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Symmetry and Iteration RM-MSMSP Geometry Project By Lyndell Kay Thompson Summer 2006

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Consider Daisies in Nature What symmetry do you see?

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Symmetry is inspirational.

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Describe the intersection points.

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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.

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Creating Daisies Congruent circles with centers on intersection points

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Why does it work? The daisies have 60 degree rotational symmetry. Using the same radius creates equilateral triangles with all interior angles = 60 degrees

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Which daisies are different? Explain.

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Fill the plane with daisies. Repeat the pattern again and again or create a tool.

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Seashells in Nature

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What shapes do you see?

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Natural Shell Patterns

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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

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Creating the seashell

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Repeat to create the shell.

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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.

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A Model of an Iterative Shell

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Symmetry and Iteration In Nature and Geometry Happy Hunting! Patterns and Inspiration True Mathematics

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

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