Generating Pascal's Triangle

 
  Pascal's Triangle is a well known mathematical pattern. Although this pattern was actually discovered in China, it has been named after the first Westerner to study it. According to Yunze He, "Pascal's" triangle was first developed during the Song Dynasty by a mathematician named Hue Yang.  
 



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).

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.

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:

1h 1 = 1

= 1

2h1 = 2 + 1

= 3

3h1 = 3 + 2 + 1

= 6

4h1 = 4 + 3 + 2 + 1

= 10

Start 4h2 =
step 1 4h1 + 3h1 + 2h1 + 1h1 =
step 2 10 + 6 + 3 + 1 =
step 3 20
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
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:

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.

 

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)

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:
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.
Pascal's Triangle Links
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Other Number Theory Pages at this Site:
Goldbach's Conjecture
the golden mean
measuring compositeness
factor patterns
spatial number patterns
binary lessons
division by 0 lesson
base 12 chart lesson
 
 


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