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Pascals Triangles (aka Sierpinski Triangles) are named for Blase Pascal, born in France in 1623. These mathematical triangles display a pattern of numbers where each number is the sum of the two numbers above it on the previous row. The first number to begin the triangle is a 1. The first number on each row is also a 1, which is the sum of the two numbers above it (0 + 1). The second row of the triangle would be "1 1", each 1 the sum of the the numbers above it, "0 + 1" and "1 + 0". The third row would be "1 2 1", which is calculated as 1 (0 + 1) then 2 (1 + 1), then 1 again (1 + 0). The numbers would continue to grown indifintely. To display the pattern, you could display all even numbers. This is a simple pattern generally used. The even numbered patterned displays all numbers divisible by 2 as one color, and all that are not divisible by two as another color. This pattern is the same pattern displayed in the Demonstration under "Base 2". Using Base 2, all even numbers end in a binary "0" and all odd numbers end in a binary "1". To make the math simple, whenever you add two zeros (0 + 0) or two ones (1 + 1), you get an even number. Otherwise you get an odd number (1 + 0) or (0 + 1). To make it even simplier, whenever the sum is "2" (1 + 1), subtract the base (2) from the sum (2) to get the zero, which represents all even numbers. These same calculation rules apply when using other number bases. Whenever the sum of the two numbers is greater or equal to the base, subtract the base. What is left is the Mode or remainder. This remainder is then displayed as a different color. In my Demonstration, remainder 1 (Mode 1) is represented by red, and remainder 0 (Mode 0) is represented by white. In base two, all sums will be either red (Mode 1) or white (Mode 0). In Base 2, all numbers are either odd (red - Mode 0) or even (white - Mode 1). Using other numbering bases, Red is still used to display Mode 1, white for Mode 0. Other colors are used to display other remainders. Select ALL Modes to display the every point in the triangle and a color representing the Mode (or remainder) for that point, using the selected base number. You can also display individual modes by using the Mode controller at the right. Once you have viewed the triangle formed by that mathematical base, you can then select each individual Mode for display. You cannot display a mode equal or higher than the base number. Obviously, if you are using base 5, you cannot have a remainder of 6, 7, or 8 etc. If you select a mode which is not available in that base, then all modes display. Below are three examples of Pascal's Triangles, each displaying the first 7 rows. NO BASE displays the actual values in Arabic numbers, without adjustment for base. The math involved is clearly seen. In the Base 2 example, any sums of 2 or greater, are adjusted by subtracting the base (2). The result is zeros. Notice the triagle of zeros beginning on the 5th row and extending to the 7th. This upside down triangle is prevalent throught Pascal's Triangles, because 0 + 0 will always results in 0 in any numbering base.
NO BASE BASE 2 BASE 10
1 1 1
1 1 1 1 1 1
1 2 1 1 0 1 1 2 1
1 3 3 1 1 1 1 1 1 3 3 1
1 4 6 4 1 1 0 0 0 1 1 4 6 4 1
1 5 10 10 5 1 1 1 0 0 1 1 1 5 0 0 5 1
1 6 15 20 15 6 1 1 0 1 0 1 0 1 1 6 5 0 5 6 1
In the third example, Base 10 is used, and no value can be greater than 9. This allows values to be displayed as single digits. The upside down triange here does not begin until row 6. A full "upside down triangle" results on the line number one higher than the base number. Line 11 for Base 10, Line 3 for base 2. This also occurs at the line number of the base2 plus 1. Understandable, whenever you have for example, 1000, or 10,000,000 in base 10, there is no remainders when divided by the base. Pascals Triangles are more than just cool mathematical art. There is a lot of related patterns contained within the numbers. Please visit the websites listed at the bottom for more about Pascal's Triangles. For More on Pascal's Triangles
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Interactive Pascal's Triangle is a creation of Bob Cozby. ©2004. All Rights Reserved. |