November 2024
Welcome to EFM's November Newsletter!
It is essential that every caregiver in the world reads books and does math with their young children!
EFM believes in every child’s mathematical right to equity, opportunity, and personal fulfillment.
News
Dragon Curve: A Magical Math Journey – I am pleased to announce a new, free EFM annotated storybook version of Dragon Curve by Alicia Burdess. This charming story describes a girl’s fractal adventure exploring the shapes created by folding a long piece of paper. As this newsletter is all about fractals, the timing couldn’t be better.
Celebrating EFM Volunteers
Everyone at Early Family Math is an unpaid volunteer. Everyone. Last month I started recognizing some of our wonderful volunteers, and I would like to continue that tradition this month.
Ingrid, Diana, Gaby, and Maria
These four formed a powerhouse of Spanish translators for EFM. They were our first translators in any language, and they translated the bulk of our storybooks and the Activities for Families. They are from different Spanish-speaking countries, and it was fun for me to eavesdrop on their lively discussions about making word choices that would be universal across all Spanish-speaking countries.
Vasilica
He is a teacher in France who teaches students who work on translating English to French. Over two years, his classes translated the majority of the EFM storybooks into French.
Fractals
Fractals are geometric figures that have fine-grained inner structural elements that resemble the overall figure. This inner detail typically continues indefinitely, allowing one to keep zooming in further and further, continuing to see something that looks like the whole figure.
Many things in nature have fractal shapes: snowflakes, lightning bolts, nautilus shells, spiral galaxies, sunflowers, ferns, broccoli, mountains, and shorelines. There are also many human-designed shapes that have fractal characteristics. Computers have made visualizing these mathematically-defined figures easier and more awe inspiring.
Mandelbrot set
This is perhaps the most famous example of fractal behavior. This set is defined in terms of complex numbers, so I won’t go into the math for creating it. Part of what has made it so well know is the beautiful imagery, both in still pictures and movies, created by so many people showing portions of the set colored in extraordinary ways. Go exploring all over the internet for creative art done by a lot of people. Here is an image from Wikipedia made by Wolfgang Beyer with the program Ultra Fractal 3.
Koch snowflake
This shape is one of the easiest fractals to make with your child. Start with a regular triangle (all sides the same length). Break each side into 3 pieces, place a new smaller triangle on each middle piece, and then erase the middle piece. Once done, keep repeating this process with the new sides. Below are the first four steps given in an illustration from Wikipedia done by Chas_zzz_brown,Shibboleth:
In addition to enjoying the beauty of the results, there are some natural questions to ask about this curve.
How many sides are there after a certain number of steps? You start with 3. After one step you have 12. After the second step you have 48. Each step increases the number of sides by a factor of 4.
How long is the curve after a certain number of steps? This has a very surprising answer. Suppose the starting side length is 1. Then the original triangle has length 3. After one step, there are 12 sides, each of length 1/3, so the total length is 12 x 1/3 = 4. After another step, the length become 48 x 1/9 = 16/3. At each step the length increases by a factor of 4/3. If one were able to complete the task by doing an infinite number of steps, the result would be a curve of infinite length that can fit inside a 1 by 1 box!
The Lévy C curve is another fractal curve that has a construction similar to the Koch snowflake.
Dragon Curve and Alicia Burdess’ book
I will leave you to download and read Burdess’ book, Dragon Curve: A Magical Math Journey, to discover interesting properties of this curve. Here is an image of a collection of dragon curves tiling the plane contributed by Prokofiev for Wikipedia.
Sierpinski triangle, Sierpinski carpet, Sierpinski tetrahedron, and Menger Sponge
These four figures have similar constructions.
For the triangle, start with a filled in regular triangle. For the first step, divide this triangle into four equal pieces and remove the central triangle. For the second step, divide each of the three remaining triangles into four equal triangles and remove the central triangle from each of those. Continue indefinitely, or until you get tired, in this way. Here is a rendering of it done by Beojan Stanislaus contributed to Wikipedia.
The carpet starts out as a filled in square. For the first step, divide it into 9 equal squares and remove the central square of the nine. For the second step, divide each of the remaining squares into 9 even smaller squares, and remove the central square from each one. Continue in this way forever.
One surprising property of the triangle and the carpet is that they have area that is 0. In the case of the triangle, at each step the area that remains is ¾ of the previous step’s area. If you keep multiplying by ¾ over and over, the remaining area gets arbitrarily small. The same is true for the carpet, only in this case you are multiplying by 8/9 each time.
The Sierpinski tetrahedron and Menger Sponge are three dimensional versions of the triangle and carpet. They have the surprising property of not having any volume or surface area!
Here are images from Wikipedia of the first four steps in creating the Menger Sponge.
Fractional dimensions
Infinite processes can lead to some surprising results. Our intuition can often be surprised by the infinite. A great example of this was how the mathematical community was taken aback when, in 1874, Georg Cantor showed that there are different sizes of infinite sets, and that there are actually an infinite number of sizes of infinite sets. This was so controversial at the time that he was called a fraud and a charlatan by some of the most famous, established mathematicians of his time.
We have already seen that we can have an infinitely long curve in a finite region (e.g. Koch snowflake) and that we can take a sparse set of pieces out of a solid region and end up having no area left (e.g. Sierpinski figures).
People analyzing fractal figures realized that those figures didn’t act like typical figures in terms of their dimension. For example, people analyzing the coast line of Great Britain realized that the length of the coast kept getting longer the more they zoomed in and included more detail. One important idea to emerge from this work was a new idea about what “dimension” might mean.
Suppose you have a one, two, or three-dimensional figure and want to make a new version of it whose sides are twice as big.
If you take a line segment, you can use two copies of it to make it twice as big. If you start with a square, you will need to put four squares together to make a new square with sides twice as large. And if you have a cube, you will need to assemble 8 cubes together to make a new cube with sides twice as large. People decided that a natural notion of dimension was determined by a calculation involving the power that was needed for the number of pieces to increase the sides by the desired amount. A line is one dimensional because 2^1 = 2; two pieces are needed to double the length. A square is two dimensional because 2^2 = 4; four pieces are needed to double the length. A cube is three dimensional because 2^3 = 8; 8 pieces are needed to double the length.
This definition of dimension is what is used to define what is called the fractal dimension (or Hausdorff dimension) of a shape. Looking back at the Sierpinski triangle, it takes 3 copies of the triangle to make a new one with sides twice as big. To get 3, we need to raise 2 to the 1.585 power. For the carpet, it takes 8 copies of the carpet to make a new one with sides three times as big. To get 8, we need to raise 3 to the 1.8928 power!
Imagine, figures with dimension 1.585 and 1.8928!
Fractals in movies and pictures
A wonderful application of fractals is having efficient ways of creating computer-generated realistic looking images for photos and movies. Using small data sets and algorithms, realistic images of mountains, countrysides, and flames can be produced, and these can be manipulated much more easily than actual images with comparable levels of detail.
Wrapping Up
I hope you have enjoyed these beautiful images and patterns, and I hope your horizons have been expanded by these surprising ideas involving infinity. I realize this topic involves relatively little for your child to play with directly, so next month I hope to give your child a bit more to play around with. I hope you all have a wonderful Thanksgiving!
If you have any questions or comments, please send them our way! We would enjoy the opportunity to chat with you. Also, if you are interested in collaborating with us or supporting us in any fashion, we would love to talk with you about ways we can work together!
November 18, 2024
Chris Wright
Chris@EarlyFamilyMath.org
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Early Family Math is a California 501(c)(3) nonprofit corporation, #87-4441486.