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New Algorithm Unzips Complexity

NSF Award:

Conformal Mapping  (University of Washington)

Loewner Evolutions and Quasiconformal Mappings  (University of Washington)

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University of Washington mathematicians Donald Marshall and Steffen Rohde have developed a powerful tool to approximate smooth regions, such as a circle in a plane, using many-sided polygons. Known as the zipper algorithm, this series of mathematical equations is a way to compute angle-preserving mappings, which are ubiquitous in science and engineering.

The algorithm provides answers to problems that are inherently too complex to allow for precise, simple mathematical analysis. Now for the first time, chemists, physicists and others can respond very accurately to such questions as: What is the shape of a large molecule such as DNA? How does water percolate through soil?

To understand the zipper algorithm think about how a polygon can fit the shape of a circle. First, a square is drawn inside a circle, placing the angles of the square on its circumference. However, the sides of the square do not touch the circumference of the circle. If an octagon--an eight-sided polygon--replaces the square it will come much closer to fitting on the circumference of the circle. Adding other polygons with more sides and angles will create an even closer approximation of the circle. Fitting polygons from both the inside and the outside of the circle creates something very similar to a zipper. Repeated applications of the procedure close in upon the circle, preserving the angles of the polygons.

Images (1 of )

  • zipper flake
  • before zipper
  • after zipper
Zipper algorithm approximation of a snowflake fractal curve.
Donald Marshall
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The algorithm solves "conformal welding" problems.
Donald Marshall
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The circular form in the previous graphic becomes the black jagged curve with the zipper algorithm.
Donald Marshall
Permission Granted

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