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The story

Where the machine comes from

In 1973 Manfred Mohr made the twelve edges of a cube into an alphabet. Here is what he built, how the system works, and where the same rules can go now.

Mohr documented in his own captions or statements Ours our extension, not Mohr's work Unconfirmed not yet established from a source

This is a reconstruction and an extension of a published system. It is not an original Mohr work and is not affiliated with or endorsed by Manfred Mohr or his gallery.

01 · 1961–1969

Before the machine

Mohr came to the computer through ideas, not equipment. In 1961 he read the philosopher Max Bense on "rational aesthetics", which, he says, "changed my artistic world view completely." In 1967 in Paris he met the composer Pierre Barbaud, a pioneer of computer music, who showed him how a work could be structured as an algorithm.

In October 1969 he wrote his first program: "a random number generator (of course, with an error)."

Source: Mohr's own timeline

1961Reads Max Bense
1967Meets Pierre Barbaud
Oct 1969First program
02 · 1970–1973

An alphabet of lines

Before the cube, Mohr reduced shapes to their edges and treated those edges as letters. "I used the edges of squares, triangles, and cubes as an alphabet in a number of different ways."

Diagrams are ours, drawn to show each alphabet's elements. They are not Mohr's drawings. Works: P-105 Squares (1972), P-128 Sphereless (1972), P-147 Triangles (1973). Source: Edges as Alphabet, pre-cube drawings.

03 · 1970 onward

The machine, at night

Early in 1970 Mohr plotted a drawing on a Zuse plotter driven by an IBM 7040 at the University of Darmstadt. Then, on 17 June 1970, the Météorologie Nationale in Paris gave him permission to use its computer and plotter on nights and weekends: "a dream come true."

"At that time I had to write everything myself, there was no software."Mohr, interview with Alice Hattrick, The White Review, 2012

Film was slower still. Of his 16mm film Cubic Limit: "I had to use microfilm. The film is three minutes and it took six months to make."

ComputerCDC 6400
PlotterBenson flatbed, ink pen
Plot area1 × 1.30 m
Step0.1 mm
AccessNights & weekends
LanguageUnconfirmed

Hardware and dates from Mohr's timeline. The programming language is often reported but we have not yet confirmed it in a Mohr source.

04 · 1973

The cube

"The 12 edges of the cube became my alphabet."Mohr, timeline
"I saw a fantastic alphabet, three dimensions projected into two dimensions."Mohr, The White Review, 2012

Mohr called Cubic Limit "the first time I used a fixed structure as a system on which to apply rules for composition." A sign is the cube, rotated and projected, with only some of its edges drawn.

How Mohr parameterised rotation, and whether his projection was parallel or perspective, are Unconfirmed. Our reconstruction uses parallel projection, and Euler angles X→Y→Z as our representation of a rotation.

The alphabet: 12 edges. Our numbering, for the controls below.
05 · 1973–1977

The programs

Cubic Limit is a family of programs, each with a P-number, drawn as plotter works in ink on paper. Mohr: "In some works, statistics and rotation are used in the algorithm to generate signs. In others, combinatorial, logical and additive operators generate the global and local structures of the images."

The original drawings are not reproduced here. Follow each link to see the work in the artist's own archive. Captions are Mohr's, from his 1975 and 1977 Galerie Weiller catalogues as published on emohr.com.

ProgramYearMohr's descriptionTry
06 · The system

Twelve switches

Each edge is either drawn or not. That makes every sign a 12-bit word, and the rules act on those words: take the complement, enumerate every sign with n edges, sample by chance, combine two signs with OR and AND, or replace one cube's edges with another's, one at a time.

Mohr's alphabet is the 12 edges. Everything the programs draw is built from them.

Terms used on this site

AlphabetThe 12 edges of the cubeMohr
SignOne configuration of drawn edges at a rotationMohr's word
Sign spaceAll 212 = 4,096 subsets of the edgesOur arithmetic
Rotation classes218, if cube rotations count as the sameOur arithmetic
Symmetry classes144, under all 48 cube symmetriesOur arithmetic
07 · The space

A twelve-dimensional room

Give each edge its own axis and the 4,096 signs become the corners of a 12-dimensional hypercube. Mohr drew this space one sheet at a time and never showed it whole. P-161 walks one layer of it: all signs with n edges, for n from 0 to 12. A Gray code is a path that visits every corner while changing one edge per step. Ours plays the whole sign space as one continuous film.

Mohr  1973–746 monthsto make Cubic Limit, a 3-minute 16mm film
Ours  2026≈ 68 secondsto play all 4,096 configurations at 60 frames a second, computed live

The second card is our extension. It is not evidence that Mohr wanted to make this film.

08 · The extension

What survives the technology?

Some of Mohr's constraints came from the machine: night-only access, batch plotting, months of work for minutes of film. Lifting those continues his system. Others were his choices: one fixed structure, rigid rules, edges that are either on or off. Changing those extends the system into territory that is ours, and we label it every time.

"It's a rigid system. To start it you need a random number so I use the day's date."Mohr, The White Review, 2012
OursContinuous edges: an edge can be half-present
OursPerspective, color, depth and line weight
OursGrids up to 64 × 64; XOR and difference logic; smooth morphs
OursHigher cubes. Mohr turned to the hypercube himself from 1977; this implementation is ours.