Wednesday, April 16, 2014

Enter the dragon

After all the fun I had making fractal snowflakes, I wanted to try out some different ideas with the “Transform Effect” in Illustrator.  Here I typed the word “Dragon,” and applied an effect that made three copies, rotated in multiples of 42º (I went for an arbitrary angle other than 60), and scaled up and shifted a little each time.
dragon fractal (1)
Applying the same effect again makes a kind of spiral, as you might imagine.

dragon fractal (2)dragon fractal (3)
But because the rotations are done with respect to the bottom-left corner (rather than the center of the spiral), a strange pattern begins to emerge: 
dragon fractal (4)
dragon fractal (5)
Here I tried it again, with a transformation that makes one copy rotated 80º about the lower left corner, and shifted right.
dragon fractal (6)
We might expect to see four “dragon”s when the transformation is applied again, but one of them falls perfectly on the first copy so we see only three.
dragon fractal (7)dragon fractal (8)
After the third iteration, the dragons have curled around enough to be farther left than the original, which moves the point of rotation.  Soon the fractal magic begins to occur.
dragon fractal (9)dragon fractal (10)
dragon fractal (11)dragon fractal (12)
dragon fractal (13)dragon fractal (14)
dragon fractal (15)dragon fractal (16)
dragon fractal (17)

Sunday, April 6, 2014

Frozen fractals all around

Watching the movie Frozen with my kids the other day, I was happy to hear the term “frozen fractals” in the acclaimed song, “Let It Go.”  It made me look closer at the snowflakes and ice shapes depicted in the movie – but I didn’t notice much in the way of fractal structure there. 

The archetypical snowflake images tend to exhibit “branches on branches,” indicating fractal self-similarity, though the six-fold symmetry appears to be the defining characteristic of iconic snowflake shapes. 

01f

I wondered about making my own fractal snowflakes using a simple iterative process…but I wanted something different than the Koch snowflake.
06-round1
Starting with a random shape, I repeated the following process (in Adobe Illustrator, which made it easy):

Create 5 more copies, rotated in multiples of 60º about the lower left corner of the bounding box.

Here are the first four iterations:
07-round2 08-round309-round4 10-round5
After that my computer was slowing down, but you get the idea.  I tried the same idea again, but this time the copies were rotated about one vertex so they overlapped and ended up making a cool tile pattern:

12-pointy1 13-pointy214-pointy3 15-pointy4

Wednesday, March 26, 2014

Happy birthday, Paul Erdős!

Here’s a caricature I made of “the man who loved only numbers” for his 101st birthday. 
Paul Erdos caricature

Wednesday, March 12, 2014

Pi Day Challenge

Here’s a fun site for your lateral thinking students (who also know some geometry).  In their words,

A team of logicians adapted or created these puzzles - some require research, some require mathematics, some require pure savvy.

Pi Day Challenge

pi day challenge

A lot of the problems don’t have anything to do with pi, and some of the presentation is a little janky, but it’s still fun.

Tuesday, March 4, 2014

Pi art ideas

I recently came across some cool ideas for blending up art and math for younger kids at whatdowedoallday.com – just in time for Pi Day next Friday!
math-art-pi-starting

Sunday, March 2, 2014

Permutation of the Bells

I came across this picture I took while visiting the bell tower of the Old Post Office in Washington, D.C. a couple of years ago.  It describes how the bells are occasionally rung in a “full peal,” consisting of all possible permutations of their tones.  With 6 bells, you get 6! = 6*5*4*3*2*1 = 720 permutations, which could be rung in under an hour, but with 7 bells, you get 7! = 7*720 = 5040 patterns to ring!

I thought maybe this would involve listing the set of permutations in increasing numerical order:

123456
123465
123546
123564
123645…

But In the example they showed, it seems that the pattern is to first swap neighbors in 3 groups:

12 34 56
21 43 65

And then to keep the first and last positions in place but swap neighbors in the 2 groups inside:

2 14 36 5
2 41 63 5

--

I can’t tell if that’s all there is to the method or not, and I wonder if there’s a proof that the method produces all permutations.  A quick test with 4 bells shows that you don’t get all 24 permutations before it repeats.

1234
2143
2413
4231
4321
3412
3142
1324
1234