Pascal's triangle and Father Brown: Difference between pages

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[[Image:Father Brown.JPG|right|thumb|175px|The ''Father Brown'' stories by G.K. Chesterton, Penguin Books edition 1981]]

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'''Father Brown''' is a fictional detective created by English novelist [[G. K. Chesterton]], who stars in 51 [[short story|short stories]], later compiled in five books. Chesterton based the character on Father John O'Connor (1870 - 1952), a [[parish priest]] in [[Bradford, Yorkshire]], who was involved in Chesterton's conversion to Catholicism in 1922. The relationship was recorded by O'Connor (by then [[Monsignor]]) in his 1937 book ''Father Brown on Chesterton''.

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<math>

\begin{matrix}

&&&&&1\\

&&&&1&&1\\

&&&1&&2&&1\\

&&1&&3&&3&&1\\

&1&&4&&6&&4&&1

\end{matrix}

</math>

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The first five rows of Pascal's triangle</div>

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In [[mathematics]], '''Pascal's triangle''' is a geometric arrangement of the '''[[binomial coefficient]]s''' in a [[triangle]]. It is named after [[Blaise Pascal]] in much of the western world, although others studied it centuries before him in [[History of India|India]], [[History of Iran|Persia]], [[China]], and [[Italy]].<ref>[http://www.jstor.org/view/0024094x/ap050069/05a00460/0]</ref><ref>[http://education.uncc.edu/cmste/summer/2006%20History%20of%20Mathematics/Andrew.doc ''Pascal’s Triangle'' by Andrew Samuels]</ref>

== Character ==

The rows of Pascal's triangle are conventionally enumerated starting with row zero, and the numbers in odd rows are usually staggered relative to the numbers in even rows. A simple construction of the triangle proceeds in the following manner. On the zeroth row, write only the number 1. Then, to construct the elements of following rows, add the number directly above and to the left with the number directly above and to the right to find the new value. If either the number to the right or left is not present, substitute a zero in its place. For example, the first number in the first row is 0 + 1 = 1, whereas the numbers 1 and 3 in the third row are added to produce the number 4 in the fourth row.

Father Brown is a short, stumpy [[Roman Catholic Church|Catholic]] [[priest]], "formerly of Cobhole in [[Essex]], and now working in [[London]]", with shapeless clothes and a large umbrella, but an uncanny insight into human evil.

He makes his first appearance in the famous story "[[The Blue Cross (fiction)|The Blue Cross]]" and continues through the five volumes of short stories, often assisted by the reformed criminal [[Flambeau (character)|Flambeau]]. Father Brown also appears in a story "The Donnington Affair" that have a rather curious history. In the October 1914 issue of obscure magazine ''The Premier'' Sir [[Max Pemberton]] published the first part of the story inviting the number of detective story writers, including Chesterton, to use their talents to solve the mystery of the murder described. Chesterton and Father brown's solution followed in the November issue. The story was first reprinted in the ''Chesterton Review'' (Winter 1981, p.1-35) and in the book <ref name="Thirteen detectives">{{cite book |author=G.K.Chesterton| title=Thiteeen Detectives |editor=Smith, Marie|origyear=1987||publisher= Xanadu|___location=London|id= ISBN 0-947761-23-3}} </ref>.

This construction is related to the binomial coefficients by [[Pascal's rule]], which states that if

:<math> {n \choose k} = \frac{n!}{k! (n-k)!} </math>

is the ''k''th binomial coefficient in the [[binomial expansion]] of (''x''+''y'')^''n'', then

Unlike his more famous co-[[detective]] [[Sherlock Holmes]], Father Brown's methods tend to be intuitive rather than deductive. He explains his method in "The Secret of Father Brown": "You see, I had murdered them all myself... I had planned out each of the crimes very carefully. I had thought out exactly how a thing like that could be done, and in what style or state of mind a man could really do it. And when I was quite sure that I felt exactly like the murderer myself, of course I knew who he was."

:<math> {n \choose k} = {n-1 \choose k-1} + {n-1 \choose k}</math>

Father Brown's abilities are also considerably shaped by his experience as a priest and [[confessor]]. In "The Blue Cross", when asked by Flambeau, who has been masquerading as a priest, how he knew of all sorts of criminal "horrors", he responds: "Has it never struck you that a man who does next to nothing but hear men's real sins is not likely to be wholly unaware of human evil?". He also states a reason why he knew Flambeau was not a priest: "You attacked reason. It's bad theology." And indeed, the stories normally contain a rational explanation of who the murderer was and how Brown worked it out.

for any nonnegative integer ''n'' and any integer ''k'' between 0 and ''n''.<ref>The binomial coefficient <math>\scriptstyle {n \choose k}</math> is conventionally set to zero if ''k'' is either less than zero or greater than ''n''.</ref>

Despite his devotion, or perhaps, because of it, Father Brown always emphasises [[rationality]]: some stories, such as "The Miracle of Moon Crescent" and "The Blast of the Book", poke fun at initially [[Scepticism|sceptical]] characters who become convinced of a [[supernatural]] explanation for some strange occurrence, while Father Brown, despite, or rather because of, his religion and his belief in [[God]] and [[miracle]]s, easily sees the perfectly ordinary, natural explanation. In fact, he seems to represent an ideal of a devout, yet considerably educated and "civilised" clergyman. This can be traced to the influence of [[Neo-Scholasticism|neo-scholastic]] thought on Chesterton.

Pascal's triangle has higher [[dimension|dimensional]] generalizations. The three-dimensional version is called ''[[Pascal's pyramid]]'' or ''Pascal's tetrahedron'', while the general versions are called ''[[Pascal's simplex|Pascal's simplices]]'' &mdash; see also [[pyramid]], [[tetrahedron]], and [[simplex]].

==Interpretations and criticism==

==The triangle==

Father Brown was the perfect vehicle for conveying [[G. K. Chesterton|Chesterton]]'s view of the world, and of all of his characters, is perhaps closest to Chesterton's own point of view, or at least the effect of his point of view. Father Brown solves his crimes through a strict reasoning process more concerned with spiritual and philosophic truths rather than scientific details, making him an almost equal counterbalance with [[Sherlock Holmes]], which Chesterton read and admired, the stories of which had been discontinued just a couple of years before.

==Father Brown in other media==

Here are first seventeen rows of Pascal's triangle:

*[[Walter Connolly]] starred as the sleuthing priest in the [[1934]] film ''Father Brown, Detective'', based on "The Blue Cross." Interestingly, Connolly would later be cast as another famous fictional detective, [[Rex Stout]]'s [[Nero Wolfe]], in the [[1937]] film, ''[[The League of Frightened Men (1937 film)|The League of Frightened Men]]''.

*A [[Mutual Broadcasting System]] radio series, ''[[The Adventures of Father Brown]]'' (1945) featured [[Karl Swenson]] as Father Brown, Bill Griffis as Flambeau and Gretchen Douglas as Nora, the rectory housekeeper.<ref name="Radio Programs, 1924-1984">{{cite book |author=Terrace, Vincent| title=Radio Programs, 1924-1984:A Catalog of Over 1800 Shows|origyear=1999||publisher= McFarland|___location=Jefferson, NC|id= ISBN 0-7864-0351-9}} </ref>

*A [[1954 in film|1954]] [[Father Brown (film) (1954)|film of Father Brown]] (released in the USA as ''The Detective'') which had a formidable cast, with Sir [[Alec Guinness]] playing the part of Father Brown, is widely regarded as a minor classic. Like the 1934 film starring Connolly, it was based on Chesterton's first Brown short story, "The Blue Cross." The experience of playing the character prompted Guinness's [[conversion]] to [[Catholicism]].<ref> [http://www.catholicculture.org/docs/doc_view.cfm?recnum=6679 How Father Brown Led Sir Alec Guinness to the Church] (Como el Padre Brwon llevo a Sir Alec Guinness a la Iglesia)</ref><ref name=guardianfilm>{{cite news

1

| last =Sutcliffe

1 1

| first =Tom

1 2 1

| coauthors =

1 3 3 1

| title =Sir Alec Guinness obituary

1 4 6 4 1

| work =[[The Guardian|Guardian]]

1 5 10 10 5 1

| pages =

1 6 15 20 15 6 1

| language =

1 7 21 35 35 21 7 1

| publisher =

1 8 28 56 70 56 28 8 1

| date =[[2000-08-07]]

1 9 36 84 126 126 84 36 9 1

| url =http://film.guardian.co.uk/News_Story/Guardian/0,4029,351452,00.html

1 10 45 120 210 252 210 120 45 10 1

| accessdate = 2007-02-28 }}</ref>.

1 11 55 165 330 462 462 330 165 55 11 1

1 12 66 220 495 792 924 792 495 220 66 12 1

1 13 78 286 715 1287 1716 1716 1287 715 286 78 13 1

1 14 91 364 1001 2002 3003 3432 3003 2002 1001 364 91 14 1

1 15 105 455 1365 3003 5005 6435 6435 5005 3003 1365 455 105 15 1

1 16 120 560 1820 4368 8008 11440 12870 11440 8008 4368 1820 560 120 16 1

*[[Heinz Rühmann]] played Father Brown in two [[Germany|German]] adaptations of Chesterton's stories, ''Das schwarze Schaf'' (The black sheep) ([[1960 in film|1960]]) and ''Er kanns nicht lassen'' (He can't stop to do it) ([[1962 in film|1962]]) with both music-scores written by composer [[Martin Böttcher]]. Later he played ''Operazione San Pietro'' ([[1967 in film|1967]]) as Cardinal Brown. <!-- http://www.imdb.com/title/tt0061324/ -->

==Uses of Pascal's triangle==

*A German television series superficially based on the character of Father Brown, ''Pfarrer Braun'', was launched in [[2003 in television|2003]]. Pfarrer Guido Braun, from [[Bavaria]], played by [[Ottfried Fischer]], solves murder cases in the (fictitious) island of Nordersand (Northsea-island) in the first two episodes. Later other German landscapes like the [[Harz]], the [[Rhine]] or [[Meißen]] in [[Saxony]] became sets for the show. Martin Böttcher again wrote the score and he got the instruction by the producers to write a title-theme hinting at the theme of the cinema-movies with [[Heinz Rühmann]]. To date nine shows have been made, which ran very successfully in German [[ARD (broadcaster)|ARD]].

Pascal's triangle has many uses in binomial expansions. For example:

*In [[1974 in television|1974]], [[Kenneth More]] starred in a 13-episode ''Father Brown'' TV series, each episode adapted from one of Chesterton's short stories. The series, produced by Sir [[Lew Grade]] for [[ATV]], was shown in the United States as part of [[PBS]]'s ''[[Mystery!]]''.

:(''x'' + ''y'')² = ''x''² + 2''xy'' + ''y''² = '''1'''''x''²''y''⁰ + '''2'''''x''¹''y''¹ + '''1'''''x''⁰''y''².

*An American TV movie, ''[http://www.imdb.com/title/tt0079850/ Sanctuary of Fear]'' (1979), starred [[Barnard Hughes]] as an Americanized, modernized Father Brown in [[Manhattan]], [[New York City]]. The film was intended as the pilot for a series but critical and audience reaction was unfavorable, largely due to the changes made to the character, and the mundane [[Thriller (genre)|thriller]] plot.

Notice the coefficients are the third row of Pascal's triangle: 1,&nbsp;2,&nbsp;1.

In general, when a binomial is raised to a positive integer power we have:

*An italian television series entitled ''I racconti di padre Brown'' ("The Tales of Father Brown") starred the well-known italian comedian [[Renato Rascel]].

:(''x'' + ''y'')^''n'' = ''a''₀''x''^''n'' + ''a''₁''x''^{''n''&minus;1}''y'' + ''a''₂''x''^{''n''&minus;2}''y''² + &hellip; + ''a''_{''n''&minus;1}''xy''^{''n''&minus;1} + ''a''_''n''''y''^''n'',

*A series of 16 Chesterton stories has been produced by the Colonial Radio Theatre in [[Boston]]. J T Turner plays Father Brown, all scripts are written by British radio dramatist [[M J Elliott]].

where the coefficients ''a''_''i'' in this expansion are precisely the numbers on row ''n''&nbsp;+&nbsp;1 of Pascal's triangle. Thus:

===Trivia===

:<math>a_i = {n \choose i}.</math>

While on ___location in [[Mâcon]], France, filming the Father Brown movie, actor [[Alec Guinness]] was approached by a young French boy who, seeing him in costume, exclaimed 'Mon père!' and, hanging on his arm, chatted away in French. Eventually, the boy said goodbye and left. Guinness, amazed and impressed that the [[cassock]] of a priest could inspire such trust and happiness in a child who was such a complete stranger, started to investigate the [[Roman Catholic]] faith, and subsequently joined the [[Catholic Church]].<ref name=guardianfilm/>

== Compilation books ==

This is the [[binomial theorem]].

{{col-begin}}

{{col-2}}

1. ''The Innocence Of Father Brown'' (1911)

:#The Blue Cross

:#The Secret Garden

:#The Queer Feet

:#The Flying Stars

:#The Invisible Man

:#The Honour of Israel Gow

:#The Wrong Shape

:#The Sins of Prince Saradine

:#The Hammer of God

:#The Eye of Apollo

:#The Sign of the Broken Sword

:#The Three Tools of Death

2.''The Wisdom Of Father Brown'' (1914)

:#The Absence of Mr Glass

:#The Paradise of Thieves

:#The Duel of Dr Hirsch

:#The Man in the Passage

:#The Mistake of the Machine

:#The Head of Caesar

:#The Purple Wig

:#The Perishing of the Pendragons

:#The God of the Gongs

:#The Salad of Colonel Cray

:#The Strange Crime of John Boulnois

:#The Fairy Tale of Father Brown

3. ''The Incredulity Of Father Brown'' (1926)

:#The Resurrection of Father Brown

:#The Arrow of Heaven

:#The Oracle of the Dog

:#The Miracle of Moon Crescent

:#The Curse of the Golden Cross

:#The Dagger with Wings

:#The Doom of the Darnaways

:#The Ghost of Gideon Wise

{{col-2}}

4.''The Secret Of Father Brown'' (1927)

:#The Secret of Father Brown

:#The Mirror of the Magistrate

:#The Man With Two Beards

:#The Song of the Flying Fish

:#The Actor and the Alibi

:#The Vanishing of Vaudrey

:#The Worst Crime in the World

:#The Red Moon of Meru

:#The Chief Mourner of Marne

:#The Secret of Flambeau

In order to prove this theorem, one must start by considering that the entire right diagonal corresponds to the coefficient of ''x''⁰ when (''x''&nbsp;+&nbsp;1) has been raised to some power equal to the row number. The next diagonal corresponds to the coefficient of ''x''¹, and so on. Now, algebraically, we are trying to determine what (''x''&nbsp;+&nbsp;1)^''n''+1 looks like, if we start by defining (''x''&nbsp;+&nbsp;1)^''n'' as being equal to

:<math>\sum_{i=0}^n a_i x^i</math>

Now

5. ''The Scandal Of Father Brown'' (1935)

:<math>(x+1)^{n+1} = (x+1)(x+1)^n = x(x+1)^n + (x+1)^n = \sum_{i=0}^n a_i x^{i+1} + \sum_{i=0}^n a_i x^i</math>

:#The Scandal of Father Brown

:#The Quick One

:#The Blast of the Book

:#The Green Man

:#The Pursuit of Mr Blue

:#The Crime of the Communist

:#The Point of a Pin

:#The Insoluble Problem

Next we clean up the summations:

::<math>\sum_{i=0}^{n } a_{i } x^{i+1} + \sum_{i=0}^n a_i x^i = </math>

::<math>\sum_{i=1}^{n+1} a_{i-1} x^{i } + \sum_{i=0}^n a_i x^i = </math>

::<math>\sum_{i=1}^{n } a_{i-1} x^{i } + \sum_{i=1}^n a_i x^i + a_0x^0 + a_{n}x^{n+1} = </math>

::<math>\sum_{i=1}^{n } (a_{i-1} + a_i)x^{i } + a_0x^0 + a_{n}x^{n+1} = </math>

::<math>\sum_{i=1}^{n } (a_{i-1} + a_i)x^{i } + x^0 + x^{n+1}</math> (because of how raising a polynomial to a power works, ''a''₀ = ''a''_''n'' = 1)

We now have an expression for the polynomial (''x''&nbsp;+&nbsp;1)^''n''+1 in terms of the coefficients of (''x''&nbsp;+&nbsp;1)^''n'' (these are the ''a''_''i''s), which is what we need if we want to express a line in terms of the line above it. Recall that all the terms in a diagonal going from the upper-left to the lower-right correspond to the same power of ''x'', and that the a-terms are the coefficients of the polynomial (''x''&nbsp;+&nbsp;1)^''n'', and we are determining the coefficients of (''x''&nbsp;+&nbsp;1)^''n''+1. Now, for any given ''i'' not 0 or ''n''&nbsp;+&nbsp;1, the coefficient of the ''x''^''i'' term in the polynomial (''x''&nbsp;+&nbsp;1)^''n''+1 is equal to ''a''_''i'' (the figure above and to the left of the figure to be determined, since it is on the same diagonal) + ''a''_{''i''&minus;1} (the figure to the immediate right of the first figure). Inspecting Pascal's triangle, we find that this is indeed the rule at the beginning of the article.

The Vampire of the Village (outside of compilation)

One very interesting consequence of this fact is obtained by substituting the number one for ''x''. In this case, we know that <math> (1+1)^n = 2^n </math>, hence the sum of

{{col-end}}

==References==

:<math> {n \choose 0} + {n \choose 1} + \cdots +{n \choose n-1} + {n \choose n} = 2^n </math>

*[[Martin Gardner|Gardner, Martin]], ''The Annotated Innocence of Father Brown'', [[Oxford University Press]], 1987, ISBN 0-19-217748-6 (Notes by Gardner, on Chesterton’s stories).

==Citations==

That is to say, the sum of the entries in the ''i''th row of Pascal's triangle sum to the ''i''th power of 2.

<references/>

==External links==

==Properties of Pascal's triangle==

*[http://www.gutenberg.net/etext/204 Project Gutenberg text of "The Innocence of Father Brown"]

*[http://www.gutenberg.net/etext/223 Project Gutenberg text of "The Wisdom of Father Brown"]

*[http://www.cse.dmu.ac.uk/~mward/gkc/books/Incredulity.txt ''The Incredulity of Father Brown''] at Martin Ward's [http://www.cse.dmu.ac.uk/~mward/gkc ''G. K. Chesterton's Works on the Web''].

*[http://www.cse.dmu.ac.uk/~mward/gkc/books/Complete_Father_Brown/index.html ''The Complete Father Brown''] at Martin Ward's [http://www.cse.dmu.ac.uk/~mward/gkc ''G. K. Chesterton's Works on the Web''].

*[http://www.zetaminor.com/cult/father_brown/father_brown_v1.htm Review of UK DVD of the 1974 TV series]

{{DEFAULTSORT:Brown, Father}}

Some simple patterns are immediately apparent in 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.

* Moving inwards, the next pair of diagonals contain the [[triangular number]]s in order.

* The next pair of diagonals contain the [[tetrahedral number]]s in order, and the next pair give [[pentatope number]]s. In general, each next pair of diagonals contains the next higher dimensional "''d''-triangle" numbers, which can be defined as

::<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 alternate 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'').

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.

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.)

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.

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.

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:

1

1 1

1 2 1

<u>'''1''' &rarr; 3 &darr;</u> 3 1

1 4 <u>&rarr;'''6''' &rarr; 4 &darr;</u> 1

1 5 10 10 <u>&rarr;'''5''' &rarr; 1 &darr;</u>

<u>'''1''' &rarr; 6 &darr;</u> 15 20 15 6 <u>&rarr;'''1'''</u>

1 7 <u>&rarr;'''21'''</u> 35 35 21 7 1

1 8 28 56 <u>'''70'''</u> 56 28 8 1

1 9 36 84 126 126 <u>'''84'''</u> 36 9 1

1 10 45 120 210 252 210 120 <u>'''45'''</u> 10 1

1 11 55 165 330 462 462 330 165 55 <u>'''11'''</u> 1

1 12 66 220 495 792 924 792 495 220 66 12 <u>'''1'''</u>

1 13 78 286 715 1287 1716 1716 1287 715 286 78 13 1

1 14 91 364 1001 2002 3003 3432 3003 2002 1001 364 91 14 1

1 15 105 455 1365 3003 5005 6435 6435 5005 3003 1365 455 105 15 1

1 16 120 560 1820 4368 8008 11440 12870 11440 8008 4368 1820 560 120 16 1

The second highlighted diagonal has a sum of 233. The numbers 'skipped over' between the move right and the move down-right also sum to Fibonacci numbers, being the numbers 'between' the sums formed by the first construction. For example, the numbers skipped over in the first highlighted diagonal are 3, 4 and 1, making 8.

In addition, if row ''m'' is taken to indicate row <math>(n+1)</math>, the sum of the squares of the elements of row ''m'' equals the middle element of row <math>(2m-1)</math>. For example, <math>1^2 + 4^2 + 6^2 + 4 ^2 + 1^2 = 70</math>. In general form:

:<math>\sum_{k=0}^n {n \choose k}^2 = {2n \choose n}</math>

Another interesting pattern is that on any row ''m'', where ''m'' is odd, the middle term minus the term two spots to the left equals a [[Catalan number]], specifically the (''m''&nbsp;+&nbsp;1)/2 Catalan number. For example: on row 5, 6&nbsp;&minus;&nbsp;1 = 5, which is the 3^rd Catalan number, and (5&nbsp;+&nbsp;1)/2 = 3.

Also, the sum of the elements of row ''m'' is equal to 2^{''m''&minus;1}. For example, the sum of the elements of row 5 is <math>1 + 4 + 6 + 4 + 1 = 16</math>, which is equal to <math>2^4 = 16</math>. This follows from the binomial theorem proved above, applied to (1 + 1)^{''m''&minus;1}.

Some of the numbers in Pascal's triangle correlate to numbers in [[Lozanić's triangle]].

Another interesting property of Pascal's triangle is that in rows where the second number (the 1st number following 1) is prime, all the terms in that row except the 1s are multiples of that prime.

===Geometric properties===

Pascal's triangle can be used as a [[lookup table]] for the number of arbitrarily dimensioned elements within a single arbitrarily dimensioned version of a triangle (known as a ''[[simplex]]''). For example, consider the 3rd line of the triangle, with values 1, 3, 3, 1. A 2-dimensional triangle has one 2-dimensional element (itself), three 1-dimensional elements (lines, or edges), and three 0-dimensional elements ([[Vertex (graph theory)|vertices]], or corners). The meaning of the final number (1) is more difficult to explain (but see below). Continuing with our example, a [[tetrahedron]] has one 3-dimensional element (itself), four 2-dimensional elements (faces), six 1-dimensional elements (edges), and four 0-dimensional elements (vertices). Adding the final 1 again, these values correspond to the 4th row of the triangle (1, 4, 6, 4, 1). Line 1 corresponds to a point, and Line 2 corresponds to a line segment (dyad). This pattern continues to arbitrarily high-dimensioned hyper-tetrahedrons (simplices).

To understand why this pattern exists, one must first understand that the process of building an ''n''-simplex from an (''n''&nbsp;&minus;&nbsp;1)-simplex consists of simply adding a new vertex to the latter, positioned such that this new vertex lies outside of the space of the original simplex, and connecting it to all original vertices. As an example, consider the case of building a tetrahedron from a triangle, the latter of whose elements are enumerated by row 3 of Pascal's triangle: '''1''' face, '''3''' edges, and '''3''' vertices (the meaning of the final 1 will be explained shortly). To build a tetrahedron from a triangle, we position a new vertex above the plane of the triangle and connect this vertex to all three vertices of the original triangle.

The number of a given dimensional element in the tetrahedron is now the sum of two numbers: first the number of that element found in the original triangle, plus the number of new elements, ''each of which is built upon elements of one fewer dimension from the original triangle''. Thus, in the tetrahedron, the number of [[cell (mathematics)|cells]] (polyhedral elements) is 0 (the original triangle possesses none) + 1 (built upon the single face of the original triangle) = '''1'''; the number of faces is 1 (the original triangle itself) + 3 (the new faces, each built upon an edge of the original triangle) = '''4'''; the number of edges is 3 (from the original triangle) + 3 (the new edges, each built upon a vertex of the original triangle) = '''6'''; the number of new vertices is 3 (from the original triangle) + 1 (the new vertex that was added to create the tetrahedron from the triangle) = '''4'''. This process of summing the number of elements of a given dimension to those of one fewer dimension to arrive at the number of the former found in the next higher simplex is equivalent to the process of summing two adjacent numbers in a row of Pascal's triangle to yield the number below. Thus, the meaning of the final number (1) in a row of Pascal's triangle becomes understood as representing the new vertex that is to be added to the simplex represented by that row to yield the next higher simplex represented by the next row. This new vertex is joined to every element in the original simplex to yield a new element of one higher dimension in the new simplex, and this is the origin of the pattern found to be identical to that seen in Pascal's triangle.

A similar pattern is observed relating to [[square (geometry)|squares]], as opposed to triangles. To find the pattern, one must construct an analog to Pascal's triangle, whose entries are the coefficients of (''x''&nbsp;+&nbsp;2)^{Row Number}, instead of (''x''&nbsp;+&nbsp;1)^{Row Number}. There are a couple ways to do this. The simpler is to begin with Row 0 = 1 and Row 1 = 1, 2. Proceed to construct the analog triangles according to the following rule:

:<math> {n \choose k} = 2\times{n-1 \choose k-1} + {n-1 \choose k}</math>

That is, choose a pair of numbers according to the rules of Pascal's triangle, but double the one on the left before adding. This results in:

1

1 2

1 4 4

1 6 12 8

1 8 24 32 16

1 10 40 80 80 32

1 12 60 160 240 192 64

1 14 84 280 560 672 448 128

The other way of manufacturing this triangle is to start with Pascal's triangle and multiply each entry by 2^k, where k is the position in the row of the given number. For example, the 2nd value in row 4 of Pascal's triangle is 6 (the slope of 1s corresponds to the zeroth entry in each row). To get the value that resides in the corresponding position in the analog triangle, multiply 6 by 2^{Position Number} = 6&nbsp;&times;&nbsp;2² = 6&nbsp;&times;&nbsp;4 = 24. Now that the analog triangle has been constructed, the number of elements of any dimension that compose an arbitrarily dimensioned [[cube (geometry)|cube]] (called a [[hypercube]]) can be read from the table in a way analogous to Pascal's triangle. For example, the number of 2-dimensional elements in a 2-dimensional cube (a square) is one, the number of 1-dimensional elements (sides, or lines) is 4, and the number of 0-dimensional elements (points, or vertices) is 4. This matches the 2nd row of the table (1, 4, 4). A cube has 1 cube, 6 faces, 12 edges, and 8 vertices, which corresponds to the next line of the analog triangle (1, 6, 12, 8). This pattern continues indefinitely.

To understand why this pattern exists, first recognize that the construction of an ''n''-cube from an (''n''&nbsp;&minus;&nbsp;1)-cube is done by simply duplicating the original figure and displacing it some distance (for a regular ''n''-cube, the edge length) [[orthogonal]] to the space of the original figure, then connecting each vertex of the new figure to its corresponding vertex of the original. This initial duplication process is the reason why, to enumerate the dimensional elements of an ''n''-cube, one must double the first of a pair of numbers in a row of this analog of Pascal's triangle before summing to yield the number below. The initial doubling thus yields the number of "original" elements to be found in the next higher ''n''-cube and, as before, new elements are built upon those of one fewer dimension (edges upon vertices, faces upon edges, etc.). Again, the last number of a row represents the number of new vertices to be added to generate the next higher ''n''-cube.

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.]]

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==

The earliest explicit depictions of a triangle of [[binomial coefficients]] occur in the [[10th century]] in a commentary on the ''Chandas Shastra'', an [[ancient India]]n book on [[Sanskrit]] [[prosody]] written by [[Pingala]] between the [[5th century BC|5th]]&ndash;[[2nd century BC|2nd centuries BC]]. While Pingala's work only survives in fragments, the commentator [[Halayudha]], around [[975]], used the triangle to explain obscure references to ''Meru-prastaara'', the "Staircase of [[Mount Meru]]". It was also realised that the shallow diagonals of the triangle sum to the [[Fibonacci numbers]]. The [[Indian mathematics|Indian mathematician]] Bhattotpala (c. [[1068]] later gives rows 0-16 of the triangle.

[[Image:Yanghui_triangle.gif|thumb|[[Yang Hui]] (Pascal's) triangle, as depicted by the Chinese using [[Counting rods|rod numerals]].]]

At around the same time, it was discussed in [[History of Iran|Persia]] by the [[Islamic mathematics|mathematician]] [[Al-Karaji]] (953–1029) and the [[Persian literature|poet]]-[[Islamic astronomy|astronomer]]-mathematician [[Omar Khayyám]] (1048-1131); thus the triangle is referred to as the "Khayyam triangle" in [[Iran]]. Several theorems related to the triangle were known, including the [[binomial theorem]]. In fact we can be fairly sure that Khayyam used a method of finding ''n''th roots based on the binomial expansion, and therefore on the binomial coefficients.

Later, the arithmetic triangle became known to [[Chinese mathematics|Chinese mathematicians]] from the 12th century onwards; today Pascal's triangle is called "[[Yang Hui]]'s triangle" in [[China]].

Finally, in [[Italy]], it is referred to as "Tartaglia's triangle", named for the Italian algebraist [[Niccolo Fontana Tartaglia]] who lived a century before Pascal; Tartaglia is credited with the general formula for solving cubic polynomials.

In [[1655]], [[Blaise Pascal]] wrote a ''Traité du triangle arithmétique'' (Treatise on arithmetical triangle), wherein he collected several results then known about the triangle, and employed them to solve problems in [[probability theory]]. The triangle was later named after Pascal by [[Pierre Raymond de Montmort]] (1708) and [[Abraham de Moivre]] (1730).

==References==

<references/>

==See also==

* [[Pascal matrix]]

* [[Multiplicities of entries in Pascal's triangle]] (Singmaster's conjecture)

* Francis Galton's "quincunx" or [[bean machine]]

* [[Leibniz harmonic triangle]]

[[Category:Characters in written fiction]]

== External links ==

[[Category:Fictional priests and priestesses]]

* {{MathWorld | urlname=PascalsTriangle | title=Pascal's triangle }}

[[Category:Fictional amateur detectives]]

* [http://www.york.ac.uk/depts/maths/histstat/images/triangle.gif The Old Method Chart of the Seven Multiplying Squares] ''(from the Ssu Yuan Yü Chien of Chu Shi-Chieh, 1303, depicting the first nine rows of Pascal's triangle)''

[[Category:Novel series]]

* [http://www.lib.cam.ac.uk/RareBooks/PascalTraite Pascal's Treatise on the Arithmetic Triangle] ''(page images of Pascal's treatise, 1655; summary: [http://www.lib.cam.ac.uk/RareBooks/PascalTraite/pascalintro.pdf])''

[[Category:Short stories]]

* [http://members.aol.com/jeff570/p.html Earliest Known Uses of Some of the Words of Mathematics (P)]

[[Category:Fictional Catholics]]

* [http://www.cut-the-knot.org/Curriculum/Combinatorics/LeibnitzTriangle.shtml Leibniz and Pascal triangles]

[[Category:Fictional religious workers]]

* [http://www.cut-the-knot.org/Curriculum/Algebra/DotPatterns.shtml Dot Patterns, Pascal's Triangle, and Lucas' Theorem]

[[Category:Mystery!]]

* [http://binomial.csuhayward.edu Pascal's Triangle From Top to Bottom]

* [http://www.stetson.edu/~efriedma/periodictable/html/O.html Omar Khayyam the mathematician]

* [http://ptri1.tripod.com Info on Pascal's Triangle]

* [http://mathforum.org/dr.math/faq/faq.pascal.triangle.html Explanation of Pascal's Triangle and common occurrences, including link to interactive version specifying # of rows to view]

[[Category:Factorial and binomial topics]]

[[Category:Eponyms]]

[[de:Pater Brown]]

[[bg:Триъгълник на Паскал]]

[[caes:TrianglePadre de TartagliaBrown]]

[[it:Padre Brown]]

[[cs:Pascalův trojúhelník]]

[[delt:PascalschesTėvas DreieckBraunas]]

[[ja:ブラウン神父]]

[[es:Triángulo de Pascal]]

[[ru:Отец Браун]]

[[fa:مثلث خیام-پاسکال]]

[[frsv:TriangleFader de PascalBrown]]

[[ko:파스칼의 삼각형]]

[[is:Pascal-þríhyrningur]]

[[it:Triangolo di Tartaglia]]

[[he:משולש פסקל]]

[[lt:Paskalio trikampis]]

[[nl:Driehoek van Pascal]]

[[no:Pascals trekant]]

[[pl:Trójkąt Pascala]]

[[pt:Triângulo de Pascal]]

[[ru:Биномиальный коэффициент#Треугольник Паскаля]]

[[simple:Pascal's Triangle]]

[[sr:Паскалов троугао]]

[[fi:Pascalin kolmio]]

[[sv:Pascals triangel]]

[[vi:Tam giác Pascal]]

[[tr:Pascal üçgeni]]

[[zh:杨辉三角形]]