Visualising the Real Orthogonal Group

In the post Can You See in Four Dimensions?, we saw some ways of visualising functions plotted in four-dimensional 'space' in various ways. Of course, we used colour and time for two dimensions because it is a bit difficult to plot in four actual spatial dimensions!

Here is another example: the orthogonal group $\mathrm{O}_2(\mathbb{R})$ over the real numbers $\mathbb{R}$. This is the group of all matrices $A\in \mathrm{GL}_2(\mathbb{R})$ such that $AA^t = I_2$, where $I_2$ is the $2\times 2$ identity. Let

$A = \begin{pmatrix}a & b\\ c & d\end{pmatrix}$

be such a matrix. Then $a = \pm\sqrt{1 – b^2}$ and $c = \pm\sqrt{1 – d^2}$. Finally, $ac + bd =0$. As long as these conditions are satisfied, then $A$ will indeed be in the orthogonal group.

If we specify $a$, then there are two choices for $b$, and for each choice of $c$, there are two choices for the row $(c,d)$. We could specify $a$ as a time, and then for each choice of $b$, we can plot a point at each possibility for $(c,d)$. Since $b\in [-1,1]$, we can choose a colour by generating a rainbow spectrum from red to violet over the interval $[-1,1]$.

This is what we get, where the value of $a$ is displayed at the top: