Ever since I was a child, I have loved learning and been fascinated by information. That is why I decided, in Grade 12, to become a teacher. After university, I taught school for a year. Although my life and career path have taken me in other directions since then, I still love to research, learn, and impart to others what I am learning. Last night, my niece, Honey, and I, were discussing this, and she suggested that developing ideas and resources for unit studies for homeschoolers would be a good way for me to communicate about some of my interests. (Of course, the ideas could be used by classroom teachers as well, and anyone who wants to learn about whatever topic the ideas are about.) Since I love surfing the Internet anyway – let’s call this “research” – I found this idea very appealing and exciting!
Recently I have been getting interested in fractals, so I thought this would be a good topic for generating some good unit study ideas.
If you are not familiar with fractals, here is a pretty straightforward definition. I got it from Xaos – Fractal Zoomer, an application (fractal viewer) I downloaded a few days ago: “A fractal is a shape that is built from pieces each of which is approximately a reduced size copy of a whole. This process repeats itself to build the complete fractal.”
The word “fractal” was coined by Benoit Mandelbroit, who discovered fractals in the 1970s. Besides forming a new branch of mathematics (fractal geometry), fractals have a wide variety of other applications including art, economics, music, and many aspects of science.
As I have been looking at a lot of fractal art lately, I have been absolutely awestruck by their beauty, and this is the main thing I enjoy about fractals. To get a “taste” for the beauty of fractals, I recommend having a look at this video clip, which shows some fractals set to music. (More clips of fractals set to music will be included at the end of the post.)
Many examples of fractals can be found in nature, such as in broccoli, fern leaves, our lungs, forked lightning, clouds, mountain peaks, snowflakes, and coastlines. Here are two sites that show cool pictures and descriptions of fractals in nature
17 Captivating Fractals Found in Nature
This site is a really good one for introducing fractals to kids. It is well-organized and has some cool activities. Questions and teacher notes are included.
A Fractals Unit for Elementary and Middle School Students
This comprehensive site includes information on different types of fractals, 36 famous fractals, and a lot more. It also provides a very good page of reference books about fractals.
This site has a fractal math lesson on it, lots of beautiful fractal pictures, and even a link to fractal jigsaw puzzles!
On this interesting website you will find some fractal art galleries, tutorials, a page of suggestions for software to make fractals, as well as a page with a lot of links about fractals.
I would say this is the “granddaddy” of fractal sites! If you want to go deep into the mathematical aspects of fractals, this is the place to see! It includes a section of lesson plans for middle school and high school levels. (To find the lesson plans, go to the left side-bar, and select #12.)
At the above site, if you go to the left side-bar and select “3K – Fractals in Art”, and click on Decalcomania, you will find an interesting, fun activity – making fractal patterns in paint.
Here’s a cool fractal pop-up card to make:
As mentioned near the beginning of the post, here are some more video clips of fractals with music. I hope you enjoy them!
Apophysis 3D fractal animation … music Bach Badinerie BWV 1067
(This one is exceptionally beautiful – the fractals are spheres with exquisite designs on them!)
Fractal Crystal Spheres
I hope you have enjoyed looking at these websites, and that you find them useful. You may also find some helpful books about fractals at your local library or bookstore. A few that I found in the library where I live are:
Fractals – by Hans Lauwerier
The Colors of Infinity: The Beauty and Power of Fractals – with contributions by Ian Stewart, Sir Arthur C. Clarke et al.
The Fractal Geometry of Nature – by Benoit B. Mandelbroit
The Snowflake – winter’s secret beauty – by Kenneth Libbrecht; photographs by Patricia Rasmussen
If any of you would like to recommend any other helpful resources, please mention them in the Comments. Thanks!
Ever since I was a child, I have loved learning and been fascinated by information. That is why I decided, in Grade 12, to become a teacher. After university, I taught school for a year. Although my life and career path have taken me in other directions since then, I still love to research, learn, and impart to others what I am learning. Last night, my niece, Honey, and I, were discussing this, and she suggested that developing ideas and resources for unit studies for homeschoolers would be a good way for me to communicate about some of my interests. (Of course, the ideas could be used by classroom teachers as well, and anyone who wants to learn about whatever topic the ideas are about.) Since I love surfing the Internet anyway – let’s call this “research” – I found this idea very appealing and exciting!
Recently I have been getting interested in fractals, so I thought this would be a good topic for generating some good unit study ideas.
If you are not familiar with fractals, here is a pretty straightforward definition. I got it from Xaos – Fractal Zoomer, an application (fractal viewer) I downloaded a few days ago: “A fractal is a shape that is built from pieces each of which is approximately a reduced size copy of a whole. This process repeats itself to build the complete fractal.”
The word “fractal” was coined by Benoit Mandelbroit, who discovered fractals in the 1970s. Besides forming a new branch of mathematics (fractal geometry), fractals have a wide variety of other applications including art, economics, music, and many aspects of science.
As I have been looking at a lot of fractal art lately, I have been absolutely awestruck by their beauty, and this is the main thing I enjoy about fractals. To get a “taste” for the beauty of fractals, I recommend having a look at this video clip, which shows some fractals set to music. (More clips of fractals set to music will be included at the end of the post.)
Fractals to Music
Many examples of fractals can be found in nature, such as in broccoli, fern leaves, our lungs, forked lightning, clouds, mountain peaks, snowflakes, and coastlines. Here are two sites that show cool pictures and descriptions of fractals in nature
17 Captivating Fractals Found in Nature
Naturally Occurring Fractals
This site is a really good one for introducing fractals to kids. It is well-organized and has some cool activities. Questions and teacher notes are included.
A Fractals Unit for Elementary and Middle School Students
This comprehensive site includes information on different types of fractals, 36 famous fractals, and a lot more. It also provides a very good page of reference books about fractals.
Fractals Unleashed
This site has a fractal math lesson on it, lots of beautiful fractal pictures, and even a link to fractal jigsaw puzzles!
CoolMath’s Fractal Gallery
On this interesting website you will find some fractal art galleries, tutorials, a page of suggestions for software to make fractals, as well as a page with a lot of links about fractals.
FractalArts.com
I would say this is the “granddaddy” of fractal sites! If you want to go deep into the mathematical aspects of fractals, this is the place to see! It includes a section of lesson plans for middle school and high school levels. (To find the lesson plans, go to the left side-bar, and select #12.)
Fractal Geometry
At the above site, if you go to the left side-bar and select “3K – Fractals in Art”, and click on Decalcomania, you will find an interesting, fun activity – making fractal patterns in paint.
Here’s a cool fractal pop-up card to make:
Fractal Cut
As mentioned near the beginning of the post, here are some more video clips of fractals with music. I hope you enjoy them!
Apophysis 3D fractal animation … music Bach Badinerie BWV 1067
Fraktal Nr. 3
Fractal with own pictures
(This one is exceptionally beautiful – the fractals are spheres with exquisite designs on them!)
Fractal Crystal Spheres
I hope you have enjoyed looking at these websites, and that you find them useful. You may also find some helpful books about fractals at your local library or bookstore. A few that I found in the library where I live are:
Fractals – by Hans Lauwerier
The Colors of Infinity: The Beauty and Power of Fractals – with contributions by Ian Stewart, Sir Arthur C. Clarke et al.
The Fractal Geometry of Nature – by Benoit B. Mandelbroit
The Snowflake – winter’s secret beauty – by Kenneth Libbrecht; photographs by Patricia Rasmussen
If any of you would like to recommend any other helpful resources, please mention them in the Comments. Thanks!
Thanks for all the information!