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The simplest view of Pascal's Triangle is that it may be generated
by affixing a one a either end of the new row and then generating
all numbers in between by by adding together the two numbers above
it. For example, 3 = 2+1, as shown
The numbers may also be generated by using the idea of combinations
found in probability theory. To do this assign a column and row
number to each value. Then use the combinations
formula to produce the value in question. For example : to find
the 3rd number in the 5th row use: 5C3, "5 things combined 3 at
a time." 5C3 = 5!/3!(5-3)! = 5*4*3*2*1 / 3*2*1*(2*1) = 10 (red
circle).
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The Question which drove me was, "Can Pascal's Triangle be turned into Pascal's
Plane by finding the numbers that exist outside the triangle to fill the
plane?" The goal was to find an array of numbers that fulfill the requirements
of Pascal's Triangle yet fill the plane.
The Approach I took was to create a function similar to the factorial but
based on addition. I called this function a "perfect" for perfect triangle
based on an oversimplified view I had, at that time, of triangular numbers.
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The Perfect worked as such: a first perfect was the value of the
triangular number of those dimensions:
A Perfect of higher dimensions would require that each number generated
be perfected before being added. Examples:
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1h 1 = 1
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= 1
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2h1 = 2 + 1
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= 3
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3h1 = 3 + 2 + 1
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= 6
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4h1 = 4 + 3 + 2 + 1
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= 10
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| Start |
4h2 = |
| step 1 |
4h1 + 3h1
+ 2h1 + 1h1
= |
| step 2 |
10 + 6 + 3 + 1 = |
| step 3 |
20 |
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| Start |
3h3 = |
| step 1 |
3h2 + 2h2
+ 1h2 = |
| step 2 |
(3h1+2h1+1h1)
+ (2h1+1h1)
+ 1h1 |
| step 3 |
(6+3+1) + (3+1) + 1 = |
| step 4 |
15 |
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As with the combination a perfect of order 0 is defined
to be 1: Nh0 = 1 . |
This method creates Pascal's triangle in the following arrangement:
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This method never did successfully define the items outside the
triangle. It suffered the same limitations a the combinations method
- items outside the triangle are not easily defined this way.
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It did, however raise many questions about what geometrical arrangement should
be used to plot Pascal's Triangle. Notice how the angle defined by the 1's
varies from about 60' above to exactly 90' by this method. So which angle
would be best for defining the rest of the plane?
| In the end, simply reversing the original steps would
define much of the plane. Instead of adding the two numbers above
to get the number below, subtract one of the numbers above from the
number below to get the other number. From Pascal's triangle you can
easily get back to this point (shown in spread sheet
format) |
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There does not exist a single clear solution to what
numbers must appear above the red line. This part of the plane may
be chosen arbitrarily, by picking a starting number then filling in
as required. Here are two possible solutions to Pascal's plane: |
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The numbers in red were picked arbitrarily. The rest
were calculated using the definition of Pascal's Triangle. The first
solution retains the pattern of 1's but with negatives. The second
retains the symmetry of the plane. Line of symmetry
shown in blue. |
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