triangle

Pascal’s Triangle

Pascal’s triangle is filled with a wealth of interesting patterns, something I never learnt during secondary school when first introduced to this mathematical object. In this post, inspired by a Numberphile video, I want to show you some of these patterns that you might have never seen before.

Firstly, what is Pascal’s Triangle?

Pascal’s triangle is a triangular array of the binomial coefficients.

Pascal's triangle

To construct Pascal’s triangle, start with the two top rows, which are 1 and 1 1. To find any number in the next row, add the two numbers above it.

 

At the beginning and end of each row, where there is only one number above, write 1.

Symmetric

Pascal’s triangle is symmetric, a fact which follows directly from the formula for the binomial coefficient:

Screen Shot 2017-03-23 at 8.32.55 AM.png

It can also be seen by how the triangle is constructed, i.e. “by the interpretation of the entries as the number of ways to get from the top to a given spot in the triangle“.

Sum of entries in row n equals 2n

 

This fact can be proved by induction. The main point of the argument is that each entry in row n is added to two entries below. This gives us:

Screen Shot 2017-03-23 at 8.35.30 AM.png

Hence, the sum of entries in row n+1 is twice the sum of entries in row n.

Hockey Stick Pattern

In Pascal’s words: “In every arithmetical triangle each cell is equal to the sum of all the cells of the preceding row from its column to the first, inclusive”. This is easier to explain using a diagram:

Mathematically, this can be proved by induction and is denoted in the following way:

Screen Shot 2017-03-23 at 8.38.18 AM

Fibonacci Numbers

To see this, let us first rearrange Pascal’s triangle.

another form of Pascal's triangle

The successive Fibonacci numbers are the sums of the entries on the marked diagonals:

1 = 1

1 = 1

2 = 1 + 1

3 = 1 + 2

5 = 1 + 3 + 1

8 = 1 + 4 + 3

13 = 1 + 5 + 6 + 1

etc.

e

Recently, the Harlan brothers highlighted that e is hidden in Pascal’s triangle. This was discovered by considering products rather than sums.

 row products in Pascal Triangle

Denoting Sn as the product of the terms in the nth row, as n tends to infinity, we find that:

Screen Shot 2017-03-23 at 8.43.46 AM

Click here for the derivation.

Catalan Numbers

Catalan numbers can be found in Pascal’s triangle in a few ways, for example:

  • If you take each ‘middle’ element and subtract its adjacent entry, you get a Catalan number.

  • If you take a middle element and divide it by its position in the list of middle terms (e.g. divide the 5th middle term by 5), you will get a Catalan number.

 

There are many more patterns, and I encourage you to find out more about them! M x

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The Sierpinski Triangle

The Sierpinski Triangle, or Sierpinski Sieve, is a fractal described by Polish Mathematician Sierpinski in 1915, although it appeared in Italian art from the 13th century. It has an overall shape of an equilateral triangle, and is subdivided recursively into smaller equilateral triangles.

Sierpinski sieve from rule 90

Source: Wolfram Mathworld

Constructing a Sierpinski Triangle

STEP 1:

Start with an equilateral triangle.

STEP 2:

Connect the midpoints of each side, hence dividing it into 4 smaller congruent equilateral triangles.

Triangle

STEP 3:

Now cut out the triangle in the centre.

Step One

STEP 4:

Repeat steps 2 and 3 with each of the remaining smaller triangles.

[Sierpinski Triangle]

Properties

 

If we let N_n be the number of black triangles after iteration nL_n be the length of a side of a triangle, and A_n be the fractional area which is black after the nth iteration, then:

Screen Shot 2016-11-27 at 6.53.59 PM.png

M x