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this page, Arcs and Points:

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.

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Interpreting the Graphic

The unit circle is added in black.

When n=0, all a^n = 1.

When n=1, all a^n = a.

Let each line be labeled by the point it passes through at n=1. MouseOver the graph to see these points.

Within or Without the Unit Circle

8 lines:

• 4 associated with Absolute Value 1/2 are contained within the Unit Circle.
• 
• 4 associated with Absolute Value 2 are without the Unit Circle.
• 

Those within the Unit Circle approach zero, those without the Unit Circle approach infinity, as the power is raised. This quality could be known as direction (towards zero, away from zero), but I will empasize the boundary of the Unit Circle and label the quality Contained, as in, within or without the Unit Circle. Note: each line begins on the real boundary of the Unit Circle when a^0 = 1.

Spiral Direction

For the purposes of this value, if the spiral moves counter-clockwise, in the direction of increasing radians, it is positive. If the spiral moves in the direction of decreasing radians, it is negative. The lines whose angle doesn't change will have a value of zero.

8 lines:

• 2 -I values associated with negative spiral direction.
• 
• 2 real positive values associated with zero spiral, straight lines.
• 
• 4 (2 real negative and 2 positive imaginary) values associated with positive spiral.
• 

Two Dualities:

In the real plane [180 degree from 1 to -1], 2 and 1/2 are similar, being lines and balanced on one with -2 and -1/2 similiarily being spirals in the positive direction. Both of these pairs share the difference in Contained. The primary duality is positive/negative and the secondary duality is within/without.

In the imaginary plane [90 degree from 1 to +/- I], similar pairs are 2I and -2I, showing the same spiral Outward, in opposite directions, leaving 1/2I, -1/2I, the same spiral inward, in opposite directions. The primary duality is within/without and the secondary duality is positive/negative.

thus:

Real [180 degree]	Points	Commonality: Spiral Direction	Difference: Contained
Positive	{2, 1/2}	Both 0 spiral Lines	Without/Within
Negative	{-2, -1/2}	Both Positive	Without/Within
Imaginary [90 degree]	Points	Commonality: Contained	Difference: Spiral Direction
Without	{2I, -2I}	Both Without	Positive/Negative
Within	{1/2I, -1/2I}	Both Within	Positive/Negative

Interpretation in the Metaphor: is open-ended. Help.

details:

*see next segment: Timing and the Big and Small.

Back: Power Circles

this page, Arcs and Points:

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.

Next: Big and Small