Mathematics at Exhibits

A lecture and Workshop created for mathematics and science teachers of Montana.

Presented October 21, 2004.

At the lecture I will present topics that are too big, too messy or too dangerous to do in a workshop.

At the workshop we will do explorations with simple materials.

Lecture

Mathematics at the Exploratorium

Mathematics is the language of science, and so it is to be expected that a science museum needs people skilled in mathematics.

There is mathematics behind each exhibit:

Radioactive Decay Model: A large audience is needed perhaps 256 people. Each of them flips a coin. If it comes up tails they clap and sit down, heads they flip again. The sound of clapping is the sound of radioactive decay. If the coin comes up heads 5 times in a row the person flipping it stops. Eventually, 8 people or so remain standing holding coins that have come up heads 5 times in a row. What will happen when they flip the coins again?

Bouncing Ball, Bounce a ball and listen to its pattern of bounces. If each bounce takes 1/2 the time of the previous bounce then we have a series in time of 1,1/2,1/4, 1/8... which leads us to one of Zeno's paradoxes there are an infinite number of bounces in a finite time interval. Make a recording of count the bounces. Do it with a metal can, a balloon, and a marble.

Mathematics is needed for exhibitions, on the floor of the museum and on-line.

Pasta Quake, In San Francisco we are concerned about earthquakes. Break strands of spaghetti to model the energy released in an earthquake, learn the exponential nature of the earthquake magnitude scale formerly known as Richter. If the energy of breaking 1 piece of spaghetti is magnitude 5, then you have to break 30 strands to make a magnitude 6, and 900 strands to make a magnitude 7.

The Origins Exhibition about the big bang and the nature of space.

2005 is the world year of physics in honor of the 100'th anniversary of the "Marvelous" year of discovery by Einstein.

One interesting discovery of Einstein was the effect of mass on the distance between two objects in a vacuum. Here is a simple version of an experiment that has actually been done. Curved Space. Arrange four points in space in a square, all four sides of the square are the same length, perhaps a light-hour. Now bring two neutron stars next to the center of one of the sides of the square, all the angles of the square remain 90 degrees, however the side that passes between the stars is now much longer than the other two sides. Physicists are pleased that mathematicians discovered non-Euclidean geometry. Since the current description of the universe requires non-Euclidean geometry.

Expansion of Light, light is an inhabitant of the vacuum of space, when the vacuum expands, then so does light. Light that left the big bang as orange light has become microwave radiation because the universe is now 1000 times larger

Expansion of the Universe, the Universe expands so that every place appears to be the center of the expansion.

GPS and general relativity. If the atomic clocks on the GPS satellites were not corrected for general relativity then they would accumulate an error of 7 miles per day.

Mathematics for teacher exhibits

Simple materials show great science and mathematics

Gravity powered calculator, in honor of Galileo who knew that 1+3 = 4 and that 1+3+5 = 9 and so on. Drop a line with weights spaced 1,3,5,7 ... to hear a constant rhythm.

Metric Prefixes, Play around with metric prefixes.

Bouncing Balls, drop a tennis ball on top of a basketball and watch the surprising result of the collision. This models what happens when a spacecraft gets a gravitational boost to its orbit by "colliding" with a planet.

Falling Dominoes, A falling domino can knock over a domino that is 1.5 times thicker higher and wider. Then this domino can knock over a larger one. This exponential growth of a series of nine such dominos can easily knock over a domino that will release a billion times the energy or the first, small, domino.

Ringing Aluminum rod, an aluminum rod will vibrate when struck by a hammer. You can sense the vibrations with your finger, and control the patterns of vibration also with a finger. The pattern of vibration is sinusoidal, in the fundamental vibration there is one half a sine wave on the rod.

Commercials:

Interesting short Mathematics Tidbits.

The Yuki Indians of California used a base 8 counting system by counting the spaces between their fingers.

The Mayans had several calendar systems, in one they had a year with 18 months of 20 days each. this produced a nice 360 day year, alas the year actually has 365+ days so they had to add another "month" with 5 days. Each month started with day 0, so the days of the month were numbered 0 through 19.

Our Gregorian Calendar was adopted from the Julian Calendar in 1582. It has no year 0, the year 1 BC was immediately followed by 1 AD. If you think of a number line and a calendar line you will see that on the number line there are 2 units between -1 and +1 while on the calendar line there is only 1 unit between 1 BC and 1 AD. This leads to problems when computing dates. Because of the missing 0, the third millennium started in January 2001, not in January 2000 when most people celebrated. The year 2000 is called the odometer millennium, people like to see all of those zeros roll over.

My new car is a PZEV "partial zero emission vehicle" partial zero? or partial, zero emission vehicle". Punctuation matters. On a highly polluted day the exhaust from my vehicle is cleaner than the air sucked into the engine, does this give meaning too negative pollution? My cars exhaust is 10 times cleaner than the average 2004 vehicle.

Other mathematics explorations not covered in the lecture but collected here for your convenience

Bicycle Gears, Count the teeth on your bicycle's rear gear cluster. Plot the numbers on a number line (at last a practical use for the number line!), plot the logarithm of the number of teeth on a number line, take ratios of front gear numbers to rear gear numbers. Learn why bicycles have gears by seeing how the force your leg exerts compares to the distance the bicycle wheel travels.

Transit of Venus Math, When Venus passed between the sun and the earth astronomers were able to use high school geometry and trigonometry to measure the size of the solar system in 1761.

Measuring the Brightness of the Sun, use a 150 Watt light and a piece of paper with a grease spot on t to measure the power of the sun.

Multiplication blocks, If you make blocks with the correct side lengths, when the 2 block is placed beside the 3 block the two blocks will be the same length as the 6 block. The secret is to make the lengths of the sides of the blocks the logarithm of the number on the block.

Mathematics exhibits at the Exploratorium:

Pendulum Snake, make a series of pendulums which start together, drift into seeming chaos and then return to swing together again.

Workshop

Hands-on Mathematics of Science Exhibits

Counting is difficult, measuring is nearly impossible.

Make models of Mars and the Moon from clay and learn some surprising scaling laws.
It's only a
clay Mars, but it's still hard to believe.
Clay Moon. If Mars was hard to believe this will be impossible.

Spinning Cylinder, When you spin a cylinder interesting patterns occur, can you do experiments that help you figure out the origins of these patterns?

Mobius strip explorations, You've met the Mobius strip before, here's your chance to explore it further.
Mobius strip, counting is difficult.
Cutting a Mobius strip, can you predict the answer.
Mobius strip collisions, a particle physics approach to Mobius strips.
Mobius strip decay, radioactive decay of Mobius strips.

Magnetic Tape, Use small strips of magnetic recording tape and magnets to explore binary numbers, the ASCII code, and digital magnetic recording.

Other possible Explorations not done during the workshop.

Reaction time, measure your reaction time by dropping a ruler.

Take it from the Top (Snack), Make a stack of cassette boxes and find the mathematical series which describes the amount of overhang.

Mars Calendar and Clock, by inventing a clock and calendar for Mars you may appreciate the earth clock and calendar more. Learn how too deal with remainders, after all the earth year is approximately 365.25 days long.

Mathematics Publications of the Exploratorium (add some activities from these publications.)

Square Wheels, Paul Doherty, Don Rathjen

Magnetic Tape

Spinning Cylinders

Mathematics Explorer, Pat Murphy and Lori Lambertson.

Expanding and Shrinking Afterimages

Counting Like and Egyptian

Breaking the Maya Code

Mayan Calendar Rounds

 Scientific Explorations with Paul Doherty © 2004 25 May 2004