# Geometrical interpretation of pictures transforms and other “high dimensional everyday objects”

During the preparation of a general audience talk on why mathematicians use dimensions higher than three (or four) even for concrete applications, I came up with the following enjoyable observation : Taking the negative of a picture is a central reflexion of the hypercube. I'm pretty sure many people already observed this, so my question is :

Are you aware of other similar transformations for "everyday life high dimensional objects" which have a nice geometric interpretation?

To provide more details, consider a picture of $240\times 240$ pixels, where each pixel has a grey level ranging from $0$ to $255$. Each grey level can be written as a $8$ digit number in base $2$ (and now your computer understands what grey level means). With this identification, this picture is a vertex of the $$240\times240\times8=460800\mbox{-dimensional hypercube: }\quad [0,1]^{ 460800}.$$ Now, the negative of a picture is the picture you obtain by reversing the grey scale, meaning mapping a grey level $\alpha$ to $255-\alpha$ for every pixels. In binary this means you switch every zero to one and vice versa. Thus the negative picture, seen as a vertex of the hypercube, is the central reflexion of your initial picture with respect to the center of the hypercube.

I would like to see if one can find other examples of nice geometric interpretations for (colored or not) image transformations, sound effects (i.e. Fourier series of reasonable functions), etc.

• Sorry, but I don't see what is illuminating about this view of "reflection in the hypercube". The simple "255 - image" is perfectly nice for me and the reflection in the hypercube makes things much more complicated without giving any insight (well, these mathematicians surely like it complicated…). If $240^2$ is not high enough for your dimension, just talk about pictures that your smartphone takes which have about a few million pixels or movies which may have a billion voxels… – Dirk Jul 3 '17 at 17:44
• @Dirk Identifying numerical data with high dimensional objects is basic science indeed and this is not what my question is about: It is about specific transformations of numerical data which still have a nice geometric interpretation after this identification; notice I didn't say "more simple" but "nice". For instance, It was a surprise to me that reversing an image has again a simple geometric interpretation after this identification. – Adrien Hardy Jul 3 '17 at 20:21