Since developing the concept of ‘Drawing in the Fourth Spatial Dimension’ (Anderson and Corti, 2015) in collaboration with Professor Alessio Corti (Imperial College, Mathematics Department), we wanted to see if our approach could be shared through the form of a public workshop. We developed a workshop based on the following questions:

- What is the fourth dimensional space?
- How can the fourth dimensional space exist when reality is three-dimensional?
- Can 4D space be represented in a three-dimensional reality…And in a two-dimensional reality…Can it be drawn?

With these questions as a starting point Corti and I guided participants through drawing in the fourth dimensional space by analogy and intuition. We integrated the key concept of ‘dimensional promotion’ that had been useful in our research through images from our own collaborative research work.

The goal for the workshops was:

1. visualization and drawing of objects in 4 dimensions. The point is not to make art objects but to make drawings that create images in the mind.
2. consistent effort to encourage participants to share thoughts and questions.

Through this approach we invited the participants to take part in our collaboration and to potentially change its direction.

The Workshop

The morning began with my introduction to the concept of the fourth spatial dimension through drawing the hypercube. Corti joined in drawing at the point when we began to fold the drawn 3D plan up into a 4D cube. All participants managed to successfully draw the folded-up ‘hypercube’ before lunch.

After lunch, prompted by the question ‘Is it possible to draw a prism in 4D?’, Corti introduced drawing prisms in 4D using prisms in 3D as facets. The advantage of working with 3D prisms is the infinite variety of 3D shapes (polyhedrals) that can be drawn. Time was given for participants to attempt their own drawings before Corti draws in front of the group.

The workshop ended with a discussion about why Corti draws in 4D in his own work with 4D reflexive lattice polytopes from the Kreuzer-Skarke (Kreuzer 1992) classification and how this collaboration led to developing a new method to draw a tree in 4D, which led to a shift in my own art away from the naturalistic, as discussed in chapter seven. Corti also spoke of how most mathematicians only use computer-models to visualise equations. He has used drawing since he was a teenager and teaches these techniques to his first-year students. He emphasised that this is an area he would certainly like to develop as he thinks that the act of drawing encourages close observation in a way that is difficult with computer-modelling, saying ‘drawing provides a level of interpretation that is distinctly human and encourages understanding’ (Corti, 2015, personal communication).

The Space

During the workshop, a selection of artworks and objects were installed in the space to support the concepts explored in the workshop. I installed a large drawing board so that Corti and I could draw ‘live’ in front of the group throughout the workshop.

Reflections

Encouraging participants to share their questions enabled a personal and sincere engagement with the task. One participant asked a question about how to draw a prism in the fourth spatial dimension and this changed the direction of the workshop because drawing a prism requires different visualisation techniques to those of the cube. In this workshop it was very exciting to see participants attempt their own drawings of 4D prisms before being shown, as this evidenced drawings epistemological role in their understanding. Looking at participants’ drawings, we could see that all understood the main point: to build faith in the concept of a 4th dimension and to achieve some way of it being accessible through drawing.

Some participants described finding a new way into mathematics through visualization and drawing while others gained insight into the role of the imagination in science. The following quotes reflect themes from participant feedback:

I thought the maths would be challenging but actually it was surprisingly clear(Engineer)

I liked the fact that we were visualizing algebra- it makes maths suddenly tangible for me. I turn off when I see numbers. This workshop made maths interesting and relevant to how I learn best, drawing can be related to anything (Falmouth School of Art graduate)

The workshop helped me to understand the concept of the fourth spatial dimension by thinking through drawing, knowing in action (Design Professor)

I found drawing helped to understand what the forms were. It would be hard to follow if we weren’t drawing at the same time (Basket maker)

Outcomes

Corti and I have collaborated since 2010 but this was the first time we had collaborated through the medium of the public workshop, which allowed us to improvise in a new way. On reflection, Corti and I realised that to think of this workshop as a way to show ‘impact’ is not integral enough. At a minimum, this is not ‘traditional’ science communication, but an honest effort to find new ways to share research and understanding. As such, our approach has been recognized as innovative by the Science Museum ‘Mathematics Festival’ (2015) and the ‘Thinking Through Drawing’ symposium ‘We All Draw’ (2015) who have both invited us to develop this workshop as part of their programme. These invitations reinforce our belief that it is important to be open about outcomes, as the impact goes further than the workshop , and it is quite possible that further outcomes will become apparent later.

The workshop had the following impact on our collaboration:

(a) ) Corti has planned to begin serious work on the 473M+ reflexive lattice 4-topes. The workshop made him more urgently aware that his team needs to develop software for computer visualization of lattice 4-topes as a tool for mathematical research. The development cycle will start from developing conventions for foreshortening objects in 4D perspective and experiment with physical drawings with pencil on paper by Anderson.

(b) In our discussion when reflecting on the workshop, Corti discussed his original approach to the undergraduate teaching of algebraic topology by systematic drawing: many drawings on the board with coloured pens, encouraging students to develop their drawing skills by making their own drawings as a way to put the algebraic topology into ‘the part of the mind where words can't go’. This led to imagining what kind of drawing software might support this teaching methodology

Drawing is the link between Corti’s research and my own. Through this workshop, we both experimented in doing something mutually new through our collaboration. For Corti this was the first time to interact with the public in a workshop format. Both the workshop itself and Corti’s teaching of algebraic topology are instances of the role of drawing in creative teaching.

Our approach supports the argument for drawing as a way of knowing - especially drawing effected in collaboration with scientific practices and instrumentation - representationally, analytically, and in terms of interpretation, which can achieve a new understanding of for both artists and scientists.

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