I was cutting up some squash the other day
and I thought I’d experiment cutting the squash into fifths and sixths.
I thought I would do a much better job cutting the squash into equal sixths than into equal fifths. I am generally more comfortable with even numbers, and there is something quite unnatural about cutting a circle into fifths. But I’m not sure either division was especially equal.
They both tasted great, though!
Google has a nice doodle celebrating the 25th anniversary of the buckyball. (A video of the doodle can be seen here.)
“Buckyball” is the informal name of a particular kind of carbon molecule that, geometrically, resembles the geodesic dome made popular by futurist Buckminster Fuller. They are more generally known as fullerenes (again, after Fuller), and among other things, have recently been detected in space.
Viewed mathematically/geometrically/graph-theoretically, a fullerene is a solid consisting of only pentagonal and hexagonal faces. There are many different fullerenes–for example, having 20, 70, or 200 vertices–but what’s amazing is that apparently all of them have exactly 12 pentagonal faces. Only the number of hexagonal faces changes.
Apparently this fact is a direct consequence of Euler’s formula, namely V – E + F = 2, where V, E, and F are the number of vertices, edges, and faces, respectively, in a given solid. For example, a cube has 8 vertices, 12 edges, and 6 faces; note that 8 – 12 + 6 = 2, just as Euler requires.
Try verifying Euler’s formula for an octahedron! Then, when you’re done with that, prove the above remark about fullerenes.
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My summer of modest carpentry continued with the staining of this unfinished bench:
That one-pint can of Black Cherry stain claims that it will cover 75 square feet of surface. If so applied, how thick would that layer of stain be? Let’s go with inches first, and convert to microns later. Or perhaps a more reasonable question is how does the thickness of the stain compare to the thickness of a sheet of paper?
Hopefully someone will figure that out and tell me.
Whenever I spend a lot of time driving, navigating, and map-reading, I find myself making a lot of rough estimates of distances. The process reminds me of how one estimates and ultimately evaluates, using Calculus, the arclength of a curve.
To find the length of a curve, we approximate the curve with a series of line segments. It’s easy to find the length of a line segment, and so by sacrificing exactness, you turn a hard problem into an easy one. This is a fundamental technique in Calculus.
I made my line segments equal in length to 40 miles on the map. Now just add up the lengths of the line segments to approximate the length of the curve.
Each of the seven line segments is (roughly) equal to 40 miles, so the approximate length of the path from Brooklyn to Burlington is 280 miles (not a terrible estimate).
There are plenty of ways to improve the approximation, and the straightforward, but complicated, calculus approach eventually produces the arclength integral.
On the actual drive my approximations weren’t as good, as I was using an inferior distance estimator.
This is an absolutely mind-blowing idea: robotic “paper” that can fold itself into an arbitrary three dimensional object. Be sure to watch the short video accompanying the article.
http://web.mit.edu/newsoffice/2010/programmable-matter-0805.html
Tying (or folding?) all of the physics and engineering together here is the mathematics of origami. How can you fold a square sheet into a boat? A plane? A tetrahedron? A super-intelligent robotic giraffe?