Friday, January 17, 2014


On Tumblr today, geometric aesthetic had this innoculous little image...

Wow. These are so elegant.  But there were no notes! What are these from? Google image search found this composite image all over, but also tipped me off that this was Josef Albers, a (literally) Bauhaus artist who moved to the US in 1933 (when the Nazis closed the Bauhaus) and became one of the 20th century's most influential artists (geometric abstract) and educators. Check out his quotations, frothing over with wisdom.  As much as I love art, there are these huge, gaping holes in my art education... I was even mooning over some of his paintings in the last year at the Hirschorn Museum in Washington! (By the way, the Hirschorn had a great retrosprective exhibition on Albers, and still has a lot online.)

The piece in the bottom left, ‘Structural Constellation, Transformation of a Scheme No.12′, particularly got my GeoGebra-juices flowing. (I found a nice image of it in a nice ArtBlart blogpost covering the Hirschorn retrospective.)  There is so much geometry to notice. But then to start to think about how to dynamicize it was the real mathart fun.

My first idea was to just make two overlapping rectangles, and see which of these proportions are consequences of that. It was much harder to make than I thought at first! This version is pretty robust; robust to me is about durability, keeping the constructed properties regardless of how you move the free points and whether you can make all the possible varieties. (For example, if you make a chevron, is it always concave, always a kite, and can it make any chevron?)

Play with the blue points to change the image. (This is an example of GeoGebra's new HTML5 embedding. Easy from GeoGebraTube on any platform that lets you edit the HTML. [I.e. not, unfortunately.])

Here's the direct link to GeoGebraTube. I made the overlap by starting with one rectangle, then determining the angle of intersection for the second rectangle by specifying the intersection points. Hmm, now reflecting, I should have continued that! But I wanted the rectangle controls to be at the vertices, so it's probably unnecessarily complicated. I do like how it lets you play to find some of Albers proportions. In particular I got an appreciation for how the vertices of the colored parallelograms come in collinear sets of three, and the symmetries that makes.

My next effort was making a version that built outward from the innermost parallelogram. I thought that would still allow for some dynamic variation, but capture more of the symmetry that makes Albers picture so gorgeous.

I used the center of the parallelogram as a symmetry point for the corners of the rectangles. This one is at GeoGebraTube, too.

Because of adding in the rotational symmetry, only two extra points are needed to determine the rest of the figure. But the control over that inner parallelogram gives a large amount of variety still. (Especially if you turn it inside out.)

Lastly I wanted to figure out what exactly were the proportions that Albers used. As I looked, I realized that you could either build it from two squares divided into thirds, or build the whole thing outward from a single isosceles triangle. I also noticed his exquisite framing... just a flawless image in design and proportion. This one is also on GeoGebraTube, though it's not very dynamic.

Art is a such a good entry into some wonderful mathematics. Look at all the angles, shapes, similarity, proportions and symmetry in this Albers design. I hope you take the time to explore more of Albers work - you won't regret it!


  1. Interesting! Also, I didn't know you could embed an interactive geogrbra object in a blog. Might need to change my latest blog!

  2. It used to be slow and buggy when it was javascript. I've been impressed with how fast it loads and smoothly it works as HTML5.

  3. Wow! I never would have thought to play like this!