Wednesday, December 1, 2010
Day Dreaming on the Highway
Here are some of the drawings I alluded to in my Monday post. While driving through the gorgeous canyon of Indian Creek on my way to Greenville, I began daydreaming of a geometry lesson. With the four drawings above, we were able to launch discussions about optical illusions, the golden mean, Fibonacci numbers, and "magical" properties of polygons. The students discovered that with compass and straight-edge, they were able to create some beautiful images even if they have little or no free-hand drawing ability. In fact, after completing a few of these, they tend to develop an interest in drawing. Here are some our "discoveries" - that is, some things the Greeks discovered many years BCE. [If your kids are bored with math, try some of these with them.]
From the top: What I call "Squares form curves," starts with a square. This one was around 6" on a side. Then, working counter-clockwise (although you can go the other way if you're consistent), mark a point around 1/2" from each corner. Connect these four points and you create a slightly smaller square inside the original one, tilted slightly counter-clockwise. Then mark four points inside this second square in a similar manner and connect them. Repeat this process until it doesn't work any more. If you are extremely accurate, which I wasn't, the innermost square should be an actual square. Mine is a bit tweaked.
The second drawing starts with two straight lines of equal length, crossed at a random angle. Try different angles for different results. Starting at the point where the lines cross, mark off each line in 1/4"-inch intervals (or some other interval as you experiment). Study my drawing carefully to see what points are to be connected with straight lines. The drawing consists solely of sraight lines but yields a beautiful illusion of curves. Fun variations include: changing the angle between the initial two lines, varying the spacing of the intervals along each line, coloring the polygons created in a checkerboard pattern, or other patterns. SOme of you may remember when "string art" was all the rage. Same principle.
The next drawing shows the construction of a regular pentagon using only compass and straight-edge. Oh, also, paper and pencil. It's a tricky construction, so it really helps if you have a nice $10 compass and keep your leads sharp. Too many geometry classes provide crappy $1 compasses that will not hold their position for long so complex drawings usually fail. Look for instructions in any good geometry book or on line. When you connect the vertices to form a five-pointed star, it turns out that many pairs of lines within the figure are in the golden ratio, approximately 1.618:1. If you keep your pencil sharp, it is fun to connect the vertices of the inner pentagon to create another star, and continue the process as far as you can.
The last drawing is of a 15-agon. Anyone know a prefix for 15? Connect every vertex with every other one - or, draw all possible diagonals. You will find there are 90 diagonals. My students construct many regular polygons, from equilateral triangles to duodecagons and beyond, draw all possible diagonals, then look for patterns. Now, they can predict, or compute, how many diagonals can be drawn in a regular polygon of any number of sides. For instance, in a 100-sided polygon, it is possible to draw 4,750 diagonals! But I skipped an import part: How does one draw a regular, 15-sided polygon in the first place? I leave you to discover that like my students did.
During yesterday's class, a culmination of blending geometry with art, music, and physical skill, we watched a number of videos of the most amazing juggler, Michael Moschen. Check him out. You'll be enthralled.
It may seem that I've forgotten this is a natural history blog. Have no fear. Coming soon: Geometry in plant and animal anatomy.