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Thursday, August 12, 2010

Sculpture



I saw this sculpture at the University of Michigan, just outside the Law quad. Its been around for a while, and it bears a striking resemblance to components of my Meme-Gene/Body-Soul sculpture design I did a year or two ago. (Edit: the sculpture was done by Chuck Ginnever in 1972).

The angles look really close to being the same; I wonder if the similar mathematical considerations motivated the designer of this sculpture. As I recall, my method was this:
  • Take an equilateral triangle with side length S as the base
  • Copy the triangle and rotate it by 22.5 degrees (which completes a 360 degree turn in 16 turns)
  • Move the copied triangle up on the x axis by .225*S back on the z axis by the same. (I chose .225 just as an amusing reference to the 22.5 degree turn; I think both are the result of choosing pieces per turn, somewhat arbitrarily)
  • Connect the sides of the copied triangles and make parallelograms out of them, and repeat till you complete a circle (four circles, in my case)
The parallelograms that result make shapes that fit the description of these pictures very closely. I wish I had had a measuring tape on me; it would be cool to see if it obeys the proportions.

More generally, they probably fit into the formula for a helix somehow: x=cos(t), y=sin(t), z=t. Which, of course, is the message encoded in the sculpture I designed.

But given the two arbitrary parameters I made (using 16 pieces per segment, and making the segment depth equal to 1/100th of the degrees per turn times the side length), its remarkable that someone else evidently chose to make the same arbitrary parameter choices.

Of course, they're not completely arbitrary, just guided by what seem like arbitrary aesthetic motivations. Sixteen is the fourth power of two, everyone knows powers of two are cool, and the other two nearest options (8 and 32) would have made it too elongated or too squished, respectively. As for the second choice: turning 22.5 degrees into .225 and using it as the ratio of the sides to the depth just seemed funny.

So we could express it's height H in terms of the side length S, Segments per turn G and number of turns T:

H=GT(S*((360/G)/100)


So, if
G=16, S=1, and T=4, it would stand 14.4 feet tall.

If it were made on the scale of the sculpture at U of M, where I'll estimate the S at five feet, it would be exactly seventy two feet tall. Is that significant? Not sure, I was just surprised that it came out to an integer value.

Curious about this, and in a procrastinative mood, I'll test how common this is. Given integer values of S, it looks like H will be an integer 1/5th of the time:
Now, its true that S=5 is the first of these integer values... and it remains true independant of the values of G and T even if they're not integers themselves. Unless T is an irrational number. It doesn't seem to matter what G is.

Cool stuff.

But wait, there's more! I found the sculpture on U of M's website, and it has a beautiful description and a meaning entirely other than what I associated with it.

Though, I note that there happen to be five parallelograms! Ahhh! Am I glimpsing deep mathematical truths, or merely numerological coincidences?

1 comment:

Chuck Ginnever said...

I just saw this while browsing. I made " Daesalis" without any math skills. I don't think I even knew of the existence of a protractor.I measured and eyeballed all my worke for visual effect.

I bent the forms for stability and told the fabricators to open the angles 10% for each section for this purpose. I'm intrigued by your mathamatical take on the work, someting I could never achieve, and am curious about the sculpture you made that seems to be similar.