Iron Science Teacher

Valentine's Day 2005

Secret Ingredient: Valentine's Day Gifts

As a mathematician and scientist it is important to find a compatible life companion.

One way to do this is by choosing the correct Valentine's Day card like this one:

This will identify you to those not fluent in mathematics as a member of the mathematics Mafia.

(Mathematics Mafia: An organization that makes you an offer you cannot understand.)

The appropriate recipient will immediately know that this card says " I love you"

Where the equation R = 1-sin Q is the equation for a cardioid in polar coordinates. And a cardioid is a heart.

Of course the card needs to be accompanied by a gift.

In search of a gift of a cardioid I knew that I could make a cardioid by rolling a circle around another circle, however it was hard to keep the circles together and rolling without slipping. Then I thought about magnets.

Two disk shaped magnets can be arranged to attract each other. Then one will easily roll around the other. A point on the rim of the outer magnet can be traced to make a cardioid. Or the point can be replaced with a bright light, like that from a small light emitting diode, LED.

Construction. Find two large disk magnets from someplace like Forcefield, or wondermagnet.com, or All Magnetics, or Dowling Miner magnetics. Get an LED flashlight that can be switched on and a small binder clip. Hot melt glue the binder clip to the edge of one magnet. clip the light into the binder clip. Now when the light-containing magnet is rolled around the other magnet's rim it will trace out a cardioid and produce a cardioid after image on the retina of the observer.

Counting is Difficult

Question, Does the outer magnet make one rotation or two as it rolls around the inner magnet? If the light starts out touching the top of the inner magnet the next time it touches the inner magnet again is when it is at the top. so it seems that it makes just 1 rotation, however if it starts in the same position touching the inner magnet and so at the bottom of the outer magnet, then as it rolls around it reaches the bottom of the outer magnet twice in one complete revolution, when the outer magnet reaches the bottom of its revolution about the inner one.

This is an important question because as the moon orbits the earth each month does it rotate once or not at all. Relative to the earth the moon keeps one face toward the earth at all times and so does not seem to rotate at all, yet viewed from the moon the sun makes a complete revolution about the moon each month (or so) so does the moon rotate or not?

Caustics

Cup

Then one day while drinking coke in a coffee cup I noticed a pattern of light in the bottom of the cup.

The light on the bottom of the cup made a cardioid shape.

It's great to do this with a point of light such as a minimag light, but it works in the sun or with any other concentrated light source.

It's better with incandescent bulbs than with large fluorescent fixtures.

The light pattern is concentrated into a point.

This sharp point is called a caustic,

Another way to make it is with a

Paper cup lined with mylar mirror.

But of course for valentines day it's best to make it with jewelry such as a

Golden Ring

or golden earring

or even a shiny hematite ring

It works best if the jewelry has a smooth flat inner surface that is circular.

But no matter what, don't forget to include the card with your cardioid gift.

At last a gift worthy of St. Valentine

Gary, lives of the Saints St. Valentine, pierced by arrows.

Like the points of a caustic.

Scientific Explorations with Paul Doherty

© 2005

12 February 2005