Pascal's triangle Construct a "d"-dimensional triangle (a 3-dimensional triangle is a tetrahedron) by placing additional dots below an initial dot, corresponding to P(1) = 1. Place these dots in a manner analogous to the placement of numbers in Pascal's triangle. To find P("x"), have a total of "x" dots composing the target shape. P("x") then equals the total number of dots in the shape. A 0-dimensional triangle is a point and a 1-dimensional triangle is simply a line, and therefore P("x") = "1" and P("x") = "x", which is the sequence of natural numbers. The number of dots in each layer corresponds
Wythoff symbol and formula_7, represent the corners of the Schwarz triangle used in the construction, which are formula_8, formula_9 and formula_10 radians respectively. The triangle is also represented with the same numbers, written formula_11. The vertical bar in the symbol specifies a categorical position of the generator point within the fundamental triangle according to the following: In this notation the mirrors are labeled by the reflection-order of the opposite vertex. The formula_18 values are listed "before" the bar if the corresponding mirror is active. The one "impossible" symbol formula_19 implies the generator point is on all mirrors, which is only possible if