MATH IS FOR THE BIRDS! Part 2

Crow photo on pixabay.com

 

A Murmuration of Starlings

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A photo that I took of crows in January

As I sat on my back porch, I noticed a group of about fifteen crows flying around the top of a tall tree and making a lot of noise. I know that they do this when a racoon is up in that tree. But today, I heard the sound of a bald eagle. I wondered if the crows were working together to chase away the eagle. Sometimes a large group, or flock, of crows is called a murder of crows.


This made me think of another grouping of birds, a murmur of starlings. Starlings fly together in large numbers for safety; they are less likely to be prey for other larger birds. 

This video is taken near Vancouver by Pacificnorthwestkate. 

The flight velocity and direction of one bird affects 7 around it which multiplies throughout the group.

Like the movement of schools of fish, murmuration is an example  of scale free behaviour correlations. 

No matter how large the flock of birds is, the action of each bird will affect and will be affected by the action of all the other birds.

This means that each bird's range of perception is much larger than if they relied on direct interactions.

Murmuration has been compared mathematically  to magnets.

A three dimensional computer model shows an attempt to understand the mathematics in this flight patterning.

Dunlin birds can also form murmurations.  This video was created near Vancouver, BC for DVWildlife.

publicdomainpictures.net

Another reason that birds form patterns in flight is for migration. For example, Canada geese fly together in a V shape when travelling north in the spring and south in the fall.

JOURNAL ACTIVITY:

Observe groupings of birds. Record what you see with drawings, photos, or videos.
Why do these birds form groups?
Do all birds form flocks?
Do all birds form mating pairs?

Continue to identify and add more kinds of birds to your journal.

Do people form groups that are similar to flocks or murmurations? 
How has social distancing changed how people interact in groups?

MATH IS FOR THE BIRDS! Part 1

Black Capped Chickadee

An Invitation to 
Urban Birdwatching

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Sitting on the back steps of my home in East Vancouver earlier this week, I was visited by four different kinds of birds. I identified them using a guide to birds in Vancouver.

European Starling

Bushtit

House Finches

The house finches appear in a pair. The male has salmon coloured feathers. The female is mainly brown and grey to help her go unseen when she sits on a nest. The two of them like to sit and feed on the top of my kale plants that are going to seed. 

I listen to the birds sing and call and when I see them. I want to try to identify them by their sounds even when I can’t see them.

ACTIVITY: START A BIRD JOURNAL

Start with three or four birds that you can see from your home or on walks close to your home. Using drawings and writing, record what you notice. 

Which birds do you see?

What do they look like?

Are the male and female different?

Identify them by name? 

What do you notice them eating?

What calls and songs do you hear?
Do they move about alone, in pairs, in groups?


Start a routine of recording when you see or hear these birds. How will you record over time?

Identify and describe more birds as you notice them.

Outside Listening: Mapping Soundscapes

In May the hummingbirds arrive on Haida Gwaii. One afternoon my good friend and I sat out on her deck and were entertained for hours by the hummingbirds. We counted 17 hummingbirds sipping the nectar from two feeders stationed on the deck corners. Listen … can you hear their calls? Can you hear the beating/buzzing sounds of their wings? 

 image: Amanda Kariella on Unsplash

Soundscapes, like landscapes, tell us a story about the place. Soundscape ecologists like Bernie Krause and Hildegard Westerkamp have recorded sounds in the natural environment for almost 50 years. Over time they have mapped how the sounds of the environment have changed. In some cases the hum or buzz of urban life on land, shipping traffic on sea, and air-traffic have transformed our soundscapes.

You can participate in being a soundscape ecologist by listening to and mapping the sounds outside in your backyard, or at a nearby park, or wherever you are.

Here’s one activity you can try by yourself or with others. You will need a piece of paper, pencil, and clipboard.

Find a spot outside to sit or stand. Draw a big circle that fills your page and place a dot in the centre. The dot represents you. Now along the edge of the circle mark nearby objects in front of you, to the right and left and to the back of you as a way to record your position. In this example the tree is front the building behind, mountains to the left and a deciduous tree to the right. 

Now close your eyes and listen. Listen. Listen. Mark on your map what you hear. How far from you is the sound and from what direction? Maybe you hear the wind rustling in the trees, or a bee nearby, or the sound of bicycle, or the call of an eagle. Mark both the distance from you and the direction of your sounds.

Try making soundscape maps at the same place over different days. What do you notice? Try making soundscape maps at the same place over different times of the day. What do you notice? What do you wonder?

Unexpected shapes

The Möbius^∞ strip

Source

   Consider a strip of plain old paper like, say, a long shopping receipt. Mathematically speaking, it’s kind of boring; it’s got the simple measurements of length and width from which we can calculate the area and a right angle at each corner but not much else. However, when we crumple it up into a ball, that same piece of paper becomes something different. All the new shapes and angles that are created make it far more interesting. The study of such twisted and scrunched up objects is called topology:

Topology is the mathematical study of the properties that are preserved through deformations, twistings, and stretchings of objects 1

wonderopolis.org/

   A simple and fun example of this is the Möbius strip. If you take a long strip of paper and make one half-twist (180°) and join the ends together (glue is best for this, but tape works too), then you’ve made a Möbius strip! So what? Well, first of all, let me tell you that the surface of the strip has only one side. To prove this, put a pen anywhere on the strip and draw a line down the centre; you’ll need to rotate the strip as you go. You’ll notice that you eventually make your way back to where you started without ever taking your pen off the paper; this is because the surface has only one side – try doing this with a regular piece of paper and you’ll find it can’t be done.

   Next, take a pair of scissors and cut down this central line all the way around the strip. It seems like you’re going to cut the strip in half and end up with two strips, right? But what happens?!

  

   You still end up with one strip! But, there’s a catch: it’s not a Möbius strip anymore. How can you tell? Well, take a pen of a different color and draw down the central line again. You’ll find your way back to where you started, but this time you’ll only have marked one side – this new strip has two sides!

But, wait, there’s more…

whatdowedoallday.com

    Make another Möbius strip but this time cut it along the point one-third of the width of the strip; see the example on the left. Keep cutting all the way around. What’s going to happen this time?

You now have a short Möbius strip intertwined with a longer twisted loop!

If that was a little hard to follow, check out this explainer video:

Cutting a Möbius strip in half  | AnimatedTopology |

Source

For more mindblowing unexpected shapes watch professor Tadashi Tokieda make a square out of two circles and some interesting conjoined love hearts.

Unexpected Shapes (Part 1) - Numberphile

For older learners, check out Dr. Tokieda's lectures on Topology & Geometry

This nifty Möbius Music Box is worth a look and this Möbius strip story telling demonstration is awesome.

 BONUS: Check out the Möbius Wall

∞  Named after German mathematician, and elderly Bilbo Baggins look-alike,  August Ferdinand Möbius

Fractals 1 of 5: Beauty, fun, and more

Jitze Couperus

A fractal is any shape that repeats itself over and over and over again at different sizes and scales.You can zoom in and find the same shapes forever.

Besides being beautiful and fun, fractals are very useful! For example, they can be used to predict or analyze parts of the human body, such as the inside of our lungs or the pattern of nerve dendrites. Another important use of fractals is in image compressing, so that some of our photos don't take up too much space in our computers! It’s mostly used if an image has patterns that are repeated (fractals!) .

Drawing, weaving, beading, building…

Indigenous people have used fractals since long before modern-day mathematicians:

Dreamcatchers can have fractal designs, like

https://shop.slcc.ca/learn/the-dreamcatcher/

Look at the fractal nature of parts of our universe in artist Margaret Nazon's beading work:
http://robertthirsk.ca/2018/10/19/the-spirit-of-innovation/
https://www.nwtarts.com/artist-profile/margaret-nazon

Many African cultures have long used fractals in their architecture, art, and other aspects of their culture, as explained in Ron Eglash’s book “African Fractals - Modern Computing and Indigenous Design” https://monoskop.org/images/f/fc/Eglash_Ron_African_Fractals_Modern_Computing_and_Indigenous_Design.pdf

Fractals 2 of 5: Drawing Fractals

We can draw fractals!

https://www.youtube.com/watch?v=PU1dwwB_Im0

Try drawing these... then, see if you can come up with your own fractal 'seeds' and draw them out to make your own original fractals... and share them with us in the comments!

www.study.com

   

https://en.wikipedia.org/wiki/Pythagoras_tree_(fractal)

Can you find some fractals outside? Or maybe even inside your fridge?

More references:

https://www.wikihow.com/Draw-the-Koch-Snowflake
https://www.es.com/Shows/FantasticFractals/FantasticFractals_EducatorsGuide.pdf

Fractals 3 of 5: Fractals in plants

I went outside fractal-hunting today. I was looking at plants, and here are a few of my findings:

A red cedar branch. It's Y's all the way down!

A camelia flower, its petals going round and round.

Each branch of this shrub comes from another (thicker) branch and so on until the trunk. Below ground, its root system is like a branch system that we could draw as a fractal.

Look at this really cool photograph (done by an artist) of an autumn leaf CLOSE UP:

PAUL OOMEN / FINEARTAMERICA.COM

Are there fractals in the plants around where you live, too? 

I wonder why plants grow with repeating patterns like this...does Nature just find fractals pretty or is there more to it? What do you think?

Fractals 4 of 5: What is the shape of a cloud?

Cloud photo credits: Susan Gerofsky

 Here's a lovely thing to do together, especially on a day when the sky is blue and those fluffy cumulus clouds are moving around in the wind. Try looking up (or better yet, lying on your back in the grass) and watching the clouds.

You might try photographing or drawing them, as I've done here. But how to draw a cloud? What shape is a cloud?

We have learned the names of shapes with straight-line edges like triangles, rectangles and octagons, and curved shapes with continuous line edges like circles, ellipses and parabolas. But none of these really describes the ragged-edged (and ever-changing) shapes of clouds.

You might notice that some clouds look a lot like the maps of islands, continents and shorelines. (The bottom photo here reminds me of a map of Europe and the Mediterranean...) Shorelines are also ragged-edged and changing, more slowly, because of forces of erosion and tectonic shift. So if we can figure out what shape clouds are, we might also figure out what shape islands, continents and shorelines are!

Try looking at a small bit of a cloud you have photographed. Is there a small section of the cloud that looks almost like a miniature version of the whole cloud? If so, then you may have discovered a fractal 'seed' that can be copied bigger and bigger (and/or smaller and smaller...) to create the shape of the whole cloud! That is what is meant by a fractal: a shape that repeats itself in more or less the same way on different scales ( really tiny, small, medium, large, really large...) to create a ragged-edged and evolving shape without straight-edged boundaries.

Here is a good explanation by a meteorologist at the University of Bonn, Germany about the ways that clouds are at least partly fractals (and not always completely so, as the big central parts of clouds are not as affected by wind turbulence as the edges).

 And here is a very nice film about natural fractals, featuring the mathematician who 'discovered' fractal geometry, the late Benoît Mandelbrot!

Try drawing the shape of bumpy or ragged edge of a cloud. It might be easiest to draw it fairly large on your page. Then, on each of the 'bumps' try drawing smaller bumps that have the same shape only smaller. Do that again, three or four times. Are you getting a shape that looks a bit like the edge of a cloud -- or a shoreline?