Saturday, March 28, 2009

Art Project 1 - Pyramids

I decided to try something that caught my eye. There was a lamp shade at a restaurant that had an inner ring of 4 wooden panels, surrounded by an outer ring of 8 panels. The panels were all the same size, and the outer ring looked like an extension of the inner one. They were also examples of powers of two. So, I wondered what would happen if I did something similar, using construction paper, and adding a pyramid on top as a hood.

Initially, I just planned on stopping at 16 sides, and when I discovered how much work 32 sides took, I vowed to stop there. But, I figured that 64 sides couldn't be that much worse, so I went for it. My mistake. Took over 3 hours just to cut the 64 pieces of paper for it.

First, I needed to find a way to make sure that all of the pyramids would be 3 cm tall, regardless of the number of sides. That took about an hour, including sketching it out and verifying the math via an Excel spreadsheet. Initially, I got the math wrong, but I did eventually discover my mistake. The edge pieces are 3 cm square, connected to isosceles triangles of varying lengths, and the resulting assembled cones themselves are 3 cm tall.

The last pyramid is 64 strips, and the tips taper off to be so thin as to be really hard to work with (it's physically impossible to cut the paper thinly enough for 128 sides, and each strip would be 3 feet long - longer than the construction paper sheets themselves). The paper collapses under its own weight and requires a supporting skeleton underneath (unlike the smaller cones). It's over 1 meter across, took over 3 large sheets of construction paper, 30 feet of tape, and more than 5 hours to assemble.

Both the 32 and 64-side shapes are hypnotic when you spin them, and the bigger one is actually hard to look at when it's standing still since my eyes can't focus on any one spot on it. The 16-sider can be worn like a pillbox hat, and the 64-sider is big enough to act as a parasol. They'd just have to be made out of stronger materials in order to be practical.

I don't have space for storing these pyramids in the apartment as-is, so I compacted them at the end.

This art project is also an excellent example of calculus in action. By taking thinner and thinner strips (or in this case, keeping the strip width the same and making the cone significantly bigger), I'm getting closer to approximating a circle. This is the heart of integrating over a curve, using an infinite number of infinitely narrower slices to determine the area (or volume) of the curve. At 64 slices, if I multiply the area of the triangles by their number I get within 0.16% of the area of the respective projected circle. At 8192 sides, I'd be within 0.0000016%. However, even at only 16 sides, the circumference is already looking pretty circle-like. So, the point is that just by using little triangles, I can calculate the area of a circle without needing to know how to calculate the value of pi. And that's cool.

(The binary data compression process.)

(The final art project, after binary compression.)

(The stack of paper triangles required at the beginning for making the project. It's at least half an inch thick.)


Shiroibara said...

That was really cool looking. Too bad you didn't have anywhere to store it :(

TSOTE said...

Thanks. Fortunately, it is something that's easily repeatable, so I could always build another one if I want it. What I failed to take into account originally was how much any single color would dominate over the others. Next time, I'll make yellow the dominant color of the 64-panel cone.