Mathematical
Geometries

Impossible forms in three dimensions

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01

The 120-Cell

4D Hyper-Dodecahedron

A projection of a 4D polytope composed of 120 dodecahedral cells into our 3D space. It appears as a complex, bubble-like structure where the cells warp and shrink toward the center.

3D Print Intricate lattice impossible with traditional molding
Method Stereographic projection from 4D to 3D
Parameters
Drag to rotate
02

Klein Bottle

Non-Orientable Surface

A surface with no distinct "inside" or "outside." Trace a path along its surface, and you'll return to start—but upside down. It exists fully only in 4D space.

3D Print Voronoi mesh reveals internal self-intersection
Method Parametric equations with u,v from 0 to 2π
Parameters
Drag to rotate
03

Menger Sponge

Fractal Volume

A fractal that lives in 3D: a cube with square holes drilled through every face, repeated infinitely. It has infinite surface area but zero volume.

3D Print Stress test for complex overhangs and bridging
Method Recursive: divide into 27 cubes, remove centers
Parameters
Drag to rotate
04

Gömböc

Self-Righting Shape

A convex 3D homogeneous body with just one stable and one unstable equilibrium point. It rights itself purely through geometry—no weighted mass required.

3D Print High-precision SLA—surface must be uniform
Method Surface curvature differential equations
Parameters
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05

Riemann Sum

Calculus Visualization

A physical visualization of integration: representing a smooth volume using a series of distinct rectangular bars approximating the area under a curve.

3D Print Educational tool for teaching calculus
Method Generate cubes based on f(x,y) function
Parameters
Drag to rotate
06

Möbius Strip

One-Sided Surface

A surface with only one side and one boundary curve. Travel along its surface and you'll cover both "sides" without ever crossing an edge—a topological marvel.

3D Print Continuous loop with half-twist connectivity
Method Parametric surface with twist parameter
Parameters
Drag to rotate
07

Trefoil Knot

Seifert Surface

The simplest non-trivial knot in topology. Its Seifert surface reveals the orientable membrane bounded by the knot—a bridge between knot theory and surface topology.

3D Print Tubular knot with smooth continuous curvature
Method Parametric: (sin t + 2 sin 2t, cos t − 2 cos 2t, −sin 3t)
Parameters
Drag to rotate