Monday, August 16, 2010

Pythagorean Puzzle

I like wooden toys, puzzles, and mechanical devices. The other day, I was looking at a simple proof of the Pythagorean Theorem. The idea behind the proof was incredibly simple, but the simplicity was obscured with the mathematical notation and formalism. But I noticed it could be directly translated into a mechanical device. So, I thought I'd make a toy out of it. The idea is it might make the concepts a little more clear, without the clutter of notation.

The general idea behind the proof is this:

Suppose I have a rug and place it flat on the floor of an empty room. If the rug is too small for my tastes, can I cover more of the floor by moving the rug around (on the floor)? Not really. Regardless of where I put the rug, the same total area of floor will be covered, the same total area will be uncovered.

So, imagine four rugs that are all the same size, and shaped like right triangles. The black space is the uncovered floor:

Now, I can rearrange the rugs, in another pattern:

But either way, the same area of floor is uncovered, regardless of how I move the rugs. So, the black area a2 + b2 has the exact same area as black area c2. So,

a2 + b2 = c2

But also, the axa, bxb, cxc squares are formed directly with the sides of the triangles. In other words, a and b are the length of the triangle's legs, and c is the length of the hypotenuse.

So, this is a visual, mechanical demonstration of the Pythagorean Theorem, for an arbitrary right triangle. :)

Below are the steps I used in making the prototype:

First I started with a large sheet of 1/8" hardboard, and cut a 1" strip. Though, in hindsight, it would have been easier to just find a precut, plained 1" x 1/4" strip.

Now, there will be three basic layers: top middle and bottom.

The top layer can all be cut from the long 1" strip. I used 2"x1" triangles. So the inside of the frame will be 3" and the outside 5". The four small 1"x1" squares will be corner supports between the bottom and top layers. The three small 1/2" x 1" rectangles will be used for sliding tabs (handles).

The middle layer (left bottom) will need to be slightly thinner so that it can slide freely between the corner supports. I just took off a hair with a flat file. The bottom layer (right bottom) will need a slot cut that intersects points (0,0) and (2,1), if the origin is at the lower left. I just used a drill and coping saw. Any thickness for the bottom is fine.

Roughly assembled:

The yellow triangle will slide along the slot in the bottom layer. The back tab will connect to the yellow triangle with screws (passing through the slot). These will need to be attached at the same angle as the triangles:

Glue the green and yellow triangles to their middle layer supports. Use two small screws to prevent rotation. It's probably best to drill pilot holes for these small pieces.:

Also, I cut a thin strip of felt, and glued to the insides of the slot. This smooths the slide mechanism, and looks a little better. After dry, then cut flush with a razor blade.

The sliding yellow triangle can be attached first. This will determine the position of the frame and other pieces. (backside)

Then add corner supports (which will be cut to size at the end). The only thickness that really matters, is that the middle layer is slightly thinner than the corner supports.

The side tabs and red blue triangles will sit on top of the middle layer, and allow the pieces to slide vertically and horizontally. Glue both tabs and triangles to the middle layers.

Personally, I like the natural wood toys better. But, since this is a wooden toy for kids, I picked bright simple colors. :)

For a children's toy, I think the squares should have been purple, and the frame should have been black. Since I think it's easier to focus on purple than black (a void), and is often a favorite color. But I didn't have any purple felt on hand. Though, black doubles as a chalk board.

(I actually painted everything at the first step, but had to touch up scratches afterwards.)

After painting everything, use a flat bastard file so that edges are clean and sharp.

Then, apply drips of glue to the corner supports, and adjust the frame so that both triangle positions are flush. Cut excess off when dry. I rounded the tabs off with a Dremel tool, and painted the white labels on the black background. These labels are obscured by the resting position of the triangles. A hole can be drilled in the back for hanging somewhere.

And that's it. Now, even a five year old can demonstrate this fundamental theorem of mathematics. :)


KW said...

Wow, this is cool and interesting, and I actually understand something about math now. I like seeing all the inner workings of the gizmo.

This reminds me, I have and idea I need to talk to you about. Remind me to tell you about the 3 yardsticks.

sevkeifert said...

Thanks! Glad you liked it. :)