Back: Arcs and Points

this page, Big and Small:

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!

Next: Other Numbers

Interpreting the Visual

Zoom Levels:

Upper right: Unit Circle, small. Lower right: smaller, radius = n^(-4) = 2^(-4) = 1/(2^4) = 1/16th.

Upper left: radius = n^4 = 2^4 = 16, big. Lower right: radius = n^8 = 2^8 = 256, bigger.

Time:

The animation shows the power to which a (2) and its reflections are being raised. It goes through two cycles of 4 powers from a^n; n = 0 to n = 8.

Rate of Rotation

This animation shows that the spirals associated with real negative numbers rotate (cross the positive real axis) 2 times for every one rotation of the imaginary numbers. Thus we can add a column to the last table:

Repetition: Big and Small

Back: Arcs and Points

this page, Big and Small:

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!

Next: Other Numbers