Geometry

3-D Illusion Application

This is a clever (and creepy) application of optical illusions–a painting, drawn on the pavement, that appears to on-coming drivers as a child playing in the middle of the street.

http://reviews.cnet.com/8301-13746_7-20016169-48.html

The purpose of this, presumably, is to catch the attention of drivers so that they will slow down.

The illusion is strikingly effective, and by watching the video you can see how long the image on the ground is. I wonder if the painting is equally stretched out at all points–is it just a dilation of a normal drawing?–or is it more of a distorted projection–like Greenland on a flat map? That is, is the head of the painting five-times normal size, while the shoes of the painting are twice normal size? My gut feeling is that the painting needs to be distorted like that, but I’m not really sure.

Now, whether this strategy will prevent accidents or actually cause more accidents remains to be seen.

By patrick honner, 14 years9 years ago

Challenge Geometry

Lobster-nomics

One of the benefits of travelling around New England is enjoying lobster in its many forms: steamed, on a roll, in a bisque, biting your toe. But while shopping at a fishmonger one evening, I was somewhat baffled to see a sign like the one at the right. It seems that the price per pound for lobster varies depending on the lobster’s size!

Why is this? Why should a larger lobster cost more per pound than a smaller lobster? Large or small, apples are still $1.69 per pound; the same goes for onions, chicken, and most other things.

What’s different about lobsters? My assumption is that, in a lobster, the ratio of meat to non-meat (shell, antennae, veins, etc) is constant, that is, the same for any size lobster. Thus, since you are paying for meat, you should then pay the same price per pound for any lobster. But maybe that’s not the case. Maybe in a small, one pound lobster, there is, say, 8 oz of meat and 8 oz of shell, but in a lobster twice the size, there is 20 oz of meat and 12 oz of shell.

I guess I assumed that the geometry of the lobster is essentially the same regardless of the size; in other words, that all lobsters are geometrically similar. Thus, the price-per-pound should be lobster-independent. But maybe I’m wrong. Is it really the case that, as lobster size increases, the amount of meat and the amount of non-meat increase at different rates?

Anyone have any other theories?

By patrick honner, 14 years9 years ago

Application Geometry

Limits and Oar Making

A boat-builder once described to me one procedure for making oars.

Start with a long piece of lumber with a square cross-section. Take off a certain amount from each of the four corners. Now their are eight corners. From each of the eight corners, take off a smaller amount. The progression of cross-sections looks something like this:

Through practice, the builder knows how much wood to remove at each stage. You can continue to repeat this process, but some sanding at this point will probably get you pretty close to what you are looking for: a circular cross-section.

This naturally brings to mind the hallmarks of Calculus: approximations and limits. At each stage of the process, the cross-section of the oar becomes a better and better approximation of a circle. Indeed, the limit of such a process is indistinguishable from a circle.

I’m not sure if anyone making oars is thinking about Calculus, but sometimes it’s hard for me not to think about it!

By patrick honner, 14 years7 years ago

Appreciation Geometry

More Math and Fruit (Vegetables?)

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!

By patrick honner, 14 years9 years ago

Challenge Geometry

More on Buckyballs

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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