Root system inner product is the Killing form. 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 formula_32 and formula_33 are both integers, by assumption, and formula_34 Since formula_35, the only possible values for formula_36 are formula_37 and formula_38, 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 2"α" or −2"α", are
Platonic realism translated (or transliterated) as "idea," but secondary literature now typically employs the term "form" (or occasionally "kind," usually in discussion of Plato's "Sophist" and "Statesman") to avoid confusion with the English word connoting "thought". Platonic form can be illustrated by contrasting a material triangle with an ideal triangle. The Platonic form is the ideal triangle — a figure with perfectly drawn lines whose angles add to 180 degrees. Any form of triangle that we experience will be an imperfect representation of the ideal triangle. Regardless of how precise your measuring and drawing tools you will never be able to recreate