Symmetry and Iteration

RM-MSMSP Geometry Project

By Lyndell Kay Thompson

Summer 2006

Consider Daisies in NatureWhat symmetry do you see?

Symmetry is inspirational.

Describe the intersection points.

Steps to Create Daisies

• Draw a circle.

• From a point on the circle, draw another circle using the same radius.

• From the point of intersection of the circles, draw another circle with the same radius.

• Repeat to create daisies.

Creating Daisies Congruent circles with centers on intersection points

Why does it work?

• The daisies have 60 degree rotational symmetry.

• Using the same radius creates equilateral triangles with all interior angles = 60 degrees

Which daisies are different? Explain.

Fill the plane with daisies.

Repeat the pattern again and again or create a tool.

Seashells in Nature

What shapes do you see?

Natural Shell Patterns

Spirals have an iterative pattern.

• Construct a segment.• Construct a perpendicular to

the segment at an endpoint• Construct a triangle.• Construct a perpendicular to

the hypotenuse at point A.• Construct a circle with radius

AP.• Use the point of intersection

as a vertex to form a new triangle.

A

P

Creating the seashell

Repeat to create the shell.

Iterating to Create the Shell

• Create a segment.• Create the right

triangle and one point.

• Select the points and map the iteration.

• Set number of repetitions and iterate.

A Model of an Iterative Shell

A Model of an Iterative Shell

Symmetry and Iteration

• In Nature and Geometry• Happy Hunting!

• Patterns and Inspiration• True Mathematics

Au revoir mes amis! Dommage! C’est la fin!