What is a torus?

The math

One hole, Euler number 0, seven colors

A torus is the simplest surface made by spinning a closed curve. MathWorld defines a toroid as “A surface of revolution obtained by rotating a closed plane curve about an axis parallel to the plane which does not intersect the curve. The simplest toroid is the torus.” The torus itself is “a surface having genus one,” “shaped like a donut.” MathWorld adds that the single-holed ring torus “is known in older literature as an ‘anchor ring.’”

You can make one from a sheet of paper, in theory. A torus “can be constructed from a rectangle by gluing both pairs of opposite edges together with no twists.” Glue the top edge to the bottom and you get a tube; bend the tube round and glue its two ends and you get the donut. That is why a game screen that wraps around on both sides is, mathematically, a torus.

Area and volume. With R from the centre of the hole to the centre of the tube and r for the tube’s radius, the surface area is 4π²Rr and the volume is 2π²Rr² (MathWorld writes c and a for R and r). Both make sense if you unroll the tube: the area is the tube’s rim, 2πr, times the distance its centre travels, 2πR. The calculator above uses exactly these formulas. They hold while R ≥ r; once r is bigger, the spinning circle crosses the axis and the surface passes through itself (a spindle torus).

Euler characteristic 0. For any surface, corners minus edges plus faces gives the same number, χ = 2 − 2g, where g counts the holes. A sphere has g = 0 and χ = 2; the torus has g = 1 and χ = 0, the value MathWorld’s table lists. The grid on the spinning torus above checks it: 432 − 864 + 432 = 0.

Four special circles through every point. MathWorld: “Exactly four distinct circles lying entirely on a ring torus pass through each point P. Two are familiar: the circle in a plane parallel to the equatorial plane and the circle in a plane through the axis of the torus. The other two are Villarceau circles.” They are named after Villarceau, who wrote about them in 1848. Press Villarceau circles above: one slanted plane cuts the torus in two of them at once.

A map on a donut can need 7 colors. On a flat map or a sphere, 4 colors always do. On a surface with g holes, the Heawood conjecture gives the number as ⌊(7 + √(48g + 1)) / 2⌋: 4 for g = 0 and 7 for g = 1. MathWorld: “A set of regions requiring the maximum of seven regions is shown above for a normal torus.” That the number is also needed was “proved by Ringel and Youngs (1968) with two exceptions: the sphere ... and the Klein bottle.” Press 7-color map above: seven regions, each one touching all six others, so no two can share a color.

The word

A cushion, then a column

“Torus” entered English as a building word. Etymonline: “torus (n.) 1560s, in architecture, ‘large, rounded molding at the base of a column,’ from Latin torus ‘a swelling, bulge, knot; cushion, couch,’ which is of uncertain origin.” So before it was a donut in geometry, it was the fat ring of stone at the foot of a classical column.

Real tori

Fusion machines and ring coils

The best-known working torus is the tokamak. ITER, the international fusion project, says: “The heart of a tokamak is its doughnut-shaped vacuum chamber.” The word itself comes from a Russian acronym for “toroidal chamber with magnetic coils.”

Ring-shaped coils are everyday electronics. Wikipedia: “Toroidal inductors and transformers are inductors and transformers which use magnetic cores with a toroidal (ring or donut) shape.”

The Rodin coil. The Rodin coil, the winding at the centre of vortex math, is also wound around a torus. It is described only by enthusiast and self-published sources; we found no patent, journal paper or news report on it.

What the tradition holds

The torus in sacred geometry

The modern spiritual reading of the torus was made famous by the 2011 film Thrive, by Foster Gamble. The film’s website put it this way: “The code may be our most powerful key to thriving. It is a pattern in nature that exists everywhere from atoms to galaxies. It’s called a ‘torus.’ You can see it in a seed, a fruit, a hurricane, even the electromagnetic field around a person or the earth.” The same page promised to show “how extraterrestrials may be using it to travel between star systems; how the code is being communicated through crop circles; how new energy inventors are using the code to power revolutionary devices.” It linked to The Resonance Project, “founded and directed by Nassim Haramein.”

A newspaper review summed up the claims. Good Times Santa Cruz (2012) wrote that the film opens on a shape “which mathematicians call a ‘torus,’ and which Foster Gamble believes holds vast significance and power,” and then claims that “the torus can be used to create a perpetual motion machine and deliver ‘free energy’” and that “the torus is a code delivered to humanity by aliens via UFO.” The movement behind the film has since closed: “After an amazing 20 years, we have closed operations of our companies, ThriveOn and Thrive Movement.”

What measurements show about Earth. Earth’s magnetic field is a dipole. NASA says it can be “represented as a dipole magnet field,” and UCAR says it has “almost the same shape as the magnetic field around a bar magnet, so it is called a ‘dipole field.’” We found no reliable source that describes Earth’s field as a torus. The torus shape is the hard math above and the tokamak; the rest is the tradition, and you can weigh it on the sacred geometry page with the other figures.