Coronavirus (Covid-19): Latest updates and information
Skip to main content Skip to navigation

Hamish Todd

Visual Cortex: Looking into the Klein bottle (through the lens of empirical modelling)

This looks like a really good theme for a WEB-EM submission. I like the combination of ideas from many different sources (biology, mathematics, computing) and feel that you have selected a topic that is admirably suited to an EM treatment. The most significant questions concern the practical feasibility of making the visualisations you propose with our current tools. Hopefully you will be able to find ways to realise your proposals that are workable even if not ideal. Certainly, you will not be penalised for models that work in principle but are slow to update. (There are plenty of precedents for excellent models that initially ran very slowly but now respond in a perfectly acceptable manner.) Even if you have to focus on feasibility, and can implement very limited prototypes that point to ways in which implementation would be possible with enhanced tools, this need not detract from the value of your project as a contribution to understanding and illustrating EM principles. And though you stress that you will put the primary emphasis on the modelling study, there are clearly rich resources (such as Bret Victor and Swindale) on which you can draw in your written component that would represent a significant original contribution to writing on EM.

The EM model that comes to mind when reading your proposal is graphicspresHarfield2007 - this is a collection of learner-guided animations that is supported using the EMPE and can be used as an adjunct to reading about 3D to 2D transformations in a standard graphics textbook. In making that the graphics-related animations for that model, I found myself inadvertently exploring more by way of visualisation of mathematical and graphics concepts than was explicit in the textbook. You may be able to do the same in your context - thinking about how your model might help to acquaint the learner with the topology of the Klein bottle itself, or potentially highlight pathologies in the visual processing.

Your proposal is quite original, and there may be no EM references that are directly relevant. You may find it useful to review EM papers #103: Visualisation using Empirical Modelling principles and tools and #022: Programming Principles for Visualisation in Mathematical Research just in case they are helpful. As far as practical techniques that may be relevant, it may be good to bear in mind a very little used "locus" feature of the line-drawing definitive notation Donald, which makes it possible to display the trajectory of a shape defined by a single geometric element:

%donald
circle c
point x
x = {200,200}
c = circle(x, 5)

%eden
A_c = "locus=true,fill=solid,color=blue";

This will display the circle c in all the positions it visits when the point x is assigned different values. This offers an alternative to declaring observables for each individual geometric element in a line drawing display and - some experiment suggests - may also allow control over the colour of each instance displayed.

I think it is important to preserve the capitalisation in Empirical Modelling, though this is unfortunate in some ways. I also noted one or two typos in passing, as below:

empirical modelling →  Empirical Modelling

swindale →  Swindale

the Klein bottle is has a manifestation →  the Klein bottle has a manifestation

depency →  dependency