Posted by Jason Polak on 09. June 2013 · Write a comment · Categories: group-theory · Tags: ,

9-soframe-21

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.
More »

Posted by Jason Polak on 08. June 2013 · Write a comment · Categories: number-theory · Tags: , ,

frame8

Have you ever tried to visualise the graph of a complex function $ f:\mathbb{C}\to\mathbb{C}$? The problem with complex functions is that usually we graph a complex number as an ordered pair $ (x,y)$ on a Euclidean plane, which corresponds to $ z = x + iy$. Unfortunately, this means that if we want to graph complex functions as we do real functions, we need to draw the graph in four-dimensional space! Some people have actually claimed the ability to visualise this, but I do not!

However, if we use time as a dimension, we could represent four dimensions as a moving three-dimensional image in time, like a movie. Sometimes, it’s hard to draw three dimensions in two dimensions, though we don’t actually lose too much because we can only see a two-dimensional picture of a three-dimensional scene at any given time.

There are a few animations in this post; they may take a few seconds to load, or a few minutes on a slower connection.
More »