Geometry

More Triangle Appreciation

It’s been a great week for special triangles.

Six days ago was a rare Equilateral Triangle day, four days ago we appreciated the 10-12-10 triangle, and today, 10/16/10, gives us another triangle to admire.

It’s actually very closely related to the 10-12-10 triangle, which we saw is just two right triangles pasted together.

Just cut along the dotted altitude:

Now rotate the two pieces:

Stick them back together along their common side of length 6:

Now flip, and voila! Another triangle made by gluing two congruent right triangles together!

I’m running out of good triangles, so we may not appreciate another until December 10th.

Geometry

Today in Triangle Appreciation

As it’s 10/12/10, I thought it would be the appropriate day to appreciate the 10-10-12 triangle!

Of course, it’s nice that the 10-10-12 triangle is isosceles. But what’s really cool is what happens when you drop the altitude from the top vertex!

It’s well known that in an isosceles triangle, the median and the altitude from the vertex are the same–this means that not only does this segment make a right angle (it’s the altitude), but it also divides the opposite side at its midpoint (it’s the median). So that segment creates two right triangles with hypotenuse 10 and side 6. Of course, the other side must be eight, since

$6^2 + 8^2 = 10^2$

Thank you, Pythagorean Theorem. So the 10-10-12 triangle is just two right triangles pasted together.

It’s not quite equilateral, but the 10-10-12 triangle is still pretty cool.

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, 16 yearsOctober 4, 2010 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, 16 yearsSeptember 30, 2010 ago

More Triangle Appreciation

Today in Triangle Appreciation

Ode to Equilateralism

Math at the Boundary

The Art of the Ellipse

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