Polynomiography: From the Fundamental Theorem of Algebra to Art

The author introduces polynomiography, a bridge between the Fundamental Theorem of Algebra and art. Polynomiography provides a tool for artists to create a 2D image—a polynomiograph—based on the computer visualization of a polynomial equation. The image is dependent upon the solutions of a polynomial equation, various interactive coloring schemes driven by iteration functions and several other parameters under the control of the polynomiographer's choice and creativity. Polynomiography software can mask all of the underlying mathematics, offering a tool that, although easy to use, affords the polynomiographer infinite artistic capabilities.

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Activities for Students: A Graphical Approach to Understanding the Fundamental Theorem of Algebra

Mathematics Teacher ◽

10.5951/mt.94.9.0749 ◽

2001 ◽

Vol 94 (9) ◽

pp. 749-756

Author(s):

Sudhir Kumar Goel ◽

Denise T. Reid

Keyword(s):

Fundamental Theorem ◽

Polynomial Equation ◽

Complex Numbers ◽

Multiple Roots ◽

Complex Root ◽

Graphical Approach ◽

Fundamental Theorem Of Algebra ◽

Complex Coefficients

The fundamental theorem of algebra states, Every polynomial equation of degree n ≥ 1 with complex coefficients has at least one complex root. This fact implies that these equations have exactly n roots, counting multiple roots, in the set of complex numbers.

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Relative equilibria of point vortices and the fundamental theorem of algebra

Proceedings of The Royal Society A Mathematical Physical and Engineering Sciences ◽

10.1098/rspa.2010.0580 ◽

2011 ◽

Vol 467 (2132) ◽

pp. 2168-2184 ◽

Cited By ~ 15

Author(s):

Hassan Aref

Keyword(s):

Fundamental Theorem ◽

Relative Equilibrium ◽

Polynomial Equation ◽

Relative Equilibria ◽

Point Vortices ◽

Generating Polynomial ◽

Second Derivatives ◽

Fundamental Theorem Of Algebra ◽

Vortex Systems ◽

Derivatives Of

Relative equilibria of identical point vortices may be associated with a generating polynomial that has the vortex positions as its roots. A formula is derived that relates the first and second derivatives of this polynomial evaluated at a vortex position. Using this formula, along with the fundamental theorem of algebra, one can sometimes write a general polynomial equation. In this way, results about relative equilibria of point vortices may be proved in a compact and elegant way. For example, the classical result of Stieltjes, that if the vortices are on a line they must be situated at the zeros of the N th Hermite polynomial, follows easily. It is also shown that if in a relative equilibrium the vortices are all situated on a circle, they must form a regular N -gon. Several other results are proved using this approach. An ordinary differential equation for the generating polynomial when the vortices are situated on two perpendicular lines is derived. The method is extended to vortex systems where all the vortices have the same magnitude but may be of either sign. Derivations of the equation of Tkachenko for completely stationary configurations and its extension to translating relative equilibria are given.

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