Pascal's triangle: Difference between revisions

Revision as of 14:46, 23 May 2007

Revision as of 15:15, 23 May 2007

Line 96:

In other words, the sum of the entries in the ''n''th row of Pascal's triangle is the ''n''th power of 2.

==Patterns and properties==

==Properties of Pascal's triangle==

Pascal's triangle has many properties and contains many interesting patterns of numbers.

Some simple patterns are immediately apparent in Pascal's triangle:

===The diagonals===

Some simple patterns are immediately apparent in the diagonals of Pascal's triangle:

* The diagonals going along the left and right edges contain only 1's.

* The diagonals next to the edge diagonals contain the [[natural number]]s in order.

Line 106 ⟶ 110:

::<math> \textrm{tri}_1(n) = n \quad\mbox{and}\quad \textrm{tri}_{d}(n) = \sum_{i=1}^n \mathrm{tri}_{d-1}(i). </math>

An alternatealternative formula is as follows:

<math>\textrm{tri}_d(n)=\begin{cases} 1 & \mbox{if } d=0 \\ n & \mbox{if } d=1 \\ \displaystyle \frac{1}{d!}\prod_{k=0}^{d-1} (n+k) & \mbox{if } d\ge 2\end{cases}</math>

The geometric meaning of a function tri_''d'' is: tri_''d''(1) = 1 for all ''d''. Construct a ''d''-[[dimensional]] [[triangle]] (a 3-dimensional triangle is a [[tetrahedron]]) by placing additional dots below an initial dot, corresponding to tri_''d''(1) = 1. Place these dots in a manner analogous to the placement of numbers in Pascal's triangle. [[Image:SierpinskiTriangle.PNG|thumb|Sierpinski triangle]]To find tri_''d''(''x''), have a total of ''x'' dots composing the target shape. tri_''d''(''x'') then equals the total number of dots in the shape. A 1-dimensional triangle is simply a line, and therefore tri₁(''x'') = ''x'', which is the sequence of natural numbers. The number of dots in each layer corresponds to tri_{''d''&nbsp;&minus;&nbsp;1}(''x'').

[[Image:SierpinskiTriangle.PNG|thumb|Sierpinski triangle]]

The pattern obtained by coloring only the odd numbers in Pascal's triangle closely resembles the [[fractal]] called [[Sierpinski triangle]], and this resemblence becomes more an more accurate as more rows are considered. More generally, numbers could be colored differently according to whether they are multiples of 3, 4, etc.; this results in other patterns and combinations.

===Other patterns and properties===

* The pattern obtained by coloring only the odd numbers in Pascal's triangle closely resembles the [[fractal]] called [[Sierpinski triangle]], and this resemblence becomes more an more accurate as more rows are considered. More generally, numbers could be colored differently according to whether or not they are multiples of 3, 4, etc.; this results in other patterns and combinations.

The sum of the numbers in each row is a power of 2 (specifically, 2^''n'', where ''n'' is the is the number of the row, beginning with the first row as row zero, then the second as 1, etc.)

* Imagine each number connectedin the triangle is a node in a grid towhich thoseis adjacentconnected to itthe (foradjacent example,numbers aabove [[Plinkoand (Thebelow Priceit. isNow Rightfor pricingany game)|Plinko]]node gamein board).the Startinggrid, atcount the topnumber 1,of withoutpaths backtrackingthere orin goingthe sideways,grid try(without tobacktracking) getwhich toconnect anotherthis node viato thesethe gridtop pathsnode as(1) manyof waysthe as possibletriangle. The answer is whateverthe Pascal number theassociated nodeto hasthat node. The interpretation of the valuenumber in a node of Pascal's Triangle as the number of paths to that nodenumber from the tip means that on a [[Plinko (The Price is Right pricing game)|Plinko]] game board shaped like a triangle, the probability of winning prizes nearer the center will be higher than winning prizes on the edges.

The value of each row, if each number in it is considered as a "place" and numbers larger than 9 are carried over accordingly, is a power of 11 (specifically, 11^{''n''&nbsp;&minus;&nbsp;1}, where ''n'' is the number of the row). For example, Row 3 reads '1, 2, 1', which is 11^{(3&nbsp;&minus;&nbsp;1)}, or 11² (121). In row 6, '1, 5, 10, 10, 5, 1' is translated to 161051 after carrying the values over, which is 11⁵, or 11^{6&nbsp;&minus;&nbsp;1}. In the algebraic interpretation of the triangle, in which each row is simply the coefficients of the polynomial (''x''&nbsp;+&nbsp;1)^{row number}, setting ''x'' = 10 and adjusting the values to fit in the decimal number system gives the above result.

* The value of each row, if each number in it is considered as a "decimal place" and numbers larger than 9 are carried over accordingly, is a power of 11 (specifically, 11^{''n''&nbsp;&minus;&nbsp;1}, where ''n'' is the number of the row). For example, Rowrow 3two reads '1, 2, 1', which is 11^{(3&nbsp;&minus;&nbsp;1)}, or 11² (121). In row 6five, '1, 5, 10, 10, 5, 1' is translated to 161051 after carrying the values over, which is 11⁵, or 11^{6&nbsp;&minus;&nbsp;1}. InThis theproperty algebraicis interpretationeasily ofexplained theby triangle,setting in''x'' which= each10 rowin isthe simplybinomial the coefficientsexpansion of the polynomial (''x''&nbsp;+&nbsp;1)^{row number}, setting ''x'' = 10 and adjusting the values to fit in the decimal number system gives the above result.

Imagine each number connected in a grid to those adjacent to it (for example, a [[Plinko (The Price is Right pricing game)|Plinko]] game board). Starting at the top 1, without backtracking or going sideways, try to get to another node via these grid paths as many ways as possible. The answer is whatever number the node has. The interpretation of the value in a node of Pascal's Triangle as the number of paths to that node from the tip means that on a Plinko board shaped like a triangle, the probability of winning prizes nearer the center will be higher than winning prizes on the edges.

===More subtle patterns===

There are also more surprising, subtle patterns. From a single element of the triangle, a more shallow diagonal line can be formed by continually moving one element to the right, then one element to the bottom-right, or by going in the opposite direction. An example is the line with elements 1, 6, 5, 1, which starts from the row 1, 3, 3, 1 and ends three rows down. Such a "diagonal" has a sum that is a [[Fibonacci number]]. In the case of the example, the Fibonacci number is 13:

Line 185 ⟶ 192:

In this triangle, the sum of the elements of row ''m'' is equal to 3^{''m''&nbsp;&minus;&nbsp;1}. Again, to use the elements of row 5 as an example: <math>1 + 8 + 24 + 32 + 16 = 81</math>, which is equal to <math>3^4 = 81</math>.

===The matrix exponential===

[[Image:Exp_binomial_grey_wiki.png|thumb|frame|right|70px|Binomial matrix as matrix exponential (illustration for 5&times;5 matrices). All the dots represent 0.]]

===The matrix exponential===

Due to its simple construction by factorials, a very basic representation of Pascal's triangle in terms of the [[matrix exponential]] can be given: Pascal's triangle is the exponential of the matrix which has the sequence 1, 2, 3, 4, &hellip; on its subdiagonal and zero everywhere else.

{{see also|Pascal matrix}}

Due to its simple construction by factorials, a very basic representation of Pascal's triangle in terms of the [[matrix exponential]] can be given: Pascal's triangle is the exponential of the matrix which has the sequence 1, 2, 3, 4, &hellip; on its subdiagonal and zero everywhere else.

''For a fuller description of this, see the article [[Pascal matrix]].''

==History==