From considering a construcion for deriving the area of a circle using its diameters and radii i arrived at an approximation 2√3 for pi.
The method requires construction of equilateral triangles using the diameter and the radius. In the large triangle 4 equlateral triangles interweave the semicircle. The total area of the large triangle is thus 8*1/2*√3/2*r*r
Inspecting the circle the construction is too large by two curious shapes that are almost triangular. This prompted a process of iteration to greater and greater accuracy. the curious delta shape occurs 6 times within the smaller triangles within the circle disc. Thus one third of the area of the circle is within one quarter of the count of smaller triangles roughly. these proportions frame the design of the pattern
While the pattern above is not directly derived from this process, it is inspired by it and the allure of pattern . It fails to converge on the calculated value of pi because it is not an attempt to, but an attempt to find a memorable pattern that gets as close as it can using the numerals of these proportions..
Iteration is a fundamental aspect of interacting with space. We use it and then ignore it for less dynamic forms, for so called exactness. This is the ultimate still life the eternal frozen moment of undeniable”truth”. But it is a lie that has taken until the 1970’s for Mandelbrot to draw together the threads of a dynamic geometry called fractal geometry.
Fractal geometry as is clear is not the creation of one mind, but seems to be the universal mind and method of apprehension. Let us explore this dynamic spaciometry.
The rituals and inspirations of the wise seem deep until you realise how distracted we have become by technology/ The simplest conclusion we can come to is that everything revolves around everything else in the most amazing convolutions. Those Pythagoreans that were called Mathematikos realised and relied upon this to frame their philosophy. In latin it was called recursion: to run the course again.
No athlete or engineer find the concept difficult. One repeats hoping for a slightly different result! Yet mathematicians of the 20th century made out it was “impossible” to understand. For that reason i use the Greek Iter which means road or path or course. The notion of iteration is to run the course again expecting a different result. To get a different result you have to build on the old result, taking that as your new measure or standard. The process of doing that makes for dynamic consequences that even in simple iterations produce unpredictable results!
The Philosophy of Mechanics attempted to derive through form, predictable outcomes. To do so rigidity or fixity, is introduced. That is dynamic changes are cut out so that the initial conditions for each and every rerun are the same. But even this rigidity conforms through mechanical processes of Wear to the dynamics of the iteration process, eventually reducing viscosity to a limit that no longer can contain the bound energies within the iteration process. Panta Rhei.
We have this anxiety driven notion that if we do not control and predict everything we are doomed! But it is precisely that attitude that dooms us! The Truth is iteration , that is rotational motion, sets us free! It is the “truth” that does actually set us free.