Happy Halloween!

Some Pumpkin Pi for you, of course! This week we had a festive episode with a spooky set and costumed crew. We even investigated strange apparitions in the studio. We still did lots of math, as usual, and we had a winner for our contest.

Sailplanes and Gliders

Aviation in general involves plenty of mathematics, but we recently talked to Bret at Sky Sailing about how math comes into play in the piloting of sailplanes and gliders. This week we showed our interview with Bret and he explained many different instances of how math and mathematical thinking are necessary skills for flying these amazing crafts. He talked about making sure that the center of gravity of the plane with its passengers is within a certain range; otherwise the plane can be out of balance and can't fly. He also mentioned the "glide ratio" of a plane which is very similar to the algebraic concept of slope. Wind affects a plane's ground speed and navigation, so the speed of the wind must be factored into distance computations and navigation decisions. Brett also mentioned that he is constantly evaluating angles to locations on the ground while in the plane, especially when landing. It's probably no surprise that he also deals with conversions between units as well.

It is well worth visiting Sky Sailing in Warner Springs and going for a ride!

More information: Wikipedia entry for sailplanes and gliders

new contest

We have a contest running right now as outlined in the graphic above. You can't change the position of the digits, but you can use as many of those mathematical symbols as you like. You can even put parentheses around digits, like ( 5 4 ), and consider that "54." And if you haven't seen that symbol "^" used before, that is used for exponents. (So 4^3 means 4 cubed.) We know of one way to do it, but there are probably many correct answers. If you find one, you can be our official winner by calling us and giving us the result during the live broadcast this Wednesday, October 22, 5-6 PM. The phone number is 1-888-762-1489. Good luck!

Archimedes Screw

The Archimedes Screw was invented over 2,000 years ago and is still in use today! Its purpose is to transfer water to a higher elevation, and it works by turning a large, tilted screw so that the blades of the screw scoop up water, the water sits between the blades as it rides upward, and it is spit out at the top. The photo here was taken at SeaWorld in San Diego where they use two of these devices to push water uphill for their Shipwreck Rapids ride. We spoke with an engineer at SeaWorld about the Archimedes Screws in use there and the mathematics involved in them. 

The screw only works if it is tilted within certain angles, so the concept of slope comes into play, both for the cylindrical screw and the blades themselves. The blades form a mathematical shape called a helix, and if you look at the blades from the side, the contours of the helix match the graphs of sine waves. These sine waves must have a downward slope as they cross the axis of the screw in order to hold water as the screw turns.

During the broadcast we mentioned a research paper (link below) that determined the optimal design for an Archimedes Screw using lots of calculus and 3D graphs. Don't worry, we talked about the highlights on the program which can be viewed once it is archived at www.collegemathline.com. 

This week's links:

Wikipedia entry for Archimedes Screw

Animation of an Archimedes Screw pushing a ball uphill

research paper by Chris Rorres (pdf document)

f/stops in Photography

When you take a photograph, there are mainly two elements controlling how much light is sent to the film or digital sensor. One is the shutter speed, which is simply how long the sensor or film is exposed to the incoming light, and the other is the aperture or "f/stop." The f/stop is a measure of the lens opening itself. The larger the opening, the more light comes in (and the faster the shutter can be). The f/stop is actually a ratio of the focal length of the lens (the distance between the lens and the film or sensor) to the diameter of the lens opening. So the larger the opening, the smaller the f/stop. Some of the fancier cameras, like SLRs, let you choose these settings yourself if you wish. Even the pocket digital cameras are using these settings, they are just done automatically for you. In fact, many photo viewing software applications can tell you what shutter speed and f/stop the camera used when it took the photo. 

Each time you take a photo, you or your camera must make a choice between a larger opening and a shorter shutter time or a smaller opening and a longer shutter time. Either way you can get enough light for a good picture. So what is the difference? If you use a long shutter time, a moving object will look blurry. If you use a large opening, the depth that can remain in focus is much shorter. For instance, the first photo of the watering can uses an f/stop of 5.6. The chair behind it is quite fuzzy. The second photo, with an f/stop of 16, has the chair in sharper focus.

If you look at the available f/stop numbers for a particular lens, you might find the numbers mysterious. You will often see a progression like 2, 2.8, 4, 5.6, 8, 11. They almost seem random. In fact, there is logic to it! These are the values of the f/stop ratio where the lens opening doubles in AREA (not diameter!) each time. This lets in twice as much light. Remember that the area of a circle is based on the square of the radius, so if you want the area to double, you can't double the radius. In fact, you would need to multiply the radius (or diameter) by the square root of two. And that multiple, root 2, is exactly where the progression of f/stop numbers comes from. 

Here are some links if you want to learn more:

Wikipedia entry for f/stop numbers

A very detailed (and good) explanation of f/stops and the math involved

Math Questions Fall 2008

You can leave math questions for us here, as comments to this post, and we will (hopefully!) solve them on the broadcast each Wednesday.

Mathematics and Legos

We're back for our 8th semester broadcasting! And during our first episode, we showed an interview with one of the Lego model builders, Eric, at Legoland California. His whole job is building things out of Legos, how is that for a cool job? If you visit Legoland, you will see "Miniland," where they have recreated cities out of Legos in amazing detail.
Eric described how he has to think mathematically every day at work, including determining scales, creating designs and blueprints, and estimating how many Lego bricks to order for a project. He showed us his most recent project: a Lego version of the extraordinary Burj Dubai skyscraper in the United Arab Emeriates which will be finished next year. 
The image at the left shows the Lego version with the actual building in the background.

This week's links:

Miniland at Legoland California

Burj Dubai official website

Wikipedia entry for Burj Dubai