One of the most basic cognitive processes is to compare and contrast. This course aims to foster an
appreciation for the subtlety and beauty of “true” mathematics by implicitly comparing and contrasting
it with art and design. Many people would consider art and mathematics to be diametrically opposed. As
a mathematician – to the extent that I understand “true” art – I see many more similarities between the
two and just a few ways in which they are opposites. I suspect that many artists – to the extent that they
understand “true” mathematics – would feel similarly.
The most basic notion in mathematics is that of the set. The fundamental achievements in mathematics
are to make definitions and to produce characterizations. The simplest of all characterizations is to
count the number of objects of a particular type. Possibly the most important notion from mathematics
for a lay person to understand is that of parameters – numbers that by themselves or as part of a group
characterize a set. These concepts and activities will be infused throughout the course.
First (and only) definition – “pleasing”:
Throughout the semester we will try to characterize when something is “pleasing”. Essentially,
this type of definition is the diametric opposite of the type of definition used in mathematics.
Mathematical definitions are completely rigid. They provide the structure necessary to create logically
rigorous arguments. Broad characterizations and intuitive notions often flow from these precisely
constrained starting points. By contrast, “pleasing” is an entirely intuitive notion that is idiosyncratic to
the individual and which will defy encapsulization. The course does not aim to mathematically
characterize the notion of “pleasing”. (However, if some students are inspired to work in that direction,
within certain confines I believe there is the potential for engaging discussion and greater mathematical
understanding.)
Explicit contrast – creative vs. formulaic:
(I would argue there exist shades of gray between these extremes.)
Implicit contrast – original vs. random:
(To me, these represent direct opposites, as will be illustrated in Rothko vs. Julia contrast
described below. However, rather than “shades of gray”, I feel there is some “meta” originality possible
that combines the two. This would consist of designing a parameter set which – when used with random
selection – produces objects that exhibit or tend to exhibit certain properties such as being “pleasing”.)
Potential examples of “meta” originality:
Randomized Haiku
Design Decks
Line Design “throws”
Julia Sets lying on certain paths in ℂ
Mondrian-esque Randomizations
Rothko-ette Progressions
Ironically, the way by which many people would consider math to overlap with art is to me an example
of opposites. (“Beautiful math? Oh, you must mean fractals.”)
Rothko vs. Julia –
Dull vs. Strikingly Beautiful
Minimal vs. Infinitely intricate
Deliberate vs. Random
Human’s Cognitive process vs. Nature’s inherent beauty
Mathematical skills developed:
In addition to exposure to the more subtle notions described above, students would explicitly
learn the following mathematical skills – (a list of 5 - 10 skills, already hinted at in other document).
Appreciation for and facility with software environments:
A software “program” performs a specific task. A software “environment” allows the user to
create their own applications and thus gives the user the ability to address a variety tasks.
As the technology evolves, tasks typically are made simpler and simpler for the user. Obviously
this constitutes a type of improvement but it tends to isolate the user from the creative power of the
computer. In the early stages, the only “environments” available were programming languages. Now,
many software programs – such as Excel and Geometer’s Sketchpad – have enough flexibility that they
can properly be considered environments.
The course will entail the exploration of three different areas. Each area will have a different
software environment associated with it – the first two are the programs mentioned above, the third is
a high level programming language. Students will gain a minimal proficiency in each of the three then
choose one in which to specialize.