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:
“Listen. We converse as we liveby repeating, by combining and recombining a few elements over and over again just as nature does when of elementary particles it builds a world.”
—William Gass (b. 1924)
“Without being forgiven, released from the consequences of what we have done, our capacity to act would ... be confined to one single deed from which we could never recover; we would remain the victims of its consequences forever, not unlike the sorcerers apprentice who lacked the magic formula to break the spell.”
—Hannah Arendt (19061975)
“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 (18561924)
“Society cannot share a common communication system so long as it is split into warring factions.”
—Bertolt Brecht (18981956)
“I tell you the solemn truth that the doctrine of the Trinity is not so difficult to accept for a working proposition as any one of the axioms of physics.”
—Henry Brooks Adams (18381918)