Fractals, Fibonacci numbers in Nature 17 mai 2015

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1 1 Sommaire 1 Sommaire Presentation Fractals in nature The Von Koch curve The Sierpinski triangle The Sierpinski carpet Hilbert s fractal Cantor set The fractal of the dragon Pythagoras tree Fun with fractals Fractals in augmented reality Create an aura for your fractal overlaying a video With AurasmaStudio With your smartphone Example of a fractal created with aurasma QR code Trigger image More ideas to work on Fibonacci numbers in flowers Fibonacci numbers in the world of bees Fibonacci numbers in the rabbit s reproduction Fibonacci numbers in greek architecture Presentation Main idea This activity will lead you in the nature, to find mathematics objects. Organisation / Methodology Group of two students, at least one of them owning a smartphone!

2 This activity will use the augmented reality to present your work. Definition : Augmented reality (AR) is a live direct or indirect view of a physical, real world environment whose elements are augmented (or supplemented) by computer generated sensory input such as sound, video, graphics or GPS data. Let s have a look on the example Romeo drew last week. To work properly, you need a special application to be downloaded on your smartphone : Aurasma. Once installed, it is rather easy to use!

3 3 Fractals in nature Introduction Fractal geometry provides a new way for mathematicians and scientists to explore. Euclidean geometry many regularly shaped natural phenomena such as cubic crystals, spherical planets, and elliptic orbits. Fractal geometry models irregular objects such as coastlines, mountains,, plants, and the human brain. Models in nature are only finite approximations of. A fractal is a geometric figure resulting from beginning with an initial figure and iterating a process an number of times. This procedure is called. A fractal has irregular (rough, crinkled) edges and fractal (fractional). Clouds/nature/self similar/infinite/fractals/models/dimension/recursion Oral 3.1 The Von Koch curve The Koch Snowflake was discovered by Helge Von Koch ( ). Draw the Koch Snowflake in the following way : 1. Start with an equilateral triangle. 2. Let the length of each side be 9 units (For example, one unit of measure such as inches). 3. Remove the middle third of each side and replace each with two segments (outside the original triangle). 4. Repeat the process an infinite number of times. Activity 3.2 The Sierpinski triangle The Sierpinsky triangle is named for Waclaw Sierpinsky ( ). The following procedure describes one of many ways to generate it. 1. Start with an equilateral triangle. 2. Remove the center triangle formed by the segments joining the midpoints of the sides of the original triangle. 3. Repeat the process with the remaining three triangles. Repeat again forever. 3.3 The Sierpinski carpet

4 3.4 Hilbert s fractal Cantor set The fractal of the dragon Pythagoras tree Fun with fractals

$4 Fractals in augmented reality 4.1 Create an aura for your fractal overlaying a video 1. Each group must choose a fractal different from the other groups. Use google and the links above. 2.$

5 4 Fractals in augmented reality 4.1 Create an aura for your fractal overlaying a video 1. Each group must choose a fractal different from the other groups. Use google and the links above. 2. Create a slide for your presentation, using powerpoint (or equivalent). The slide must contain only the title of the fractal, its initial shape, the dimension if known, a photo of the mathematician and of course your name and class. 3. Example of TRIGGER IMAGE : 4. Create an aura triggered by this shape with a video of you presenting the steps of the construction of the fractal, starting from the beginning to the final shape showing the full fractal, and explaining the dimension in the case of it is easy to understand. (the steps below are illustrated in the next page with screenshots) 4.1. Make a video of you talking about the fractal (30s to 1 min), save into your GoogleDrive Make your trigger image (see the example above) Launch AurasmaStudio on your computer, sign up with your gmail address Click on Create New Aura to create your aura Upload your trigger image Add the hashtag #sectioneurobazin and make your aura or channel public Copy the link of your aura and generate a QR code : code generator.com/ 4.8. Share both your aura and your QR code by to your teacher Print your poster and your QR code.

6 4.2 With AurasmaStudio Fractals, Fibonacci numbers in Nature

8 4.3 With your smartphone Fractals, Fibonacci numbers in Nature

$5 Example of a fractal created with aurasma 5.$ 2 Trigger image Refresh your auras from the explore menu of

9 5 Example of a fractal created with aurasma 5.1 QR code Scan the QR code to download the aura 5.2 Trigger image Refresh your auras from the explore menu of the app Aurasme on your smartphone and scan this trigger image to be able to listen to Louise.

10 6 More ideas to work on Fractals, Fibonacci numbers in Nature 6.1 Fibonacci numbers in flowers sites/r.knott/fibonacci/fibnat.html#petals 6.2 Fibonacci numbers in the world of bees sites/r.knott/fibonacci/fibnat.html#bees 6.3 Fibonacci numbers in the rabbit s reproduction sites/r.knott/fibonacci/fibnat.html#rabbits 6.4 Fibonacci numbers in greek architecture orange.fr/re%20site/th%e9atre%20d%27epidaure.htm

Fractals. Materials. Pencil Paper Grid made of triangles

Fractals. Materials. Pencil Paper Grid made of triangles Fractals Overview: Fractals are new on the mathematics scene, however they are in your life every day. Cell phones use fractal antennas, doctors study fractal-based blood flow diagrams to search for cancerous