Activity 4.4.1. What’s in a Name.
Mathematics can be described as the study of patterns. For this reason, we often look for similar structure in both nature and mathematical systems. Mathematics can certainly be used to study patterns in the physical world, but we can also look for patterns within mathematics itself. Are there times when one mathematical system is, for all practical purposes, “identical” to another mathematical system and therefore governed by the same properties and relationships? If this is so, then one system can give us quite a bit of information about another system. In fact, if one system is easier to operate on, we can use it instead and then deduce information about the other system without having to do more difficult computations.
Recall that in Activity 4.3.6 we used the set of functions \(T=\left\{e, f, g, h, j, k\right\}\) under the operation of function composition where the functions are defined as follows:
\begin{equation*}
e\left(x\right)=x, f\left(x\right)=\frac{1}{x}, g\left(x\right)=\frac{x}{x-1}, h\left(x\right)=1-x,
j\left(x\right)=\frac{1}{1-x}, k\left(x\right)=\frac{x-1}{x}
\end{equation*}
Further, recall that you found this set and operation to form a group and you constructed the Cayley (operation) table given below.
We will now consider a different set and operation. Take your equilateral triangle and mark the vertices 1, 2, and 3 on both faces (you will be flipping them over and will need to identify the same vertex from both faces of the triangle). Orient the triangle as shown in Figure 4.4.2.
(a)
Keeping the vertices labeled, how many different ways can you orient the triangle so that one side lies along the horizontal with the opposite vertex pointing upward? For example,
are different orientations.
(b)
Using the orientations you found in part (a) as the “basic” moves for the triangle, describe each orientation as a movement such as a flip or rotation from the original position given in Figure 4.4.2. For example, the movement described in part (a) could be considered a \(\frac{1}{3}\) or 120˚ counter-clockwise rotation or a \(\frac{2}{3}\) or 240˚ clockwise rotation.
(c)
In order to communicate with each other, we will decide on a common notation for moves of the triangle. Rotations will be clockwise. We will let \(r_0\) stand for no rotation, \(r_1\) denote a 120˚ rotation, and \(r_2\) represent a 240˚ rotation. For the flips, we have three axes about which we can flip the triangle (vertical and two diagonals). We will use \(v\text{,}\) \(d_1\text{,}\) and \(d_2\) to denote them as shown below.
We will claim that these are the “basic” moves for the triangles so that it comes to rest back in the same “space” as it started. We will denote this set of moves as \(M=\left\{r_0, v, d_1, d_2, r_1, r_2\right\}\text{.}\) Explain how you know that these are the only “basic” moves. (think about how many ways you could re-label the triangle).
While we can use physical triangles to do these manipulations, we can also do it virtually by using matrix transformations. To help with performing the moves, we will use a set of matrices on the TI-Nspire CX CAS. These six matrices will take the original vertices and map them to their new locations. For your convenience, the matrices have been given names corresponding to our original names for the triangle moves as follows (see TI-Nspire document page 1.2).
\begin{equation*}
M=\left\{
\begin{array}
"r_0=\begin{bmatrix} 1 \amp 0\\ 0 \amp 1 \end{bmatrix}
\amp
r_1=\begin{bmatrix} -\frac{1}{2} \amp \frac{\sqrt{3}}{2}\\ -\frac{\sqrt{3}}{2} \amp -\frac{1}{2} \end{bmatrix}
\amp
r_2=\begin{bmatrix} -\frac{1}{2} \amp -\frac{\sqrt{3}}{2}\\ \frac{\sqrt{3}}{2} \amp -\frac{1}{2} \end{bmatrix}
\\
v=\begin{bmatrix} -1 \amp 0\\ 0 \amp 1 \end{bmatrix}
\amp
d_1=\begin{bmatrix} \frac{1}{2} \amp \frac{\sqrt{3}}{2}\\ \frac{\sqrt{3}}{2} \amp -\frac{1}{2} \end{bmatrix}
\amp
d_2=\begin{bmatrix} \frac{1}{2} \amp -\frac{\sqrt{3}}{2}\\ -\frac{\sqrt{3}}{2} \amp -\frac{1}{2} \end{bmatrix}
\end{array}
\right\}
\end{equation*}
Here the triangle vertices are stored in a \(2 \times 3\) matrix, named \(t\) for “triangle”, where each column represents the \(x-\) and \(y-\)coordinates of vertices 1, 2, and 3 respectively as labeled on the triangle. The effect of matrix multiplication is that the vertices trade locations. For example, applying \(r_1\) to the triangle vertices matrix gives
\begin{equation*}
r_1 \cdot t=
\begin{bmatrix} -\frac{1}{2} \amp \frac{\sqrt{3}}{2}\\ -\frac{\sqrt{3}}{2} \amp -\frac{1}{2} \end{bmatrix}
\cdot \begin{bmatrix} 0 \amp \frac{\sqrt{3}}{2} \amp -\frac{\sqrt{3}}{2}\\ 1 \amp -\frac{1}{2} \amp -\frac{1}{2}
\end{bmatrix}=
\begin{bmatrix} \frac{\sqrt{3}}{2} \amp -\frac{\sqrt{3}}{2} \amp 0\\ -\frac{1}{2} \amp -\frac{1}{2} \amp 1
\end{bmatrix}
\end{equation*}
which is the same as rotating the triangle 120˚ clockwise. Notice that the coordinates of the vertices underwent the mappings \(\left(0,1\right) \mapsto \left(\frac{\sqrt{3}}{2}, -\frac{1}{2}\right)\text{,}\) \(\left(\frac{\sqrt{3}}{2}, -\frac{1}{2}\right) \mapsto \left(-\frac{\sqrt{3}}{2}, -\frac{1}{2}\right)\text{,}\) and \(\left(-\frac{\sqrt{3}}{2}, -\frac{1}{2}\right) \mapsto \left(0, 1 \right)\text{.}\) To see the changes from the original position displayed in the “Before” pane appear in the “After” pane of the screen, we perform the calculation and store the result in the matrix \(tt\) for “transformed triangle” (see screen images below). We can also perform multiple moves together. Consider applying the \(d_1\) flip followed by the \(v\) flip. This combination of moves yields an equivalent scenario as performing the \(r_2\) rotation (this has been filled in for you in the following operation table given in part (d)).
(d)
Build a table for the composition of the moves on the triangle of the form \(A \circ B\) where move \(B\) is performed first followed by move \(A\text{.}\) In the space below, record the result as shown.
(e)
In general, is the set of moves on the triangle closed under the operation of composition of moves? If not, what elements yield an element not in the original set of moves?
(f)
In general, do the elements of the set commute with each other? If not, do some of the elements commute with some of the other elements? Explain.
(g)
How might you check associativity? How many different permutations of these moves would you need to check to be certain of associativity? Devise a plan to check associativity. Your plan might include other groups from the class (divide and conquer is often very effective). Is there any pattern that you have noticed that might allow you to claim associativity without checking all possible combinations (consider your table of functions under composition from Figure 4.4.1)?
(h)
Is there a move from the set that acts like an identity element? If so, what is the move and explain how you know?
(i)
Does every element have an inverse? If so, list all elements and their inverses. If not, list any elements that do have an inverse along with their inverse element.
(j)
Again, recall that if a set along with an operation meets the four criteria of closure, associativity, an identity element, and all elements have inverses, then we call the set a group. Does this set of six moves on a triangle under the operation of composition of moves form a group?
(k)
Now consider the Cayley table you have just created. What similarities do you notice in relation to the earlier Cayley table of the group of six functions (see Figure 4.4.1)? What differences do you notice?
(l)
Given your observations about each set and operation, explain how you could use the table for function composition on \(T\) to answer questions about set \(M\) and its composition of movements on the triangle. (Think about ideas such as element orders and inverses).
(m)
When looking at these two groups, you might have noticed similarities for the behavior among the elements of each group. We can think of these two groups as parallel universes where the relationships among the “people” of the two universes are the same within each world. We can therefore give a mapping between the worlds that tells who in one world corresponds to their similar person in the other world. Let’s call this mapping \(\varphi\) where if the person, \(a\text{,}\) in the first world behaves like person, \(a'\text{,}\) in the other world, then \(\varphi\left(a\right)=a'\)(see the diagram below).
