Polynomiography Animations
© 2003 Bahman Kalantari All Rights Reserved Polynomiography™
- www.polynomiography.com
- Gives links to articles on the subject of polynomiography. Here we demonstrate some animations with this technique. This site will be upgraded periodically and will eventually contain many more animations and detailed explanations. Some of these animations can be used for educational purposes while others can be viewed as visual art. Please forward your comments by visiting the above web site.
- Dance
- This is polynomiography animation with a fourth degree polynomial.
- Sensitivity
- This is polynomiography animation with the polynomial (x-1)(x-2)(x-3)(x-4)(x-5)(x-6) showing the sensitivity of the roots to changes in the coefficient of x^6 as it is decreased, some becoming complex numbers.
- Polynomiography of polynomials in a problem of Knuth
- This is polynomiography animation with the polynomial
p(x)=summation of a_k * (n choose k) z^k (1-z)^{n-k} where the vector
a=(a_0, ..., a_n) is a point in (n+1)-dimensional hypercube. In this example we have n=5 and we take a range from (0,1,0,1,0,1) to (1,0,1,0,1,0). This amounts to going from one corner of the 6D hypercube to the opposite corner.
- Rotation
- This is polynomiography animation of the roots of x^4-1 under rotation of the roots, induced by the change of variable x with e^it x, where t goes from 0 to pi/2. (e^it=cos t+ i sin t)
- Zoom
- This is a polynomiography animation with zooming.
- A Problem of the Monthly
- This is a polynomiography animation with x^m(x^4-1) for different values of m.
- Playing with fourth root of unity
- This is polynomiography animation corresponding to approximation of
roots of (x^4-1) under different iteration functions.
- Spiral
- Billiard
- Voronoi Regions - Roots of Unity (x^4-1)
- This is a polynomiography animation with basins of attraction of (x^n-1) for different iteration functions.
- Voronoi Regions - Random Points
- This is a polynomiography animation with basins of attraction of a polynomial under different iteration functions.