This document discusses fractals and their properties. It defines a fractal as an object that can be broken down into parts, where each part is a reduced copy of the whole (self-similarity). Fractals can be simple, like the Koch curve, or complex, like the Sierpinski carpet. Fractals are found everywhere in nature, in patterns like ferns, coastlines, and snowflakes. The document also discusses how fractals are used in computer graphics and digital effects to generate realistic landscapes and textures.

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.

8. Fractals are
found
everywhere

9. Fractals are found everywhere

10. Fractals are found everywhere

11. Fractals are
found
everywhere

12. Fractals are
found
everywhere

13. Fractals are
found
everywhere

14. Fractals are found everywhere

15. Fractals are found everywhere

16. Fractals are found everywhere

17. Fractals are found everywhere

18. Fractals are found everywhere

19. Fractals are found everywhere

20. Fractals are found everywhere
http://paulbourke.net/fractals/googleearth/

21. Fractal Art

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

51. 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”

56. Assessing Your Work