by Scott Crass (Author)
Working
out solutions to polynomial equations is a mathematical problem that
dates from antiquity. Galois developed a theory in which the obstacle to
solving a polynomial equation is an associated collection of
symmetries. Obtaining a root requires "breaking" that symmetry. When the
degree of an equation is at least five, Galois Theory established that
there is no formula for the solutions like those found in lower degree
cases. However, this negative result doesn't mean that the practice of
equation-solving ends. In a recent breakthrough, Doyle and McMullen
devised a solution to the fifth-degree equation that uses geometry,
algebra, and dynamics to exploit icosahedral symmetry.
Polynomials,
Dynamics, and Choice: The Price We Pay for Symmetry is organized in two
parts, the first of which develops an account of polynomial symmetry
that relies on considerations of algebra and geometry. The second
explores beyond polynomials to spaces consisting of choices ranging from
mundane decisions to evolutionary algorithms that search for optimal
outcomes. The two algorithms in Part I provide frameworks that capture
structural issues that can arise in deliberative settings. While
decision-making has been approached in mathematical terms, the novelty
here is in the use of equation-solving algorithms to illuminate such
problems.
Features
- Treats
the topic―familiar to many―of solving polynomial equations in a way
that’s dramatically different from what they saw in school
- Accessible to a general audience with limited mathematical background
- Abundant diagrams and graphics.
