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.
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