Pascal's Fractals

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• Pascal's Fractals

By Penny Boyd
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Introduction to Pascals Triangle

• Number triangle which represents regularities in
  the structure of arithmetic.

• Studied by French mathematician and philosopher
  Blaise Pascal (1623-1662).

• Originally described about 500 years earlier by
  Chinese mathematician Yanghui (known as the
  Yanghui Triangle in China).

• Left and right hand borders are all 1.

• Each number is the sum of the two immediately
  above it.

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Features of Pascals Triangle

• The kth number of the nth row is known as the
  BINOMIAL COEFFICIENT C(n,k). These numbers occur
  as the coefficients in the expansion of (1x).

n
4
4
For example (14) 1 4x 6x² 4x³ 1x
corresponds to row 4.
Row 0 (n0) Row 1 (n1) Row 2 (n2) Row 3
(n3) Row 4 (n4) Row 5 (n5) Row 6 (n6)
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Sign of Binomial Coefficients

• How do we determine the sign of the entries in
  the triangle?

• Is there a pattern?

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Sierpinskis Gasket

• When Pascals Triangle is coloured like this, it
  closely resembles another important triangle from
  the world of mathematics.

• The Sierpinski Gasket belongs to a group of
  geometrical objects called FRACTALS.

• No matter how many times the triangle is
  magnified, it always looks the same. It goes on
  forever.

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Construction of Sierpinskis Gasket

• Start with a black equilateral triangle.

• Draw an upside down white triangle inside it.

• Do the same for each remaining black triangle.

• Continue in this way forever!!

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Construction of Sierpinskis Gasket
Black area 1
x 3/4
x 3/4
x 3/4
x 3/4
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Back to.
Sign of Binomial Coefficients
.in Pascals Triangle

• We want to know the probabilities of finding an
  even (white) number and finding an odd (black)
  number.

• For a Pascals Triangle with a very large number
  of rows, these two proportions can be
  approximated by those of a Sierpinski Gasket.

• So using our Sierpinski Gasket with a black area
  0
• and a white area 1

• We can see that ALMOST ALL NUMBERS ARE EVEN and
  in very large Pascals Triangles, odd numbers
  occur with a probability 0.

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Binary Notation
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Lucass Theorem

• k implies n
• (every binary digit of the bottom number k) lt
  (every binary digit n above it)
• k implies n if you NEVER get 0 only get 1
  or 1
• 1 0 1

Notation
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Lucass Theorem
C(n,k), the kth entry of the nth row of Pascals
Triangle is even if k n odd if k n
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Other Patterns

• Odd and even numbers are special cases of
  arithmetic to a modulus. We call it Mod 2. All
  even numbers are 0(mod 2) and odd numbers are 1
  (mod 2).

• What does Pascals triangle look like when we use
  different moduli?

• Lets look at an example Mod 5.

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Other Patterns
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Other Patterns
Mod 5
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