Gallery

Surfaces rendered in real time with Surface Explorer

Parametric surfaces

Equation-driven geometries, including 4D shapes projected down to three dimensions.

Klein bottle immersed in three dimensions — 3D Klein Bottle the bottle passes through itself: no boundary, no inside

Klein bottle immersed in three dimensions, second view — 3D Klein Bottle another view of the same self-intersecting immersion

Calabi-Yau quintic, cross-section of the six-dimensional manifold — Calabi-Yau Quintic cross-section of the quintic threefold

Tubes around the coordinate great circles of the 3-sphere, opening into one another where they meet — Clifford Tori Labyrinth tubes around the great circles of S³ in the coordinate planes, opened into one another where the circles meet

Coiled coil, a slice of the three-torus — Coiled Coil a coil wound on a coil: a slice of S¹×S¹×S¹

Double cone twisted into spiral fins — Cone Distortion a double cone whose horizontal slices turn through one full revolution from bottom to top

Cuspid Koranyi sphere in Heisenberg geometry — Cuspid Korányi Sphere the Korányi sphere of the Heisenberg group, its profile sharpened into cusps at the poles

Figure-eight torus — Eight Torus a torus whose profile is a figure-eight

Boundary curve of Enneper's surface drawn as a tube — Enneper Border the edge of Enneper's minimal surface as a tube, with the surface itself made invisible

Four stacked epitrochoids drawn as tubes — Epicycloids a point on a small circle rolling around a larger one traces 3, 4, 5 and 6 lobes, one curve per layer

Figure-eight knot drawn as a tube — Figure-Eight Knot four crossings: the simplest knot after the trefoil

Concentric waves with fractal detail — Fractal Wave concentric ripples with fractal detail, seen up close

Lobed surface whose radius is a root of a cubic, textured with the surface of Mars — Fresnel its radius is one root of a cubic with the symmetry of the cube, perturbed by noise

Helicoid with a border along its edges — Helicoid swept by a straight line that turns as it rises; a minimal surface, like a soap film

Five intertwined helices drawn as tubes — Helix Bundle five helices around one axis, each a fifth of a turn behind the next

Ten linked great circles of the 3-sphere — Hopf Fibers ten circles of the Hopf fibration, five on the Clifford torus and five on a thinner one; any two are linked once

Eight linked circles fanned around a common axis — Hopf Fibers Ring eight Hopf fibers on the Clifford torus, in stereographic projection: every pair is linked once

Nested Hopf tori in the 3-sphere, seen from close range, some solid and some as wireframe — Hopf Tori tori of the Hopf fibration nested in S³, each drawn from its own mesh — seen from within the innermost one

Slice of the H2xR geometry — H²×R Slice a slice of H²×R, one of the eight Thurston geometries

Catenoid in hyperbolic space with a fractal texture — Hyperbolic Catenoid the Euclidean minimal surface pulled into hyperbolic space

Ellipsoid in hyperbolic geometry — Hyperbolic Ellipsoid the ellipsoid as hyperbolic geometry reshapes it

Enneper surface mapped into the Poincaré ball, textured with the surface of Venus — Hyperbolic Enneper the Euclidean minimal surface pulled into the hyperbolic ball — texture: Venus surface map by Solar System Scope

Helicoid of hyperbolic space in the Poincare ball model — Hyperbolic Helicoid a helicoid in hyperbolic space, drawn in the Poincaré ball: its turns shrink toward the boundary sphere

Mobius band in hyperbolic space, nested transparent shells — Hyperbolic Möbius Band one side and one edge, carried into hyperbolic space — texture “Squished Coordinates” by pyBlob — shadertoy.com/view/W3KcWd

Torus in hyperbolic space — Hyperbolic Torus the torus carried into hyperbolic space

Twisted ribbon in hyperbolic space — Hyperbolic Twisted Ribbon a ribbon given a half-twist and carried into hyperbolic space

Hyperboloid with a coordinate grid — Hyperboloid the coordinate grid follows the curvature of the surface

Hyperboloid in hyperbolic geometry with a fractal boundary — Hyperboloid in Hyperbolic Geometry a fractal texture breaks the edge into filaments

Hyperboloid drawn in the Poincare ball model — Hyperboloid in Poincaré's Ball the hyperboloid seen inside the ball model of hyperbolic space

Slice of the Kerr geometry textured as a star field — Kerr Space Slice texture by nayk — shadertoy.com/view/tXKfz1

Lissajous curve in space drawn as a tube — Lissajous Curve one sine wave per axis, at different frequencies, traced together until the curve closes

Mobius band rendered as a solid surface — Möbius Band the classic one-sided surface, drawn solid

Single boundary curve of a Mobius band drawn as a tube — Möbius Border the Möbius band has a single edge: with the band hidden, it is one closed loop

Four nested ellipsoids, transparent and in wireframe — Nested Ellipsoids four ellipsoids one inside another, each with its own colour, transparency or wireframe

Torus deforming in Nil geometry — Nil Deforming Torus a torus carried through the Heisenberg group

Geodesic sphere of the Nil geometry in Cartesian coordinates — Nil Geodesic Sphere geodesics of the Heisenberg group, all stopped at the same distance

Oloid, the convex hull of two perpendicular circles — Oloid the convex hull of two perpendicular circles

Slice of the Poincare ball model — Poincaré Ball Slice layer upon layer, each one visible through the next

Catenoid drawn in the Poincare ball model — Poincaré's Catenoid the catenoid compressed into the ball model of hyperbolic space

Hyperboloid drawn in the Poincare ball model — Poincaré's Hyperboloid a ruled surface inside the hyperbolic ball

Twisted catenoid with curling fins — Pseudo Catenoid a catenoid wrung around a crosswise axis into curling fins

Torus built from the Iwasawa decomposition — Pseudo-Iwasawa Torus from the Iwasawa decomposition of SL(2,R)

Pseudo-wormhole surface — Pseudo-Wormhole two mouths joined through a throat

Pulsating torus in the S2xR geometry — Pulsating Torus in S²×R a tube around the equator, its section pulsing three times per turn

Pulsating torus in the S2xR geometry — Pulsating Torus Inside the same torus of S²×R, seen from within

Torus seen with the geometry of Sol — Sol Torus area preserved, shape stretched exponentially

Tube following a spherical cycloid traced by one cone rolling on another, in the looped regime — Spherical Cycloid Tube a cone rolling on another traces this curve on the sphere; past the cusp threshold each petal opens into a loop

Torus in S2xS1 whose sections spiral over a sphere — S²×S¹ Loxodrome a surface, not a path: each section spirals over a sphere, a Clelia curve rather than a true loxodrome

Sliding spring in S2xS1: a torus whose section turns into the fourth dimension — S²×S¹ Sliding Spring winding W = v + 3u: three turns of the section into the fourth dimension for every lap

Ellipsoid built inside the 3-sphere — S³ Ellipsoid an ellipsoid living in the 3-sphere

Slice of the three-sphere — S³ Slice a slice of the three-sphere brought down to 3D

Symmetrized double Enneper surface forming two tiers of arches around the axis — Symmetrized Double Enneper two Enneper ends joined and closed into a ring of arches

Geodesics of the 3-sphere leaving a trefoil knot, in wireframe with their border — Toroidal Knot great circles of S³ leaving a trefoil knot wound around a great circle, with the border along both ends

Three trefoil knots in red, yellow and blue — Trefoil Trio three copies of one trefoil, each turned a third of a turn and lifted; the middle one half as thick

One-sided ribbon of straight segments with three half-twists — Tri-Twisted Ribbon straight segments through a circle making three half-turns on the way round: a one-sided band whose single edge is a trefoil knot

Four concentric bands with their edges drawn as tubes — Twisted Bands four nested bands with one to four half-twists; with an odd number the band is one-sided and its edge a single loop

Funnel of revolution twisted through the fourth dimension — Twisted Funnel a funnel of revolution twisted through the fourth dimension, in stereographic projection

Ring of straight segments turning ten times around a circle — Twisted Torus Ribbon geodesics of flat space are straight lines: segments through a circle whose direction turns ten times on the way round

Villarceau circles drawn as tubes — Villarceau Tubes the oblique circles of the torus, thickened

Torus warped into a tunnel, seen from inside — Warping Torus stereographic projection, camera inside the surface

Clifford torus with a wave running along it — Wavy Clifford Torus the flat torus of S3, rippled

Quarter great circles of the 3-sphere fanned out from a great circle, in wireframe — Wobbling Ribbon quarter great circles of S³ fanned out from one great circle, their slant wobbling three times along it

Implicit surfaces

Shapes defined by an equation and rendered by ray marching, with procedural displacement and textures.

Merging implicit blobs — Blobs separate fields merge into one smooth surface

Ergosphere of a rotating black hole — Ergosphere the surface where nothing can stay still

Gyroid with a fractal texture — Fractal Gyroid the periodic surface carrying fractal detail

Genus-three surface rendered by ray marching — Genus-Three Surface three handles from a single implicit equation

Gyroid minimal surface — Gyroid a triply periodic minimal surface, no straight lines and no symmetry planes

Gyroid minimal surface, second view — Gyroid another view of the same periodic labyrinth

Gyroid minimal surface, third view — Gyroid the two interpenetrating labyrinths it separates

Gyroid showing ray-marching artifacts — Gyroid Artifact where the ray marcher runs out of budget the surface breaks up

Three-dimensional slice of a circle bundle over the Klein bottle, a four-dimensional object — Klein Bundle in R⁵ a circle attached to every point of a Klein bottle: what is shown is one three-dimensional slice of it

Solid gyroid seen from inside its channels — Solid Inside Gyroid the thickened gyroid, seen from within

Tangle cube, a quartic implicit surface — Tangle Cube a quartic surface with the symmetry of the cube

Two tori welded into a genus-three surface, rendered by ray marching — Triple Torus two tori welded together: the lens where they overlap opens a third hole

Smooth torus obtained from the implicit equation of the Klein bottle, with a fractal texture — Unknotted Klein Bottle the sextic of the Klein bottle, moved off its zero level: the crossing opens and a smooth torus remains

TOP