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
Napoleon's theorem outer Napoleon triangle has a length of The relation between the latter two equations is that the area of an equilateral triangle equals the square of the side times formula_5 The centers of regular n-gons constructed over the sides of an n-gon P form a regular n-gon if and only if P is an affine image of a regular n-gon. Napoleon's theorem In geometry, Napoleon's theorem states that if equilateral triangles are constructed on the sides of any triangle, either all outward or all inward, the centres of those equilateral triangles themselves form an equilateral triangle. The triangle thus formed