Visuals and such regarding the exponential function in the complex plane applied to the unit circle and reciprical relationship.

I. Bootstrapping a metaphor

We see natural numbers as representing real, distinct, objects. Negative numbers, which may be taken to represent absense or lack, are chosen to be the subjective, interior, invisble, internal representation of the external objects. This leaves zero as a boundary and point of reflection, the real reflected within the mind.

II. PowerCircles.html

It is shown that the Unit Circle is formed in the Complex Plane by applying the exponential function to the values -1 or +/- I.

III. ArcsAndPoints.htm

With the Unit Circle in place, the exponential function is applied to 2: +,2 -2, +2I, -2I; and the interior recipricals, Absolute Value = 1/2: +1/2, -1/2, 1/2 I, -1/2 I.  These values are selected because they are the second, reflected across zero into the negatives, than rotated orthogonally to the Imaginary axis. All of these are exactly two units from the origin, so their Absolute Value is two. The reciprical to two, one-half, is also so reflected and turned.

Several qualities in the exponential graph are examined: positive and negative seem to be enfolded into the within and the without. The period between crossing the real positive ray begins at zero for the positive values, moves to two for the negative numbers and is four for all of the imaginary numbers. The positive imaginary values rotate in the counter-clockwise, positive radian, direction. The negative imaginary values rotate in the clockwise, negative radian direction.

Points originating within the unit circle (1/2) move towards the center, and points originating without the unit circle (2) move outward indefinately. All points appear at the real boundary to the Unit Circle when the exponent is zero, and are at their origination point when the exponent is one.

IV. BigAndSmall.htm

Animation showing four perspectives on the function. This shows the rate of rotation, times of crossings, and demonstrates that the shape repeats at every level, cycle of 4 powers, as the points spiral inward and outward.  Great graphic!

V. OtherNumbers.htm

Some still images of the pattern formed by sets of numbers other than two and its recipricals and reflections.