Applications of group theory to geometry

What are the applications of group theory to geometry? Where can I know more about these applications?

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4 Answers

You might be interested to know there is a specialization in mathematics called Geometric Group Theory. The Wikipedia article (see link) elaborates on the domain of this specialization.

The article also gives the history of the relatively new field of math, and examples of how it is applied, and/or the topics studied.

See also this fascinating page: Symmetry and Group Theory, with links to:

The Aesthetics of Symmetry, essay and design tips by Jeff Chapman.

Symmetry web, an exploration of the symmetries of geometric figures.

Antipodes.

Cognitive Engineering Lab, Java applets for exploring tilings, symmetry, polyhedra, and four-dimensional polytopes.

Convex Archimedean polychoremata, 4-dimensional analogues of the semiregular solids.

Crystallography now, tutorial on the seventeen plane symmetry groups.

Escher's combinatorial patterns.

Figure eight knot / horoball diagram.

Fractal patterns formed by repeated inversion of circles.

Investigating Patterns: Symmetry and Tessellations.

Visualizing Cayley graphs of Coxeter groups as symmetric 4-dimensional polytopes.

Kaleidoscope geometry.

Mirror Curves.

Origami: a study in symmetry.

Platonic solids

• quaternion groups,
• Platonic spheres,
• Associating the symmetry of the Platonic solids with polymorf manipulatives.
• Platonic tesselations of Riemann surfaces.

Symmetry and Tilings.

• Rhombic tilings
• Rhombically tiled 48-gon
• MagicTile. Klein's quartic meets the Rubik's cube, by Roice Nelson.

Symmetries of torus-shaped polyhedra

Transformational geometry.

Wallpaper groups. - An illustrated guide to the 17 planar symmetry patterns. - Wallpaper patterns,

A word problem. Group theoretic mathematics for determining whether a polygon formed out of hexagons can be dissected into three-hexagon triangles, or whether a polygon formed out of squares can be dissected into restricted-orientation triominoes.

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@amWhy: Good to see you on! :-) – Babak S. Mar 10 '13 at 19:51
@amWhy , i wrote a comment at 5:40 on my time but when i revised it , i found many tyops ! , so i deleted it , i said that i apologise for being late for 40 minutis , but being late was because of some problem in electricity which required some time to be fixed , so i apologise about that . i hope that you accept my apologise . – Maths Lover Mar 13 '13 at 0:20
Yes, of course I accept your apologies, MrWhy! Some things happen that mess with our best of intentions! No need to worry! – amWhy Mar 13 '13 at 0:27

Group theory can be considered as the "Mathematical Theory of Symmetries". And since the most natural place to encounter symmetries is in geometry, there is a deep connection between these two areas.

Actually, geometry is one of the historic origins of group theory: Given some geometric object, which symmetries does it have? The resulting group can be quite complicated. For example, the the symmetry group of an icosahedron is the simple group $A_5$ of order 60.

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+1 for the historical remark. It is quite important to recoqnize that Klein's Erlangen program was one of the main motivations for the notion of a group. The other main motivation, for the abelian case, came from algebraic number theory. – Martin Brandenburg Mar 9 '13 at 16:44

There is even an amazing book so called "Groups & Symmetry" by M.A. Armstrong containing lots of examples on the interaction between groups and geometry. If you want one concrete and "easy looking" example, have a look at The Platonic solids.

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This is a bit of a plug for a book that you might buy or read for free: Group Theory: Birdtracks, Lie's, and Exceptional Groups. The book is one of many answers to your question, formulated this way:

Suppose someone came into your office and asked, “On planet Z, mesons consist of quarks and antiquarks, but baryons contain three quarks in a symmetric color combination. What is the color group?” The answer is neither trivial nor without some beauty (planet Z quarks can come in 27 colors, and the color group can be E6).

The previous answers are mostly about discrete symmetries, but continuous symmetries can be pretty cool too.

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