Blaise Pascal (19 June 1623 – 19 August 1662) was a French mathematician, physicist, inventor, philosopher, and Catholic writer.
https://en.m.wikipedia.org/wiki/Blaise_Pascal
In mathematics, Pascal’s triangle is an infinite triangular array of the binomial coefficients which play a crucial role in probability theory, combinatorics, and algebra. In much of the Western world, it is named after the French mathematician Blaise Pascal, although other mathematicians studied it centuries before him in Persia, India, China, Germany, and Italy.
https://en.m.wikipedia.org/wiki/Pascal's_triangle
Pascal’s Triangle is an infinite triangular array of numbers beginning with a 1 at the top. Pascal’s Triangle can be constructed starting with just the 1 on the top by following one easy rule: suppose you are standing in the triangle and would like to know which number to put in the position you are standing on. Look up and to the left, then up and to the right, sum the numbers and you have the entry of Pascal’s Triangle corresponding to your current location. Rows 0 thru 12 of Pascal’s Triangle look like.
Notice if there is not a number either on the left or the right in the row above an entry then the missing number is replaced with a zero
https://pi.math.cornell.edu/~araymer/Puzzle/PascalsTriangle.pdf
Pascal’s Triangle Formula
Pascal’s Triangle is a triangular array of numbers where each number is the sum of the two numbers directly above it. The triangle starts with a “1” at the top, and each subsequent row contains the coefficients of the binomial expansion.
Here is an example of Pascal’s Triangle:
The formula to calculate any element in Pascal’s Triangle is based on binomial coefficients, denoted as:
Pascal’s triangle is a beautiful concept of probability developed by the famous mathematician Blaise Pascal, which is used to find coefficients in the expansion of any binomial expression. It is a method to know the binomial coefficients of terms of binomial expression (x + y)n, where n can be any positive integer and x, and y are real numbers.
This triangle is used in different types of probability conditions.
https://www.geeksforgeeks.org/maths/pascals-triangle-formula/
Pascal’s Triangle Pattern
Pascal’s triangle has various patterns within the triangle which were found and explained by Pascal himself or were known way before him. A few of the Pascal triangle patterns are:
The sum of values in the nth row is 2n. For example, in the 4th row 1 4 6 4 1, sum of the elements is 1 + 4 + 6 + 4 + 1 = 16 = 24.
If a row has the second element a prime number, then all the following elements in the row are divisible by that prime number (not considering 1s). ex. 1 5 10 10 5 1.
By adding the different diagonal elements of Pascal’s triangle, we get the Fibonacci series.
https://www.cuemath.com/algebra/pascals-triangle/
In 13th century, Yang Hui [杨辉] (1238–1298) presented the arithmetic triangle that is the same as Pascal’s triangle. Pascal’s triangle is called Yang Hui’s triangle in China. The “Yang Hui’s triangle” was known in China in the early 11th century by the Chinese mathematician Jia Xian [贾宪] (1010–1070).
https://www.sinosplice.com/life/archives/2012/10/15/pascals-triangle-and-chinese
Omar Khayyam, the Persian mathematician, poet, and philosopher, used a version of Pascal’s Triangle in the 11th century. He referred to it in the context of solving problems related to binomial expansion, much like Yang Hui and Jia Xian. Khayyam’s triangle, often known in Persian as Khayyam’s Triangle, was used to find binomial coefficients and was part of his work in algebra. In fact, Omar Khayyam’s contributions to both mathematics and astronomy were highly influential, and his version of Pascal’s Triangle predated Pascal’s work by several centuries.
In ancient India, a similar triangular arrangement appeared in the Meru Prastāra, a method described by Pingala, a Sanskrit grammarian and mathematician, around 300 BCE. Pingala’s work was primarily concerned with prosody—the study of poetic meters and rhythms—but he used the same triangular pattern to compute combinations of syllables in poetry. The Meru Prastāra was essentially a form of Pascal’s Triangle, although it was not used for binomial expansions in the same way as it was in China or Persia.
Fractals and Sierpinski Triangles
Pascal’s Triangle also has connections to fractals, particularly the Sierpinski Triangle, a fractal pattern named after the Polish mathematician Wacław Sierpiński. If you shade in the odd numbers in Pascal’s Triangle, a fractal pattern resembling the Sierpinski Triangle emerges. This relationship between Pascal’s Triangle and fractal geometry illustrates its relevance in modern mathematical research and its connections to other advanced fields of study.
https://www.historymath.com/pascals-triangle/
The Sierpinski Triangle is a well known example of a “large” compact set without interior points. It is defined by the following construction: Start with an equilateral triangle and subdivide it into four congruent equilateral triangles. Remove the middle one. Subdivide the remaining triangles again and remove in each the middle one. Repeat this procedure. Each step reduces the area by a factor 3/4.
https://www.virtualmathmuseum.org/Fractal/Sierpinski_triangle/Sierpinski_triangle.html
What’s odd about Pascal’s triangle?
I asked my students to use their mathematical intuition to determine whether they thought that there were more or less (or the same number of) even numbers than odd numbers in Pascal’s triangle.
Are there more or less even numbers than odd numbers, or are there the same number? Look at it and draw on your mathematical intuition: what do you reckon?
There are three possible conclusions that you might have reached:
“There are the same number of even and odd numbers.” Perhaps you’re thinking that evens and odds are in the same ratio in the set of natural numbers and so they should appear in the same ratio in the triangle.
“There are more odd numbers than even numbers.” Potentially you’ve spotted that every row starts and ends with an odd number and so you’ve surmised that that is going to skew the results in favour of the odd numbers.
“There are more even numbers than odd numbers.” This is what a lot of my students thought. Their reasoning? “Why else would you ask this question?” Since this happens to be the right answer, I guess that they know me too well. However, from the information that we currently have, this doesn’t seem like the right answer.
To gain more confidence in our answer we need more information. So, I wrote a short program that would tabulate the first twenty rows. (If you want to play with the numbers yourself, you can find my code at github.com/graeme-f/Pascal). By running the code, I was able to see that the percentage of even numbers gradually grows: by the 18th row, the total number of even numbers has exceeded the total number of odd numbers. This raised a number of questions: does the percentage of even numbers continue to increase? Does it tend to a steady value? And if so, what is that value?
I ran the code again, this time looking at more rows. By the time I had reached the hundredth row the percentage of even to odd numbers was in excess of 75%. I reached 80% on the 164th row and by the 870th row, 90% was reached.
I extended this to just over 4000 rows and by then, over 93% of the numbers were even. It was difficult to see if the trend was still growing because it wasn’t smooth, rather it came in small bumps which slowly rise and then dip down:
In order to understand the behaviour as the number of rows gets much bigger, we need to understand these bumps. The bumps can be illustrated by showing the even numbers in one colour and the odd numbers in a second colour.
This diagram has the even numbers shown by yellow shapes. Some readers may think this shape seems familiar and indeed they’d be right. The triangles here mimic those seen in the Sierpinski triangle. These inverted triangles get gradually larger and, as the triangle starts to taper off, other (smaller) triangles will appear to the side.
The diagram provides some clues as to why there are so many even numbers. If there is a row of even numbers then the next row will have one fewer even number, because an even number plus an even number will result in another even number, and the numbers on the next row are created from the sum of two numbers in the previous row. It will be one fewer because of the two edges which are always one. Conversely, if there is a row of odd numbers, then the next row will have a row of even numbers, because an odd number plus an odd number will result in an even number. Obviously, if there is an odd and an even number then that will produce an odd number, but because the edge is always one then an odd number that is generated next to the edge will become an even number in the next row. This starts to be propagated into the centre of the triangle and, as the triangle gets larger, the ones (which are odd) that appear on the edge of the triangle are less significant to the total count.
Moving down the centreline, it can be seen that there are five distinct triangles, each significantly larger than the previous one. The side length of each triangle is 1, 3, 7, 15 and 31. These are all one short of powers of two. With a little work, you can show that the size of the triangles obey the formula 2n−1 where n is the triangle number starting from one.
So that answers our question: Pascal’s triangle has more even numbers than odd numbers. But what about multiples of other numbers? Well, I’m glad you asked. Colouring the multiples of 2 yellow gave us this pattern:
Instead, colouring the multiples of 3 yellow gives:
Where the 2-triangle had triangles of size 1, 3, 7, 15 and 31, the 3-triangle has triangles of size 2, 8 and 26. This follows a pattern of 3n−1.
What is more, these come in groups of three. The number 3 is many things, but the property I will highlight is that it’s the 2nd triangle number.
Colouring the multiples of 5 yellow gives this:
The 5-triangle has triangles of size 4 and 24, following the pattern of 5n−1, and these are grouped in batches of 10—the 4th triangular number.
The multiples of 7 look this this:
The 7-triangle continues the pattern with triangles of size 6 and 48 (or 7n−1) and they come in batches of 21, the 6th triangular number.
In fact, all prime numbers appear to follow this regular pattern: triangles are all of size pn−1 , and are grouped in batches of the (n−1)th triangle number.
Composite numbers are, however, more fragmented. The 6- and 4-triangles illustrate two different characteristics of triangles based on composite numbers. First, let’s look at the 6-triangle:
The 6-triangle is a combination of the 2-triangle and the 3-triangle. Where these two overlap, a multiple will appear.
Now, look at the 4-triangle:
4 is a power of 2, so the 4-triangle has the same structure as the 2-triangle, but with more gaps. And yet, these gaps themselves form a regular pattern. This can be explored further by comparing additional powers of 2:
Looking at these triangles we see that, as the exponent of 2 increases, each yellow triangle bigger than one shape is split up. In fact, it’s split in a similar Sierpinski pattern as our multiple-of-2-triangle, but with the colours inverted. So, going from 2 to 4, a single triangle will disappear, a 3×3 triangle becomes just 3 single triangles each at the apex, a 7×7 triangle becomes three 3×3 triangles with a single triangle in the centre, and the 15×15 triangle becomes three 7×7 triangles with the centre triangle being replaced with a single 3×3 and three single triangles. This same pattern is then replicated going from 4 to 8 and 8 to 16.
Now we do what mathematicians do, and check to see if a similar pattern can be seen when looking at powers of 3. And it can! The only difference is that, now, the triangles are reduced in blocks of 2×2 triangles.
Returning to my students (remember them?), they were in awe. We had uncovered these patterns embedded deep within the triangle. Even more exciting was when they realised that the triangle itself was independent of any number-base system. Yet, it was able to highlight these characteristics of prime numbers. We saw a regular structure, a fingerprint if you will, of each prime number. It’s certainly not a meaningful way to find prime numbers, but it is a way to recognise them, a way that is independent of our number system, and perhaps most importantly, a beautiful way to discover more mathematics.
And so, as the curtain fell on another school year, I requested my students, as I request you now, to keep asking questions…because you will never know what is there to be discovered.
https://chalkdustmagazine.com/features/whats-odd-about-pascals-triangle/
Pascal’s Triangle
Kazuyo Doi
Size : 78.5 in x 85.5 in (199 cm x 217 cm)
Year : 2019
Geometric Expressions (SAQA Virtual Gallery)
I designed pascal’s triangle replacing the numbers to alphabets, such as “1“ to “A“, “2 “ to “B” , “12,870” to “SAZ” and “0” to “Z”. Each block is colored by its mathematical property. Ex. red means prime number and pink means a multiple of three. I hope you enjoy the mathematical beauty.
Materials : Cotton
Techniques : Machine pieced, machine quilted