Technical background to the video

The theory

At present (August 2023), most digital images are displayed on screens that uses about 16 million distinct colours. The colour of each pixel is produced from three shining dots. There is one dot for each of the additive primary colours—red , green and blue . A dot can shine at a limited number of brightness levels. Each of the three dots can be set at any one of 256 levels: 0 (off), 1, 2, 3… 255 (full on), which gives about 16 million possible colours.

We can now think of the colours of an image on a screen as points inside a 256×256×256 cube. Each colour is made by setting the brightness levels (r,g,b) of three dots. The three values (r,g,b) are the coordinates of a point in the cube. We use these rgb axes instead of the usual xyz axes and call this 3-D space The RGB cube.

The symmetries

There are 24 ways to orient a cube if it’s to occupy a given space. Think of choosing which of the six square faces to put at the bottom, and then which of the four sides to put at the front; 6×4 = 24. This accounts for the 24 rotational symmetries of a cube.

Now imagine that we could turn the cube inside out, like turning a right-hand glove inside out to make a left-hand glove. That way we could get the mirror image of each of the previous 24 orientations.

That accounts for all 48 symmetries of the cube.

The algebra

Each of the 48 symmetries corresponds to a 3×3 matrix of a transformation. For example:

$(\begin{array}{r} 1 & 0 & 0 \\ 0 & 0 & - 1 \\ 0 & 1 & 0 \end{array})$ is a quarter turn (a rotation of order 4);

$(\begin{array}{r} 0 & 1 & 0 \\ - 1 & 0 & 0 \\ 0 & 0 & 1 \end{array})$ is a quarter turn about a different axis;

$(\begin{array}{r} - 1 & 0 & 0 \\ 0 & - 1 & 0 \\ 0 & 0 & - 1 \end{array})$ turns things inside out
(it’s a reflection in the origin);

In fact you can get any of the 48 symmetries from these three using matrix multiplication: it’s a matter of choosing the right sequence of rotations for each one.

The process

First I designed a short computer program in the Python language. The program processed every pixel in an image. It treated each pixel as a point in the RGB cube, and used a matrix to transform it. The program repeated this process for each of the 48 symmetries of the cube, producing 48 distinct images from one original image.

Next I arranged these images in a sequence I thought would result in smooth transitions.

Finally I used commercial software^† to render this sequence as a digital video, connecting the images with long slow dissolves.