Appreciation

Math at the Boundary

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

Shoreline

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, ago

The Art of the Ellipse

ellipse -- conicThis 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:

Ellipses

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, ago

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)!

Strange Circles

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

By patrick honner, ago

Insignificant Digits

No U TurnsAs someone with an affinity for numbers, I’m always conscious of how they appear around us.  And I’m always trying to decode the numbers I see:  what do they mean?  Why are they here?

It didn’t take much to figure out these numbers below these No U-Turns signs along the highway.  (It’s a bit blurry, but if you squint you can make out the 55.39 in yellow.)

After a couple of miles of confirmation, it was clear that the 55 corresponds to the highway’s mile marker.  So the 55.39 tells emergency vehicles the exact location of the U-Turn, 55.39 miles up the highway.

But is the .39 really necessary?  How much accuracy do we need in these measurements?  I can see wanting to avoid the confusion of naming it 55 (is it between 54 and 55, or 55 and 56?), but couldn’t we just call it 55.5 and save ourselves the trouble of the .39?

If the Highway Department has extra money to spend, how about new speed limit signs?  Rounded the nearest integer, please!

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By patrick honner, ago
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