Like a wheel within a wheel

**Introduction**

A spirograph can be used to create artistically interesting
patterns.

The patterns can be used to show:

the mathematics of least common multiples,

of clock arithmetic (aka modular arithmetic)

and the fundamental theorem of mathematics.

**Material**

A spirograph

paper and colored pens or pencils

optional 2 common pins and some corrugated cardboard.

**The Spirograph**

**To Do and Notice**

Place one of the solid disks with gear teeth on its outside,
inside one of the circular holes with gear teeth on its inside.

Roll it around until the inner wheel is at the top center of the
outer circle.

Place a pen or pencil inside one of the holes in the inner wheel.

Roll the inner wheel around creating a pattern.

Stop when the pen returns to its starting spot.

Examine the pattern, what does it remind you of?

(I see a flower with many petals.)

Explore the patterns made by wheels and circles with different
numbers of teeth.

Notice how the patterns are similar and different.

What happens if you start the pattern with the wheel at the bottom
center of the circle?

(Be careful to use the same hole for the pen that you used the first
time.)

**What’s Going On?**

The number of teeth on the wheel and the circle determine the
shapes of the patterns made by the spirograph.

The numbers of teeth on the two wheels determines the number of
cycles the smaller wheel wil make before the line drawn by the pen
returns to its starting point.

**etc.**

A point on a wheel rolling inside a circle traces out a
hypocycloid.

A point on a wheel rolling on a flat surface traces out a curve
called a cycloid.

A point on a wheel rolling outside another wheel traces out an
epicycloid.

Starting at different places produces the same pattern just rotated around from the original.

**Counting**

**To Do and Notice**

Count the number of teeth on your spirograph wheels and in the
spirograph circles.

Stick a piece of tape onto your wheel and write a letter of the
alphabet on this wheel such as "A."

Write the letter A on a piece of paper and next to it write the
number of teeth you counted.

Give the wheel to a different person and ask them to count the number
of teeth and write it on a piece of paper.

Keep your number a secret from them.

After they have counted the number of teeth and written it on their
paper, compare your answers.

Are they the same. If not decide who is right.

Then label each gear with its number of teeth.

My spirograph has three large open circles with 72, 96, and 105
teeth.

It has 3 small plastic toothed disks with holes in them. There are
small wheels with 36, 52, and 63 teeth.

**Question**

Are the number of gaps between the teeth the same as the number of teeth?

**What’s Going On?**

Caution, counting is not easy.

In the Pooh stories there is a forest with either 40 or 41 trees no
one can tell which, even if they tie a ribbon around each tree as it
is counted.

One good way to count is to pin the wheel down so it cannot move. Insert two pins through holes in the wheel then through a piece of paper, into a sheet of cardboard. Use a pencil to make a mark on the paper next to your starting tooth, number this mark 1. Make a mark next to tooth number 10, 20 and so forth.

**Answer**

The number of gaps equals the number of teeth.

**Mathematical patterns**

**To Do and Notice**

Find a wheel with a number of teeth that evenly divides the number of teeth in a circle, that is, the smaller number divides into the larger with no remainder. For example, our 36 tooth wheel divides into the 72 tooth circle exactly twice. Use a pen and trace the pattern created by rolling the wheel inside the circle. Notice the pen returns to its starting point after just one circuit of the larger circle by the smaller wheel.

Find a wheel with a number of teeth that does not evenly divide the number of teeth in the larger circle. Use a pen and trace the pattern created when you roll the wheel inside the circle. For example use our 52 tooth wheel inside our 72 tooth circle and the pattern closes after 13 circuits of the circle by the wheel.

**What’s Going On?**

**The mathematics behind the spirograph.**

Consider a small wheel with 10 teeth,inside a larger circular hole
with 40 teeth and 40 gaps.

Assign the teeth on the wheel numbers from 0 to 9. Starting at 0 will
make our mathematics easier later.

Assign the forty gaps in the large circle numbers starting at 0 and
ending at 39.

As it rolls around, the small wheel measures off the larger
wheel.

Every 10 holes, the marked tooth, number zero, fits into a hole in
the rim.

It returns to the rim at hole number 10,and again at 20, 30,and then
after hole 39 it returns to hole 0.

So after one “roll-around” things are back as they started
and the pattern made by the pen in the spirograph repeats the same
pattern exactly.

But what would happen if there were 45 holes in the outer
circle?

In this case hole 0 would be the same as hole 45.

The disk would roll around once hitting, 10,20,30,40, then 50 = 45 +
5, the next time around the pen would make a different pattern as
tooth zero hit holes 15,25,35 and then 45 aka 0, returning to its
start.

What if there were 42 holes?

then we would get:

cycle 1: 10,20,30,40, 50= 42 + 8.

cycle 2: 18,28,38,48 = 42 + 6

cycle 3: 16,26,36,46 = 42 + 4

cycle 4: 14,24,34,44 = 42 + 2

cycle 5: 12,22,32,42= 42 + 0

and after 5 complete circles the pattern will repeat.

There are several bits of mathematics here:

Least Common Multiples

The fundamental theorem of arithmetic

cycloids and their relatives

modular mathematics and time telling

Let's look at each of these in turn.

**Least common multiples.**

**What’s Going On?**

The least common multiple of 10 and 40 is 40

any pair of gears whose larger gear has a number of teeth which is an
exact multiple of the number of teeth on the smaller gear will result
in a pattern which will be complete after one revolution.

With a 45 tooth circle and 10 tooth wheel the least common
multiple is 90.

To go 90 teeth the inner disk must make two complete revolutions (2 =
90/45) of the 45 tooth gear so the pattern repeats after two
revolutions.

With 42 teeth the least common multiple of 10 and 42 is 210.

the inner wheel must make 5 revolutions (5 = 210/42) before the
pattern repeats.

We can now ask how many revolutions will be made with gears of 10
teeth and 41 teeth.

The least common multiple of 10 and 41 is 410 and the inner disk must
make 10 revolutions before the pattern repeats. The maximum number of
revolutions before a repeat is equal to the number of teeth on the
inner gear.

**So What?**

It seems surprising, at least to me, that the patterns made by the
spirograph relate to the mathematics of adding fractions!

to add 2/5 and 1/2 find the least common multiple of the
denominators

5 and 2 which is 10

convert each fraction to this denominator

2/5 = 4/10 and 1/2 = 5/10

then add

9/10

**More Least Common Multiples
The fundamental theorem of Arithmetic.**

(Every number is either prime or else can be expressed as a product of primes in one unique way.)

To find the least common multiple you must decompose each number
into its prime factors.

So when we wanted the least common multiple of 10 and 42 we found

2 x 5 = 10 and 2 x 3 x 7 = 42

To make the least common multiple assemble all the prime factors for
each number.

In the above numbers 2 appears in both numbers once so include it
only once.

The least common multiple is:

the product of 2,3,5,and 7 = 210.

Some numbers contain a prime number more than once such as 8 =
2x2x2. The LCM must include the larger number of appearances of each
prime.

With numbers such as 8 and 18 the least common multiple is

2x2x2 = 8

2x3x3 = 18

the least common multiple must contain 2x2x2 from the 8, it can
ignore the single 2 from the 18 since it is already included in these
three 2's.

The least common multiple is thus

2x2x2x3x3 = 72

**Canceling Prime factors
**Another way to find the number of cycles before a repeat is to
decompose each number into its prime factors

e.g. 96 = 2,2,2,2,2,3

36 = 2,2,3,3

Then cancel out of the smaller number

the prime factors which appear in the larger one

both of the 2’s in 36 appear in 96 and one of the threes

This leaves one three behind

the remaining number,3, means that the inner wheel must make 3
circles before the pattern repeats.

**Modular arithmetic what is the remainder?**

Modular arithmetic also allows us to calculate the number of revolutions of the spirograph that are needed to close the pattern. Modular arithmetic also allows us to calculate time on a 12 or 24 hour clock.

When you divide two numbers you sometimes get a remainder.

8/2 = 4 exactly, but 9/2 = 4 with a remainder of 1.

The remainder is what is important when figuring out the repeating
pattern of the spirograph.

Consider the hole with 72 teeth and the wheel with 36 teeth.

72/36 = 2 no remainder. With no remainder the pattern will repeat
exactly after just one revolution of the inner disk within the outer.
No remainder means that the marked tooth will return to its starting
position after one cycle.

However if there were 96 teeth in the outer ring then 96/36 = 2
remainder 24. The marked tooth misses returning to its original
position by 24 teeth!

One its second time around it rolls around 2 x 96 = 192 teeth and
192/36 = 5 remainder 12. It still misses

On the third time around it rolls around 3 x 96 teeth = 288 teeth for 288/36 = 8 no remainder and the pattern repeats after 3 revolutions.

There is a whole branch of mathematics devoted to division with
remainders called modular arithmetic. 8 mod 2 = 0 means: What is the
remainder when you divide 8 by 2?

In the spirograph, when the modulus is zero the pattern repeats
exactly. That is, with an 8 tooth circle and a 2 tooth wheel the
pattern will repeat after the first cycle.

However, for a 9 tooth circle, 9 mod 2 = 1 There is a remainder of 1
the pattern does not repeat after this first cycle. The second time
around however we get 18 mod 2 = 0 and the pattern made by the
spirograph repeats.

Similarly for a 36 tooth wheel inside a 72 tooth cicular hole: 72
mod 36 = 0 and the pattern repeats on the first time around.

While 96 mod 36 = 24 and the wheel does not repeat the first time
around.

2 x 96 mod 36 = 12

3 x 96 mod 36 = 0

**The arithmetic of clocks**

Modular arithmetic is the arithmetic of clocks. Consider our usual
12 hour clock, but ignore AM and PM. If you want to know what 6 hours
after 8 is then use modular arithmetic, mod 12.

8 + 6 mod 12 = 14 mod 12 = 2 and it is 2.

This is particularly interesting when crossing time zones. England
is 8 hours later than California. so to find the time in England when
it is 11 in California

11+8 mod 12 = 19 mod 12 = 7

You can also do modular arithmetic with 24 hour clocks, mod
24.

And with months mod 28, 29, 30 or 31 depending on the number of days
in the month.

**Etc.**

**Cycloids**

When a circle rolls along a surface a point on the rim of the
circle traces out a curve known as a cycloid.

A pen placed at the rim of your rolling wheel as it moves along the
straight-toothed edge of the plastic is a cycloid.

Put a light on the rim of the bicycle tire and the path it traces out
as the bicycle rolls is a cycloid.

**Illusion** **of the rolling wheel**

Many people see the light going in a circle because they mentally
follow the moving bicycle, and see the wheel just rotating, but the
actual path involves both the circular motion of the wheel plus the
linear displacement of the wheel as it rolls along the ground. The
result of the combination of the two motions is a cycloid.

The axle of the wheel moves in a straight line.

And points on the wheel between the axle and the rim move in paths
that are also cycloids.

The classic question is, Is there a point on a train that is moving
backward? The answer is yes, a point on the flange of the wheel
beneath the top of the rail moves backward.

If you slide a bead down a wire shaped like a cycloid the bead will
slide from point A to point B in the shortest time. This problem was
solved by John Bernoulli. Galileo thought the least time shape was
the arc of a circle, he was wrong. This problem of least time is one
of the most important problems of physics. It is called the
brachistochrone problem.

Copernicus modeled the planets as moving in circular orbits about the sun. This was heresy at the time. As more accurate measurements of planetary motion became available Kepler added other circles, epicycles to the circles in order to match the model the the motion of the planets. An epicycle is drawn by a circle rotating on another circle just similar to what you are drawing here. Eventually, he had to add so many circles that he abandoned epicycles and shifted to conic sections, ellipses, for planetary orbits. This is one of the greatest moments in all of science.

**Spirographs**

Spirographs are available in many models, Search through toy stores and garage sales. Don't miss the large toothed versions for children.

# Spirograph Scientific Explorations by Paul Doherty 1/14/00