Matrizenmultiplikation Serie 29 (Matrix Multiplication)

Basic Information

Title: Matrizenmultiplikation Serie 29 (Matrix Multiplication)
Artist(s): Frieder Nake
Date Created: 1967
Framed Dimensions: 27.5 x 21.5 in.
Unframed Dimensions: 18.75 x 18.75 in.
Medium: ZUSE Graphomat Z64 plotter drawing on paper
Inventory ID: Nake-1967-02

Description

artist’s name, computer identifier (TR4), plotter machine identifier (Z64), title, and date printed lower left

hardware: Telefunken-Rechner 4 Computer
output machine: Zuse Z64 Graphomat Plotter

Description of the artwork is courtesy of the artist:
“Nine states of a sequence of “powers” of an initially randomly determined so-called stochastic diagonal matrix are drawn and shown in Matrix Multiplication Series 29. A matrix is a rectangular array of numbers. The array can also be square, e.g., have 20 rows and 20 columns. You can multiply matrices together, resulting again a matrix. If you multiply two square matrices, you get a square matrix again. This is the mathematical basis of the artwork. A square matrix chosen at random by the programme, (let’s call it A), is multiplied by itself. We get A2. If we multiply this result by A again, we get A3. We continue this simple principle: we thus form the powers of A, one after the other. The matrices of the programme are also of a special kind: they are so-called “stochastic” matrices. This means that all the numbers lie between 0 and 1, and that the sum of the numbers in each of the rows is 1.

In order to create the picture, some of the matrix powers are selected. So that the matrices selected for the picture are no longer number schemes but become visible picture components, the numbers matrix must be translated into colours. In concrete terms, colours are assigned to intervals of numbers. This is done as follows. If, for example, six colours are to be used, the number interval from 0 to 1 is divided into six intervals. This can be done in many ways: from 0 to 0.2, from 0.2 to 0.3, from there to 0.45, then to 0.6, to 0.8, finally from 0.8 to 1.0. The number 0.512, for example, is then in the fourth interval (from 0.45 to 0.6). One colour is assigned to each number interval. Our number 0.512 lies in the fourth partial interval, so the fourth colour is assigned to it. At the point in the square number scheme where the 0.512 is located, a small square is then filled with the fourth colour.

According to the algorithmic scheme indicated here (which becomes the “pro- gram”), the selected powers (we can say “the states”) of the sequence of matrices are translated into small images that appear materially arranged in the graphic. A purely mathematical process becomes the source and cause of an image process. We could say that the mathematically determined sequence of “stochastic” matrices is “visualised”. The visible appearance of the visualisation is strongly influenced by the determination of the intervals in the division of the entire interval from 0 to 1 into its sub-intervals. The artist’s contribution to this drawing shrinks to this arbitrary division and the selection of colours. Of course, these two specifications could also be programmed. The artist would then hand over even more of their decisions to the machine, the computer. Even more of their own work would have to be incorporated into the programmed specifications. We can mark the pictures of the programme “Matrix Multiplication” like this:

Strictly determined but only weakly transparent translations of a mathematical process into an aesthetic one.”
— Frieder Nake