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SUNFLOWER BLUEBIRD Math Teachers’ Circle

Fractal Dimension

of Pomo Baskets

May 2023

By Donna Fernandez and Maria Droujkova

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Dimensions: SPACE

Move up and down

Move the whole body or one hand. Jump or stretch or make up your own motion.

Move forward and backward

Slide, step, jump, or ride your wheeled chair.

Move left and right

Again, make up your own motions.

Image: Math in Your Feet

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Dimensions: SPACE

🍏🍏🍏 When we move in space, we boost our spatial and visual mathematical reasoning.

Move up and down

Move the whole body or one hand. Jump or stretch or make up your own motion.

Move forward and backward

Slide, step, jump, or ride your wheeled chair.

Move left and right

Again, make up your own motions.

Move up and down

Move the whole body or one hand. Jump or stretch or make up your own motion.

Move forward and backward

Slide, step, jump, or ride your wheeled chair.

Move left and right

Again, make up your own motions.

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Dimensions: Our 3D Space and 3D Objects

Pick and hold an object, any object

Anything you can hold is 3-dimensional, like our space.

Imagine a point inside your object. From it, imagine moving:�up-down�forward-backward�left-right�while you remain inside your object.

Image: Apple.com

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🍏🍏🍏 Ask your students: Does everything have to be 3D? What other numbers of dimensions are possible? This is not a hard question for most, and it gets them talking!

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Dimensions: 2D Space and 2D Objects

Move up and down

Imagine that you can only:

Move forward and backward

Move left and right

Look around you. What can you use to model 2D objects?

Image: Flatland the Movie https://youtu.be/C8oiwnNlyE4

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2D: Does it Have to be FLATland?

🍏🍏🍏 Students often come up with the floor, a book cover, a tabletop, or a screen to model 2D objects. They believe 2D=FLAT. How would you help students see why mathematicians think of any surface as 2D?

Images: Wolfram Math World

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Dimensions: 1D Space and 1D Objects

Move up and down

Move forward and backward

Imagine that you can only:

Move left and right

Look around you. What physical objects can you use to model 1D lines and curves?

Images: Wolfram Math World

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Spatial Dimensions: 1D, 2D, 3D… Any Other Number?

Images: Robert Webb, London Institute for Mathematical Sciences, Marvel

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Problem: Hard to Visualize >3D in Space

All Disney's money couldn't buy a good picture of a tesseract.

That's only 4D, four spatial dimensions!

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Data Dimensions: Mr. Potato Head

Data dimension: a category for values

Dimension	Values
Ears	Pink, blue
Top	Hat, hair
…find more dimensions	…

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Data Dimensions: Features, Creatures, Experiences…

Easy to imagine any number of data dimensions
Visual, social, or contextual reasoning

Images: Chernoff faces from Wolfram MathWorld, ePic

41D

12D

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Data Dimensions: Your Example

Choose something you like, such as:

An experience
A thing
A place
An art piece
A math object
…

Think of a few data dimensions you could notice, evaluate, or measure in your thing. Put your example in chat.

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Dimensions so Far

Spatial dimensions ✔

Data dimensions ✔

1D, 2D, 3D, 4D, 5D… 13D… 41D… = natural numbers ✔

Can dimension be any other number?

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Inspiration: Pomo Baskets

Photo: Donna Fernandez; beginning of Donna's basket; a Pomo basket from California Academy of Sciences museum

Pomo baskets come in many beautiful styles and variations. They are complex, both artistically and mathematically. In each activity below, you will make a mathematical model called a fractal. Each fractal relates to a Pomo design.

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Model: Dragon Curve Fractal

Images: CutOutFoldUp.com

Cut a strip of paper about an inch wide. Fold the strip in half.
Fold it in half again in the same direction—that is, iterate.
Then iterate again, and again: 4 times total.
Unfold and crease every angle to be 90° (a right angle).
(Optional) Combine several folded pieces to make a big model!

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Imagine: Dragon Curve Fractal

Imagine iterating the dragon curve model again and again. Then watch animations.

https://en.wikipedia.org/wiki/Dragon_curve

https://youtu.be/UBuPWdSbyf8

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Dragon Curve Fractal Dimension

Dragon curve is a space-filling curve.

Weird and awesome fact:

this curve fills a 2D shape entirely!

The dragon curve is two-dimensional.

But we had already seen a lot of 2D things.

Let's try more fractals and hope for even weirder dimensions.

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Model: Koch Snowflake Fractal

Start with a triangle. It's made of 3 segments. Let's say each is 1 unit long.
Divide each segment into thirds.
Replace the middle third with a bump: two sides of a smaller triangle.
How many segments do you have?
How long is your new shape?

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Imagine: Koch Snowflake Fractal

Imagine infinite iterations of Koch Snowflake.
Do you feel this is a space-filling curve, like Dragon Curve?

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Make: A Koch-Style Fractal of Your Own

Design a fractal curve of your own.
What to try:

Different numbers of small triangles
Squares or other shapes instead of triangles
Turn your small shapes inside or outside

Pomo basket inspiration: Annie Burke, California State Library

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Make: A Sierpinski-Style Substitution Fractal

Start with an equilateral triangle or a square.
Split it into some smaller, equal triangles or squares.
Color some of your smaller shapes.

Pomo basket inspiration: So'-kah-dam, UC Davis; Oakland museum of California

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Make: A Sierpinski-Style Substitution Fractal

Start with an equilateral triangle or a square.
Split it into some smaller, equal triangles or squares.
Color some of your smaller shapes.
What's the linear scaling factor between your small and big shapes?
How many smaller shapes did you color?

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Imagine: A Sierpinski-Style Substitution Fractal

Imagine replacing each of your colored smaller shapes with the scaled-down copy of the big shape. Iterate again and again.
How "filled in" is your triangle or square?

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A Fractal Way to Calculate Dimension

[scaling factor]^D= [number of small copies]

Let us try with a completely filled square, split into smaller squares.

Scaling factor =

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A Fractal Way to Calculate Dimension

[scaling factor]^D= [number of small copies]

Scaling factor = 7

Number of small copies =

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A Fractal Way to Calculate Dimension

[scaling factor]^D= [number of small copies]

Scaling factor = 7

Number of small copies = 49

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A Fractal Way to Calculate Dimension

[scaling factor]^D= [number of small copies]

Scaling factor = 7

Number of small copies = 49

7^D=49

D =

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A Fractal Way to Calculate Dimension

[scaling factor]^D= [number of small copies]

Scaling factor = 7

Number of small copies = 49

7^D=49

D =2 So, a square is a two-dimensional shape.

(We knew that!)

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A Fractal Way to Calculate Dimension

[scaling factor]^D= [number of small copies]

Let us try with a completely filled triangle!

Scaling factor =A Fractal Way to Calculate Dimension

Image: Panopticonopolis

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A Fractal Way to Calculate Dimension

[scaling factor]^D= [number of small copies]

Scaling factor = 2

Number of small copies =

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A Fractal Way to Calculate Dimension

[scaling factor]^D= [number of small copies]

Scaling factor = 2

Number of small copies = 4

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A Fractal Way to Calculate Dimension

[scaling factor]^D= [number of small copies]

Scaling factor = 2

Number of small copies = 4

2^D=4

D =

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A Fractal Way to Calculate Dimension

[scaling factor]^D= [number of small copies]

Let us try with a completely filled triangle!

Scaling factor = 2

Number of small copies = 4

2^D=4

D =2 So, a triangle is a two-dimensional shape.

(We knew that!)

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A Fractal Way to Calculate Dimension

[scaling factor]^D= [number of small copies]

Let us try with a line and a cube, too.

In these pictures, the linear scaling factor is 3 for every shape.

What is D?

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Calculate the Fractal Dimension of Your Own Fractals

[scaling factor]^D= [number of small copies]

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Fractal Dimension of the So'-kah-dam Pomo Basket Fractal

[scaling factor]^D= [number of small copies]

5^D= 16

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Fractal Dimension of the Pomo basket = 1.72270623229…

[scaling factor]^D= [number of small copies]

5^D= 16

log(5^D) = log(16)

D = log(16)/log(5)

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Fractal Dimension of the Pomo basket = 1.72270623229…

🍏🍏🍏 Younger students who don't know logarithms can still play guess-and-check with a grapher, spreadsheet, or calculator.

[scaling factor]^D= [number of small copies]

5^D= 16

log(5^D) = log(16)

D = log(16)/log(5)

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