1. Fractal Geometry Fractals are important because they reveal a new area of mathematics directly relevant to the study of nature. - Ian Stewart

2. Euclidean Geometry Triangles Triangles Circles Circles Squares Squares Rectangles Rectangles Trapezoids Trapezoids Pentagons Pentagons Hexagons Hexagons Octagons Octagons Cyclinders Cyclinders

3. Can nature be described in terms of Euclidean Geometry?

4. Try drawing nature using Euclidean Geometry: A tree using cylinders??? A tree using cylinders??? A mountain range using triangles and pyramids??? A mountain range using triangles and pyramids??? Clouds using circles??? Clouds using circles??? Leaves??? Leaves??? Rocks??? Rocks??? Humans and animals with rectangles and circles??? Humans and animals with rectangles and circles???

5. Look outside… Do you see any shapes in Euclidean Geometry?

6. If so, they were more than likely man made. For example…

7. The point is this… Our world is fashioned with rough edges and non-uniform shapes. Our world is fashioned with rough edges and non-uniform shapes. Euclidean geometry describes ideal shapes which rarely occur in nature. Euclidean geometry describes ideal shapes which rarely occur in nature. Why then do we even bother with Euclidean geometry? Why then do we even bother with Euclidean geometry?

8. 1. Historically… Plato believed he could explain nature with five regular solid forms. Plato believed he could explain nature with five regular solid forms. Astronomers believed that our orbit around the sun was circular. Astronomers believed that our orbit around the sun was circular. Scientists now know that Plato’s shapes are particles and waves. Scientists now know that Plato’s shapes are particles and waves. Astronomers now know that our orbit is elliptical not circular. Astronomers now know that our orbit is elliptical not circular.

9. 2. The mathematics is relatively easy. Perimeter Perimeter Area Area Surface area Surface area Volume Volume Relationships between two shapes Relationships between two shapes For example: Square vs. shape of Mississippi

10. 3. Men and women consider a house with smooth edges and uniform shapes more beautiful than a house with rough edges and non- uniform shapes. Why is that nature is more beautiful with rough edges and non-uniform shapes and man made objects are less beautiful with rough edges and non-uniform shapes? And vice versa? Why is that nature is more beautiful with rough edges and non-uniform shapes and man made objects are less beautiful with rough edges and non-uniform shapes? And vice versa?

11. Fractals Defined. Geometry of irregular shapes which are characterized by infinite detail, infinite length, and the absence of smoothness. Geometry of irregular shapes which are characterized by infinite detail, infinite length, and the absence of smoothness. Let’s first see what a fractal is not. Let’s first see what a fractal is not.

12. A rectangle is not a fractal. When we look through a microscope at the rectangle do we see any new details. When we look through a microscope at the rectangle do we see any new details.

13. The teacher may want to put fractal pictures here. The teacher may want to put fractal pictures here.