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:
“If men as individuals surrender to the call of their elementary instincts, avoiding pain and seeking satisfaction only for their own selves, the result for them all taken together must be a state of insecurity, of fear, and of promiscuous misery.”
—Albert Einstein (1879–1955)
“The middle years are ones in which children increasingly face conflicts on their own,... One of the truths to be faced by parents during this period is that they cannot do the work of living and relating for their children. They can be sounding boards and they can probe with the children the consequences of alternative actions.”
—Dorothy H. Cohen (20th century)
“Today, supremely, it behooves us to remember that a nation shall be saved by the power that sleeps in its own bosom; or by none; shall be renewed in hope, in confidence, in strength by waters welling up from its own sweet, perennial springs. Not from above; not by patronage of its aristocrats. The flower does not bear the root, but the root the flower.”
—Woodrow Wilson (1856–1924)
“... the yearly expenses of the existing religious system ... exceed in these United States twenty millions of dollars. Twenty millions! For teaching what? Things unseen and causes unknown!... Twenty millions would more than suffice to make us wise; and alas! do they not more than suffice to make us foolish?”
—Frances Wright (1795–1852)
“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)