Donuts, Sprouts, Maps and Möbius strip drawings

Topology was a regular feature of morning drawings. All the staff approached topology as a key to a deeper understanding of form, but each had a different predilection for the subject and therefore as a topic it was tackled as always from a variety of approaches that were designed to on the one hand introduce a fundamental visual problem but on the other hand open out the poetry and humour within it.

These are some of the drawing sessions as I remember them, but of course there were many more that I either didn't take part in or have simply forgotten.

Topology was introduced to students as a design tool. One of the key design principles we always introduced was that of visual variation. (I'll come back to this again, as variation is key to visual thinking) However there were different ways to come to the exploration of variation and one was to consider relative changes within fixed points of information.

Harry Beck's London Underground map would be introduced as a wonderful example of topographical thinking. All vital information is there and unnecessary details are removed. Scale, distance and direction have been changed and varied according to use value, however the relationship between the points has been maintained. Everything is relative, the most important issue is that stops are in the same order and interconnections are clear.  

Students would be asked to now go out into the world and solve similar problems, perhaps to think about colour coding their way through a walk to and from the supermarket, finding similar forms or concepts that linked and then making a map for them. An emotional map of walking to college, a fast food map, a where you can have interesting conversations map, whatever seemed interesting. The point being that you could eliminate unnecessary detail and find connections and relationships of importance that helped the viewer find their way into and around the concept. 

Sprouts and variation.

Everyone would play the topographical game sprouts on teams of two. 

Start with some dots on the paper. The more dots you have the longer the game takes so we would just start with two or three. Players take turns either connecting two of the dots with lines or drawing a line that loops back and connects a dot with itself. The lines can be straight or curved but they can’t cross themselves or any other lines. Each dot can have at most three lines connecting it. When you draw a line put a new dot in the middle. The first player who can’t draw a line loses and the game is finished. The important issue is that the precise positions of the dots is unimportant; it is only the pattern of connections between them that counts. The final drawings were of course all similar but full of variations, depending on choices made by the players. 

This concept would then be applied to simple drawings of object outlines made by connecting lines and dots. Each student would make an individual drawing, then again working in teams of two, students would make game play decisions on one of the drawings as to where and how different points could be connected, then they would play the game again with the second drawing. Each drawing now had a series of further possible forms arriving, taking the original drawing back, each student would then proceed to either rubout and highlight new forms emerging, or take from the drawing's gameplay a new set of forms in order to arrive at a series of small drawings that were seen to belong to a family of form. 

Family of form alphabet

A topographical description of the alphabet was given to the students, as so; one hole two tails, two holes no tail, no holes, one hole no tail, no holes three tails, a bar with four tails, one hole one tail, and no holes four tails. Students are asked to then separate the individual letters into formal categories. It would usually end up like so: (AR) (B) (CGIJLMNSUVWZ) (DO) (EFTY) (HK) (PQ) (X) Now without compromising the category, students were to consider, length, width, consistency, weight, direction, counter, etc. and explore the possible variations. 

The Four Colour Theorem.

It is mathematically proven that no more than four colours are required to colour in a map so that no two adjacent areas have the same colour. Students were asked to pick up an old discarded drawing, quickly draw five vertical pencil lines across it and 5 horizontal lines. Then using a rubber convert the drawing into a series of discrete shapes that had clear boundaries. Next of course they had to choose four colours and set out to prove the theorem. 

Stretching and squeezing using grids
Although topology was of great interest in relation to flat surfaces and shape, other types of variation thinking and the use of geometry were also important such as those derived from the theories of  D'Arcy Thompson and laid out in his book, 'On Growth and Form'. He had pointed out that as rates of growth vary and proportions change the configuration of an overall form alters accordingly. Thompson pointed out deep correlations between biological forms and mathematical principles. In particular we often directed students towards his interest in the Fibonacci sequence and it's relationship to the Golden Rectangle.  It was here that grids were now used to stretch and reform one form into another, so that there was an underlying order to the formal change. A drawing might be squared up and another made from it using for instance the Fibonacci series as a way of determining a new grid with the same number of divisions. (Again using relative changes within fixed points of information) 

Other mathematic principles such as those underlying anamorphic projections were also used for this and these could be linked with a more three dimensional approach such as applying distortions to perspective forms. (The teaching of perspective was another issue and again hopefully I'll return to that and how it fitted into the mix) 

Topographical thinking was also of course applied to 3D. My favourite moment was Colin Cain introducing the torus form to a group of students. Colin was very interested in the torus and was trying to introduce the group to the fact that within a topological framework a donut and a coffee cup are seen as the same type of form. (Homeomorphic) He explained that the torus was a closed surface that could be seen as the product of two circles and in his own work he had recently been exploring variations on slices and sections of similar forms. He left them to get on with the exploration of slices through his donuts. He came back to find everyone drawing donuts, putting jam in some and icing on others, he was totally misunderstood. Very funny at the time, but a useful reminder that students were often totally lost as to what we were going on about. At some points this confusion could lead to wonder and excitement; at others to loss of faith and disillusionment. 

Another torus form we were all interested in was was the Möbius strip. All the students would make one and be asked to make a continuous drawing on it. Of course the drawing eventually covered both sides of the paper proving it had a continuous surface with a hole in the middle, just like the torus. I think it was as much as anything the fact that everybody should at one time in their life make and consider the principles of a Möbius strip that lay behind this, I can't remember where this led to next, except for the fact that we could all talk about space being twisted. 

Other related things that we used were often isometric forms and their tendency to have an ambiguity of reading. These included the Penrose Illusion, (the impossible 3D cube), Schroeder's reversible staircase illusion, (as used by Escher) and most Roman mosaics, where cubes were often formed of isometric shapes that popped forwards and backwards in space as you switched read. 

One over-riding principle was that any and all of these systems were an aid to giving a form life. If a form was constantly switching in the mind from one thing to another (E.g. the duck/rabbit ambiguity) it was therefore optically alive. If a form had been stretched it was in tension and had a visual memory or retention of its former self which again caused the brain to see the form as active, therefore alive.  The flat surface was something to be played with and manipulated constantly, but of course you were never to illustrate within it, or forget that the reality was in the optical interaction between the surface and the viewer.