Dimensions
I bumped into Pascal’s Triangle whilst thinking about dimensions.
I was thinking about the minimum number of points, vertices, faces, objects that were required to construct an entity in a given number of dimensions.
For instance:
A point can exist in zero dimensions.
A line consists of two points and one vertices and exists in one dimension.
A triangle consists of three points, three vertices and one face and exists in two dimensions.
A tetrahedron consists of four points, six vertices, four faces, and has one volume.
The table below should help to clarify:
| |
Entity |
Dimensions |
Points |
Edges |
Faces |
Objects |
Hyper Object |
 |
Point |
0 |
1 |
0 |
0 |
0 |
0 |
 |
Line |
1 |
2 |
1 |
0 |
0 |
0 |
 |
Triangle |
2 |
3 |
3 |
1 |
0 |
0 |
 |
Tetrahedron |
3 |
4 |
6 |
4 |
1 |
0 |
| ? |
Hyper Tetrahedron |
4 |
5 |
10 |
10 |
5 |
1 |
Notice that with the exclusion of the ‘Dimensions’ column, that the value of any number square that you pick is the sum of the number in the square above and the square above and to the left.
The numbers in the table above which are bold, form half of Pascal’s triangle. I was amazed when I realised this!
Hyper Tetrahedron
Using the table above, reading the values from the ‘Hyper Tetrahedron’ row, we can predict that a four dimensional 'Hyper Tetrahedron' would have 5 points, have 10 edges, 10 faces and have 5 objects!
It is rather difficult to visualise a 'Hyper Tetrahedron' because in our world we seem only able to perceive entities only with a maximum of three dimensions (excluding time).
The Hyper Tetrahedron is also known as a ‘Pentatope’ and as a ‘Simplex’.
Here is another image I rendered to help in the visualisation.
Pascal's Triangle
Pascal’s Triangle is pictured below:
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Number Patterns
Pascal’s Triangle contains many amazing patterns:
Powers of Two
1) The sum on the numbers in each row is equal to a power of 2.
| 1 |
=1 |
=20 |
| 1+1 |
=2 |
=21 |
| 1+2+1 |
=4 |
=22 |
| 1+3+3+1 |
=8 |
=23 |
| 1+4+6+4+1 |
=16 |
=24 |
| 1+5+10+10+5+1 |
=32 |
=25 |
| 1+6+15+20+15+6+1 |
=64 |
=26 |
| 1+7+21+35+35+21+7+1 |
=128 |
=27 |
| 1+8+28+56+70+56+28+8+1 |
=256 |
=28 |
Powers of Eleven
2) The sequence of numbers in each or the first 5 rows is equal 11 raised to a power.
| 1 |
110 |
1 |
| 11 |
111 |
11 |
| 121 |
112 |
121 |
| 1331 |
113 |
1331 |
| 14641 |
114 |
14641 |
Fibonacci
3) Adding the numbers in Pascal’s Triangle reveals the Fibonacci sequence.
Triangular Numbers
4) The ‘Triangular’ numbers are here.
The ‘Triangular’ numbers may be visualised by examining the table below. As can be seen they are effectively (1), (1+2), (1+2+3), (1+2+3+4) etc. To calculate the
nth ‘Triangular’ number, see
Sum of Series.
Square Numbers
5) Summing pairs of numbers reveals the ‘Square’ numbers.
The ‘Square’ numbers may be visualised by examining the table below.
Notice that the ‘Square’ number
n is the products of the sum of the ‘Triangular’ numbers
n +
n-1 i.e. (0+1), (1+3), (3+6), (6+10), (10+15) etc.
It is interesting to notice that nth ‘Square’ number is also the sum of the series of
n odd numbers. i.e. (1), (1+3), (1+3+5), (1+3+5+7) etc.
Sierpinski Triangle
6) If you colour the odd and even numbers in Pascal’s triangle you get this.
Here is a Pascal’s triangle with a few more rows, so you can see the pattern emerging.
The pattern produced is very similar to the
Sierpinski triangle.
Binomial Theorem
The number sequences in the rows of Pascal’s Triangle are used as the coefficients in
Binomial Theorem.