Unity of Geometry – Root Power : by Jonathan Quintin

**Unity of Geometry Root-Power – the Triangle – excerpt**

Unity of Geometry – Root Power : by Jonathan Quintin

**Unity of Geometry Root-Power – the Triangle – excerpt**

"We shall therefore borrow all our Rules for the Finishing our Proportions, from the Musicians, who are the greatest Masters of this Sort of Numbers, and from those Things wherein Nature shows herself most excellent and compleat." Leon Battista Alberti (1407-1472).

These pointers discuss triangles and their higher-dimensional generalizations (*simplices*). I am particularly interested in *triangulation* by which I mean partitioning regions into triangles, tetrahedra, or higher dimensional simplices, for various applications including finite element mesh generation and surface interpolation. (The other meaning of triangulation involves determining locations and distances from certain measurements.) For more material on the first type of triangulation, see the mesh generation section of Geometry in Action or the list of my own triangulation papers. For other kinds of partitions, see the page on dissection.

Plato considered geometry and number as the most reduced and essential, and therefore the ideal, philosophical language. But it is only by virtue of functioning at a certain ‘level’ of reality that geometry and number can become a vehicle for philosophic contemplation. Greek philosophy defined this notion of levels, so useful in our thinking, distinguishing the ‘**typal**‘ and the ‘**archtypal**‘. Following the indication given by Egyptian wall reliefs, which are laid out in three registers, an upper, a middle and a lower, we can define a third level, the ‘**ectypal**‘, situated between the archtypal and typal.

To see how these operate, let us take an example of a tangible thing, such as the bridle of a horse. This bridal can have a number of forms, materials, sizes, colours, uses, all of which are bridals. The bridal considered in this way, is typal; it is existing, diverse and variable. But on another level there is the idea or the form of the bridal, the guiding model of all bridals. This is an unmanifest, pure, formal idea and its level is ectypal. But yet above this there is an archtypal level which is that of the *principal* or *power-activity*, that is a *process* which the ectypal form and typal example of the bridal only represent. The archtypal is concerned with universal processes or dynamic patterns which can be considered independently of any structure or material form