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
Equilateral pentagon relations. The equilateral pentagon described in this manner will be unique up to a rotation in the plane. When the equilateral pentagon is dissected into triangles, two of them appear as isosceles (triangles in orange and blue) while the other one is more general (triangle in green). We assume that we are given the adjacent angles formula_1 and formula_2. According to the law of sines the length of the line dividing the green and blue triangles is: The square of the length of the line dividing the orange and green triangles is: According to the law of cosines, the cosine