Geometry in Nature

By

Rebecca Dow, Sara Howard, Julie Russell, Jessie Buchheim, and

Jordann Tomasek

Symmetry

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

• Transformations

• Types of Symmetry

Symmetry in Nature

Symmetry Can Be Found All Around Us.

Reflective Symmetry

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

Reflective Symmetry

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

Reflective Symmetry

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

Reflective Symmetry

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

Rotational Symmetry

• Radial symmetry is one kind of rotational symmetry.

Rotational Symmetry

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

Rotational Symmetry

• Snowflakes also provide an example of radial symmetry.

• They have hexagonal symmetry around an axis.

Rotational Symmetry

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

Attraction to Symmetry

• (Wo)man is naturally attracted to symmetry.

• A face is considered beautiful when the features are symmetrical.

Attraction to Symmetry

• Many animals choose their mates on the basis of symmetry.

• The same goes for humans.

Polygons

• Polygons are closed plane figures made up by 3 or more connecting line segments.

• Simple/Non-Simple

• Convex/Non-Complex

Polygons in NaturePolygons in Nature

There are polygons found in nature everywhere, you just have to take a closer look!

Have you ever stopped to consider how many inanimate things that we see in nature that are

geometrically arranged?

Polygons found in FruitPolygons found in FruitIf you slice a kiwi in half, you will see that the core forms a six-sided shape, also known as a hexagon.

Look closely at a pineapple and you will see that all pineapples have the same skin, they are tessellations of trapezoids.

This is also true for an apple, except it is a slightly different version of a pentagon, it becomes a star.

Polygons found in Polygons found in PlantsPlantsThis is an example of a polygon found within a plant. Each leaf is a triangle, 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.

Animals andAnimals and PolygonsPolygonsMost polygons found on animals are repetitive, but slightly altered. On these two cheetahs, almost all of the polygons are present, except a solid triangle.

Have you ever looked closely at a honeycomb? Each cell wall stands at a correct 120 angle with respect to one another to form a tessellation of regular hexagons.

A giraffe’s body is completely covered in regular polygons.

On your own time: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!

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.

Examples of Tessellations

Semi-regular Tessellation                        

Regular Tessellation

                  

                 

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

Examples of Tessellations in Nature

• Division of Cells

More Examples

• Honey Comb• Fish scales• Shell of a turtle• Pineapple• Ear of corn

Spheres in Nature

• Geometry– Geo-Earth– Metry-measurement

• Angles

• Great Circle

The Earth

• Sphere• Lines of latitude and

longitude• Equator

Other Examples of Spheres

• Sun• Moon• Planets• Oranges