Geometry

Ode to Equilateralism

Today, 10/10/10, is Equilateral Triangle Day! This day doesn’t come around that often, so I thought I’d expound a bit on the virtues of equilateralism.

There is so much to appreciate about the equilateral triangle–its uniform shape, its simple area formula, its presence in regular hexagons, its decomposition into 30-60-90 triangles–and most of these properties are a consequence of the equilateral triangle’s abundance of symmtery.

Consider this plain old non-equilateral triangle.

In any triangle, from any vertex there are three important segments you can draw–the angle bisector, the altitude, and the median. Each of the segments defines a kind of symmetry for that part of the triangle.

Notice how these three segments are all quite different in our plain old non-equilateral triangle. But, if we were to make the two sides from that vertex the same length (i.e., make the triangle isosceles), all those segments become equal!

The sides being the same length force all those lines of symmetry to become one. Now, the beauty of the equilateral triangle is that it’s like three isosceles triangles rolled into one!

It’s an isosceles triangle from every vertex! So, all those lines of symmetry are the same no matter how you look at it.

Enjoy the next 397 days, until the next Equilateral Triangle Day.

Appreciation Geometry

Math at the Boundary

While in Maine, I took some nice photos of the boundary between the beach and the sea:

It made me think of something I saw a long time ago (maybe on 60 Minutes?) about a scientist who thought deeply about coffee spills on his countertop. The power of the internet helped me locate Sidney Nagel, a physicist who studies the physics of drops, why things get “jammed”, and why a coffee spill leaves a dark ring after it dries.

Is there any way to predict the kind of edge this water will make as it crawls up the beach? Is there any order in this chaos? If this inspires you to great scientific accomplishment, please remember where you got your start.

By patrick honner, 14 years9 years ago

Appreciation Geometry Uncategorized

The Art of the Ellipse

This article, the first in a series about drawing, is about how important the ellipse is to the artist.

http://opinionator.blogs.nytimes.com/2010/09/23/the-frisbee-of-art/

The author gives a nice, if long, explanation about the significance of the ellipse, but it basically boils down to this: circles are everywhere. And often, when we are looking at circles, we’re looking at them atilt. We see projections of the circle, and projections of circles are ellipses.

Think of it this way: suppose you have a hula hoop and you hold it parallel to the ground. The shadow you see is circular, but if you tilt the hula hoop, the shadow will change–into an ellipse.

I don’t have a hula hoop, so I made do with a key ring:

As the circular key ring is rotated, it becomes less parallel to the ground; the shadow becomes less circular and more elliptical. And at the end, the ellipse vanishes–an ellipse eclipse!

By patrick honner, 14 years9 years ago

Challenge Geometry Student Work Teaching

An Impossible Construction

I enjoy offering impossible problems to students as extra credit, although I usually don’t tell them the problems are impossible. Such tasks usually engage them, confuse them, and make them suspicious of me. It’s a win-win-win.

While discussing some three-dimensional geometry, I offered extra credit to anyone who could build a model of a Klein bottle. The Klein bottle is a hard-to-imagine surface that has neither an inside nor an outside. It’s like a tube where one end meets the other and makes a seal, but somehow got turned inside out in the process. If you are familiar with the Mobius strip, the Klein bottle is basically a higher-dimensional Mobius strip.

One reason that the Klein bottle is hard to visualize is that it can’t be observed in three dimensions: it needs a fourth dimension in order to see it turn itself inside-out. This is analogous to the standard construction of the Mobius strip: we take a long strip of paper, give one end a half-twist, and tape the ends together. We think of the paper itself as being 2-dimensional, but we need that third dimension to twist through.

So, I was pretty impressed with the student who made this.

Not bad at all, for someone who is dimensionally challenged. Here’s a nice representation for comparison, although it’s still a cheat. The Klein bottle doesn’t really intersect itself.

A nice example of impossibly creative student work!

By patrick honner, 14 years9 years ago

Appreciation Geometry

Some Strange Circles

I was trying to construct a simple, introductory intersection problem for the first day of Calculus class, so I started with a well-behaved circle:

This is a circle centered at (3,2) with radius 5. I picked two points on the circle, (0,6) and (7,-1), found the equation of the line between them, and put together my system of equations:

So I had successfully reverse-engineered my circle-and-line intersection problem with two nice solutions: (0,6) and (7,-1).

Unfortunately, I made a typo on the handout. At the end of the left side of the circle equation I wrote ” + 12″ instead of ” – 12″.

So all my work was for naught. Or so I thought. Turns out, at least two amazing things happen:

First, the new circle still ends up having a nice radius, namely 1. What’s even more amazing is that the new circle also ends up having two nice intersections with the given line, (3,3) and (4,2)!

I wish my intentional work always turned out as well as this mistake!

By patrick honner, 14 years9 years ago

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