Visualizing the pascal’s triangle
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Pascal's triangle presents a fascinating number pattern constructed by adding the two numbers directly above each position, with each row revealing unique sequences like counting numbers, triangular numbers, and powers of 11. Notably, odd and even numbers reveal a pattern similar to the Sierpinski triangle, and the arrangement of the triangle can also identify the Fibonacci sequence through specific diagonal sums. Additionally, the triangle exhibits symmetry and has higher dimensional generalizations known as Pascal's pyramid and simplices.
Visualizing the pascal’s triangle
- 2. AN INTRODUCTION … One of the most interesting Number Patterns is the Pascal's Triangle (named after Blaise Pascal, a famous French Mathematician and Philosopher).
- 3. CONSTRUCTION To build the triangle, start with "1" at the top, then continue placing numbers below it in a triangular pattern. Each number is just the two numbers above it added together (except for the edges, which are all "1").
- 4. ACCORDING TO THE NCR FORMULA
- 5. DIAGONALS The first diagonal is, of course, just "1"s, and the next diagonal has the Counting Numbers (1,2,3, etc). The third diagonal has the triangular numbers (The fourth diagonal, not highlighted, has the tetrahedral numbers.)
- 6. ODDS AND EVENS If you colour the Odd and Even numbers, you end up with a pattern the same as the Sierpinski Triangle.
- 7. THE SIERPINSKI TRIANGLE The Sierpinski Triangle is an ever repeating pattern of triangles. On colouring the various odd & even numbers in the Pascal’s Triangle, we obtain a pattern similar to this
- 8. HORIZONTAL SUMS There is also a pattern in the horizontal sum of each row in the Pascal’s Triangle. It doubles each time (exponent of 2).
- 9. EXPONENTS OF 11 Each line is also the powers (exponents) of 11: • 110=1 (the first line is just a "1") • 111=11 (the second line is "1" and "1") • 112=121 (the third line is "1", "2", "1") But what happens with 115 ? Simple! The digits just overlap, like this: The same thing happens with 116 etc.
- 10. SQUARES For the second diagonal, the square of a number is equal to the sum of the numbers next to it and below both of those. Examples: • 32 = 3 + 6 = 9, • 42 = 6 + 10 = 16, • 52 = 10 + 15 = 25, • …
- 11. THE FIBONACCI SEQUENCE The Fibonacci Sequence starts "0, 1" and then continues by adding the two previous numbers, for example 3+5=8, then 5+8=13, etc If we arrange the triangle differently, it becomes easier to detect the Fibonacci sequence. The successive Fibonacci numbers are the sums of the entries on sw-ne diagonals: 1= 1 5= 1 + 3 + 1 1= 1 8= 1 + 4 + 3 2= 1 + 1 13= 1 + 5 + 6 + 1 3= 1 + 2
- 12. PRIME NUMBERS If the first element in a row is a prime number (remember, the 0th element of every row is 1), all the numbers in that row (excluding the 1's) are divisible by it. For example, in row 7 (1 7 21 35 35 21 7 1) 7, 21, and 35 are all divisible by 7.
- 13. SYMMETRY The triangle is also symmetrical. The numbers on the left side have identical matching numbers on the right side, like a mirror image. Thus, the triangle can be rewritten from left to right and it will remain identical, all due to the vertical line of symmetry along the central column.
- 14. IN THE THIRD DIMENSION Pascal's triangle has higher dimensional generalizations. The three-dimensional version is called Pascal's pyramid or Pascal's tetrahedron, while the general versions are called Pascal's simplices.