Question
A. Explain the general relationship between abstract algebra and geometry.
B. Discuss how the symmetries of a square can be conceptualized as the dihedral group of order 8.
Solution
As almost anyone can remember, classical geometry as pioneered by the Greeks, which starts with the study or shapes, congruence, lines, conics, quickly becomes extremely cumbersome when it comes to proving even simple statements. A great leap forward consisted in the introduction of analytical geometry and linear algebra:
• Lines and planes are represented by linear equations, and their intersection properties studies algebraically. This generalizes the notion of parallelism through the...

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