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

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

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

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

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

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

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

Twisted geodesic sphere in Nil geometry — Nil Twisted Geodesic Sphere points at fixed geodesic distance in Nil

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

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

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

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

Smooth torus obtained from the implicit equation of the Klein bottle, with a fractal texture — Implicit Klein Bottle the sextic of the figure-8 bottle, moved off its zero level: the crossing opens and a smooth torus remains

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