Fundamental Polygon - Explicit Form For Standard Polygons

Explicit Form For Standard Polygons

Explicit expressions can be given for the standard polygons. One of the more useful forms is in terms of the group associated with the standard polygon. For a genus oriented surface, the group may be given by generators . These generators are given by the following fractional linear transforms acting on the upper half-plane:

a_k= \left( \begin{matrix} \cos k\alpha & -\sin k\alpha \\ \sin k\alpha & \cos k\alpha \end{matrix} \right) \left( \begin{matrix} e^p & 0 \\ 0 & e^{-p} \end{matrix} \right) \left( \begin{matrix} \cos k\alpha & \sin k\alpha \\ -\sin k\alpha & \cos k\alpha \end{matrix} \right)

for . The parameters are given by

and

and

It may be verified that these generators obey the constraint

which gives the totality of the group presentation.

Read more about this topic:  Fundamental Polygon

Famous quotes containing the words explicit, form and/or standard:

    “Like dreaming, reading performs the prodigious task of carrying us off to other worlds. But reading is not dreaming because books, unlike dreams, are subject to our will: they envelop us in alternative realities only because we give them explicit permission to do so. Books are the dreams we would most like to have, and, like dreams, they have the power to change consciousness, turning sadness to laughter and anxious introspection to the relaxed contemplation of some other time and place.”
    —Victor Null, South African educator, psychologist. Lost in a Book: The Psychology of Reading for Pleasure, introduction, Yale University Press (1988)

    “Thought is the seed of action; but action is as much its second form as thought is its first.”
    —Ralph Waldo Emerson (1803–1882)

    “As long as male behavior is taken to be the norm, there can be no serious questioning of male traits and behavior. A norm is by definition a standard for judging; it is not itself subject to judgment.”
    —Myriam Miedzian, U.S. author. Boys Will Be Boys, ch. 1 (1991)