boundary has dimension ≈ 1.0934.
Golden-Mean Siegel Disk Julia Set
Tuning f(z) = e^(2πi/φ²) z + z² to the golden-mean rotation angle traps a neutral Siegel disk filled with smooth invariant orbits, ringed by a self-similar chain of pre-image tadpole basins.
Gosper Island Boundary
Each edge of a regular hexagon is replaced by 3 segments scaled by 1/√7, creating a crinkly rep-tile coastline of dimension log(3)/log(√7).
Herman Ring Rational Map Julia Set
Unlike simple quadratic maps z² + c, degree-3 rational maps can trap smooth rotating ring-shaped bands ("Herman rings") nested inside a fractal web of smaller inverted pre-image rings.
Basilica Julia Set (z² − 1)
For f(z) = z² − 1, the superattracting period-2 cycle 0 ↦ −1 ↦ 0 carves a symmetric chain of pinched cathedral domes whose boundary curve has Hausdorff dimension ≈ 1.1615.
Airplane Julia Set (z² − 1.75488)
At the superattracting period-3 root c ≈ −1.7548777 on the real antenna of the Mandelbrot set, the Julia set forms a chain of winged airplane fuselages strung along the real axis.
Dendrite Julia Set (z² + i)
Because c = i is preperiodic under z ↦ z² + c (i ↦ −1+i ↦ −i ↦ −1+i), its Julia set has empty interior and forms a pure branching lightning-bolt skeleton of dimension ≈ 1.20.
60° Fibonacci Word Fractal
Generated by reading the infinite Fibonacci word 01001010… and turning left or right by 60° at every "0".
San Marco Cathedral Julia Set (z² − 3/4)
At the parabolic neck c = −3/4 where the main Mandelbrot cardioid meets the period-2 bulb, the Julia set resembles St. Mark’s Basilica in Venice reflected across water.
Loop-Erased Random Walk (SLE₂)
Whenever a 2D random walk crosses its own trail, erasing the closed loop it just formed turns a tangled path into a simple, non-intersecting curve of exact Hausdorff dimension 5/4 = 1.2500.
Uniform Spanning Tree (Wilson’s Algorithm)
Connecting every point of a 2D grid into a random branching maze without closed loops (built from loop-erased random walks) makes the unique path between any two corners scale with dimension 5/4.
Hénon Strange Attractor
Repeatedly folding and stretching the plane via (x, y) ↦ (1 − 1.4x² + y, 0.3x) produces a strange attractor that is locally a smooth curve crossed with a Cantor dust.
Koch Curve & Snowflake
Replacing the middle third of every line segment with an equilateral triangle peak creates N = 4 segments of scale r = 1/3.
Four-Corner 2D Cantor Dust
Dividing a square into a 3×3 grid and keeping only the 4 corner sub-squares (r = 1/3) leaves a disconnected cloud of points with the exact same Hausdorff dimension as the continuous Koch Snowflake.
Chirikov–Taylor Standard Map & KAM Cantorus
In a periodically kicked rotor at the critical threshold K ≈ 0.9716, smooth phase-space orbits break apart into a porous fractal chain of islands ("cantori") surrounded by a sea of chaos.
Apollonian Gasket
Inscribing mutually tangent circles inside every curvilinear triangular gap via Descartes’ theorem leaves a residual set of dimension ≈ 1.3057.
Penrose Rhomb Aperiodic Fractal Gasket
Deflating a 10-fold decagonal wheel of golden Robinson triangles by scale 1/φ while excising one interior sub-triangle at every generation carves a five-fold quasicrystalline lace.
2D Self-Avoiding Polymer Walk (SLE₈/₃)
A random walk on a 2D grid that is forbidden from ever visiting the same site twice pushes itself outward like a long polymer chain, converging to a fractal curve of exact dimension 4/3.
Planar Brownian Frontier (External Hull)
Although a 2D Brownian walk crosses itself so densely that its interior fills an area (D = 2), its outer coastline accessible from the outside has exact Hausdorff dimension 4/3—matching a self-avoiding walk.
Terdragon Curve Boundary
Replacing every segment with three segments of length 1/√3 at angles +30°, −90°, +30° builds a three-fold symmetric dragon rep-tile whose crinkly outer perimeter has Hausdorff dimension ≈ 1.3496.
Critical 2D Ising Spin-Cluster Interface
At the critical Curie temperature T_c = 2/ln(1 + √2) of a 2D ferromagnet, the domain walls separating spin-up and spin-down islands converge to conformal SLE₃ curves of exact Hausdorff dimension 11/8.
Symmetric 5-Fold Pentagram Star Fractal
Replacing every outer vertex of a regular five-pointed star with a child pentagram scaled by the golden ratio conjugate while leaving the central pentagon open carves nested five-fold star corridors.
Douady Rabbit Julia Set
The Julia set of f(z) = z² − 0.12256 + 0.74486i has a period-3 superattracting basin shaped like a fractal rabbit with boundary dimension ≈ 1.3934.
Clifford Strange Attractor
Iterating the trigonometric map (x, y) ↦ (sin(ay) + c cos(ax), sin(bx) + d cos(by)) (with a = −1.4, b = 1.6, c = 1.0, d = 0.7) folds the plane into translucent, silk-like curtains with sharp density caustics.
Newton’s Method Fractal (z³ − 1 = 0)
Iterating z ↦ z − (z³ − 1)/(3z²) partitions the complex plane into three cube-root basins whose shared Julia boundary has the Wada property: every boundary point touches all three colored basins simultaneously.
Burning Ship Fractal Boundary
Taking absolute values of the real and imaginary parts before squaring—z_{n+1} = (|Re z_n| + i|Im z_n|)² + c—folds the complex plane along both axes, sculpting a fleet of jagged masts, rigging, and flames.
Vicsek Cross Fractal
Subdividing a square into a 3×3 grid and keeping only the 5 cross squares (center plus 4 cardinal neighbors) gives D = log(5)/log(3).
Gumowski–Mira Particle-Accelerator Attractor
Formulated at CERN to model nonlinear transverse proton-beam instabilities in circular storage rings, the Gumowski–Mira map weaves a winged, marine-like strange attractor of resonant islands and filaments.
Peter de Jong Trigonometric Strange Attractor
Iterating (x, y) ↦ (sin(ay) − cos(bx), sin(cx) − cos(dy)) (for a = 1.4, b = −2.3, c = 2.4, d = −2.1) folds the plane into twisted Möbius ribbons and sharp hyperbolic cusps.
Minkowski Sausage (Quadratic Koch Curve)
Replacing every straight segment with N = 8 orthogonal links of scale r = 1/4 (a symmetric square wave that neither loses nor gains area) produces a self-similar curve of exact Hausdorff dimension log(8)/log(4) = 3/2 = 1.5000.
Weierstrass Nowhere-Differentiable Function
Adding cosine waves where each harmonic doubles in frequency while shrinking by 1/√2 in height creates a continuous curve that is rough at every magnification, with Hausdorff dimension 3/2.
2D Gaussian Free Field Zero Contour (SLE₄)
Tracing the zero-elevation contour lines across a random 2D Gaussian Free Field landscape produces nested fractal loops of exact Hausdorff dimension 3/2.
Heighway Dragon Boundary
While the interior of the paper-folding Dragon curve fills 2D area, its intricate outer perimeter has fractal dimension ≈ 1.5236.
Ikeda Laser-Cavity Strange Attractor
Modeling laser pulses circulating inside a nonlinear optical ring resonator via z_{n+1} = 1 + 0.9 z_n exp(i(0.4 − 6/(1 + |z_n|²))) folds phase space into a swirling whirlpool of fractal filaments.
Barnsley Fern (Affine IFS Attractor)
Four contractive 2D affine maps—a stem map, a main successively smaller frond map (85% probability), and left/right pinna maps—drive the Chaos Game onto a self-affine botanical frond of Hausdorff dimension ≈ 1.55.
Berry–Goldberg Curlicue Fractal (Quadratic Weyl Sum)
Adding unit steps whose angle turns by n²√2 π at step n coils a single chain into nested Cornu spirals where every spiral arm is built out of smaller spirals.
T-Square Fractal Boundary
Placing a half-scale square on every exposed corner spawns 3 new outward corners at scale r = 1/2 at each step, giving its fractal boundary exact dimension log(3)/log(2) ≈ 1.5850.
Sierpiński Triangle Gasket
Every triangle splits into N = 3 corner sub-triangles at half the linear scale (r = 1/2), yielding D = log(3)/log(2) ≈ 1.5850.
Ballistic Deposition & KPZ Crystal-Growth Horizon
Particles raining vertically onto a substrate and sticking upon first contact with an occupied neighbor build a porous columnar crystal forest whose roughened upper horizon obeys Kardar–Parisi–Zhang (KPZ) scaling.
Golden Dragon Curve
Replacing every segment with two unequal legs scaled by r₁ = (1/φ)^(1/φ) ≈ 0.74274 and r₂ = r₁² ≈ 0.55167 satisfies r₁^φ + r₂^φ = 1/φ + 1/φ² = 1, giving a curling asymmetric dragon of exact Hausdorff dimension φ.
Pascal’s Triangle Modulo 3
Coloring the entries of Pascal’s triangle not divisible by 3 packs 6 sub-triangles into every 3×3 block: log(6)/log(3) = 1 + log₃ 2.
90° Fibonacci Word Fractal
Tracing the infinite Fibonacci substitution word 01001010… with 90° orthogonal turns at every "0" transforms the 60° snow-crystal curve (1.2083) into a dense, self-similar labyrinth of dimension ≈ 1.6379.
Harter–Heighway Twindragon & Dragon Tiling
Joining two Heighway dragon curves head-to-tail forms the central-symmetric Twindragon rep-2 tile, which tessellates the entire plane with interlocking dragon territories.
Pascal’s Triangle Modulo 5
Coloring the entries of Pascal’s triangle not divisible by the prime p = 5 packs p(p+1)/2 = 15 sub-triangles into every 5×5 block: log(15)/log(5) = 1 + log₅ 3 ≈ 1.6826.
Diffusion-Limited Aggregation (DLA)
Brownian particles wandering randomly until they stick to a seed crystal grow into branching lightning-like dendrites with dimension ≈ 1.70.
Conway’s Pinwheel Fractal
Subdividing a 1 : 2 : √5 right triangle into 5 congruent sub-triangles of scale r = 1/√5 and removing the central tile at each step produces an aperiodic fractal whose triangles rotate through infinitely many irrational angles.
Diagonal Hexagonal Slice of the Menger Sponge
By Marstrand’s slicing theorem, cutting a 3D Menger Sponge (D = log₃ 20 ≈ 2.7268) along the diagonal plane x + y + z = 0 reduces its dimension by 1 and reveals six-pointed hexagonal stars instead of squares.
Critical Percolation Hull (SLE₆ Boundary)
While the interior of a 2D critical percolation cluster has dimension 91/48 ≈ 1.8958, its outer coastline (including deep fjords whose necks have not yet pinched shut) is a fractal curve of exact dimension 7/4 = 1.7500.
Hele–Shaw Viscous Fingering (Saffman–Taylor Instability)
Injecting a low-viscosity fluid into a high-viscosity fluid confined between two parallel glass plates triggers repeated Laplacian tip-splitting instabilities, forming a radial Hele–Shaw rosette of dimension ≈ 1.75.
Hexaflake
Replacing every regular hexagon with N = 7 hexagons at scale r = 1/3 carves negative-space voids shaped like Koch snowflakes.
Cesàro Torn Square Fractal (85° Sweep)
Replacing all four sides of a square with inward-pointing 85° Koch spikes (scale r = 1/[2(1 + cos 85°)] ≈ 0.4599) carves four-way diagonal lightning fissures that almost touch across the interior.
Indra’s Pearls (Kleinian Circle Limit Set)
When Kleinian circle generators become mutually tangent at parabolic cusps, their limit set transforms from disconnected Schottky dust (0.4400) into interlocking spiral chains of glowing circles.
Nutrient-Limited Bacterial Colony Rosette
When colony growth is governed by local nutrient diffusion and surface-tension tip splitting, competing lobes thicken into a dense coral rosette sitting between thin DLA (1.7000) and critical percolation (1.8958).
Optimal River Drainage Network (Hack’s Law Basin)
Erosion-driven competition between tributaries carves a dendritic watershed where channel width scales with upstream catchment area (A^0.45) and the space-filling river network reaches dimension ≈ 1.855.
Golden Pentaflake
Replacing every regular pentagon with N = 6 pentagons scaled by r = 1/(1 + φ) = (3 − √5)/2 carves nested five-pointed golden stars into the negative space.
Sierpiński Carpet
Subdividing a square into 9 sub-squares of scale r = 1/3 and punching out the middle one leaves N = 8 copies: log(8)/log(3) ≈ 1.8928.
2D Critical Percolation Cluster
At the critical site occupation threshold p_c ≈ 0.592746 on a 2D square lattice, the incipient spanning cluster forms a scale-invariant fractal archipelago of exact conformal dimension 91/48.
Self-Avoiding Moore Maze Loop
Arranging four rotated Hilbert curves into a closed ring that leaves narrow non-touching corridors between branches winds a single loop through the square with fractal dimension ≈ 1.912.
Lévy C Curve
Replacing each segment with two 45° legs scaled by 1/√2 builds a self-overlapping canopy whose intricate outer boundary has dimension ≈ 1.9340.
Peano–Gosper Flowsnake Curve
Replacing every directed segment with 7 segments on a triangular grid winds a single crinkly path through every sub-hexagon of the Gosper Island, packing the plane just shy of the D = 2 space-filling limit.
Planar Dielectric Breakdown (Lichtenberg Figure)
In high-voltage planar dielectric breakdown with weak Laplacian screening (η < 1), branching discharge channels feather out across every angular sector to nearly fill the 2D disk.
2 ≤ D < 3 — Strange Attractors, Surfaces & Sponges
Sheet-like chaotic attractors, crumpled surfaces, and porous 3D cages
Boundary of the Mandelbrot Set
By Shishikura’s theorem, the filaments along the perimeter of the Mandelbrot set branch so densely that its 1D boundary has full Hausdorff dimension 2.
3D Sierpiński Tetrahedron (Tetrix)
Subdividing a 3D regular tetrahedron into N = 4 half-scale corner tetrahedra (r = 1/2) and excising the central octahedron produces a 3D polyhedral sponge whose Hausdorff dimension is exactly log₂ 4 = 2—a three-dimensional object with the scaling weight of a flat plane.
Pythagoras Tree (a² + b² = c²)
Branching each square into two child squares along the legs of a right triangle (scales r₁ = cos θ and r₂ = sin θ) satisfies r₁² + r₂² = 1 directly from the Pythagorean theorem, giving exact Hausdorff dimension 2.
Rössler Strange Attractor
Driven by a single quadratic nonlinearity z(x − c), the Rössler ODE spirals outward in a flat 2D disk before kicking upward along z and folding back onto itself, creating a folded-band strange attractor of dimension ≈ 2.013.
Lorenz Strange Attractor
Crossing past dimension 2 into 3D phase space, the chaotic butterfly trajectory weaves an infinite stack of two-dimensional sheets.
Chua’s Double-Scroll Strange Attractor
Generated by Chua’s nonlinear electronic circuit with a piecewise-linear negative-resistance diode, trajectories wind around two flat coplanar vortex rings linked by a twisted 3D bridge.
Thomas’ Cyclically Symmetric Attractor
Driven by cyclically symmetric sine-wave feedback across x, y, and z, chaotic trajectories braid through a 3D lattice cage of three mutually perpendicular vortex tubes.
Dadras Four-Wing Butterfly Strange Attractor
While the Lorenz attractor has two wings and Halvorsen has three, the 3D Dadras system couples quadratic cross-terms across all three axes to spin trajectories through four symmetric butterfly wings.
Halvorsen Three-Wing Strange Attractor
Driven by three-fold symmetric quadratic coupling, trajectories spiral outward through three tetrahedral funnels before folding over into the neighboring wing.
3D Tetrahedral Koch Star Surface
Subdividing each equilateral triangle face into 9 sub-triangles of scale r = 1/3 and erecting a 3-sided regular tetrahedral spike over the central triangle replaces 1 face with N = 11 sub-triangles: log(11)/log(3) ≈ 2.1827.
Aizawa Torus-Sphere Strange Attractor
A 6-parameter 3D polynomial ODE weaves chaotic orbits across the latitudes of a glowing sphere before plunging vertically through its polar core like a planetary magnetic dynamo.
Cartesian Product Fractal (Koch × Interval)
By the product dimension theorem D_H(A × B) = D_H(A) + D_H(B), extruding a Koch Snowflake curve (log₃ 4 ≈ 1.2619) along a 1D vertical interval (D = 1) forms a crinkly 3D prism wall of exact dimension 2.2619.
Chen’s Double-Helical Tornado Attractor
As the topological dual to the Lorenz system, Chen’s attractor braids its two chaotic wings into a tall, tightly twisting 3D tornado column of dimension ≈ 2.292.
3D Fractal Square Pyramid
Subdividing a 3D square pyramid into N = 5 half-scale pyramids (4 at the base corners plus 1 at the apex) gives D = log(5)/log(2) ≈ 2.3219.
3D Dodecahedral Polyflake
Replacing a regular dodecahedron with N = 20 sub-dodecahedra at its vertices scaled by r = 1/(2 + φ) ≈ 0.2764 builds a five-fold polyhedral cage of dimension log(20)/log(2 + φ) ≈ 2.3297.
3D Quadratic Koch Surface (Type 1)
Subdividing each square into a 3×3 grid and erecting a 5-sided cube turret over the middle square replaces 1 square with N = 13 sub-squares at scale r = 1/3: log(13)/log(3) ≈ 2.3347.
3D Cuboctahedral Space-Frame Truss
Subdividing a 3×3×3 cube and keeping the N = 12 edge-midpoint sub-cubes plus 2 opposite corner sub-cubes at scale r = 1/3 carves an open 3D diagonal space-frame truss of dimension log(14)/log(3) ≈ 2.4022.
3D Mandelbox (Box-Fold & Sphere-Inversion Fractal)
Alternating cubic box-reflections across [−1, 1]³ with spherical inversions inside a ball of radius 1/2 (at scale s = −2.0) carves an architectural 3D sponge of struts, grilles, and recursive courtyards.
3D Apollonian Sphere Packing
Packing mutually tangent spheres into every tetrahedral gap between four base spheres leaves a residual 3D fractal foam of dimension ≈ 2.4739.
Fractional Brownian Surface (H = 0.5)
A 2D random Brownian landscape has Hausdorff dimension 3 − H = 2.5, sitting halfway between a smooth 2D sheet and a 3D volume.
3D Diffusion-Limited Aggregation (3D DLA Cluster)
Extending Brownian particle aggregation from 2D (D ≈ 1.70) into three-dimensional space raises the cluster dimension to D ≈ 2.50, forming a 3D dendritic coral bush screened by the 3D harmonic potential.
Jerusalem Cube
Drilling Jerusalem-cross tunnels scaled by the silver ratio r = √2 − 1 through every face leaves 8 corner cubes of scale r and 12 edge cubes of scale r², satisfying 8r^D + 12r^(2D) = 1.
3D Icosahedral Golden Polyflake
Replacing a 20-face regular icosahedron with N = 12 sub-icosahedra at its twelve vertices scaled by r = 1/(1 + φ) ≈ 0.3820 completes the 3D Platonic fractal family at dimension log(12)/log(1 + φ) ≈ 2.5819.
3D Octahedron Fractal
Replacing a regular octahedron with N = 6 half-scale octahedra at its six cardinal vertices carves octahedral cavities that project along the axes into Sierpiński gaskets.
Mosely Snowflake Sponge
Removing the 8 corner sub-cubes and the central sub-cube from every 3×3×3 block leaves N = 18 sub-cubes at scale r = 1/3, sculpting a 3D crystalline snowflake polyhedron of dimension log₃ 18 ≈ 2.6309.
3D Turbulent Vorticity Filament Tangling
In the β-model of intermittent high-Reynolds-number fluid turbulence, vortex stretching concentrates energy dissipation into a braided 3D tangle of vortex tubes and hairpin eddies of dimension ≈ 8/3.
Menger Sponge
Drilling tunnels through every face of a 3×3×3 cube removes 7 sub-cubes and leaves N = 20 sub-cubes at scale r = 1/3: zero volume, infinite surface area.
3D Cortical Sulci & Gyri Buckling Surface
Tangential expansion of a 2D cortical sheet constrained inside a 3D cranial sphere buckles into nested convoluted ridges (gyri) and deep branching fissures (sulci) with surface fractal dimension ≈ 2.765.
Romanesco Golden-Angle Cone Fractal
Arranging self-similar 3D paraboloid buds at the golden divergence angle Δθ = 2π/φ² ≈ 137.5077° generates interlocking Fibonacci parastichy spirals where every bud is a miniature cone of smaller buds.
3D Kleinian Schottky Sphere Foam
Lifting Möbius circle inversions into three-dimensional space across six octahedral generator spheres packs nested 3D spheres inside spheres into a volumetric Kleinian foam.
3D Quaternion Julia Set (q ↦ q² + c)
Slicing the 4D hypercomplex Julia set of q ↦ q² + c (for quaternion c = −0.2 + 0.6i + 0.2j + 0.2k) across 3-space reveals a sculpted fractal body of folded ribbons and self-similar tendrils.
3D Mandelbulb (Power-8 Spherical Map)
Lifting z ↦ z⁸ + c into spherical coordinates (r, θ, φ) ↦ (r⁸, 8θ, 8φ) produces a 3D fractal body adorned with recursive crown-like bulbs, canyons, and spirals whose boundary approaches dimension ≈ 2.88.
3D Multi-Scale Gyroid Minimal-Surface Sponge
Layering triply periodic Schoen gyroid minimal surfaces across nested spatial scales forms a curved 3D labyrinth of channels that packs surface area densely into volume.
3D Space-Filling Bronchial Tree
Mammalian lung airways and vascular networks bifurcate across ~23 generations with Murray’s scaling factor r ≈ 2^(-1/3), creating a 3D canopy whose terminal surface packs 3D volume with dimension ≈ 2.97.
D = 3 — Space-Filling Volume
A 1D curve folded so densely that it visits every point of three-dimensional space
3D Hilbert Space-Filling Curve
Folding a 1D curve into N = 8 sub-cubes at half scale (r = 1/2) visits every voxel of three-dimensional space, reaching D = log(8)/log(2) = 3.