Elementary Consequences of The Root System Axioms
The cosine of the angle between two roots is constrained to be a half-integral multiple of a square root of an integer. This is because and are both integers, by assumption, and
Since, the only possible values for are, corresponding to angles of 90°, 60° or 120°, 45° or 135°, 30° or 150°, and 0 or 180°. Condition 2 says that no scalar multiples of α other than 1 and -1 can be roots, so 0 or 180°, which would correspond to 2α or -2α are out.
Read more about this topic: Root System
Famous quotes containing the words elementary, consequences, root, system and/or axioms:
“When the Devil quotes Scriptures, it’s not, really, to deceive, but simply that the masses are so ignorant of theology that somebody has to teach them the elementary texts before he can seduce them.”
—Paul Goodman (1911–1972)
“There is a delicate balance of putting yourself last and not being a doormat and thinking of yourself first and not coming off as selfish, arrogant, or bossy. We spend the majority of our lives attempting to perfect this balance. When we are successful, we have many close, healthy relationships. When we are unsuccessful, we suffer the natural consequences of damaged and sometimes broken relationships. Children are just beginning their journey on this important life lesson.”
—Cindy L. Teachey. “Building Lifelong Relationships—School Age Programs at Work,” Child Care Exchange (January 1994)
“Black creeps from root to root,
each leaf
cuts another leaf on the grass,
shadow seeks shadow,
then both leaf
and leaf-shadow are lost.”
—Hilda Doolittle (1886–1961)
“There are obvious places in which government can narrow the chasm between haves and have-nots. One is the public schools, which have been seen as the great leveler, the authentic melting pot. That, today, is nonsense. In his scathing study of the nation’s public school system entitled “Savage Inequalities,” Jonathan Kozol made manifest the truth: that we have a system that discriminates against the poor in everything from class size to curriculum.”
—Anna Quindlen (b. 1952)
“The axioms of physics translate the laws of ethics. Thus, “the whole is greater than its part;” “reaction is equal to action;” “the smallest weight may be made to lift the greatest, the difference of weight being compensated by time;” and many the like propositions, which have an ethical as well as physical sense. These propositions have a much more extensive and universal sense when applied to human life, than when confined to technical use.”
—Ralph Waldo Emerson (1803–1882)