Toroid

For other uses, see Toroid (disambiguation).

A toroid using a square.

A torus is a type of toroid.

In mathematics, a toroid is a surface of revolution with a hole in the middle. The axis of revolution passes through the hole and so does not intersect the surface. ^[1] For example, when a rectangle is rotated around an axis parallel to one of its edges, then a hollow rectangle-section ring is produced. If the revolved figure is a circle, then the object is called a torus.

The term toroid is also used to describe a toroidal polyhedron. In this context a toroid need not be circular and may have any number of holes. A g-holed toroid can be seen as approximating the surface of a torus having a topological genus, g, of 1 or greater. The Euler characteristic χ of a g holed toroid is 2(1-g). ^[2]

The torus is an example of a toroid, which is the surface of a doughnut. Doughnuts are an example of a solid torus created by rotating a disk, and are not toroids.

Toroidal structures occur in both natural and synthetic materials. ^[3]

Equations

A toroid is specified by the radius of revolution R measured from the center of the section rotated. For symmetrical sections volume and surface of the body may be computed (with circumference C and area A of the section):

Square toroid

The volume (V) and surface area (S) of a toroid are given by the following equations, where A is the area of the square section of side, and R is the radius of revolution.

${\displaystyle V=2\pi RA}$

${\displaystyle S=2\pi RC}$

Circular toroid

The volume (V) and surface area (S) of a toroid are given by the following equations, where r is the radius of the circular section, and R is the radius of the overall shape.

${\displaystyle V=2\pi ^{2}r^{2}R}$

${\displaystyle S=4\pi ^{2}rR}$

Pappus's centroid theorem generalizes the formulas here to arbitrary surfaces of revolution.

See also

• Toroidal inductors and transformers
• Toroidal propellers
• Annulus
• Solenoid
• Helix

Notes

1. Weisstein, Eric W. "Toroid". MathWorld .
2. Stewart, B.; "Adventures Among the Toroids:A Study of Orientable Polyhedra with Regular Faces", 2nd Edition, Stewart (1980).
3. Carroll, Gregory T.; Jongejan, Mahthild G. M.; Pijper, Dirk; Feringa, Ben L. (2010). "Spontaneous generation and patterning of chiral polymeric surface toroids". Chemical Science. 1 (4): 469. doi:10.1039/c0sc00159g. ISSN   2041-6520.

External links

• The dictionary definition of toroid at Wiktionary