Parabolic Telescopes

Last week as I was going through a whirlwind tour of conics to wrap up my Math Analysis class, I wanted to illustrate some real-life examples. Rather than just saying "these things are parabolas, these things are ellipses [etc]" I wanted to have the kids DO something with conics.

I'm not sure I succeeded, but I did come up with one activity that, with further work, could be a nice problem for future Advanced Algebra / Analysis classes.

I had a nice high quality picture of the Very Large Array in New Mexico, since my sister and I visited there on a cross-country drive. I wondered if it was possible to find an equation that would actually model the parabolic shape of the telescopes. So I popped the picture into Geogebra and constructed a parabola via the locus tool (which I just recently learned how to use).

The stated goal of the assignment is "find an equation for the telescope." I told them not to worry about the rotation. We could handle that later. (we didn't handle it, but perhaps next time, if this comes after matrices we could multiply by a rotation matrix... find the angle by inverse trig based on the slope of the dish)

But I think if I had used Geogebra with the kids more this year, they might have the capability to construct the locus themselves instead of me giving them focus / directrix. Basically, if they did what I did, I feel they'd get a good understanding of what a parabola is from a locus standpoint.

I didn't put this on the mathlet, but the diameter of one dish is 85 feet. Two ways to handle that: (1) rescale it so the geogebra numbers match it. (2) have the kids find the scaling factor (put points on the locus, measure the diameter using a segment). I like option 2, but that does require them to be well versed in Geogebra (goal: verse my kids in Geogebra next year!)

Well, I appreciate any comments / critique.

http://scottfarrar.googlepages.com/VLA.html

News Media

This XKCD comic reminded me of why I hate most of the News Media. The panel on the left is a pretty accurate rendition of a full screen graphic one of the 24 hour networks had up:

AIG
Bailout: $170 billion
Bonuses: $165 million

At first glance, one might arrive at the conclusion that AIG was using almost its ENTIRE bailout for these bonuses. "How irresponsible! What an outrage! Despicable!" But upon closer inspection, one notices the difference between the numbers. What if we actually took the difference?

$170,000,000,000

- $165,000,000

= $169,835,000,000

There's $165 million is a drop in the hat. Its about 0.1% of the bailout money. It pales in comparison! It should not be compared. I'm not defending AIG. That's not the issue. The issue is the News Media is presenting information in a misleading format. I also assert the News Media is doing so purposely. They want the outrage, because outraged viewers will watch more. Everything is sensationalized and distorted.

The new propoganda isn't for political gain. The Media wants your viewership and your money. They will distort the facts for it.

Our financial crisis is a result of people misrepresenting information about loans rates, feasibility of real estate transactions, investment options and the market in general. People would bend the truth to gain more money. Now, the News Media is now misrepresenting information... to gain more money. They mimic the actions of the very organizations they codemn.

"Why do we have to take math?" students ask me. Look at the world. Those who understand the system will take advantage of those who do not. A simple concept like orders of magnitude (billions vs. millions) will be lost on many viewers who have been in the habit of non-mathematical thinking.

The next time stats or data is thrown in your face, by the Media or anybody else, pause and think: "what does it really mean?"

Symmetry!

What do these words have in common?
dollop, suns, noon
Ok, try turning the screen upside down and reading them... (for 'noon', try reading it from behind your screen... oh wait... doh!)

I was over at Tom's the other day. We were talking about Blu-ray and HDDVD and how movies will inevitably be re-released forcing consumers to either cope with the fact that they do not have the pinnacle releases of their favorite movies, or to take cash against a sea of discs, and by conceding, buy them. Sidenote: how much clearer can a movie like the Simpsons get?

Whilst I was making fun of Tom for having multiple copies of some movies in his collection already (Princess Bride, Star Wars... ) he mentioned that there was yet another release of Princess Bride. I was ready to add another $40 onto the running tally (yeah, I keep a tally of close friends' leisure expenditures in my head, you wanna fight about it?) he mentioned how the cover of this latest release was quite interesting.

If you rotate the cover 180 degrees, the title reads the same way. Thus they have created an rotational Ambigram. Just like the words suns and dollop, it is symmetric under a rotation on the z axis. (z positive is coming out at you, z negative is looking directly into the screen).

The Princess Bride, however, is not a natural ambigram like dollop and suns. It took some nice artistic work for that one. Perhaps there is a job for math majors. "Oh yes I graduated with a B.S. in Math and an emphasis in ambigram synthesis."

In reading the Wikipedia article it saw examples of other ambigrams. noon, for example, is also a 180 degree rotational ambigram, but you rotate it around the y axis (between the two o's)

Then there's the rather impressive x-axis symmetry of
HICKOK DIED DEC 3 1883 -- DOC BEECH DECIDED HE CHOKED. Imagine the 1 has no little doohickeys on it. I actually saw this at Asilomar during Harold Jacob's presentation.

Ambigrams have actually become more popular recently with Dan Brown's cover for Angels and Demons. Dan Browns books are nice for inciting a little pop-math fever, they aren't exactly rigorous... but they did lead to a re-release of John Langdon's (he being the basis for the Da Vinci Code's protagonist) book Wordplay.

Anyway, this got me thinking... is this a key to more hidden elements in Princess Bride? The 180 degree flips of Wesley and Inigo's dominant hands during their duel suddenly seem more mathematically significant. This scene is actually called "swordplay/wordplay" on DVD chapter selections. Dun dun dunnnnnnnnnn....

Pop culture data representations

Many people have seen, or even made, drawings or illustrations depicting song lyrics either sincerely or sarcastically. But how many have made accurate data representations of pop culture? I will share with you some examples.

I like the cleverly inactive "Touch this"

Layers upon layers.










Biggie never said anything about it being linear though... he merely asserted that for the function f:money->problems, f'(x) > 0 for all x.

Triangles from folding

Using a square piece of paper, label each side 1 unit long. Find the midpoint of the top side by folding in half. Take the bottom right corner and fold it up to touch the midpoint of the top side. This creates three triangles. ABC, CDE, EFG. Determine the side lengths of all three triangles.

This was a nifty little problem given by Harold Jacobs at Asilomar 2007.

e^(i*pi)+1=0