Dodecagon

Dodecagon is a twelve cornered, twelve-sided shape (a twelve-gon) — in this case a regular dodecagon under consideration and experimentation to diagram systems components related to time- and population-completeness.

Insights From Compass-and-Straightedge Construction (CSC)

The dodecagon can be drawn using a CSC in two ways: at a given circumcircle or from a side length.

At A Given Circumcircle

At any point on a line, a circle is drawn of any size. The Matrix / Mandorla / Vesca Piscis is drawn using the diameter of this circle. This is then bisected to create a perpendicular line intersecting to the original line. From the four intersections of these two lines and the circle, four semi-circles of the same radius of the original circles are drawn. This creates twelve equi-distant intersections on the circle (4 original + 8 new) which may be connected in sequence to create the twelve points and sides of the dodecagon.

This construction is allegorically useful when considering time and community within a system. Notice the original lines and circle form The Compass which can symbolize human identity, especially the Imago Dei, but also Earth.¹ Also interesting is the construction of the Matrix around the circle — visually showing that the finished dodecagon exists within the confines of an evolving Complex Dynamic System, centered on the mid point of that system’s evolution.

Continuing to think allegorically, the dodecagon is completed from the intersection of the original circle and four over-lapping circles, or a world intersected by four other worlds. In the case of the the Imago Dei interpretation of The Compass, the central world would be the individual, and the four intersecting worlds would be heaven, positive, earth, negative (clockwise from the top).

From A Side Length

Any line segment is chosen as a side length of one side of the eventual dodecagon. This serves as the radius for two circles that will create the Matrix / Mandorla / Vesca Piscis which allows the finding of a line perpendicular to the original. A line that intersects both an end of the original line and this line bisecting the Matrix finds the center of the larger circle upon which the dodecagon will be constructed. The length of the original line is now measured out across the circle resulting in the twelve points and sides of the dodecagon.

Allegorically, this could be used when a system — especially the maximum sustainable extent of its Alpha-Beta Creative Tension — is used as one in a series to build a complete time or community system. This would imply all the twelve systems in the series must construct the dodecagon at the same time in their evolution — ideally at their sustainable maximum (mid-point).

1. Earth’s planetary symbol dating from Greco-Roman astronomy is a cross within a circle (⊕) See https://en.wikipedia.org/wiki/Planetary_symbols.