Novel Method To Construct A Pentagon

NEW DISCOVERY: NOVEL METHOD TO CONSTRUCT A PENTAGON. I have used and even taught the Dürer Method of constructing a Pentagon starting from a Vesica Piscis (which is also used to construct a Hexagon). Many have known that there is a slight imperfection in method introduced by Renaissance polymath Albrecht Dürer (known by some as the 'Da Vinci' of Germany). Following several attempts to construct the Pentagon using the Dürer Method utilizing the incredible 15 decimal digit precision of Geogebra (Google's precision online Geometry software), I have now concluded that, while "close", this method, as predicted FAILS to construct a 'PERFECT' PENTAGON.


So, I set out on a quest to find a more perfect and simple solution that produces a perfect pentagon and doesn't require too many construction lines. I cannot explain why, but I honestly never felt comfortable with the notion that the proportions of the Hexagon could construct the perfect Pentagon. The reason for this method's failure is that the diagonal 'Cross' (connecting the Vesica to the upper Pentagonal angles) used in its construction, is NOT a perfect 90° relationship, therefore resulting in imperfect proportions).


Recognizing the above (and unfortunate) reality I knew that there had to be a better (and probably even simpler) way. I studied all the various approaches that exist online and while 'close' they all seemed both overtly arduous and 'forced', requiring A LOT of steps. One such method (attached below) excludes the circular intersections (excludes a critical Φ^2 circle resulting in many unecessary steps) required for the upper angles of incidence of the Pentagon. I kept asking myself: Why is it so difficult to draw a Pentagon?....


Here is a NEW method (adhering to the Ancient Greek :

1. Construct three circles with radius values of 1, Φ, and Φ^2 (matching the photo above) using the rules of construction {using only a Compass and Square} and thus adhering to the rules established by Ancient Greek Mathematicians/Geometers.

2. Now draw one more Φ circle in a Vesica Relationship with the first Φ circle from #2 above.

3. Now, connect the "dots" at each of the circular intersections to constructing the final and 'perfect' PENTAGON.


With the above, I found it MUCH easier to construct the HexaPentakis starting from the Pentagon than by starting from the Hexagon. I am not sure why this is, but it also felt like an important metaphor to me which resulted in a critical realization for me: Even when the Head and Heart achieve a 'perfect' balance, ultimately, the Heart MUST lead the Head. (The Pentagon is widely viewed as the geometry of the Heart (Pathos), and the Hexagon is the Logos).


Of course, this now means that I must redraw the Sephiroth based on the perfect HexaPentakis!




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