"Harmony" is a state of balance, an at-oneness with opposing forces. Harmony is an even-ness between two
extremes. The struggle between perimeter and surface, and surface and volume, is the beating heart of The
Geometry of Form. As light and darkness are to day, perimeter and surface, and, surface and volume are to
form.

Many years ago I conceived and created the sculptural maquette Harmony 3D as illustration of one of
Geometry's most formative principles: scale, and its necessary consequences (such as The Geometry of Sunshine).
I named this original sculpture "Harmony" for reasons more profound than it having pleasing proportions.

A curious viewer would soon notice that the diameter of the sphere was the same as the width of the cube;
and then, that the two of them together were the same height as the tetrahedron. But it is the sub-division of the
cube into a six unit edge-length that gives the whole arrangement scale.

It happens that a cube so proportioned has its surface area equal to its volume. They are in a one to one
ratio; each quality just balances the other. The sphere with its six unit diameter, and the tetrahedron with its
two times six unit height also exhibit this harmonious trait, as does any other polyhedron so scaled.

Many years later, in 1999, I discovered that in two dimensions Geometry similarly has an harmonic
preference for all polygons. Harmony 2D , and Harmonic Scaling illustrate this preference. In these images the
object is an arrangement of regular polygons ranging from the minimal-sided triangle to the infinite-sided
circle. Scale manifests only after the grid is applied subdividing the circle and revealing its two unit radius. At
this "size", commensurate polygons are in a state of balance with respect to perimeter units and surface units;
they are in a one to one ratio; they are in harmony.

Copyright P. Kasprzycki 2003