By Kristopher Tapp

This textbook is ideal for a math path for non-math majors, with the objective of encouraging potent analytical pondering and exposing scholars to dependent mathematical rules. It contains many themes generally present in sampler classes, like Platonic solids, Euler’s formulation, irrational numbers, countable units, diversifications, and an explanation of the Pythagorean Theorem. All of those themes serve a unmarried compelling aim: realizing the mathematical styles underlying the symmetry that we notice within the actual international round us.

The exposition is attractive, distinctive and rigorous. The theorems are visually influenced with intuitive proofs applicable for the meant viewers. scholars from all majors will benefit from the many appealing issues herein, and should come to higher savour the robust cumulative nature of arithmetic as those issues are woven jointly right into a unmarried interesting tale in regards to the ways that items might be symmetric.

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**Extra resources for Symmetry: A Mathematical Exploration**

**Sample text**

But these irrelevant notational differences do not matter – Swahili readers will learn the same things as English readers. It is a single group represented in two different notational systems. The dictionary between the English and Swahili notational systems is like an isomorphism. It translates true equations into true equations. For example, when the Swahili symbol for H is composed with the Swahili symbol for R90 it had better equal the Swahili symbol for D; otherwise the translator needs to be fired.

To investigate this similarity, let us first review the familiar algebraic properties of multiplication and addition of numbers. Î The order in which a pair of numbers are added (or multiplied) does not affect the result. This is called the commutative property. In symbols: A+B = B+A and A × B = B×A Î The order in which a pair of additions (or a pair of multiplications) is performed does not affect the result. This is called the associative property. In symbols: (A+B)+C = A+(B+C) and (A × B) × C = A×(B×C) Î Adding 0 to a number has no effect.

Similarly, you will now study the algebraic operation of composition by building a table that exhibits the result of composing any pair of symmetries. Let us start with a square. Its eight symmetries are: {I, R90, R180, R270, H, V, D, Dcc} where H means horizontal flip, V means vertical flip, D and Dcc mean the two diagonal flips, and R means a counterclockwise rotation by the subscripted angle. The illustrations below show the effect of these eight symmetries on a square whose corners are labeled A, B, C, D and whose center is decorated with a picture of a gnome.