College Math Teaching

August 19, 2016

A fun question concerning projections

Filed under: geometry, popular mathematics — Tags: — collegemathteaching @ 11:34 am

The semester is about to start. I decided to have some fun on Facebook. I took some office shots (yes, my office is messy) and decided to retouch the photos by adding joke photos from Facebook; I wanted to see which friends noticed (without being specifically notified). One did.

projection1

projection2

Of course, the color alone will give away which “on the wall” photos are genuine and which were put in (via Paint).

But a fun question is: if those photos were to be genuine, what size and shape would they be in real life (to appear the way that they do in those shots). One would give the genuine dimensions of the genuine photos on the wall to help the student solve the problem.

A more sophisticated 3-d version of the problem can be obtained from this cool video:

March 21, 2014

Projections, regressions and Anscombe’s quartet…

Data and its role in journalism is a hot topic among some of the bloggers that I regularly follow. See: Nate Silver on what he hopes to accomplish with his new website, and Paul Krugman’s caveats on this project. The debate is, as I see it, about the role of data and the role of having expertise in a subject when it comes to providing the public with an accurate picture of what is going on.

Then I saw this meme on a Facebook page:

These two things (the discussion and meme) lead me to make this post.

First the meme: I thought of this meme as a way to explain volume integration by “cross sections”. 🙂 But for this post, I’ll focus on this meme showing an example of a “projection map” in mathematics. I can even provide some equations: imagine the following set in R^3 described as follows: S= \{(x,y,z) | (y-2)^2 + (z-2)^2 \le 1, 1 \le x \le 2 \} Now the projection map to the y-z plane is given by p_{yz}(x,y,z) = (0,y,z) and the image set is S_{yz} = \{(0,y,z)| (y-2)^2 + (z-2)^2 \le 1 which is a disk (in the yellow).

The projection onto the x-z plane is given by p_{xz}(x,y,z) = (x,0,z) and the image is S_{xz} = \{(x,0,z)| 1 \le x \le 2, 1 \le z \le 3 \} which is a rectangle (in the blue).

The issue raised by this meme is that neither projection, in and of itself, determines the set S . In fact, both of these projections, taken together, do not determine the object. For example: the “hollow can” in the shape of our S would have the same projection; there are literally an uncountable. Example: imagine a rectangle in the shape of the blue projection joined to one end disk parallel to the yellow plane.

Of course, one can put some restrictions on candidates for S (the pre image of both projections taken together); say one might want S to be a manifold of either 2 or 3 dimensions, or some other criteria. But THAT would be adding more information to the mix and thereby, in a sense, providing yet another projection map.

Projections, by design, lose information.

In statistics, a statistic, by definition, is a type of projection. Consider, for example, linear regression. I discussed linear regressions and using “fake data” to teach linear regression here. But the linear regression process inputs data points and produces numbers including the mean and standard deviations of the x, y values as well as the correlation coefficient and the regression coefficients.

But one loses information in the process. A good demonstration of this comes from Anscombe’s quartet: one has 4 very different data set producing identical regression coefficients (and yes, correlation coefficients, confidence intervals, etc). Here are the plots of the data:

And here is the data:

Screen shot 2014-03-20 at 8.40.03 PM

The Wikipedia article I quoted is pretty good; they even provide a link to a paper that gives an algorithm to generate different data sets with the same regression values (and yes, the paper defines what is meant by “different”).

Moral: when one crunches data, one has to be aware of the loss of information that is involved.

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