## Twisted Toroids

A toroid is a surface of revolution with a hole in the middle, created by rotating a curve (the generatrix) around an axis of rotation. The generatrix can be a polygon such as a square (an n-gon with n = 4):-

n = 4
If the square is rotated by 90° (2π/n) about an axis perpendicular to its own centre while rotating 360° (2π) about the axis of rotation, the result is a twisted toroid (the twist factor t is the multiple of 2π/n, s is the number of sides):-
n = 4, t = 1, s = 1
Like a Mobius strip, this has only one side:-
n = 2, t = 1, s = 1
If the square is rotated by 180° (2 × 2π/n) about an axis perpendicular to its own centre while rotating 360° about the axis of rotation, the resulting twisted toroid has two sides:-
n = 4, t = 2, s = 2
If the square is rotated by 270° (3 × 2π/n) about an axis perpendicular to its own centre while rotating 360° about the axis of rotation, the resulting twisted toroid again has only one side:-
n = 4, t = 3, s = 1
More examples follow. These GIFs were created in Python using the NumPy, surf2stl, and PyVista libraries. True toroids are perfectly smooth, but these are approximated by 100 segments facetted with triangles.
n = 3, t = 1, s = 1
n = 3, t = 2, s = 1
n = 3, t = 3, s = 3
n = 3, t = 4, s = 1
n = 5, t = 1, s = 1
n = 6, t = 1, s = 1
n = 6, t = 4, s = 3
The 3D model of a twisted torus can be saved as an STL file and sent to a 3D printer.

n = 3, t = 1, s = 1
n = 3, t = 3, s = 3
Both models can stand stably on a triangular facet one of whose edges in parallel to the axis of rotation of the toroid. A smooth twisted toroid without facets would presumably not be stable to stand like this.
PyVista can use Physically Based Rendering to render a 3D model with a metallic finish, placing it in a simulated environment:-

For a torus, where r is the radius of the circular generatrix, and R is the radius of revolution, the two circles in a cross-section touch if R = r, and the hole vanishes. If R < r, the axis of revolution passes through the surface, which then intersects itself.
For some twisted toroids, R can be less than r (the radius of the circumcircle of the polygon) without the polygons in a cross-section intersecting. In general, regular n-gons just touch when the twist t is odd and the radius of revolution is the mean of the radii of the circumcircle and the incircle, when R = (1 + cos(π/n))r/2.
For an equilateral triangle with odd twist t, and R = 3r/4, for example, there is still a hole and the rotating cross-section is shown by:-

n = 3, t = 1, s = 1
n = 3, t = 3, s = 3
Christopher B. Jones 2022-12-17