The document discusses Sierpinski's triangle, a fractal pattern named after the Polish mathematician Waclaw Sierpinski. It is constructed by recursively subdividing an equilateral triangle into smaller equilateral triangles. As the construction continues, the pattern is self-similar and can be reproduced at any magnification. Tables and examples are provided showing how the number of green and black triangles increases predictably according to the level or figure number in the Sierpinski pattern. Links are given for additional information on the mathematical properties and generation of the Sierpinski triangle fractal.

1. P A T T E R N S
1.)
Sierpinski's Triangle or the Sierpinski gasket
figure 3 figure 4
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figure 2
figure 1
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2. How many green triangles are in the fourth figure?
How many black triangles are in the fourth figure?
Figure Green Triangles
(B.)
Figure Black Triangles Is there a pattern???
1 1
2 2 3.) How many black triangles are in the 3rd figure? 4 CORRECT!
3 3
4 4
5 5
6 6
7 7
4.) If the pattern continued, how many green triangles are in the nth figure? 2 SUPER!
( 1 ) 2^(n-1)
( 2 ) 3^(n-1)
Clear Tables
History: ( 3 ) 3 x (n-1)⅓ *above answer with numericals 1-4
( 4 ) 4 x (n-2)
*Answer the table questions and watch the graphs grow!
The Sierpinski triangle, also called the Sierpinski gasket or the Sierpinski Sieve, is a fractal and
attractive fixed set with the overall shape of an equilateral triangle, subdivided recursively into
smaller equilateral triangles. Originally constructed as a curve, this is one of the basic examples
of self similar sets. Self similar sets are mathematically generated patterns that can be
reproducible at any magnification or reduction. It is named after the Polish mathematician
Waclaw Sierpinski but appeared as a decorative pattern many centuries prior to the work of
Sierpinski. For more information visit: http://en.wikipedia.org/wiki/Sierpinski_triangle
1.2
1
0.8
0.6
0.4
0.2
0
(A.) Green Triangles
0 2 4 6 8
1.2
1
0.8
0.6
0.4
0.2
0
(B.) Black Triangles
0 2 4 6 8

3. Tr ia ngul a r Number s
Complete the table :
Number Dots
1
2
3
4
5
6
7
1 2 3 4 5
Clear table
a Math anxiety PowerPoint Click here!
YouTube video (How to insert a hyperlink)
https://www.youtube.com/watch?v=rEu2K9JQzzA
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Triangular Numbers
1 2 3 4 5 6 7 8

4. Math for Teachers
• http://en.wikipedia.org/wiki/Sierpinski_triangle#mediaviewer/File:Animated_construction_o
f_Sierpinski_Triangle.gif
All Excel by: Ruth Ann Dapkus

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