Nature. The great outdoors! Great enough for ... math! Geometrical patterns are found in nature more than you might think. Geometry is a part of everything you see, outdoors and beyond, from the flowers to the trees to the solar system.
Remember, patterns are a big part of geometry. Weather follows a pattern. Temperatures follow a pattern. The petals of flowers follow a pattern. The planets in orbit follow a pattern. In nature, there are lots of patterns!
INVESTIGATE!
Why are there so many examples of the Fibonacci pattern in nature?
fractal: a repeating or mirroring pattern that gets more complex the closer you look at it.
Can you picture the patterns you have seen on animals? Think of a snake’s skin or a lizard’s tail. Think about a raccoon tail or a lemur! What about a zebra’s stripes or a giraffe’s spots? These are all patterns found in nature.
More patterns can be found in snowflakes, clouds, and even seashells! Let’s take a look.
SNOWFLAKES AND FRACTAL PATTERNS
Have you ever looked at snowflake through a magnifying glass? What do you see? Each snowflake has its own unique pattern. The frozen water in snowflakes follows a fractal pattern.
A fractal is formed from a pattern that spirals or mirrors itself. A fractal gets more and more complex and unique the closer you look at it.
jagged fractal: a pattern that does not follow a set pathway.
When water crystallizes, it makes patterns that repeat. This is true for frost, too. Have you ever seen frost on a car windshield or on your bedroom window? These patterns are made from repeating images. Each one is its own design. They are unique, like you and your friend, your teacher, and your family. Each one is beautiful in its own way.
Where else can we find such dynamic patterns in nature? Next time there is a thunderstorm, peek out of your window. You might see a lightning bolt in the sky. Lightning is another fractal! The pathway of the lightning bolt is a jagged fractal. It zigs and zags! Each bolt is made up of jagged lines. It does not travel in straight lines.
TRY THIS!
Fractals are very easy to create. Draw a Y shape in your math journal. Then, on every branch of your Y, draw another Y. And then, on every branch of those Ys, draw more Ys. Keep doing this as long as you’d like! This is a fractal. You are repeating a pattern over and over to create a design.
meteorologist: a scientist who studies the weather.
Even the thunder that happens along with the lightning bolt has a fractal sound. The sound is caused by the superheating of the air. The time it takes the thunder to reach you ear can depend on distance. The thunder rumbles in fractal patterns.
Using your sight and your hearing, you can experience fractals in nature. What about clouds? Yes! They are fractals, too. Clouds are made up of liquid water droplets.
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DID YOU KNOW? In 1887, a giant snowflake fell in Montana! It was the largest snowflake that was ever recorded—15 inches wide! |
When the cloud gets full of water, rain will fall. The clouds form in fractal patterns in the sky. Clouds might look soft and fluffy, but when a meteorologist uses a special camera to zoom in close, they can see the water droplets that make up the cloud and spot the fractal design at work!
WILSON BENTLEY’S SNOWFLAKES
The man who first said, “No two snowflakes are alike,” was Wilson Bentley. He was born in the small town of Jericho, Vermont. He was a farmer who loved to take photos of snowflakes. Farmer Bentley paid close attention to the crystals. He was the first person to ever photograph a single snowflake, way back in 1885!
Take a look at Wilson Bentley’s snowflakes. Can you find the fractal patterns?
KEYWORD PROMPTS
Wilson Bentley’s snowflakes 
Fibonacci sequence: a series of numbers where each number is the sum of the two preceding numbers: 1, 1, 2, 3, 5, 8, 13....
FIBONACCI AND THE GOLDEN RATIO
Almost 1,000 years ago, Leonardo Fibonacci (c. 1170–1240) figured out a pattern called the Fibonacci sequence! He found a number pattern that goes like this: 1, 1, 2, 3, 5, 8, 13, 21....
Notice that 1 + 1 = 2. And 2 + 1 = 3. And 3 + 2 = 5. And 5 + 3 = 8. And 8 + 5 = 13. Do you see how the series continues? If this pattern is graphed on graphing paper, it makes a very cool spiral. The Fibonacci sequence is also called the “Golden Ratio.”
This spiral is found in nature. Notice the numbers of petals on flowers. Take a close look at leaves on a tree. Look at the outside of a pineapple. Peek at the rows of spiraling seeds in the center of a sunflower. Slice a banana and closely look at the center of the banana. Where else can you find a spiral in nature?
Sometimes, the Fibonacci sequence happens in places we can’t see with our eyes. Let’s think about the honeybee family tree. There are three kinds of honeybees—the queen bee, the worker bees, and the drone bees. The drone bees do not work. But the family tree of the drone bee follows a sequence!
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DID YOU KNOW? Sea creatures called mollusks have shells that are shaped like tubes and coils. More geometry in the sea! |
ALWAYS EAT YOUR FIBONACCI!
The Fibonacci sequence appears in vegetables! There is a vegetable called a Romanesco that is much like broccoli. It has many bumps. And on each bump, there is a spiral. There are more spirals that appear on top of those spirals! Artichokes, pineapples, and Brussels sprouts are more foods that display the Fibonacci sequence. Does this mean we can even eat geometry? Delicious!
CREDIT: JON SULLIVAN
A drone bee is a male bee with one female parent. The drone bee has two grandparents because his mother had two parents. He has three great-grandparents. His grandmother had two parents, but his grandfather only had one. How about great-great-grandparents? Five! And great-great-great-grandparents? Eight! There is the sequence so far: 1, 1, 2, 3, 5, 8. And it goes on!
We can find the Fibonacci sequence in other places, too. The nautilus shell has a special spiral that shows us Fibonacci’s sequence at its finest. The top of a conch shell also has this swirl of geometry!
molecule: a group of atoms, which are the smallest particles of matter.
crystal: a solid with its molecules arranged in a repeated pattern.
geode: a hollow piece of stone lined with crystals.
rover: a slow-moving vehicle used to explore planets.
TETRAHEDRONS IN NATURE
A tetrahedron is a 3-D shape you learned about in Chapter 2. It is a kind of pyramid with four triangular faces, six straight edges, and four vertices, or corners.
Like the Fibonacci sequence, this shape appears in nature. There are certain molecules that are in the shape of a tetrahedron! Ice crystals take the shape of tetrahedrons, as do the crystal structures in rocks and geodes. The tetrahedron shape is another form of a fractal, because it is made up of many repeating triangles.
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DID YOU KNOW? NASA has used robotic rovers to explore the planet Mars. These rovers are often in the shape of a tetrahedron. This allows for a safe landing and better movement. |
Geometry is more than just shapes and angles. Geometry is found in our bodies and on our bodies. We can make shapes and angles with our bodies. Geometry is in the classroom, the backyard, the neighborhood, and nature. Always observe the world around you. There is geometry waiting to be discovered ... and measured!
CONSIDER AND DISCUSS
It’s time to consider and discuss: Why are there so many examples of the Fibonacci pattern in nature?
FRACTAL FINDER
SUPPLIES
* magnifying glass
* binoculars
* math journal and pencil
Fractals are found in nature, and not just in snowflakes. There are many things in nature that show us fractals at their finest.
Caution: Never look directly at the sun.
1Use a magnifying glass to look at a snowflake on your jacket. Can you see a fractal pattern? Draw the pattern in your math journal. Is it easier to draw the outside edge of the snowflake or to figure out the pattern and draw the repeating parts?
2Find a pinecone and look for its fractal patterns. Draw the patterns you discover in your math journal.
3Use binoculars to peer at the clouds in the sky. In your math journal, draw the patterns you see along the edges of the clouds
4The next time a storm happens, look at a lightning bolt through binoculars. Watch how jagged it is! Then listen for the fractal rumble of thunder.
5Ask an adult to cut a pineapple in half for you. Can you find fractal patterns inside? Draw them in your math journal.
TRY THIS! Keep a weather journal. Look outside and fractal hunt every week, or even every day. Notice fractals in weather patterns and how they change depending on the weather. Are you seeing fractals or jagged fractals?
SUPPLIES
* shells with spirals, such as conch or nautilus
* graph paper
* pencil
The Golden Ratio is present in nature, and it can be reproduced easily on graph paper. When constructing the sequence, follow the number pattern carefully. Be sure to turn your paper so you can follow this sacred swirl.
1Locate shells with spirals, such as conch shells or snail shells.
2Draw the Fibonacci pattern on graph paper. Follow the pattern: 1, 1, 2, 3, 5, 8, 13, 21....
3Compare and contrast your drawing on the graph paper with a conch shell. What are some of the similarities? What is different about the two images?
4Find other examples of the Golden Ratio in nature and compare them to your diagram. Are some patterns closer to your drawing than others? Why might this be?
TRY THIS! At an aquarium, or even at the beach, you can find many diferent sea creatures. Do you notice the Golden Ratio on any parts of these creatures? Remember the pattern when you look at the number of legs!
SUPPLIES
* small sticks
* large leaves
* a cozy spot under the stars
* sunflower seed
Tents are tetrahedrons, or pyramids, we can sleep in! Tents are great for camping. They are structures that keep you warm, dry, and safe. Have you ever set up a tent from a store? Or have you ever made your own? Make a miniature tent!
1Prop the sticks up in triangular shapes. Lean the sticks together to form a pyramid. The base, or bottom, is the ground.
2Now design another tent in the shape of a tetrahedron. What are the differences between the two? Which one is taller? Which one is sturdier? Which one will hold more sleepers?
3Place leaves on the outside of the sticks of one of the tents. These are the faces of the tent!
4Put a little snack in the tent. Maybe a sunflower seed? A small animal might find a bite to eat!
TRY THIS! Make your own tent to sleep in! What materials can you use to make the tetrahedron or pyramid shape that will work best for you? What can you use as your base? How will you get your tent to stay standing when the breeze blows?
Use words from the text and the glossary to create a silly story.
•noun: a person, place, or thing
•plural noun: more than one person, place, or thing
•adjective: a word that describes a noun
•verb: an action word
•adverb: a word that describes a verb
Kelvin was observing a thunderstorm from his (NOUN) . He noticed the large (PLURAL NOUN) coming from the (ADJECTIVE) clouds above. The sky was (ADJECTIVE) and the storm was getting (ADJECTIVE) . There were large rain (PLURAL NOUN) that were (VERB WITH -ING) from the sky. The wind was roaring and the (NOUN) shook. This make Kelvin feel (ADJECTIVE) ! But he wanted to (VERB) anyway.
Suddenly, a loud clap of (NOUN) rang in Kelvin’s ears. Then, there was a (ADJECTIVE) bolt of lightning. Kelvin (ADVERB) the curtains. The wind (VERB WITH -ED) . He was done (VERB WITH -ING) out of the window. Instead, he felt like (VERB WITH -ING) under his (NOUN) ! Just when he went to his (NOUN) , his mother called, “Kelvin, the (NOUN) is over. Come have (NOUN) at the table, please.” Kelvin was (ADJECTIVE) !
