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.

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.