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California Standards Addressed
Code CCSS Math
N-Q Number and Quantity – Quantities
A-CED Algebra – Creating Equations
Modeling Modeling
G-MG Geometry – Modeling with Geometry
G-SRT Geometry, Similarity, Right Triangles, and Trigonometry
Strand CA Visual Arts Standard (Proficient)
Creative Expression 2.1 Solve a visual arts problem that involves the effective use of
the elements of art and principles of design
2.3 Develop and refine skill in the manipulation of digital
imagery
Historical and
Cultural Context
3.2 Identify and describe the role and influence of new
technologies on contemporary works of art
Connections &
Applications
5.4 Demonstrate an understanding of the various skills of an
artist, art critic, art historian, art collector, art gallery
owner, and philosopher of art (aesthetician).
Fractals are mysterious
• A structure defined by fine
and small scales and/or
substructures
• A structure demonstrating
repetition and/or iteration
(repetition of a process)
• Shape cannot be defined
by Euclidean geometry.
• Infinitely complex (appear
similar in all levels of
magnification)
What is a fractal?
• A fractal is an object that can be broken down
into parts; each part is a reduced-size copy of
the whole. This is called “self-similarity.”
What is self-similarity?
• An object is self-similar
if it is exactly (or
approximately) similar
to itself.
• The means the whole
object has the same
shape as one or more
of its parts.
Fractals can be simple or complex
• A fractal is an object that can be broken down
into parts; each part is a reduced-size copy of
the whole.
A Koch Curve fractal
Fractals can be simple or complex
Sierpinski Carpet Fractal
The Sierpinski carpet is a fractal first described by Waclaw
Sierpinski in 1916.
Fractals are
found
everywhere
Fractals are found everywhere
Fractals are found everywhere
Fractals are
found
everywhere
Fractals are
found
everywhere
Fractals are
found
everywhere
Fractals are found everywhere
Fractals are found everywhere
Fractals are found everywhere
Fractals are found everywhere
Fractals are found everywhere
Fractals are found everywhere
Fractals are found everywhere
http://paulbourke.net/fractals/googleearth/
Fractal Art
Where do we use fractals?
• Many computer and digital game designers
use fractals to create life-like imagery
Using Fractals to generate CGI
Using Fractals to generate CGI
Using Fractals to generate CGI
The more complicated
the fractal gets, the
smoother (more
realistic) the shape
appears in CGI
Using Fractals to generate CGI
• In the following slides we will see some landscapes
that are progressively more complex. These
landscapes are NOT drawings. They are created
entirely by a computer using “fractal
interpolation.”
• This is the procedure that is used by special effects
artists to create computer generated scenes for
the big screen.
Fractal Landscapes
From: https://goo.gl/y5DjFv
This is the crudest version. “Nature made of triangles.”
From: https://goo.gl/y5DjFv
In this picture, the grid of triangles is a bit finer
From: https://goo.gl/y5DjFv
The triangles are finer still and the mountains in the distance
begin to look a bit better. From: https://goo.gl/y5DjFv
The next step. The mountains in the background
look like real mountains. From: https://goo.gl/y5DjFv
Now we add some texture to the triangles.
From: https://goo.gl/y5DjFv
Here we add some crude clouds and improve the landscape.
From: https://goo.gl/y5DjFv
Here we add some crude trees, and improve the
landscape and the clouds. From: https://goo.gl/y5DjFv
From: https://goo.gl/y5DjFv
Your project:
• Synthesize what you know of squares, fractals,
and animation to create an animated “fractal
tiling.”
Square Tiling Option 01
Sierpinski Carpet
Square Tiling Option 02
Square Tiling and Sierpinski Carpet fractals
Square Tiling Option 01
Sierpinski Carpet
Square Tiling Option 02
Project How-To Video
https://goo.gl/stIvN3 https://goo.gl/NgHnMb
https://goo.gl/YLPOSg Difficulty: 1
Difficulty: 2Difficulty: 3
Square Tiling Option 01
Sierpinski Carpet
Square Tiling Option 02
Project Exemplar and “Cheat Sheet”
Assessing Your Work

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Teacher guide animation fractals

  • 1.
  • 2. California Standards Addressed Code CCSS Math N-Q Number and Quantity – Quantities A-CED Algebra – Creating Equations Modeling Modeling G-MG Geometry – Modeling with Geometry G-SRT Geometry, Similarity, Right Triangles, and Trigonometry Strand CA Visual Arts Standard (Proficient) Creative Expression 2.1 Solve a visual arts problem that involves the effective use of the elements of art and principles of design 2.3 Develop and refine skill in the manipulation of digital imagery Historical and Cultural Context 3.2 Identify and describe the role and influence of new technologies on contemporary works of art Connections & Applications 5.4 Demonstrate an understanding of the various skills of an artist, art critic, art historian, art collector, art gallery owner, and philosopher of art (aesthetician).
  • 3. Fractals are mysterious • A structure defined by fine and small scales and/or substructures • A structure demonstrating repetition and/or iteration (repetition of a process) • Shape cannot be defined by Euclidean geometry. • Infinitely complex (appear similar in all levels of magnification)
  • 4. What is a fractal? • A fractal is an object that can be broken down into parts; each part is a reduced-size copy of the whole. This is called “self-similarity.”
  • 5. What is self-similarity? • An object is self-similar if it is exactly (or approximately) similar to itself. • The means the whole object has the same shape as one or more of its parts.
  • 6. Fractals can be simple or complex • A fractal is an object that can be broken down into parts; each part is a reduced-size copy of the whole. A Koch Curve fractal
  • 7. Fractals can be simple or complex Sierpinski Carpet Fractal The Sierpinski carpet is a fractal first described by Waclaw Sierpinski in 1916.
  • 9. Fractals are found everywhere
  • 10. Fractals are found everywhere
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  • 20. Fractals are found everywhere http://paulbourke.net/fractals/googleearth/
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  • 38. Where do we use fractals? • Many computer and digital game designers use fractals to create life-like imagery
  • 39. Using Fractals to generate CGI
  • 40. Using Fractals to generate CGI
  • 41. Using Fractals to generate CGI The more complicated the fractal gets, the smoother (more realistic) the shape appears in CGI
  • 42. Using Fractals to generate CGI
  • 43. • In the following slides we will see some landscapes that are progressively more complex. These landscapes are NOT drawings. They are created entirely by a computer using “fractal interpolation.” • This is the procedure that is used by special effects artists to create computer generated scenes for the big screen. Fractal Landscapes From: https://goo.gl/y5DjFv
  • 44. This is the crudest version. “Nature made of triangles.” From: https://goo.gl/y5DjFv
  • 45. In this picture, the grid of triangles is a bit finer From: https://goo.gl/y5DjFv
  • 46. The triangles are finer still and the mountains in the distance begin to look a bit better. From: https://goo.gl/y5DjFv
  • 47. The next step. The mountains in the background look like real mountains. From: https://goo.gl/y5DjFv
  • 48. Now we add some texture to the triangles. From: https://goo.gl/y5DjFv
  • 49. Here we add some crude clouds and improve the landscape. From: https://goo.gl/y5DjFv
  • 50. Here we add some crude trees, and improve the landscape and the clouds. From: https://goo.gl/y5DjFv
  • 52. Your project: • Synthesize what you know of squares, fractals, and animation to create an animated “fractal tiling.” Square Tiling Option 01 Sierpinski Carpet Square Tiling Option 02
  • 53. Square Tiling and Sierpinski Carpet fractals Square Tiling Option 01 Sierpinski Carpet Square Tiling Option 02
  • 54. Project How-To Video https://goo.gl/stIvN3 https://goo.gl/NgHnMb https://goo.gl/YLPOSg Difficulty: 1 Difficulty: 2Difficulty: 3 Square Tiling Option 01 Sierpinski Carpet Square Tiling Option 02
  • 55. Project Exemplar and “Cheat Sheet”