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Wednesday, January 21, 2009

Ellipses, Circles, Epicycloids, Pi, and Unity

(1) The red line on the ellipse is the same length where it is touching a point anywhere on the ellipse. The endpoints of that line are called the foci of the ellipse.

(2) The diameter of the larger circle is exactly twice the diameter of the inner circle that is rotating around inside it.

(3) The diameters of the dynamic circles that are spinning around the outside of the stationary circles is equal to (3a)the stationary circle diameter, (3b)half the stationary circle diameter, (3c) a third of the stationary circle diameter, and (3d) a fourth of the stationary circle diameter, respectively. The flower pattern is called an epicycloid.

(4) This shows that the circumference of a circle, when unrolled is equivalent to pi times the diameter.

(5) This shows that as the number of vertexes on a perfect shape increases, that shape conforms more and more to the shape of a circle. It also shows the roots of unity along the imaginary unit circle, where the formula for the roots of unity:

X(x,n) = 1^(1/n) = exp(x*2*Pi*i/n) = cos(x*2*Pi/n) + i*sin(x*2*Pi/n)


X(x,n)=> the xth root of the n roots of unity;

x=> the root number (going from 1 to n);

n=> the number of roots;

(Note: All Gif images are a courtessy of Wolfram's Mathworld which is the best resource for studying math around next to getting a personal tutor.)