Math Monday: The twisted torus

By George Hart for the Museum of Mathematics

A twisted torus is a donut which twists as it closes on itself. It is a visually engaging mathematical design used by many artists through the ages. The origins of the idea are unknown, but below is a woodcut, published by the German graphic artist Johannes Lencker in 1567, which shows that the idea has been around for a long time.

Many twentieth century sculptors, such as Charles Perry and Heleman Ferguson, have adapted the form to modern abstract sculpture. But how can you make your own twisted torus? Below is an elegant solution from the 16th century: a twisted torus in the Schloss Ambras Museum, assembled from seventy-six identical wood components.

Each piece has a slot for the next to pass through and there is enough play in the slots for the chain to be twisted slightly before closing into a loop. It is a visual puzzle to figure out how the ring was closed. If you make many copies of the basic unit sketched below, you can try to solve the puzzle and assemble them into your own twisted torus.

Bret Rothstein of Indiana University has studied this object to understand its art historical significance and has also made a modern reproduction in walnut. If you want to make your own copy, and have access to a 3D printer, there’s more information and .STL files to make the components here.

More:

• Math Monday: Binder clip constructions
• Math Monday: Twirligami
• Math Monday: Found objects
• Math Monday: Kirigami polyhedra
• Math Monday: Mathematical lathe work
• Math Monday: Modular Kirigami
• Math Monday: Mathematical beading
• Math Monday: Nailbanger’s Nightmare
• Math Monday: Recycling soda bottles into icosahedra
• Math Monday: Two-layer geodesic spheres
• Math Monday: What to make with golf balls?
• Math Monday: Knitted cellular automaton tea cosy
• Math Monday: Whittling links and knots
• Math Monday: Magnet constructions
• Math Monday: Hexagonal stick arrangements
• Math Monday: Paper plate geometry
• Math Monday: 3D Hilbert curve from plumbing supplies
• Math Monday: Math-play with your food
• Math Monday: Mathematical art in the lava
• Math Monday: Balloon polyhedra
• Math Monday: Sierpinski tetrahedron
• Math Monday: Skewer hyperboloid
• Math Monday: Morton Bradley sculpture
• Math Monday: Tetraxis puzzle
• Math Monday: Giant burr puzzles
• Math Monday: Fractal polyhedra clusters
• Math Monday: Giant SOMA puzzle
• Math Monday: Tie your bagel in a knot!
• Math Monday: Playing card constructions
• Introducing “Math Monday”