Problems
What are the symmetries of an isosceles triangle (which is not equilateral)?
What are the symmetries of the reduce-reuse-recycle symbol?
What are the symmetries of an equilateral triangle?
What are the symmetries of a rectangle (which is not a square)?
What are the symmetries of a rhombus (which isn’t a square)?
There are six symmetries of an equilateral triangle: three reflections, and three rotations (thinking of the identity as one the rotations).
Label the three reflections \(s_1\), \(s_2\) and \(s_3\). Label the identity by \(e\), rotation by \(120^{\circ}\) as \(r_1\), and rotation by \(240^{\circ}\) clockwise as \(r_2\).
What are the inverses of \(r_1\) and \(s_1\)?
Think about the symmetries of an equilateral triangle. We label rotation by \(120^{\circ}\) as \(r_1\), and reflection in the vertical median by \(s_1\).
Is applying \(r_1\), then \(s_1\) the same as applying \(s_1\), then \(r_1\)?
Let \(n\ge3\) be a positive integer. A regular \(n\)-gon is a polygon with \(n\) sides where every side has the same length, and every angle is the same. For example, a regular \(3\)-gon is an equilateral triangle, and a regular \(4\)-gon is a square.
What symmetries does a regular \(n\)-gon have, and how many?
The set of symmetries of an object (e.g. a square) is called a group. We can formally define a group \(G\) as follows.
A is a non-empty set \(G\) with a binary operation \(*:G\times G\to G\) satisfying the following axioms (you can think of them as rules that \(G\) and \(*\) have to satisfy). A binary operation takes two elements of \(G\) and gives another element of \(G\).
Associativity: For all \(g\), \(h\) and \(k\) in \(G\), \((g*h)*k=g*(h*k)\).
Identity: There is an element \(e\) of \(G\) such that \(e*g=g=g*e\) for all \(g\) in \(G\).
Inverses: For every \(g\) in \(G\), there exists a \(g^{-1}\) in \(G\) such that \(g*g^{-1}=e\).
Closure: For all \(g\) and \(h\) in \(G\), \(g*h\) is also in \(G\).
Prove that the symmetries of the ‘reduce-reuse-recycle’ symbol form a group.
