Geometry in Nature

1. Geometry in Nature By Rebecca Dow, Sara Howard, Julie Russell, Jessie Buchheim, and Jordann Tomasek

2. Symmetry • Definition-The preservation of form and configuration across a point, line or a plane. • Transformations • Types of Symmetry

3. SymmetryinNature Symmetry Can Be Found All Around Us.

4. Reflective Symmetry • Also known as line symmetry, means that one half of an image is the mirror image of the other half.

5. Reflective Symmetry • Point symmetry - any straight cut through the center point divides the organism into mirroring halves.

6. Reflective Symmetry • Another example of this particular symmetry in nature, is a reflection on the water.

7. Reflective Symmetry • As we look at this photograph, the ground acts as the bisecting line between the two images.

8. Rotational Symmetry • Radial symmetry is one kind of rotational symmetry.

9. Rotational Symmetry • The planets, with slight variation due to chance, exhibit radial symmetry.

10. Rotational Symmetry • Snowflakes also provide an example of radial symmetry. • They have hexagonal symmetry around an axis.

11. Rotational Symmetry • All snowflakes have this sort of symmetry due to the way water molecules arrange themselves when ice forms.

12. Attraction to Symmetry • (Wo)man is naturally attracted to symmetry. • A face is considered beautiful when the features are symmetrical.

13. Attraction to Symmetry • Many animals choose their mates on the basis of symmetry. • The same goes for humans.

14. Polygons • Polygons are closed plane figures made up by 3 or more connecting line segments. • Simple/Non-Simple • Convex/Non-Complex

15. Polygons in Nature Have you ever stopped to consider how many inanimate things that we see in nature that are geometrically arranged? There are polygons found in nature everywhere, you just have to take a closer look!

16. Polygons found in Fruit If you slice a kiwi in half, you will see that the core forms a six-sided shape, also known as a hexagon. This is also true for an apple, except it is a slightly different version of a pentagon, it becomes a star. Look closely at a pineapple and you will see that all pineapples have the same skin, they are tessellations of trapezoids.

17. Polygons found in Plants This is an example of a polygon found within a plant. Each leaf is atriangle, a three-sided polygon. There are many types of flowers that form polygons. This tulip, has three triangles on top of three other triangles. This poppy makes the shape of a regular pentagon.

18. Animals andPolygons Most polygons found on animals are repetitive, but slightly altered. On these two cheetahs, almost all of the polygons are present, except a solid triangle. A giraffe’s body is completely covered in regular polygons. Have you ever looked closely at ahoneycomb? Each cell wall stands at a correct 120 angle with respect to one another to form a tessellation of regular hexagons.

19. On your own time: Next time you step outside, take a look around you and see what polygons you can find within nature itself. You might be surprised as to what you actually do find!

20. Tessellations • Definition- repeating pattern of distinct shapes • Regular Tessellations- tessellations of only one type of polygon • Semi-regular Tessellation- tessellation of more than one polygon.

21. Examples of Tessellations Semi-regular Tessellation Regular Tessellation

22. Tessellations in Nature • Random tessellations- vertices of all orders and polygons of all sizes and shapes • Quazi-symmetry- tile a surface without a repeating pattern • X-ray Crystallography- repeating arrangements of identical objects in nature

23. Examples of Tessellations in Nature • Division of Cells

24. More Examples • Honey Comb • Fish scales • Shell of a turtle • Pineapple • Ear of corn

25. Spheres in Nature • Geometry • Geo-Earth • Metry-measurement • Angles • Great Circle

26. The Earth • Sphere • Lines of latitude and longitude • Equator

27. Other Examples of Spheres • Sun • Moon • Planets • Oranges