Back: Bootstrapping

this page, Power Circles:

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

Next: Arcs and Points

So far we have defined 5 points of reference between Mathematics and Metaphysics.

Looking at Plus and Minus, Multiplication and Division.

Now, a^n, repeated Multiplication as...

What happens to 1 at infinity?

Steadfastly one.

-1 alternates between odd powers, -1 and even powers, 1.

Imaginary numbers start at (-1)^(1/2) or Sqrt[-1] = I.

Area, Sqrt[2].

Interpreting the Animation mathematically

Red: One raised to N;

Green: Negative one raised to N;

Blue: I raised to N.

Purple: -I raised to N

In the animation, N completes one cycle of four powers.

Read N off of either of the bottom graphs: the x axis.

Theta green, blue: Indicates the angle at which the point a^N is. I.E. {Cos[theta], Sin[theta]} is the point on the unit circle. Notice that the Cos relates to real and Sin relates to imaginary such that they cycle at 2Pi.

Negative one circles twice as I circles once.

Arbitrarily I was only focused on three of four right angles, 1, I, -1. Where is -I? Mirroring I across the real (x, horizontal) axis. Missing from this graphic is the path of -I, which is around the unit circle in the opposite direction and the same speed as I.

Back: Bootstrapping

this page, Power Circles:

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

Next: Arcs and Points

Notes below the line

Out of Infinite Regress

Arbitrarily I was only focused on three of four right angles, 1, I, -1. Where is -I? Mirroring I across the real (x, horizontal) axis.

Focusing on the 4 part cycle that I takes,

External Expression, I^0 = (1,0) {Objective, External WORLD, Realized Self; SEPERATE Ego}

Observation of External, I¹ = (0, I), {BOUNDARY MIRROR; Self HEART}

Internal Expression, I² = (-1, 0), {Subjective, Internal HEART: Self; SEPERATE Ego}

Observation.of Internal, I³ = (0, -I), {BOUNDARY MIRROR; Oberving Self}

1 and -1 represent expressions, I and -I represent observations,

+ side represents concrete realization, - side represents abstractions.

{

Points to make:

Positive numbers < 1 -begin "INSIDE" - as power is applied, they shrink towards zero,slowing their approach; at what rate?

Positive numbers = 1 - begin at boundary - as power is applied, remains UNITY

Positive numbers > 1 - begin "OUTSIDE" - as power is applied accelerate towards infinity; at what rate?

1 is the only stationary point;

Case of 0^N, 0^0, etc. - Center of SELF

Case of -1 - circular at 4Pi radians per 4 units

Work in how?

Case of I - circular at 2Pi radians per 4 units

Spirals: -1<n<0 - begin inside in mind

n<-1 begin outside in mind

correlation: Spiral Crossing with analagous positive value at real number line?}