Posts Tagged ‘geometry’

Justifying an addictive game: it’s GEOMETRY!

November 13, 2008

small-pearlOne of my students recently sent me an email with a link to the game Shinju. (Thanks Chris Z!) He included the note, “I thought you would enjoy this game I came across. It deals with Maxi geometry!”

The game is played on an 8×8 board that is partially covered with shells, one of which has a golden pearl inside. You click on a shell, and it either reveals the hidden pearl (You Win!) or it gives a number indicating how many steps away the pearl is, where each step can be horizontal, vertical, or diagonal. It’s fun to play the game, and I found it surprisingly addicting. The analysis of the game, including how to win, is beyond the jump.

Geometers’ Vanity Plates

June 20, 2008

A car was parked in front of our house recently:

Clearly a special order license plate for someone who fondly recalls the Angle Side Angle Theorem in geometry. And since New York switched from an ABC 123 to an ABC 1234 format a few years ago, there are 10,000 such vanity plates available for this Theorem alone. How cool is that?

Is a square a rectangle?

June 6, 2008

I like this question. My first reaction — since I get this pretty much every semester that I teach a problem-solving or geometry class — is to ask what the definition of a rectangle is. Most people respond that it’s a quadrilateral with 4 right angles, maybe they add something about the opposite sides being parallel and/or equal, and then I ask if a square fits that definition. They answer yes, and the problem is solved.

But I think the question is really a little more subtle than that. In all the children’s books that we’ve acquired on shapes, none of them show a square on the rectangle page. Years of reinforcement that squares and rectangles are different shapes is hard to overcome with a single definition.

Furthermore, when I started teaching Geometry I learned that 2300 years ago Euclid didn’t define rectangles (which he called oblongs) in quite the same way as we do. Here’s a page from Oliver Byrne’s 1847 translation of Euclid’s Elements, which is one of my favorites because Byrne sure liked his color markers. He uses oblong the way we use rectangle.

Notice that Euclid said that an oblong did not have all four fides equal: a fquare was a completely different beast, not a special kind of rectangle. Euclid kept this distinction with all his geometric figures: a rhombus couldn’t have right angles (so a square wasn’t a special kind of rhombus either), a parallelogram (rhomboid) did not have right angles or equal sides, and an isosceles triangle had exactly two equal sides, not at least two. At Euclid’s Geometric Figures party when the figures divide into teams, the squares knew EXACTLY where to go, and it wasn’t with the rectangles: it was a partition, rather than a Venn diagram.

Another place where geometric problems can occur is with triangles. I think of the stereotypical triangle [in the US — is it true in other countries as well?] as being one with a horizontal base, and probably isosceles.

But, just like the definition of rectangle, that hasn’t always been the case. In in “Words and Pictures: New Light on Plimpton 322”, Eleanor Robson explains, “if we look at triangles drawn on ancient cuneiform tablets like Plimpton 322, we see that they all point right and are much longer than they are tall: very like a cuneiform wedge in fact.”

Neither triangle is better or worse than the other, but they are different, illustrating the cultural influence on mental images of shapes. I find that intriguing.

I believe that the page of Byrne’s translation is fair to include because its over 70 years old. And an edition only sold for $300 in the ’70s — can you believe it? Not that I had more than $5 at any one time in that decade, but still, if I had and I wasn’t buying dollhouse furniture, I’m sure I would have bought it.

The Geometry Van

May 7, 2008

I taught Geometry this spring, and we spent about half the semester working through Euclid (Book I and a smattering of some others). We proved SAS (Side Angle Side), ASA, AAS, but not ASS (Angle Side Side). Because there is no ASS in Geometry.

Here <A=<A’, AB=A’B’, and BC=B’C’ but the two triangles are not congruent. All we get are bad jokes.

Except that’s not quite true! If <A=<A’, AB=A’B’, BC=B’C’, and BC≥AB, then the two triangles are congruent! In class, I referred to this theorem as ASS.

(The Hypotenuse-Leg Theorem is a special case of this, since the hypotenuses, as the sides across from the corresponding right angles, are certainly longer than the corresponding legs of the triangle).

We didn’t refer to this theorem very often, but it is, well, memorable. And so Adele, one of our majors, was thrilled when she noticed ASS written on a van this past weekend! She snuck a photo of it:

According to Adele and everyone who was with her, this is exactly how I would write the shorthand. We have no idea why it’s on this van. I like to think that this is a Secret Geometry Van, coming out to help students everywhere by providing them with extra theorems.

Friday Software Review: GeoGebra

November 9, 2007

In the first of what I hope to be a (nearly) weekly column, I’ll review GeoGebra, a free geometry software package.

GeoGebra is similar to Geometer’s Sketchpad in that you can perform all of the standard ruler-and-compass constructions (e.g., bisecting a segment, bisecting an angle), but you can also enter algebraic equations. So if you want to work with a circle, you can enter the equation directly, or you can create a point and use it as the center to draw a circle with the “circle” tool (GeoGebra will then find the equation for you, if you want). Here’s the construction of a regular pentagon in progress (click for full-size):

Pentagon

The interface is fairly intuitive. I didn’t need to refer to the documentation at all to do my first construction (the pentagon). In my attempt to construct the heptadecagon, round-off error caused the last side to be off slightly, and I haven’t tried again lately (because really, who wants to follow the instructions to construct a 17-sided polygon twice?), so I don’t know if that has been fixed.

GeoGebra is Java-based, so it’s a little slow to load (as are all Java apps), but you can download the whole thing and install it locally. It’s a nice thing to have when you’re not at school and can’t get to Geometer’s Sketchpad, and it generates very nice pictures if you need to include some in a document (LaTeX, Word, a blog).

As a big fan of free software (which will become more apparent the more I review), I like GeoGebra. If you ever need a little help with those geometry constructions, you can’t find a cheaper solution.