Definition of The Notation
The following types of Euclidean transformation can occur in a group described by orbifold notation:
- reflection through a line (or plane)
- translation by a vector
- rotation of finite order around a point
- infinite rotation around a line in 3-space
- glide-reflection, i.e. reflection followed by translation.
All translations which occur are assumed to form a discrete subgroup of the group symmetries being described.
Each group is denoted in orbifold notation by a finite string made up from the following symbols:
- positive integers
- the infinity symbol,
- the asterisk, *
- the symbol (a solid circle in older documents), which is called a wonder
- the symbol (an open circle in older documents), which is called a miracle.
A string written in boldface represents a group of symmetries of Euclidean 3-space. A string not written in boldface represents a group of symmetries of the Euclidean plane, which is assumed to contain two independent translations.
Each symbol corresponds to a distinct transformation:
- an integer n to the left of an asterisk indicates a rotation of order n around a point
- an integer n to the right of an asterisk indicates a transformation of order 2n which rotates around a point and reflects through a line (or plane)
- an x indicates a glide reflection
- the symbol indicates infinite rotational symmetry around a line; it can only occur for bold face groups. By abuse of language, we might say that such a group is a subgroup of symmetries of the Euclidean plane with only one independent translation. The frieze groups occur in this way.
- the exceptional symbol o indicates that there are precisely two linearly independent translations.
Read more about this topic: Orbifold Notation
Famous quotes containing the words definition of the, definition of and/or definition:
“The very definition of the real becomes: that of which it is possible to give an equivalent reproduction.... The real is not only what can be reproduced, but that which is always already reproduced. The hyperreal.”
—Jean Baudrillard (b. 1929)
“No man, not even a doctor, ever gives any other definition of what a nurse should be than this—”devoted and obedient.” This definition would do just as well for a porter. It might even do for a horse. It would not do for a policeman.”
—Florence Nightingale (1820–1910)
“... if, as women, we accept a philosophy of history that asserts that women are by definition assimilated into the male universal, that we can understand our past through a male lens—if we are unaware that women even have a history—we live our lives similarly unanchored, drifting in response to a veering wind of myth and bias.”
—Adrienne Rich (b. 1929)