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I like to visualize everything I study but yet I have found pretty nothing to visualize in abstract algebra.I have studied group theory upto subgroups Cyclic groups and Cosets and Lagrange's theorem.Is there any way of visualizing these things?Please suggest some good reference book/text also which discusses these things and also the motivation/idea behind different theorems and concepts.

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    $\begingroup$ Group theory can be visualized as groups of symmetries. $\endgroup$ – Michael Burr Jun 6 at 13:14
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    $\begingroup$ Cayley graphs provide a way to visualize groups. There is a textbook called Visual Group Theory that gives visual interpretations for many group theoretic properties. $\endgroup$ – André 3000 Jun 6 at 13:41
  • $\begingroup$ I seem to recall a book by Fraleigh (First Course on Abstract Algebra) which had some visuals to go with quotient groups/cosets. Also simply the vertices of a regular $n-gon$ are a starting point for visualising subgroups of the finite cyclic groups. $\endgroup$ – Mark Bennet Jun 6 at 18:07
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    $\begingroup$ There is a reason it called "abstract algebra" but there is ofcourse there are some ways to visualize it (if not all). But I am not sure if you have that enough background yet or not? $\endgroup$ – Anubhav Mukherjee Jun 6 at 21:55
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One nice visual approach to basic results on cyclic groups is using periodical curves such as Roulettes (Spirograph curves), Star Polygons, etc. Some of these have concrete implementations in toys like Spirograph - which can be employed to motivate these more abstract algebraic ideas at very elementary levels.

enter image description here

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  • $\begingroup$ Do the colors in the images have any meaning or are they there to make them pretty? $\endgroup$ – B.Swan Jun 6 at 16:35
  • $\begingroup$ So how do you use these pictures to visualize the concepts? $\endgroup$ – Don Thousand Jun 6 at 18:00
  • $\begingroup$ Tell me something more on the above pictures as Don Thousand said? $\endgroup$ – user679537 Jun 6 at 18:15
  • $\begingroup$ @KishalaySarkar Sure, I'll elbaorate in a bit. $\endgroup$ – Bill Dubuque Jun 6 at 18:45
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    $\begingroup$ If the OP wants plane curves with cyclic symmetry groups then there are simpler versions like this with a 3-fold rotational symmetry. Complex power mappings lead to simple equations. Here is a 5-fold symmetry. I didn't bother to kill the dihedral symmetry of the latter curve. Sorry about that. Can't offer curves related to toys or pieces of art either :-( $\endgroup$ – Jyrki Lahtonen Jun 6 at 20:03
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To add a bit of history here - the famous German mathematician Felix Klein in 1872 launched the so-called Erlangen Program, proposing a method of characterizing geometries based on group theory and projective geometry. This was a novel and visionary idea a the time of his writings. It preceded later developments that showed how the symbiosis of (Complex) Analysis, Geometry and Algebra led to new insights and theories.

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