Math Worksheet

Lars Eighner's Vector
''Worksheets are scratch pads where math problems, formulas, expressions are set up before they are copied to the places they belong. They may be completely wrong and may not make much sense out of their intended context. Moreover, they change as new material is worked out. What you see here will probably not be here if you return and is in no sense to be considered reliable.

Scratch pad

To show: $1+8+16+\dots+8(n -1)= (2^n -1)^2$

a. True for $n=1$

$\displaystyle{ \begin{align} 1 &= 1\\ &= (1)^2 \\ &= (2 -1)^2 \\ &=(2n-1)^2 \end{align}}$

b. if true for $n$, then true for $n+1,\ n>1$

$\displaystyle{ \begin{align}&1+8+\dots+8(n -1)= (2^n -1)^2\\&1+8+\dots+8(n -1)+8((n+1)-1)\\&= (2^n -1)^2+8((n+1)-1)\\&= 4n^2 - 4n +1 + 8n \\ &= 4n^2+4n + 1\\&=(2n+1)^2\\&=(2(n+1)-1)^2\\&\therefore 1+8+\dots+8(n-1)+8((n+1)-1)\\&=(2(n+1)-1)^2 \ \text{ Q.E.D}\end{align}}$

Subgroup of the Triangle Symmetries.

$\color{blue}{f}\color{lime}{\circ}\color{red}g;\quad f, g \in G$


If we look at just the upper left corner of Cayley table for the equilateral triangle symmetries, we discover that the rotations and the id (do nothing) form a group. Satisfy yourself that this meets the requirements of a group.


Cayley notation

$\displaystyle{\begin{array}{lll}id=\begin{pmatrix}1 &2 &3\\1 &2 &3\end{pmatrix} & r1=\begin{pmatrix}1 &2 &3\\3 &1 &2\end{pmatrix} & r2=\begin{pmatrix}1 &2 &3\\2 &3 &1\end{pmatrix} \\f1=\begin{pmatrix}1 &2 &3\\1 &3 &2\end{pmatrix} & f2=\begin{pmatrix}1 &2 &3\\3 &2 &1\end{pmatrix} & f3=\begin{pmatrix}1 &2 &3\\2 &1 &3\end{pmatrix} \end{array}}$

These are the permutations of the equilateral triangle symmetries in Cayley notation. $r2$ is read, for example, "1 goes to 2, 2 goes to 3, and 3 goes 1." Many texts use $i$ or $e$ for $id$, rho ($\rho$) for $r$ and mu ($\mu$) for $f$.


Another permutation presentation



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