Includes a nice animation of the Doppler effect; explains red shifting of distance galaxies; and how we can use geometry (parallax) to determine the distance to nearby stars.

Vi Hart has some excellent videos where she gives you things to doodle during math class (see her YouTube channel). There’s lots of wonderful geometry and algebra. The Fibonacci sequence video below is a great example:

Jupiter and Venus have been sitting near the western horizon, shining so brightly that even I have noticed them. Phil Plait explains with some back-of-the-envelope math, why Venus is brighter even though it’s smaller than Jupiter. It’s a nice example of how a little math can do a great job explaining how the world (and others) works.

The Missouri Council of Teachers of Mathematics (MoCTM) has a Math & Art Contest that focuses on Geometry. It has fairly simple expectations, and it’s aimed at Middle School students and lower. The tie between the math and the art does not require much depth, but that’s probably appropriate for students who are still developing abstract thinking.

A tessellation. Image via Wikipedia.

I’m usually a bit cautious about the utility of contests. Their primary benefit is in the work that they motivate, not the reward (or hope of a reward) at the end; although, students do need to learn to win or lose with equanimity.

Quasicrystalline ordering of a aluminum-palladium-manganese alloy. Image by J.W.Evans via Wikipedia.

Regular ordering of a halite crystal. The atoms that make up salt crystals are arranged in a cubic shape. The smaller grey atoms are sodium (Na), and the larger green ones are chlorine (Cl).

Daniel Shechtman was just awarded the Nobel Prize in Chemistry (2011). He discovered that matter can exist not only as crystals, which have a regular geometric arrangement of atoms, and amorphous non-crystals that do not, but also as quasicrystals which have a different type of atomic ordering.

The Guardian has an interesting article on Schechtman, whose discovery was roundly disbelieved by other scientists and he was ridiculed for years.

In an interview this year with the Israeli newspaper, Haaretz, Shechtman said: “People just laughed at me.” He recalled how Linus Pauling, a colossus of science and a double Nobel laureate, mounted a frightening “crusade” against him. After telling Shechtman to go back and read a crystallography textbook, the head of his research group asked him to leave for “bringing disgrace” on the team. “I felt rejected,” Shachtman said.

Geometric proof of the Pythagorean Theorem by rearrangemention from Wikimedia Commons' user Joaquim Alves Gat. Animaspar.

Elegant in its simplicity but profound in its application, the Pythagorean Theorem is one of the fundamentals of geometry. Mathematician Alexander Bogomolny has dedicated a page to cataloging 93 ways of proving the theorem (he also has, on a separate page, six wrong proofs).

Some of the proofs are simple and elegant. Others are quite elaborate, but the page is a nice place to skim through, and Bogomolny has some neat, interactive applets for demonstrations. The Wikipedia article on the theorem also has some nice animated gifs that are worth a look.

Cut the Knot is also a great website to peruse. Bogomolny is quite distraught about the state of math education, and this is his attempt to do something about it. He lays this out in his manifesto. Included in this remarkable window into the mind of a mathematician are some wonderful anecdotes about free vs. pedantic thinking and a collection of quotes that address the question, “Is math beautiful?”

Mathematics, rightly viewed, possesses not only truth, but supreme beauty — a beauty cold and austere, like that of sculpture, without appeal to any part of our weaker nature, without the gorgeous trappings of painting or music, yet sublimely pure, and capable of a stern perfection such as only the greatest art can show.

This one’s for the nerds. It’s hard to argue against the elegance of using tau instead of pi. Natalie Wolchover explains, while Kevin Houston makes the argument in video:

My students and I had a great chance to use the our recent geometry work when we figured out how long it would take to drain the new moat in front of the school.

It’s not really a moat, it’s going to be a flower bed that will soak up some of the runoff that tries to seep into the school’s doors every time a spring or fall thunderstorm sweeps through.

The hole was dug on Thursday evening and filled with rainwater with water, half a meter deep, by Friday morning’s rain. At least we know now that the new beds are in the right place to attract runoff.

But to fill the trenches with gravel, sand and soil, we needed to drain the water. With a small electrical pump it seemed like it would take forever; except that we could do the math.

The pump emptied water through a long hose that runs around the back of the building where the topography is lower. I sent two students with a pitcher and a timer (an iPod Touch actually) to get the flow rate.

They came back with a time of 18.9 seconds to fill 4 liters. I sent them back to take another measurement, and had them average the to numbers to get the more reliable value of 18.65 seconds.

Then one of the students got out the meter-stick and measured the depth of the moat at a few locations. The measurements ranged from 46 cm to about 36 cm and we guesstimated that we could model the moat as having two parts, both sloping. After measuring the length (~6 m) and width (2 m), we went inside to do the math.

Rough sketch of the volume of water in the moat.

With the help of two of my students who tend to take the advanced math option every cycle, we calculated the volume of water (in cm^{3}) and the flow rate of the pumped water (0.2145 cm^{3}/s). Then we could work out the time it would take to drain the water, which turned out to be a pretty large number of seconds. We converted to minutes and then hours. The final result was about 7 hours, which would mean that the pump would need to run until 10 pm.