Generative Geometry

The archetypal circle and square can geometrically generate many forms. Ancient cultures recognized these forms and relationships as essential and sacred, a metaphor of universal order. The circle and square in the act of self-division give us three generative roots: the square roots of 2, 3, and 5 (figs. 1A and 1B of "Generative Geometry"). These root relationships are all that are necessary to form the five regular (Platonic) solids that are the basis for all volumetric forms (fig. 1C). Also, 2, 3, and 5 are the only numbers required to divide the octave into musical scales.

Geometric Properties of Plane Figures Volumes and Surfaces of Double Curved Solids Surfaces of Solids The area of the surface is best found by adding together the areas of all the faces.

The area of a right cylindrical surface = perimeter of base x length of elements (average length if other base is oblique).

The area of a right conical surface = perimeter of base x 1/2 length of elements.

There is no simple rule for the area of an oblique conical surface, or for a cylindrical one where neither base is perpendicular to the elements. The best method is to construct a development, as if making a paper model, and measure its area by one of the methods given in the "Triangular and Circular Geometry" information.

Volumes and Surfaces of Typical Solids

Triangular and Circular Geometry

Oblique Triangles

Right Triangle

Arcs

Cords