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4. Symmetry Step by Step.

Repeat it in some way - translation, reflection over a line, rotation around a point - and you have created symmetry. Symmetry is a fundamental phenomenon in art, science, and nature that has been captured, described, and analyzed using mathematical concepts for a long time.

Inspired by the geometric intuition of Bill Thurston and empowered by his own analytical skills, John Conway, with his coauthors, has developed a comprehensive mathematical theory of symmetry that allows the description and classification of symmetries in numerous geometric environments. This richly and compellingly illustrated book addresses the phenomenological, analytical, and mathematical aspects of symmetry on three levels that build on one another and will speak to interested lay people, artists, working mathematicians, and researchers.

Subject Symmetry Mathematics Geometry. Bibliographic information. Publication date ISBN alk. Browse related items Start at call number: QA S96 C66 He is a prolific mathematician active in the theory of finite groups, knot theory, number theory, combinatorial game theory, and coding theory. He has also contributed to many branches of recreational mathematics, notably the invention of the Game of Life.

Her primary interests are educational technology and discrete geometry.

## The symmetries of things

Chaim Goodman-Strauss is a professor in the department of mathematical sciences at the University of Arkansas. His research interests include low-dimensional topology, discrete geometry, differential geometry, the theory of computation, and mathematical illustration. Sets of numbers can form groups, for example. There are more exotic forms of symmetry than the ones we learn in elementary school.

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Consider, for example, the symmetries of lattices. The simplest lattice is just a two-dimensional grid. In the plane, you can shift the lattice up, down, left or right any number of squares and end up with a lattice that looks exactly like the one you started with.

You could also reflect the lattice over any individual square in the grid. Spaces equipped with lattices have an infinite number of different lattice symmetries.

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Lattices can exist in spaces of any number of dimensions. In three-dimensional space the lattice might be made of cubes instead of squares. In four dimensions and higher you can no longer picture the lattice, but it works the same way; mathematicians can precisely describe it.

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• Symmetries of Things.
• Throughout the 20th century, mathematicians discovered these groups in many different settings — not just geometry, but in number theory, logic and computer science as well. But the combination of a group with infinite symmetries and a space with many dimensions makes it hard to determine whether or not the group applies.

When we think of symmetry, we picture a whole shape being rotated, like a square turned clockwise 90 degrees. At a granular level, though, symmetry is really about moving points. To transform a space by symmetry means to take each point in the space and move it to some other point in the space.

## Symmetry Step by Step | Klein Project Blog

In that light, rotating a square clockwise by 90 degrees really means: Take each point on the square and rotate it clockwise 90 degrees so that it ends up on a different edge from where it started. This business of moving around points can be done in more or less rigid fashion. The most familiar symmetry transformations — reflect a square over its diagonal, or rotate the square 90 degrees — are very rigid.

The Symmetries of Things

Points that were vertices before the reflection are still vertices after the reflection just different vertices and points that formed straight edges before the reflection still form straight edges after the reflection just different straight edges.