The Mathematics of the Spirograph

(Any Excuse to Play)

Spirograph

• The Spirograph typically consists of several large outer gears (also known as wheels or rings) and many smaller gears that can either roll around the outside of the rings or fit and roll inside

Spirograph

• We will be using a Java program to simulate the mechanical version of the Spirograph

The Program

• Using Internet Explorer, navigate to the following URL:

http://www.math.dartmouth.edu/~dlittle/java/SpiroGraph/

Pattern Terminology

• Think of each pattern as a star or flower

Rotation

• In this diagram, the smaller, inner gear has completed one complete rotation

• One point of the star (or petal of the flower) will be formed for each complete rotation of the smaller gear

Revolution

• One complete revolution of the smaller gear around the larger gear is called an orbit

• In this example, it took four rotations of the smaller gear to complete one revolution

Symmetry

• Each Spirograph pattern displays two kinds of symmetry:– Rotation– Mirror (Reflection)

• How does the number of petals in a pattern affect each kind of symmetry?

Mirror Symmetry

• For patterns with an even number of petals, one axis of mirror symmetry runs from the endpoint of one petal to the endpoint of the opposite petal

• Another axis goes from the joint of two adjacent petals to the joint of the opposite two petals

Mirror Symmetry

• In the case of a pattern with an odd number of petals, one axis of mirror symmetry runs from the joint of two petals to the endpoint of the opposite petal

Axes of Mirror Symmetry

• How do we determine the number of axes of mirror symmetry in a pattern?

• Is there a difference between patterns with even and odd numbers of petals?

Rotational Symmetry

• What is the smallest angle of rotational symmetry for a Spirograph pattern?

• Is the answer the same whether the pattern has an even or odd number of petals?

Symmetry

• There’s much more to this topic.

• The Four Types of Symmetry in the Planehttp://mathforum.org/sum95/suzanne/symsusan.html

• Symmetry and Group Theoryhttp://www.ics.uci.edu/~eppstein/junkyard/sym.html

96:n Combinations

• Combinations 96:32 and 96:64 both produce patterns with the same number of petals

96:n Combinations, page 2

• Likewise with 96:24 and 96:72

96:n Combinations, page 3

• Other examples include 96:36, 96:60 and 96:84, with 8 petals each, and…

96:n Combinations, page 4

• 96:40 and 96:56 both have 12 petals

96:n Combinations, page 5

• 96:30 and 96:42 have 16 points each

96:n Combinations, page 6

• 96:45, 96:63 and 96:75 top out with 32 petals each

105:n Combinations

• Wheel 150/105 allows for even more combinations of patterns with the same number of points…

• For example, combine gears 42, 63 and 84 for 5-point patterns

105:n Combinations, page 2

105:n Combinations, page 3

• Here we have patterns with 21 points each, formed with gears 40, 50, and 80

105:n Combinations, page 4

• In the Super Spirograph set, 105:56 is the only wheel/gear combo that produces a flower with 15 petals

105:n Combinations, page 5

105:n Combinations, page 6

• 32, 52, and 64 create the designs with the most numbers of petals, 105 each

Position Zero

• Try setting Position to 0 in the Java Spirograph.

• Note that the result is a perfect circle, where with other Position settings a 105-petal flower is produced.

Gear:Wheel Ratios

• Let’s look at gear:wheel ratios to see why different combinations produce such similar curves

Reduced Denominator

• These are the same arrays as on the previous slide

• Only here, the fractions have been simplified

• Notice that the number of petals in each pattern is equal to the denominator of each fraction

The Equation

• Epicycloid• Hypocycloid

Gear / Ring Combinations

Gear / Ring Combinations

Gear / Ring Combinations

Gear / Ring Combinations

Gear / Ring Combinations

Gear / Ring Combinations

Gear / Ring Combinations

Gear / Ring Combinations

Gear / Ring Combinations

Gear / Ring Combinations

Super Spirograph

Your Assignment

• Homework: Check your closet, check your little sibling’s closet, ask your parents – I’ll bet you can find a Spirograph somewhere.

• Play with it. Try some of the things we’ve discussed. See if you can think of other mathematical topics that can be illustrated with this simple tool. Color the designs, cut them out and stick ‘em on the ‘frige. Tell your parents you’re doing your homework. Really.