logarithmic spiral pendant

this is a pendant featuring a particularly interesting logarithmic spiral. part of a family of self-generating objects called autograms, this curve generates itself under the mechanism of evolutes. the evolute is the locus of all centers of curvature for a given curve. this means that the evolute of a curve is a kind of description of it, similar to how a derivative of a function describes the slope at all points, the evolute describes the curvature at all points. this spiral has the parametric equation (cos(t)e^At,sin(t)e^At) where A=2W(3pi/2)/3pi and where W(x) is the Lambert-W function. this is the largest logarithmic spiral which is its own evolute, all the others can be generated by different A values corresponding to any n value in: 2W((4n-1)pi/2)/(4n-1)pi. since this curve can be thought of describing itself, starting from the center you could pick a point and find that it is the center of curvature for the point that lies on the tangent line. then that point is the center of curvature for the point on its tangent line, and so on eminating from the center of the spiral. this behavior is quite remeniscent of the way some reasearchers believe we gain self-awareness in the human brain. being a recursive process of auto-observation and then meta-auto-observation going up an infinite ladder until it all collapses down and you are not aware of yourself being aware of yourself being aware of youself... you are just aware. hence, the spiral and the lines indicating its' self-descriptive nature as an analogy for consciousness.