Bridges 2009 Mathart show in Banff, Alberta

Here is a link showing all the great math art that will be part of this year’s Bridges Conference, “Mathematical connections in Art, Music and Science” The beautiful image I chose for this blog entry is Anita Chowdry’s “Illuminated Julia Dragon”.

개꿈 The American Mathematical Society Has Accepted “DOG DREAM” And “TEMPTATION”

I am grateful and honored that the American Mathematical Society has accepted “DOG DREAM” (above) and “TEMPTATION” (below) to the 2008 art exhibition, which is concurrent with the 2008 AMS conference which takes place in January of 2008.

Both poems are in the form of an orthogonal space poem.

Mandelbrot Rock

JoAnne Growney was kind enough to send me this youtube link to a tune by fellow geek Jonathan Coulton

Merry Christmas

Merry Christmas

Here is my Christmas fractal I constructed for you.

I wish you a Merry Christmas and a Happy New year

Fractal Vispo by Marko Niemi

Fractal Visual Poetry

I want to get back to the fractal poem that was created by Marko Niemi and the post of mine from March 25th Midwinter night's dream (click here)

To understand some of the ideas incorporated into Marko’s piece we need to understand a little about this idea of fractals. Fractals have many facets of interest however; I want to focus on two:
1.) Self-similarity
2.) Fractional dimension

You may ask yourself, “Have I experienced a fractal in nature?” The answer is yes! I believe the easiest way to recognize that you have experienced one is by asking the question: “Have I ever noticed something that looks the same viewed at different scales: In another words the look very much the same whether I am close to it or far away.

Here is a short list of fractals patterns found in nature

*the branching of tracheal tubes
*the leaves in trees
*the veins in a hand
*galaxies
*water swirling and twisting out of a tap
*a puffy cumulus cloud
*tiny oxygen molecule, or the DNA molecule
*the stock market
*sea shells

I believe out of this list the easiest one to comprehend is a cumulus cloud. These kinds of clouds look the same from 60 feet away or 6 miles away. From either distance clouds basically have the same shape.
The picture below looks very much like the runoff from my neighbor’s water sprinkler before he got his Arizona lawn seeded properly.






But the fact is that I shot this photo of the Grand Canyon from one of my airplane flights between Denver and San Diego. Here we can see another self-similar example in the fact that water erosion looks pretty much the same from different scales.

Mathematical Fractals
We have all probably seen some ‘self similar’ mathematically created fractal images on the Internet or at a bookstore. However, to get a good understanding of it I wish to focus on the image below. The figure below is called a Sierpinski triangle named after the 20th century polish mathematician.
I don’t think that I could explain self-similarity any better than the following link so I stole the text “shown in red” from that site and posted it below:

http://math.bu.edu/DYSYS/chaos-game/node5.html

Self-similarity
One of the basic properties of fractal images is the notion of self-similarity. This idea is easy to explain using the Sierpinski triangle. Note that S may be decomposed into 3 congruent figures, each of which is exactly 1/2 the size of S! See Figure 7. That is to say, if we magnify any of the 3 pieces of S shown in Figure 7 by a factor of 2, we obtain an exact replica of S. That is, S consists of 3 self-similar copies of itself, each with magnification factor 2.

Figure 7: Magnifying the Sierpinski triangle

We can look deeper into S and see further copies of S. For the Sierpinski triangle also consists of 9 self-similar copies of itself, each with magnification factor 4. Or we can chop S into 27 self-similar pieces, each with magnification factor 8. In general, we may divide S into 3^n self-similar pieces, each of which is congruent, and each of which may be maginified by a factor of 2^n to yield the entire figure. This type of self-similarity at all scales is a hallmark of the images known as fractals.

Fractional Dimension

The word fractal actually comes from the notion of an object having a fractional dimension. In other words we can have an object existing in a dimension between the first and second dimension or the second and third dimension etc. A point has dimension 0, a line has dimension 1, and a plane has dimension 2. So how is it we can have something that has a dimension of 1.6 or 2.34? Again I found a wonderful link that explains it good enough that even I (an artist) can understand it.

http://math.bu.edu/DYSYS/chaos-game/node6.html

Understanding Marko’s piece

Midwinter night's dream (click here)

Marko’s piece has 6 triangular shaped words stacked together to form a larger triangle each of the words in this structure have white letters except one that shines yellow-orange. When you first click on this piece six words appear and if you notice the yellow-orange letters also spell a word. When I first started clicking on the words I could not make out any pattern. However, once I started writing them down I noticed that the triangular word that I clicked upon was soon duplicated throughout the entire structure of the large triangle with the yellow-orange letters. So in effect the smaller triangle conceptually contained the larger structure within its smaller structure. Thus being self-similar! A fractal made of words!

If you don’t get this at first then just think of each letter in the smaller triangle as being the possibility of also being the yellow-orange letter in a larger triangle and when you click on the small triangle it pops into a large triangle made of 6 more words. The larger triangle is contained in the smaller one you just can not see it because the resolution of the screen gets in the way … there are countless levels you can go through with this poem (its not really countless there is a finite number) but the bottom being it is a very beautiful structure.

I asked Marko about his choices for the words in the piece and the following in blue is his response:

In the Midwinter piece, there's the "source text," consisting of all the six-letter words from "Midsummer Night's Dream" by Shakespeare, in their original order. When you click for instance the word "hermia," the program gathers all the words from the source text which include the first letter of the clicked word, in this case "h", the first letter of "hermia." Then the program chooses one of those words at random and, after that, the program chooses the next word after previously chosen word in the source, which includes the second letter of the clicked word, in this case "e", and so on, until six words are chosen. Those six words are then displayed on screen, and the letters "h", "e", "r", "m", "i", and "a" are higlighted. This process takes place again and again, after each clicking.

So my next thought what is the fractional dimension for Marko’s piece? Here is the link again to help you:

http://math.bu.edu/DYSYS/chaos-game/node6.html

Here is our equation:

So Marko has 6 self similar pieces with a magnification factor of three. Therefore midwinter nights dream has a fractal/fractional dimension of approximately 1.63

Now isn’t that cool!