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PATTERNS in PASCAL's
TRIANGLE
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SELF-SIMILAR
PATTERNS
Gorgeous geometric
patterns can be created from very simple ones by a
repeatitive algorithm using 'zooming' similarity.
After many repetitions, every part of such a
picture, if zoomed in, resembles the entire
picture.
If repeated infinitely,
they form fractals.
The classical examples
are
Koch's Snowflake
(Curve),
Sierpinski Carpet
(Triangle, Gasket),
Cantor's Set, and
Menger Sponge.
Amazingly, the well
known Pascal's
Triangle,which
cleverly builds up the binomial coefficients, turns
into Sierpinski Triangle - if we color all the
cells containing even numbers, as shown in the
picture in the right.
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INTERACTIVE
APPLET
Click on the picture
below for the 'big
picture'
You can color multiples
of any number, not just even numbers
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LUCIMATH
ACTIVITIES
This relationship of
Pascal's Triangle with Sierpinski' Carpet
is nicely demonstarated
in the following Units of
LUCI Math Content
Program for Teachers:
"KIDS-CAHSEE LUCI",
chapter 'PascalSierpinski'
"FUNCTIONS-1", chapter
'Sierpinski Triangle'
"FUNCTIONS-2", chapter
'Pascal's Triangle'
"FUNCTIONS-1", chapter
'Koch's Snowflake'
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EXPLORE
The picture above shows an
example of self-similarity.
Can you formulate this
rule exactly?
Instead of even numbers,
divisibles by 2, let's
color the numbers divisible by 3. Color
divisibles by 4, 5, etc... Click the buttons of the
interactive applet and enjoy the beauty of
such
generalized
Pascal-Sierpinski 'baskets'.
Try to guess the patterns you are about to see.
Multiples of prime
numbers create simple patterns. Powers of primes are slightly complex. But what
about composite
numbers? Do you see any pattern for 6?
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EXERCISES: You
can create your
own geometric fractal-like pattern starting with
some colored basic block and repeating it similarly
with increasing scale. Only one step of this
process can be done with the help of
MyFractal
Applet in the right. Self-similarity is applied
here (in a more general 'structural'
sense) with respect to the upper-left corner, using
rectangular format.
Cantor's Set...
is a special case of MyFractal (1x3, red
center).
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MyFractal
Applet
1.
Choose your 'basic block' with 'Longer' and 'Wider'
buttons.
2.
Click 'Expand" for Larger Block similar to the
basic one.
3.
Click on some cells of the basic block to
color your pattern.
Square basic
blocks with symmetric colors seem prettier.
Which
cases will correspond to
Pascal-Sierpinski patterns?
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More general and COLORFUL
patterns of Pascal's Triangle can be described by
the different remainders left after dividing by a
given number, not just divisibles only, as
above. Click here, the 'Residue'
Applet, to watch rich
rectangular 'carpets'.
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Supplement:
Find how Pascal's Triangle lesds
to famous Random Walk processes
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