:Toroid

{{Short description|Surface of revolution with a hole in the middle}}

File:Torus.png 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.{{MathWorld|Toroid|Toroid}} 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).Stewart, B.; "Adventures Among the Toroids:A Study of Orientable Polyhedra with Regular Faces", 2nd Edition, Stewart (1980).

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.{{Cite journal |last=Carroll |first=Gregory T. |last2=Jongejan |first2=Mahthild G. M. |last3=Pijper |first3=Dirk |last4=Feringa |first4=Ben L. |date=2010 |title=Spontaneous generation and patterning of chiral polymeric surface toroids |url=http://xlink.rsc.org/?DOI=c0sc00159g |journal=Chemical Science |language=en |volume=1 |issue=4 |pages=469 |doi=10.1039/c0sc00159g |issn=2041-6520}}

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.

: $V = 2 \pi R A$

: $S = 2 \pi R C$

= 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.

: $V = 2 \pi^2 r^2 R$

: $S = 4 \pi^2 r R$

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

Notes

External links

{{Wiktionary-inline|toroid}}

Category:Topology

Category:Geometric shapes