Polytwisters

What are polytwisters?

Polytwisters are a family of curved shapes that exist in four spatial dimensions. They were discovered in 2000 by Jonathan Bowers, an American mathematician known for his work on polytopes in higher dimensions. Check his original polytwisters page.

If you are entirely new to 4D Euclidean space, I recommend the following resources:

Article series by H. S. Teoh: "4D Visualization."
Video series from HyperCubist Math: "Visualizing 4D."

The 4D analogy of the sphere is known as the 3-sphere. On the ordinary sphere, a great circle is a circle with the same radius as the sphere itself, dividing it into two hemispheres. Any two distinct great circles on a sphere must intersect at exactly two points. On the 3-sphere, it is possible for two great circles on the 3-sphere to not intersect each other at all. As the 3-sphere is a dimension "bigger" than the sphere, there is far more room for these circles.

In fact, it is possible to divide the 3-sphere into infinitely many great circles so that no two of them overlap each other and the circles cover the entire 3-sphere. This is Hopf fibration, and these circles are called Hopf fibers.

Even stranger, there exists a highly symmetrical one-to-one mapping from Hopf fibers to points on the ordinary sphere. This is the Hopf map. If we have a polyhedron whose vertices are on a sphere, we can invert the Hopf map to convert that collection of vertices to a set of Hopf fibers which we call rings.

Polytwisters are a four-dimensional analogy to polyhedra which, in place of "vertices," have rings. In place of a polyhedron's edges which connect two points, two rings may be connected by a strip, which is a two-dimensional surface topologically equivalent to an open cylinder. Finally, a set of strips joined end-to-end form a twister, which is a polytwister's equivalent of a polygonal face.

Mathematical definition of polytwisters

A major part of this project was establishing a formal definition of polytwisters for the first time. This section defines convex polytwisters, and less formally describes strips, twisters, and nonconvex polytwisters. It is intentionally terse because as of October 2025, I am working on a paper which elaborates greatly on this.

Equate $\mathbb{R}^4$ and $\mathbb{C}^2$ with $(a, b, c, d)_{\mathbb{R}^4} \equiv (a + bi, c + di)_{\mathbb{C}^2}$ . Define the equivalence relation $x \sim y$ on $\mathbb{C}^2$ as true iff there exists $k \in \mathbb{C},\, |k| = 1$ such that $y = kx$ , i.e. $y$ is a phase rotation of $x$ . The partition $\mathbb{C}^2 / \sim$ divides the space into fibers. Reinterpreted in $\mathbb{R}^4$ , fibers are geometric circles centered on the origin, except for one trivial fiber comprising the origin; the rest we refer to as nontrivial fibers.

Taking only the fibers of unit radius gives us the Hopf fibration, partitioning the 3-sphere into infinitely many great circles. The Hopf map is actually not necessary to understand or define polytwisters, so I will not discuss it here.

Given $y \in \mathbb{C}^2 \backslash \{0\}$ define a log and a pipe respectively as:

$\begin{align*} L(y) = L(y_1, y_2) &= \{ x \in \mathbb{C}^2 : |\langle x, y \rangle| \leq 1\} \\ P(y) = P(y_1, y_2) &= \{ x \in \mathbb{C}^2 : |\langle x, y \rangle| = 1\} \\ \end{align*}$

where $\langle (x_1, x_2), (y_1, y_2) \rangle = x_1 y_1^* + x_2 y_2^*$ . The pipe $P(1, 0)$ viewed in $\mathbb{R}^4$ is the Cartesian product of a unit circle and a plane, and the log $L(1, 0)$ is the Cartesian product of a closed unit disk and a plane. All other pipes and logs are respectively of the form $k\mathbf{U}P(1, 0)$ and $k\mathbf{U}L(1, 0)$ where $\mathbf{U} \in \text{SU}(2)$ and $k$ is a positive real number. Crucially, all logs and pipes are unions of fibers.

We can now define a convex polytwister:

Definition. A convex polytwister is an intersection of a finite set of three or more logs, but not equal to the intersection of fewer than three logs.

The key to this is the analogy to the definition of " $\mathcal{H}$ -polytopes," which may be defined as the bounded intersection of finitely many closed half-spaces. (See Ziegler's Lectures on polytopes for an introduction.) The polytwister equivalent of a closed half-space is a log, and the equivalent of a hyperplane is a pipe.

Just as two planes in general position in 3D space intersect at lines and three planes intersect at a point, the intersection of pipes allows us to produce lower-dimensional shapes. An important fact, although not an obvious one at all, is that three pipes in general position intersect at exactly two nontrivial fibers (sometimes zero or one, but those cases are irrelevant to polytwisters). You can see a visual of this in the order 3 dyadic twister, which is the simplest convex polytwister; the solid is the intersection of exactly three logs $L(y_1) \cap L(y_2) \cap L(y_3)$ . The two fibers in the set $P(y_1) \cap P(y_2) \cap P(y_3)$ appear as "vertices" on the boundary of the figure, which we properly call rings.

Taking further set operations on pipes and logs gives us the face lattice of the polytwister. A strip can be formed as the intersection of two pipes and at least one log, such as $P(y_1) \cap P(y_2) \cap L(y_3)$ . In $\mathbb{R}^4$ it is an embedding of the cylinder, and its topological boundary is two fibers. Unlike line segments, strips are not uniquely determined by their boundary: there are infinitely many strips between a pair of fibers, and they bow out in different amounts depending on the orienation of the pipes. Finally, a twister (2-face) is formed by the intersection of one pipe and two or more logs, such as $L(y_1) \cap L(y_2) \cap P(y_3)$ , and its boundary is a set of two or more strips. (I am being informal here and leaving out some degeneracy conditions.)

Nonconvex polytwisters are defined roughly as abstract 3-polytopes with a realization that maps each facet (twister) to a pipe called its "containing pipe," each edge to a strip, and each vertex to a ring (fiber), such that each strip's incident rings are precisely its boundary, and each twister's incident strips are subsets of its containing pipe.

All polytwisters shown in this application are uniform, which means their symmetry group acts transitively on their rings and on their strips.

About the application

This application is open source, MIT license. Check the source code.

The renderer is a GLSL fragment shader which implements a classical raytracer, using closed-form expressions to compute intersections of rays and logs. A cross-section of a log is an affine transformation of an infinite cylinder, and polytwisters are formed by Boolean operations on logs, so rendering polytwisters is no more difficult than raytracing cylinders. No discrete meshes are used, although a mesh generation feature is under way.

Uniform polytwisters are generated on the fly from the symbols alone, using a variant of the Wythoff construction to determine ring locations and combinatorial structure. The internal database of polytwisters contains only their names and symbols.

Thanks foremost to Jonathan Bowers for giving me access to his original POV-Ray code and answering my many questions. I'm also grateful to members of the Polytope Wiki and Discord for their assistance, in particular galoomba, Violeta, PlanetN9ne, ThePokemonkey123, and Mecejide.

Changelog

v0.2.2 (2025-11-19): App now initially opens to the quasi ditrigonary icosidodecatwister rather than the more plain-looking tetratwister. Colors of twisters adjusted to reduce ugly combinations. Article now displays when JavaScript is disabled. Partially fix flash of unstyled content.

v0.2.1 (2025-11-02): Cross-sections are now symmetrically rotated so there is always a ring in the xy-plane.

v0.2.0 (2025-10-13): Major rework of user interface.

2025-10-04: All 222 uniform polytwisters and three infinite families are implemented.

2025-06-13: Added 142 uniform polytwisters.

2025-04-09: Total rewrite.

2024-05-04: Initial launch.