Monday, February 24, 2014

Thursday, January 9, 2014

art for maths sake

"Let none be ashamed to learn, for a good work requireth good counsel."
- Albrecht Dürer, 1520

Mathematics and Art, sometimes thought to be opposite poles of human experience and activity, can sometimes sit down together in conversation.

Sometimes we might think of the conversation as more of a lecture - with Mathematics providing some technical advice. Perspective drawing is the prime example - where mathematics allows artists to create realistic images, or else to subvert the expectations of realism (as in anamorphic art). Art does reply - for these mathematical tools, it repays the favor by providing Mathematics with inspiring teaching and learning opportunities (in the case perspective drawing, see this essay by mathematician Annalisa Crannell).

Art may ask questions of Mathematics - after all, math is not merely a tool; it's often the inspiration for Art's work. Geometry, often in the form of polygons, polyhedra, and tessellations, makes many appearances in art from the old (see for example the work of Albrecht Druer, or this essay on Raphael's famous painting The School of Athens), to the new (check out the latest Bridges mathematical art galleries). Art that springs from Math can, in turn, say something back to Math - providing motivation in math education (see most posts from mathmunch), and inspiration for working mathematicians. A big part of what gets me excited about the recreational math that I do are the pretty pictures (I've been putting pictures from this blog that I like over on this tumblr for a while).

The book Beautiful Geometry gives us a rare opportunity to listen in on an extended and fascinating dialog between Math and Art: here they are talking like old friends, sharing jokes, and discussing other subjects like history and literature through the pairing of brief and interesting essays by mathematician Eli Maor and beautiful illustrations by artist Eugen Jost.

You will likely want to join in the conversation - the essays and artwork in Beautiful Geometry are inspiring and motivating - prompting me, at least, to try to play around with their ideas. For example, the images below are from a Geometer's Sketchpad iteration based on Jost's artwork entitled 3/3 = 4/4, which accompanies Maor's essay on geometric series (Chapter 30).

These images represent some early partial sums of the series below - can you see how?

Jost's piece Pentagons and Pentagrams (Chapter 22), similarly had me reaching for GSP:

Although some of Jost's pieces allow themselves to be mimicked by us amateurs (armed with appropriate software), many are true art works, conveying an aesthetic that keeps them from being mere diagrams (while still saying something substantive about mathematics).

Several of the topics that Maor writes about are ones I've looked at before (Lissajous figures, means, hypocycloids, figurate numbers), but even in these somewhat familiar areas the essays and illustrations are nudging me to look back and explore some more. (I even learned something new about quadrilaterals: connecting the midpoints of a  quadrilateral always yields a parallelogram - Chapter 3).

Jost's art and Maor's articles are going to inspire many of us to continue experiencing mathematics through beauty, and to look at art with a greater understanding of the math that helps to make it beautiful.

Wednesday, December 18, 2013

brain curve

This closed curve looks a bit like a lateral slice of a cerebral cortex, and has a very simple parametric formula:

If you let n = 0, the curve is a circle, for n = 1, you get an epicycloid, and for n around 4 you start to get something that looks like the brain slice. I think that there is a small copy of the whole curve on the left, giving the curve some self-similarity.

I first saw this curve, and some other neat ones that have very similar formulas, when looking at the scrambler amusement park ride (see here, and here).

Update: A very nice Mathematica animation that traces out this curve can be found at Curiosa Mathematica.

Tuesday, November 26, 2013

euler spirals in Fathom and TinkerPlots

A nice post on mathen inspired me to construct some Euler spirals using Fathom and Tinkerplots. I like using these tools for this sort of playing around - they are intended for middle and high-school data management activities, but are, effectively, simple quasi-programming environments. The results are not as pretty as those from mathen, but are nice enough and very easy to generate.

To recreate images like these in either Fathom or Tinkerplots you need two sliders - I called them magnitude and delta.

You then create a collection (or card set in TP) with four attributes (n, x, y, and theta). Each attribute will have its data generated by a formula, as shown below.

Adding cases to the collection generates the data (the pictures shown have about 2000 cases). You can then graph or plot the results using the x and y attributes as your axes. Varying magnitude adjusts the gap between the points, varying delta adjusts how quickly the curvature changes.

Some values for delta suggested that you can obtain curves within curves and fractal-like images.

Update: If you want to see Euler spirals in Geogebra, checkout this post by mathhombre.

Tuesday, November 19, 2013

snowflake construction in GSP

Here are instructions for making a snowflake iteration similar to the one shown in the previous post using Geometer's Sketchpad. I generally don't put useful things like instructions on this blog, but I thought I would make an exception: these fun and easy constructions are worth doing because they are pretty, and they illustrate some important concepts associated with recursion. GSP is a great tool for playing with these sorts of things - I haven't explored Geogebra at all, so I can't comment on what similar sorts of things can be done with that tool.

1. Draw a line segment AB (using the segment tool), construct its midpoint C (with the line selected, choose Construct > Midpoint), and then construct the midpoint of AC (we'll call that D).

The important thing to note is that AB is the only thing that you will draw. Everything else will be constructed. What we are doing here is a really good example of what GSP aficionados call geometric programming: AB is your input, and everything else we construct as part of a program written in the language of geometry. A mistake that some people make with GSP when they are first playing with it is to consider it a drawing rather than construction tool - you really should draw very little, and construct a lot.

2. Mark A as the center of rotation (select A, and then Transform > Mark Center) and rotate all lines and points around A by 60 degrees, five times. You'll end up with six spokes that look like this:

3. Connect the midpoints of the spokes to form a hexagon.

4. Select all sides of the hexagon and construct the midpoints of the sides. Then connect those midpoints to the points on the interior of the hexagon to form a star, like the one shown below.

5. At this point it would be wise to hide some of the things that we don't want to include in our construction (select lines and dots and press CNTRL+H). You should leave your star, the original points A and B, and the mid and end-points of each spoke.

6. Now we are ready to iterate. To do this, select the points A and B, and then open the Iterate dialog (Transform > Iterate...). The first iteration is to map A to C and B to itself. Keep the dialog open (we need to add more maps to this iteration).

7. Repeat the same sort of mapping on each spoke: A gets mapped to the midpoint of the spoke, and B gets mapped to the endpoint of the spoke. Do this by using the Struct... > Add New Map on the iteration dialog.

8. After mapping onto all the spokes, hit the Iterate button.

9. You can now hide any points or parts of the iteration that you'd like, and increase or decrease the number of iterations (select the whole shape and use the plus (+) and minus (-) keys).

Two thing to note: when we mapped A and B, we mapped them onto points that were derived from A and B (midpoints and endpoints of the line AB, and rotations of those points), and we mapped A and B such that the distance between their images was smaller than the original distance between A and B. Mapping A and B so that they moved closer to each other means that the resulting shape will be bounded - otherwise the shape will get larger with each iteration.