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A sphere in Nil geometry, the Heisenberg group: the points at equal distance from a centre in a space where motion is confined to a plane of admissible directions and the vertical axis is reached only by spiralling. The poles are not smooth caps but conical dimples, like the stalk of an apple — a sub-Riemannian sphere fails to be smooth exactly where the geometry forbids moving directly.
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The thickened gyroid again, with a second pattern pushed into its walls: a product of three sines at about three and a half times the frequency of the gyroid, subtracted from the distance so that the surface bulges outward wherever it is positive. Some three and a half blisters fit along each cell of the labyrinth, and since the relief follows the material rather than the viewer, the swellings stay fixed to the walls as the camera passes through the channels.
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Sound: spatial audio 4 by BadWolf23 on Freesound, licensed under CC BY 4.0.
Three ellipses, each twice as long as it is wide, lying in the three perpendicular planes through the centre. No two of them are linked: take any one away and the other two fall apart. Yet the three together cannot be separated — the Borromean rings. They cannot be built from three flat round circles, which is why the classic model uses ellipses; here each one is a solid curve, with a tube drawn around it.
Only the 3D rotations move: a slow precession and spin turn the figure over, showing from every side how each ring slips past the next, held in place only by the third.
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Another immersion of the real projective plane, and the one that answers a question the Veronese surface leaves open: can the plane be placed in ordinary three-dimensional space without any singular point — no pinch, no cusp, only ordinary self-intersection? Werner Boy showed in 1901 that it can, against his supervisor Hilbert's expectation that it could not. What is drawn here is a trigonometric parametrisation of it, in the form given by François Apéry.
The figure has a threefold symmetry, visible as the three lobes that fold over one another, and a single triple point where all three sheets meet. It is one-sided and has no boundary: the surface crosses itself along a curve but never ends, and a path drawn on it returns to its start reversed. The rotation carries it through the angles where the triple point comes into view and back out again.
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Surface and background images: planetary maps by Solar System Scope, licensed under CC BY 4.0.
The true loxodrome, or rhumb line: the course a ship holds by keeping one compass bearing, crossing every meridian at the same angle. On the globe it spirals from pole to pole, tighter and tighter, without ever reaching either — and on a Mercator map it is a straight line. It is drawn as a solid curve, with a tube around it.
The breathing is its height: every three seconds or so the spiral flattens into a double spiral in the plane and opens again upside down, as the mirror image of the rhumb line it was. A slow tumble turns it over, and a rotation in four dimensions squeezes it, more slowly, across its width.
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The trefoil, the simplest true knot: a single closed curve that cannot be turned into a plain circle without cutting it. It circles its centre twice and rises and falls three times on the way, which gives it three lobes and three crossings, each passing alternately over and under. It is drawn as a solid curve, with a tube of constant thickness around it.
The breathing is the knot's height: the rise and fall swells and flattens by thirty per cent about every three seconds, while a slow tumble turns the figure over. The crossings never meet — however flat the knot becomes, it stays the same knot.
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Sound: Labyrinthus Sub Terra by RushmoreEffect, licensed under CC BY-NC-SA 3.0.
The complex surface z1^3 + z2^5 = 1, a true 4D shape projected into three dimensions. The fifteen patches are the pairs of roots of unity of the two exponents.
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A two-dimensional slice of the Fermat quintic, z15 + z25 = 1 over the complex numbers — the shape that illustrates the hidden dimensions of string theory. Each of the twenty-five patches belongs to one pair of fifth roots of unity, and they meet along the edges where the roots agree. The fourth coordinate is folded into the depth by a projection angle, here the customary 45°; the camera orbits while the figure turns.
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Background: Mount Zoomington 3D by chronos, licensed under CC BY-NC-SA 3.0.
The Schwarz P surface, cos x + cos y + cos z = 0 — a minimal surface that repeats every 2π in all three directions and divides space into two interlocking labyrinths, neither of them closed. Here cellular noise is subtracted from the equation, so the walls thicken and thin and the perfect periodicity is broken without the structure ever coming apart. The camera flies a circle that passes through the origin, rolling once as it goes, so the corridors are entered rather than observed.
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Planetary texture by Solar System Scope, licensed under CC BY 4.0.
All six great circles of S3 that lie in the coordinate planes, thickened into tubes and seen from outside. They meet three at a time at the eight points ±e1…±e4, which are the vertices of the 16-cell — so the figure is that polytope's skeleton given substance. Complementary pairs never touch at all, staying ninety degrees apart everywhere, while the others fuse at the vertices. The camera circles the structure as a 4D rotation turns it, and the stereographic projection keeps rearranging which tubes appear to pass through which.
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Planetary textures by Solar System Scope, licensed under CC BY 4.0.
A trefoil knot that lives in four dimensions. The curve turns twice in one plane of ℝ⁴ while it turns three times in the other, so it never leaves the Clifford torus — the flat torus inside the 3-sphere, made of two circles of the same radius — and winds around it as the (2,3) torus knot, which is the trefoil. It is projected stereographically into ordinary space and drawn as a solid curve, with a tube of constant thickness around it.
The 4D rotations turn the knot through planes that mix its two circles, so the projection keeps changing shape: lobes swell, shrink and swing through the centre, while the knot itself never changes. A slow 3D tumble turns the whole figure over at the same time.
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Tubes drawn around the great circles of S3 that lie in the coordinate planes, and a slider sets how many, from one to six. The first three share a coordinate and touch only at its two poles, so their tubes run clear of one another except in two chambers. Each further tube adds meeting points, and with all six the circles meet three at a time at eight vertices — the full skeleton of the 16-cell, the four-dimensional polytope whose edges these circles are. Removing the overlaps at every junction opens the passages that make the figure a labyrinth rather than a bundle of pipes.
The lines running along the walls are Hopf fibres. Each tube is parametrised along the fibration that makes its own axis a fibre, so every one of those lines is a great circle of S3 — a Villarceau circle once projected — rather than a meridian of the tube. Each tube is its own mesh, which is what lets all six carry their own fibration at once.
The camera flies inside, and this is what the run looks like from within: the walls converge to a point ahead because the tube closes on itself in S3, not because it recedes. The path threads three of the tubes in turn, passing from one to the next through the openings at the poles, and comes back to where it started without ever turning around.
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Background: Cyclic noise 2D by nimitz, MIT License.
Three tubes drawn around the three great circles of S3 that lie in coordinate planes through the same pair of poles, each parametrised along the fibres of its own Hopf structure — so every transverse line is a great circle, and the stereographic projection turns it into a Villarceau circle. The circles meet only at the two poles, so the tubes interpenetrate in just two chambers, and cutting the overlaps there opens the passages from one tube to another: that is the labyrinth. Each tube is its own mesh, which is what keeps the three from being stitched into a single grid with seams to mask.
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Surface texture by Solar System Scope, licensed under CC BY 4.0.
The flat torus that cuts the 3-sphere into two congruent halves, drawn here through a stereographic projection. Its two circles carry equal radius 1/√2, and the coupling of the angles rules the coordinate curves into Hopf fibres — great circles that never meet yet stay everywhere the same distance apart. Under the projection those fibres become the Villarceau circles of a torus of revolution, and the 4D rotation sweeps the surface through the projection pole, where it opens out and closes again.
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The same curve as in Clifford Knot Rotating — twice around one plane of ℝ⁴ and three times around the other, a trefoil on the Clifford torus — but seen through the parallel projection, which simply drops the fourth coordinate. The knot does not survive it: in ordinary space its shadow is an unknotted loop, and where the curve heads straight into the fourth dimension the shadow almost stops and turns sharply.
A 4D rotation turns the curve while a slow 3D tumble turns the figure over, and now and then the shadow passes right through itself — something no knot in three dimensions can do. In four dimensions any knot can be untied, and the projection shows only its untangled shadow.
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A sphere is attached to every point of a line, and as the line runs through the fourth dimension the centre of the sphere winds round a helix. The result is a three-dimensional object living in four-dimensional space — the side wall of an infinite spherinder, bent into a coil — and what the screen shows is its slice.
A rotation in the plane of z and the fourth coordinate tilts that slice. Cut straight across, the object is a single sphere; as the cut leans along the coil the sphere stretches into a curling worm that grows longer and longer, until at a right angle it becomes an endless spring, then shortens back into a sphere — while the whole figure tumbles in 3D.
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The Korányi gauge is the algebraic stand-in for distance in the Heisenberg group, and its unit sphere is a flattened solid of revolution, smooth and almost flat at the poles. Here its profile is raised to an exponent set by a slider: at one half it is the Korányi sphere itself, at 1 an ellipsoid, and past 2 the profile is drawn onto the axis until the poles close into sharp cusps. The cusps come from the exponent, not from the gauge.
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Planetary texture by Solar System Scope, licensed under CC BY 4.0.
A torus whose tube is itself coiled: the cross-section does not merely change size along the ring, it travels around a circle of its own, and the second turn is taken through the fourth dimension. Five of those turns fit into one lap of the ring, so the tube swells and thins five times over while its profile rotates out of ordinary space and back. The camera flies along the axis of the tube, rising and falling once per lap in the fourth coordinate while the section winds five times around it.
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Texture: Fire clouds by ygilk, licensed under CC BY-NC-SA 3.0, based on Voronoi by Inigo Quilez.
A tractrix dragged along a helix. The tractrix is the curve a weight traces when pulled by a string whose free end moves along a straight line — it approaches that line without ever reaching it. Revolve it and you get the pseudosphere; twist it into a screw instead, advancing along the axis while it turns, and you get this.
In Dini's surface what survives the twisting is the curvature: like the pseudosphere, it has constant negative Gaussian curvature at every point. The version drawn here softens the profile — one term of its height is smaller than the tractrix calls for — so the curvature no longer holds: negative towards the narrow end, it turns positive towards the wide one. The spiral tapers without end — the domain is cut short here, because the true surface would wind on forever, thinning towards an edge it never reaches.
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Texture: Squished Coordinates by pyBlob, licensed under CC BY-NC-SA 3.0.
Surface image: planetary map by Solar System Scope, licensed under CC BY 4.0.
An ellipsoid that is not in ordinary space at all. The three hyperspherical angles of the 3-sphere are driven with amplitudes in the ratio 1 : 2 : 3, so the ellipsoid lives in the space of angles; what reaches us is its image on a 3-sphere of radius 1.13, projected down to three dimensions. Every point of the surface sits at exactly that distance from the origin in four dimensions, and nothing in the equations changes with time: the shape we see swelling and contracting is rigid — it is the projection that changes as the object turns.
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Sound: spatial audio 4 by BadWolf23 on Freesound, licensed under CC BY 4.0.
A tube swept along the figure-eight knot, drawn as a curve that winds three times around the axis while its radius beats twice and its height four times — frequencies 2, 3 and 4 in the same closed loop. The fourth coordinate is where the motion happens: it rises and falls as a slow sine, lifting the curve out of ordinary space and setting it back down. The first three coordinates never change, so the knot drawn in them stays the same; in four dimensions every knot can be undone, and the lift only changes how much of the curve lies outside ordinary space.
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Sound: Neptune Racing by Dave_Hoskins, licensed under CC BY-NC-SA 3.0.
Two tori, overlapped and welded by a smooth minimum rather than simply placed together. The join costs nothing and adds something: besides the two original holes, the lens where the rings cross closes into a third one, so the surface carries three handles built out of two pieces. The camera never sees it from outside — it travels along the inside of the tube, and the ray marcher keeps resolving the wall from within, so the passage reads as a gut rather than as a solid. What little geometry the view catches beyond the wall is the far side of the same surface, arriving from the other direction.
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The gyroid, sin x·cos y + sin y·cos z + sin z·cos x = 0 — a minimal surface that repeats in all three directions and, unlike Schwarz's P and D surfaces, contains no straight line and no mirror plane at all. A point reflection turns the surface into itself, swapping its two sides, but the two labyrinths it separates are chiral: each is the mirror image of the other. The flight is a circle of radius 2.12 with a single rise and fall folded into it, so the camera crosses between levels of the maze rather than staying on one.
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Sound: Trailing the Twinkling Tunnelwisp by mrange, licensed under CC0 1.0; original music by Pestis, Cassini’s Cosmic Conclusion (demozoo).
Seen from inside. An ordinary sphere is carried into the Heisenberg group by the shear z → z + A·xy, the whole of Nil geometry in one expression: height is no longer absolute but depends on horizontal position. The camera travels within the closed shell, and the banded texture makes the distortion legible — the stripes converge into two spiral points, the poles of the parametrisation, where the twisting winds up on itself.
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Three helices around a common axis, each a third of a turn behind the next — a triple helix. All three are copies of one curve: the equations tell them apart only by their number, and each strand is a solid curve of its own, with its own texture.
The phase of the strands advances with time, so the bundle turns about its axis like a screw. On top of that the whole figure is turned in four dimensions and brought back by stereographic projection: the coil widens at one end and narrows at the other, and its proportions keep shifting as it turns.
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Eleven circles of the Hopf fibration, which fills the 3-sphere with great circles so that any two of them are linked exactly once. Six lie on the Clifford torus and five on a thinner torus nested inside it, each family spread evenly around its torus. Projected stereographically into ordinary space they stay circles, and every one of them threads once through each of the others.
The 4D rotations turn the circles through planes that reach into the fourth dimension, so they swell, shrink and slide through one another while staying round and linked; a circle that passes close to the pole of the projection grows huge and runs off the screen almost straight. A slow 3D tumble turns the whole figure over at the same time.
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Five Hopf tori, each built from its own mesh and cut in half between two Hopf fibres — which the stereographic projection renders as Villarceau circles. Unlike the equatorial cut of an ordinary torus, the two boundary fibres are linked, yet the resulting band, despite its Möbius-like look, is orientable. Nested and interlocking around the same fibration axis, the half tubes spin together under a slow 4D rotation, exposing from the inside the nested-shell structure that a full Hopf torus keeps hidden.
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The Hopf fibration sends every point of the 2-sphere to a whole circle in S3. Lift a parallel and you get a torus; lift the spiral θ = A φ, a Clelia curve that winds from pole to pole with its colatitude growing in step with its longitude, and you get this instead — a band closed around the fibre direction but bounded at both ends, its two edges being the fibres over the poles. The lift is exact: the Hopf map is constant along the fibres to within machine precision.
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Background: Trailing the Twinkling Tunnelwisp, gyroid logic adapted from Paul Karlik, CC0 1.0.
Parallel is meant in the geographic sense, not the geometric one: each surface here is the preimage of one parallel of the base sphere under the Hopf map. A parallel at colatitude θ lifts to the torus |z1| = cos(θ/2), |z2| = sin(θ/2) — a single circle on the sphere becomes a whole torus in S3, because every one of its points carries a circle of its own above it. The equator lifts to the Clifford torus, and the two poles do not lift to tori at all: they are single fibres, the two circles the whole family is nested around.
Up to nine of those lifts can be drawn, each as its own mesh, so the fibres stay visible as curves on each surface rather than being smeared into shading. The nesting is symmetric about the equator: the surfaces come in pairs at colatitudes θ and π−θ, one inside the Clifford torus and one outside, with the equator itself joining in whenever their number is odd.
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Textures, one per surface:
Cyclic noise 2D by nimitz, MIT License —
shadertoy.com/view/fcjXzh
Fiber Optics by NFX —
shadertoy.com/view/33yyDd
Quasicrystal GIF in HD by pvillano —
shadertoy.com/view/sXsGDl
Iridescent Fibers by evesira —
shadertoy.com/view/tffSDr
The last three are licensed under
CC BY-NC-SA 3.0.
Five tori of the Hopf fibration, each the preimage of a different parallel of the base sphere. A parallel at colatitude θ lifts to the torus |z1| = cos(θ/2), |z2| = sin(θ/2), so the equator gives the Clifford torus — the only one equidistant from the two polar fibres, and the one that cuts S3 into two congruent solid tori. Nested here in stereographic projection, they are rotated in three dimensions: the object in S3 is fixed, only the shadow turns.
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Texture: Fire clouds by ygilk, licensed under CC BY-NC-SA 3.0, based on Voronoi by Inigo Quilez. Cyclic noise 2D by nimitz, MIT License.
The same five Hopf tori, with the four-dimensional rotation switched on as well. Now two motions run at once: the projected image turns in three dimensions as before, while the whole family also rotates in S3 itself. The nesting is not preserved by the second motion the way it is by the first — a rotation of 4-space moves the tori through one another in the projection, and which one appears to be inside which stops being a fixed fact. Compare with Hopf Tori above, where the same five surfaces are turned in three dimensions only and the order never changes.
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Texture: Fire clouds by ygilk, licensed under CC BY-NC-SA 3.0, based on Voronoi by Inigo Quilez. Cyclic noise 2D by nimitz, MIT License.
The catenoid — the surface a soap film takes between two rings — carried into hyperbolic space and drawn in the Poincaré ball. Its flared mouths grow like cosh, so the projection crushes them against the rim: the whole infinite trumpet ends up in the last thin shell of the ball, while the narrow waist keeps its shape near the centre. The surface then orbits the origin, and since the compression depends only on distance from the centre, it swells and thins as it travels. It is moved rigidly in the flat coordinates, before the compression, and those moves are not motions of hyperbolic space: what changes is the shape itself, not just the viewpoint.
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Sound: The Sanctuary by srtuss, licensed under CC BY-NC-SA 3.0.
Enneper's minimal surface, known since 1864, carried into the Poincaré ball model of hyperbolic space. Every point is pulled inward along its own radius by the map r → tanh(r/2), which compresses infinite space into a ball of radius one: the lobes that run off to infinity in flat space fold back and stay visible inside it.
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The Möbius band carried into hyperbolic space and drawn in the Poincaré ball model, where the projection x → x·tanh(r/2)/r pulls the whole of H3 inside a unit sphere. Distances grow without bound towards the boundary, so the band curls as it approaches the rim — yet the single edge and the one-sided surface survive the change of geometry untouched.
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Texture: sun with stars by nayk, licensed under CC BY-NC-SA 3.0.
Background: Disco Sun Vortex by workingclasshacker, licensed under CC BY-NC-SA 4.0.
Sound: Cross-Galactic Ocean by Klems, licensed under CC BY-NC-SA 3.0.
The same gyroid, but thickened until it is no longer a surface. Taking the absolute value of the function and subtracting a thickness turns the thin sheet into a solid, and at this setting the solid fills about 95 per cent of space — what remains is a network of narrow channels, and that is where the camera is. The lattice also drifts while the flight proceeds, faster along one axis than the other, so the walls slide past for reasons that have nothing to do with the camera moving.
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A surface drawn inside SL(2,ℝ) itself, the group of two-by-two matrices of determinant one. Iwasawa's theorem factors every such matrix uniquely as a rotation times a diagonal times a triangular one, and each factor has determinant one on its own — so a point built from those three ingredients lies in the group by construction, not by approximation. Two of the parameters are turned into a closed loop, and what you see is that loop dragged through the group while its third coordinate rotates.
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Texture: Squished Coordinates by pyBlob, licensed under CC BY-NC-SA 3.0.
Sound: The Sanctuary by srtuss, licensed under CC BY-NC-SA 3.0.
A flight down the inside of a Jeener-Klein bottle. The tube follows a deltoid rather than a circle, and its radius breathes three times over the course of the loop, from 0.31 to 1.81 — so the corridor opens into chambers and closes to a throat as the camera advances. The path is a Fourier series of three harmonics threading the core: it holds no fixed clearance, running wide where the tube swells and passing within a hundredth of a unit of the wall at the tightest point.
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The figure-8 immersion of the Klein bottle, the version that fits in ordinary space at the price of passing through itself. A lemniscate is carried around a circle while turning through half a revolution, so after one lap it returns mirrored — the same point of space is reached twice with opposite normals, and no consistent inside survives. Set tumbling about three axes at once, it never presents the same silhouette twice within a turn.
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Planetary texture by Solar System Scope, licensed under CC BY 4.0.
A circle is attached to every point of a Klein bottle, and the whole construction is a three-dimensional object living in four-dimensional space. It cannot be seen as it is: what the screen shows is a slice, the set of points whose fourth coordinate is the one currently selected.
The equation is the Klein bottle's own, squared and set against the fourth coordinate, so each slice cuts the fibres at a different height. Moving through the fourth dimension the figure opens and closes, tubes detach and rejoin: not a shape that deforms, but successive cross-sections of something that never changes.
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The Klein bottle as it exists in four dimensions, where the neck passes through the fourth coordinate instead of through the wall: no self-intersection, and no seam to drive across. The camera stays inside the tube for the whole flight, a tenth of a unit clear of the wall and never changing side. What turns is the projection: a 4D rotation advancing at half the rate of the lap, so the same fixed course is shown from a continuously changing angle — and the walls that seem to open outwards are the shadow moving, not the camera.
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The Schwarzschild black hole seen whole. In the usual coordinates the solution appears to break down at the horizon, but that is a failure of the chart and not of the geometry: Kruskal and Szekeres found coordinates in which nothing goes wrong there, and those reveal that the solution has been describing two exterior regions all along, joined through the horizon.
What is drawn is the embedding of a slice at constant time: the two mouths are the two exteriors, and the throat between them is the Einstein–Rosen bridge. It is not a passage — the bridge pinches off faster than anything could cross it, so the two regions stay causally separate. The slider sets the mass, which fixes how wide the throat opens.
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A Lissajous curve in space: three sine waves of frequencies 4, 3 and 5, one per axis, traced together until they close up after a single turn. The phase of the first one drifts with time, so the curve keeps reshaping itself inside its box without ever opening.
Because the other two waves start in step, the curve is symmetric and passes through itself at three points, which slide back and forth with the drift. Twice in each cycle it folds completely and runs back over its own path, as a flat Lissajous figure does when its phase lines up.
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Three cosine waves of frequencies 3, 2 and 7, one per axis, with their phases slightly offset. Unlike the Lissajous Drift, this curve never meets itself: it closes up into a true knot, the three-twist knot, 52 in the knot tables, drawn as a solid curve with a tube around it. The knot is chiral — no motion in ordinary space turns it into its mirror image.
A slow 3D tumble turns it over, while a rotation in four dimensions does what ordinary space cannot: about every ten seconds the knot flattens into a plane and opens again as its own mirror image.
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A loxodrome is the path that crosses every meridian at the same angle — the course a ship holds by keeping one bearing on the compass, which on a globe spirals in towards the pole rather than following a great circle. The name is borrowed loosely here. The surface is a torus in S2×S1 whose section, at each point of the ring, is a curve on a 2-sphere: the winding W = v + 3u makes that curve a spiral whose longitude grows in step with its colatitude — a Clelia curve rather than a true loxodrome — and turns it three times for every lap around the ring.
Both angles then drift with time, and at different rates: U advances once per unit while W runs backwards at twice that. The drift of U only slides the grid along the ring; the net effect is a rigid turn of the whole surface in the plane of z and the fourth axis, so the shape never changes in four dimensions — only its shadow does.
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Surface image: planetary map by Solar System Scope, licensed under CC BY 4.0.
Sound: Xyptonjtroz by nimitz, audio by Dave_Hoskins, licensed under CC BY-NC-SA 3.0.
The Möbius band taken as a racetrack, with the camera flying just above the centre line at a fixed height. One lap does not bring it home: after a full turn around the ring the camera is running along the other face of the band, and only the second lap restores the starting position — the path closes after 4π, not 2π. The roll follows at half the rate, so the horizon turns over once per lap and the one-sided surface is felt rather than described.
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Sound: Waterfalls by P_Malin, licensed under CC BY-NC-SA 3.0.
A torus whose main radius is not constant: it carries a travelling wave, R = A + B cos(C v + t). The wave depends on v, the angle around the tube, not on the angle around the ring: it pushes each cross-section out on one side and in on the other, so the tube widens and narrows as the wave runs around it, in the same way at every point of the ring.
A second relief is laid over it, a vertical ripple that runs around the ring instead, with a frequency tied to the same slider that sets the wave depth. The two act in different directions, and where they cross the surface picks up a texture that neither produces alone.
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The circle, the torus, and then this: the three-dimensional torus, a torus of tori. It is a closed three-dimensional object that only fits in four-dimensional space, so what the screen shows is its slice — the part of it that lies in our space.
The camera flies a four-dimensional path, and as it turns it tilts the slice. Cut head-on, the 3-torus is a compact, full-bodied figure; as the slice leans it thins out and stretches, splits into two separate pieces around a quarter turn, then rejoins and fills out again, every half turn.
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A sphere of four-dimensional space — a 3-sphere — with its fourth coordinate twisted: every point is pushed along it by an amount that grows with its height and with its distance from the axis, a twist in the spirit of Nil geometry. It stays a closed object, and what the screen shows is its slice.
The camera flies a four-dimensional path that tilts the slice. Head-on, the twist squeezes the sphere at mid-latitudes into a rounded double cone; as the slice turns it flattens into a thick pillow folded along a crease, its upper half curling one way and its lower half the other, then unfolds back — always a single piece.
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Take a spherical cycloid — the curve a point of a rolling circle traces on a sphere, here closing after five lobes — and lift every Hopf fibre above it. The result is a surface in S3, and the lift is exact: the Hopf map stays constant along the fibres to within machine precision. What moves is the axis of the cycloid, tilting back and forth, so the lobes deepen on one side of the sphere while flattening on the other and the whole figure appears to nod.
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The real projective plane, the surface obtained by gluing every point of a sphere to the one opposite it. No such gluing fits in ordinary space without the surface crossing itself, so this is the Veronese immersion: the fourth coordinate carries the part that will not fit, and what is left in three dimensions folds through itself along a seam. The camera flies inside it, drifting through the fourth coordinate as it goes, and passes to the far side of a sheet that has no far side — the surface is one-sided, and the flight never reaches an edge. The view turns as it travels, rolling once about one axis of the moving frame and half a turn about another for every lap, so the same stretch of surface is met each time at a different angle.
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Sound: Trailing the Twinkling Tunnelwisp by mrange, licensed under CC0 1.0; original music by Pestis, Cassini’s Cosmic Conclusion (demozoo).
The tractrix turned about its own asymptote. Where the sphere has the same positive curvature everywhere, this surface has the same negative curvature everywhere — it is the sphere's counterpart, and the reason for the name. Beltrami used it in 1868 to show that hyperbolic geometry is no less consistent than Euclid's: here is a piece of it, made of ordinary space, where the angles of a triangle fall short of a straight angle.
Two cusps meet at the waist and the surface tapers away in both directions without ever ending. It is only a piece: no surface in ordinary space can carry the whole hyperbolic plane, as Hilbert proved in 1901. The camera circles it while rolling, so the flare of one horn passes across the view and the waist comes back into frame from a different side each time.
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Surface and background images: planetary maps by Solar System Scope, licensed under CC BY 4.0.
Two motions at once. The radius of the tube carries a wave, three crests running forward around the ring, so the surface swells and narrows as they pass — that part is a genuine deformation. The rest is not: the last two coordinates are turned rigidly into one another, a rotation of 4-space that preserves every length. The figure appears to breathe on two counts, and only one of them is really happening.
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Background: Fire clouds by ygilk, licensed under CC BY-NC-SA 3.0, based on Voronoi by Inigo Quilez.
Sound: Neptune Racing by Dave_Hoskins, licensed under CC BY-NC-SA 3.0.
A Klein bottle embedded in four dimensions, where it needs no self-intersection. The doubled angle in the first two coordinates against the single angle in the last two is what closes it with a reversal: follow u through half a turn and the fibre returns mirrored, so no consistent side survives — measured, every identified pair of parameters gives opposite normals. The animation interpolates between two configurations and back, changing the shape without ever touching the topology.
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Texture: Squished Coordinates by pyBlob, licensed under CC BY-NC-SA 3.0.
Sound: Red Cells by P_Malin, licensed under CC BY-NC-SA 3.0.
Steiner's model of the real projective plane, found in Rome in 1844 and left unpublished — Weierstrass wrote it up years later from his notes. It is the third immersion of that plane in this collection, and the most singular of them: where the Boy surface avoids pinch points and the Veronese needs a fourth dimension to do so, this one accepts them. Six of them, at the ends of three segments of self-intersection meeting at the centre.
The shape is a tetrahedral cushion: four flattened lobes, one towards each vertex of a tetrahedron, with the symmetry that follows. Its equations are quadratic in the sphere's coordinates, which is what forces antipodal points together and makes the surface one-sided, with no inside and no boundary anywhere.
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A surface built inside the 3-sphere, where every point is written as a direction in ordinary space paired with a latitude: the rule w = u + v hands each direction its own height on S3. Because that sum runs three times further than the directions themselves, the surface climbs through the poles four times over and folds back on itself, and the rings this produces are what the 4D rotation keeps turning inside out.
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Planetary texture by Solar System Scope, licensed under CC BY 4.0.
Sound: Red Cells by P_Malin, licensed under CC BY-NC-SA 3.0.
Four rose curves stacked one above the other — in polar terms r = cos(kθ), with k from 2 to 5. An odd k gives k petals and an even one twice as many, so the layers carry four, three, eight and five petals. All four are copies of one curve, told apart only by their number, and each is a solid curve with a look of its own.
Each rose spins in its own plane, and each one faster than the layer below — the top one turns four times as fast as the bottom one — so the petals keep drifting out of line and back, while a slow tumble turns the whole stack over.
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For sixty years the catenoid and the helicoid were the only minimal surfaces known; in 1834 Heinrich Scherk found new ones, and this is his second surface. It is singly periodic: it repeats in one direction only, along the line where the two planes it approaches cross, like a tower of saddles stacked on top of one another. One period of the tower is drawn here.
What it does is join two planes that cross at right angles, replacing the line where they meet with a column of saddle-shaped passages. Every point sits on a saddle: the surface curves up along one direction and down along the perpendicular one by exactly the same amount, which is what minimal means — a soap film would take this shape and no other.
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Surface and background images: planetary maps by Solar System Scope, licensed under CC BY 4.0.
The Schwarz P surface, cos x + cos y + cos z = 0, flown without the noise that perturbs it elsewhere. The path is built from hyperbolic tangents steep enough to saturate almost at once, so the camera does not glide: it holds at a corner of a 2π cube, then snaps to the next. Those corners are the centres of the chambers, the points furthest from the walls, and the route never once crosses the surface — measured along the whole loop the equation stays negative, so the flight stays inside a single labyrinth and passes from room to room through the throats that connect them.
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Planetary textures by Solar System Scope, licensed under CC BY 4.0.
An ordinary torus in its first two coordinates, with the section turned into the fourth dimension by an angle w = v + 3u + 2t. The 3u is the spring: three full turns of the section for every lap of the ring, which closes the surface after one lap all the same. The 2t makes those turns travel: the whole surface turns rigidly in the plane of z and the fourth axis, so nothing changes shape in four dimensions, while in the shadow the coil appears to slide along the tube and the ring stays where it is.
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Background: Raymarching Basic by gyabo, licensed under CC BY-NC-SA 3.0.
Sound: Trailing the Twinkling Tunnelwisp by mrange, licensed under CC0 1.0; original music by Pestis, Cassini’s Cosmic Conclusion (demozoo).
Geodesics of S3 released from a circle drawn on a sphere that is not great — one of the spheres at constant distance from a pole. They leave it at once. A sphere that is not great is not totally geodesic: its own geodesics are not geodesics of S3, just as a parallel of latitude is not a shortest route on the Earth. Only the equatorial sphere holds them, and the curves here sweep across the whole family of non-great spheres, swelling out to the great one and shrinking back. They are great circles throughout — it is the surface they started on that cannot keep them.
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Planetary texture by Solar System Scope, licensed under CC BY 4.0.
The stretch of Sol — one direction pulled out exponentially, the perpendicular one squeezed by exactly as much — applied to a sphere. Where the Sol Torus takes the height as the exponent, here it is the fourth coordinate: every value of it gives the sphere its own factor, and together they build a three-dimensional object in four-dimensional space. What the screen shows is its slice.
Three 4D rotations and a slow 3D tumble keep tilting the slice. Cut straight across, the object is a plain sphere; as the cut leans toward the fourth direction it runs through more and more of those stretches at once, and the sphere shoots out into a long band — round in the middle, flattening toward the two ends into wide, thin blades set at right angles to each other — before gathering back into a sphere, about every five seconds.
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An ordinary torus, seen with the geometry of Sol — one of the eight Thurston geometries, the only ones a three-dimensional space can be modelled on. Sol stretches one direction exponentially while contracting the perpendicular one by exactly as much: area is preserved, shape is not.
What makes it strange is that the amount of stretching depends on where the point sits — the exponent is the height. The top of the torus is pulled one way and the bottom the other, with every level between them getting its own factor, so the figure crumples rather than growing. The camera flies inside it on a circular path, rolling as it goes.
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Sound: Alps by David Hoskins, licensed under CC BY-NC-SA 3.0.
A pulsing sphere at the centre and six small tori, one at each end of the three axes — written as three tori and doubled by taking absolute values of the coordinates. The sphere breathes between radius 0.55 and 1.45 while the rings sit with their inner edge at exactly 1.00, so for about half of every cycle the sphere has swallowed them into a single body and for the other half there are seven separate pieces. The topology changes twice per breath.
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Background: Fractal pyramid by bradjamesgrant, licensed under CC BY-NC-SA 3.0.
A Lissajous figure drawn on a sphere instead of a box: the latitude swings up and down five times while the longitude goes twice around, and the curve closes after a single turn. On a surface one strand cannot pass over another, so where the curve meets itself it really crosses: at five points, all on the equator, evenly spaced around it. It is drawn as a solid curve, with a tube around it.
A slow tumble turns it over, while a rotation in four dimensions squeezes the sphere it lies on into an ellipsoid and, about every ten seconds, flattens it into a disc before it opens again as its own mirror image.
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Six rhumb lines on one sphere, each turned a sixth of the way round from the next: six courses with the same compass bearing, spiralling from pole to pole side by side. On the sphere they never touch — at every height each one sits at its own longitude. All six are copies of one curve, told apart only by their number, and each is a solid curve of its own.
As in the Breathing Loxodrome, their height breathes: every three seconds or so the ball flattens into a disc of interleaved double spirals, where the strands do cross, and opens again as its own mirror image. A slow tumble turns it over, and a rotation in four dimensions, seen in perspective, swells one side of it and shrinks the other.
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The group SL(2,R) — the 2×2 real matrices with determinant one — is itself a three-dimensional object in four-dimensional space: its four entries, suitably combined, are the four coordinates, and the determinant condition becomes the equation x² + y² − z² − p² = 1. What the screen shows is its slice, and a straight slice is an ordinary one-sheeted hyperboloid.
Three 4D rotations turn at different speeds and keep tilting the slice. For most of the time the hyperboloid holds together; whenever the tilt crosses the balance point its waist stretches without end, tears, and the surface falls apart into two separate sheets — then closes up again, over and over, while the whole figure tumbles in 3D.
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Surround a circle with a small sphere at every one of its points, instead of a small circle, and you get the four-dimensional cousin of the torus: a three-dimensional object, S²×S¹, that lives in four-dimensional space. What the screen shows is its slice, and a straight slice is an ordinary ring.
Three 4D rotations and a slow 3D tumble keep tilting the slice. As long as the cut runs along the central circle, the section is a ring; when it turns across that circle it meets it at only two points, and the ring thins on two opposite sides, snaps and splits into two separate pieces — which then swell, meet and close up into a ring again, about every five seconds.
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An ordinary torus, carried unchanged along the fourth dimension: every slice across that direction is the same ring, and together they make a three-dimensional object in four-dimensional space, T²×R — a torus times a line. What the screen shows is its slice.
The camera flies a four-dimensional path that keeps tilting the slice toward the fourth direction. Since the object never changes along it, a tilted cut spreads the same torus over a longer and longer stretch: the ring elongates, faster and faster, until the cut runs parallel to the fourth direction and the torus opens into two endless coaxial cylinders, its inner and outer walls — then shortens back into a ring, about every nine seconds.
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The torus equations with the guide radius set to zero, which collapses them into a sphere — but the domain runs twice around, so the sphere is traced two times over and the fourth coordinate, growing linearly with the angle, pulls the two passes apart into a spiral. The extent in that hidden direction is nearly six times the radius of what we can see, so the visible figure is a thin slice of something much longer, and the rate of the pull breathes as the animation runs.
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Texture: planetary map by Solar System Scope, licensed under CC BY 4.0.
Background: QuasiCrystals — Dragon Eye by mrange, licensed under CC BY-NC-SA 3.0.
Sound: Neptune Racing by Dave_Hoskins, licensed under CC BY-NC-SA 3.0.
A curve wound on a torus: twice around its axis while it goes five times around the tube. Because 2 and 5 have no common factor it closes up after a single pass, and the result is the cinquefoil, 51 in the knot tables — the (2,5) torus knot. Like every torus knot it is chiral: its mirror image is a different knot. It is drawn as a solid curve, with a tube around it.
A slow tumble turns it over, while a rotation in four dimensions presses it flat every ten seconds or so and lets it open again as its own mirror image — the swap no motion in ordinary space can make.
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Five tori arranged in a ring and welded by a smooth minimum, so the tubes merge instead of intersecting. The camera flies along a rosette curve threaded inside the tubes: the path has five petals and the ring has five tori, and it is that matching count which keeps the flight within the surface for the whole loop.
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Two tori welded into one body by a smooth minimum, which rounds the join instead of leaving a crease. Their generating circles cross at two points, so the tubes merge in two separate places, and the lens-shaped gap between those junctions becomes a third hole beside the two of the original rings — a surface of genus 3. The separation breathes as the animation runs, and the weld travels with it.
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Background: Disco Sun Vortex by workingclasshacker, licensed under CC BY-NC-SA 4.0.
A funnel of revolution: its profile climbs to a rim and then narrows toward a point. A fourth coordinate that rises on one side of the axis and sinks on the other lifts it into four-dimensional space, and the stereographic projection brings it back as overlapping sheets. As time passes the meridian rotates, so the figure turns about its axis.
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Background: Iridescent Fibers by evesira, licensed under CC BY-NC-SA 3.0.
Sound: The Sanctuary by srtuss, licensed under CC BY-NC-SA 3.0.
A flattened ellipse carried around a circle while it turns, but only by a third of a revolution for each lap — so the tube does not close after one circuit, nor after two. It takes three passes around the ring before the section returns to its starting attitude, and the surface is a single band that overtakes itself twice on the way. Three sheets pass at every angle of the ring — the same ellipse set at 0°, 120° and 240° — and since equal ellipses turned against one another meet in four points, the surface cuts through itself as it goes. The twist never reverses the section, though, so unlike a Möbius band this one keeps two distinct sides.
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The implicit equation of the Klein bottle, a polynomial of degree six in x, y and z. Nothing here is traced out point by point: the figure is whatever satisfies that single equation, and the ray marcher finds it by asking, for each point in space, which side of the equation it falls on.
At level zero the surface crosses itself along a circle, as a Klein bottle in ordinary space must. Here the level is moved slightly off zero and the crossing opens up: what remains is smooth and never passes through itself. It keeps the bottle's silhouette, but it is a torus — a closed surface that sits in three dimensions without crossing itself always has two sides. The sliders act on the coefficients and on the level, so the same equation can be pushed back towards the bottle or away from it.
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A planet tipping through the three-sphere. The surface is a great sphere of S³, drawn at full radius, and the equations swing it in the plane of the third and fourth coordinates while its equator stays fixed. The radius never changes, so nothing here is a sphere merely swelling and shrinking. What moves instead is the fourth coordinate: as the sphere tips, the polar caps drain into the direction we cannot see, and the globe flattens towards a disc before filling out again. The surface of Venus is mapped onto it throughout, and the way the texture gathers and spreads is the only visible trace of a rotation that carries one axis of ordinary space into the fourth.
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Surface image: planetary map by Solar System Scope, licensed under CC BY 4.0.
Tubes threaded along Villarceau circles — the oblique sections a torus makes with a plane tangent to it at two points, tilted here by about twenty-eight degrees. The copies are rotated about the axis by equal steps, and because the torus is unchanged by those rotations every copy is again a Villarceau circle of the same torus, in the same family. That is also why the only motion possible is a rigid turn of the whole configuration: fix the mutual distances and every angle is fixed too, up to one common rotation.
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Viviani's curve is the figure eight that a cylinder cuts out of a sphere when it touches the sphere from inside and passes through its centre: it winds twice around the cylinder and crosses itself once, at the point of contact. It is drawn as a solid curve, with a tube around it.
Here the height breathes. At every moment the curve is still the cut of the same cylinder with a sphere touching it at that point, but the sphere swells and shrinks: at its smallest the figure eight collapses onto a single circle, and the classic Viviani curve is one moment along the way. A slow tumble turns it over, and a rotation in four dimensions presses it flat every ten seconds or so, opening it again facing the other way.
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Two motions at once, and only one of them is a deformation. The radius of the tube is 0.30 + 0.15·sin(3u + 2t): three crests around the ring, travelling backwards as time advances. The rest is not the surface changing shape but turning — the equations rotate the visible height into the fourth coordinate and back, so material swells and thins on screen while nothing about it actually moves. The camera meanwhile flies inside the tube, along the circle at its core.
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Sound: Deep Space Vibrations Ambience Loop by rhodesmas on Freesound, licensed under CC BY 4.0.
The tractroid — the surface swept by rotating a tractrix, the curve a weight traces when dragged by a string. Given equal coefficients it is the pseudosphere, the classical model of a geometry with constant negative curvature; here the vertical scale is a quarter of the horizontal, so the curvature varies instead of holding, and the profile is a flattened horn. A wave runs along it, meridians displaced by a sine that travels down the figure, while a fourth coordinate lifts the whole thing obliquely out of ordinary space.
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Texture: Disco Sun Vortex by workingclasshacker, licensed under CC BY-NC-SA 4.0.
The same four-dimensional ring as the Splitting Torus — a sphere attached to every point of a circle, S²×S¹ — but with the sphere as wide as the circle itself, so the tube reaches right into the centre. The camera sits near that centre and, for most of the video, inside the object: what surrounds it is the inner wall of the slice, drawn as a wireframe grid.
Only the three 4D rotations move, and they keep tilting the slice around the camera. The walls sweep past, and the small hole in the middle of the ring closes into a solid blob and opens again, about every five seconds; with the sphere this wide, the section never breaks apart.
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A slice of SL(2,ℝ), the group of 2×2 real matrices of determinant one — each point of the surface literally is such a matrix, its four coordinates read as the entries. The parametrisation follows the Iwasawa decomposition, factoring every element into a shear, a scaling and a rotation, so the unit determinant holds by construction rather than by accident. The shape earns its name from the neck at the centre: the radius falls to a minimum midway and widens again on either side, giving the two mouths of a bridge joining separate regions.
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Texture: Disco Sun Vortex by workingclasshacker, licensed under CC BY-NC-SA 4.0, with a cosine palette by Inigo Quilez (MIT).
Background: Infinite Zoom (tension) by CriticalMammal, licensed under CC BY-NC-SA 3.0.
Sound: Everything's A Caustic by Wyatt, licensed under CC BY-NC-SA 3.0.