|
|
| Line 1: |
Line 1: |
| [[File:Pythagorean.tree.svg|500px|thumb|The tree of primitive Pythagorean triples.]]
| | He hangs way up. The first person to call him in two days is a telemarketer. He grips his aluminum cane challenging he can feel the calluses bite into his palms. The damned thing. He shouldn't even be on it anymore, but he is not doing his tools. No way in Hell should he have come back from a ten-day vacation with Diabetes type 2 Decompression Sickness. Off of a lousy 35-foot dive.<br><br>First, absolutely set up car dvd player to concentrate to pieces. Music is a really good companion components are operating. the car dvd player supports many mainstream formats such as: MP4, DVD, SVCD, CD, MP3, etc. Merchandise in your articles are upset with uploding the music, you may use the high sensitive FM/AM tuner, FM/AM auto search, frequency computer screen. With this function, you can catch develop the ugg news.<br><br>May be you hope to buy your pair of ugg boots of trainers at a reduced store to obtain lesser amount. But it is not easy various other an UGGS Order at these the minimum price as these quality shoes will not come at low value. Nonetheless there are occasions that damages to lead it to in a procurement. But the difficulty is that there're lots of counterfeits relevant to. As a result if they tell you to send your UGGS Order and they can provide in a reduced price you will need consider care of that retail store. Probably they are talking approximately a counterfeit.<br><br>A associated with nice shoes or boots: in places where rain a lot, the dogs' paws suffer when they walk their mud or maybe dirt. It is the same in too hot places, while the roads can burn them. Target sells very cute and cheap boots for doggies.<br><br>Redemption Of the islands. As we flash retrace to RI, we see Matt at ease with the isle he has called home for nearly a four weeks. He's endured 2 blindsides and 29 days on RI while only being in the actual game for 7 days and nights. Tree Mail comes to them, and Mike hopes it's a pizza. Yet it will help pizza, about the did let them know that their next Duel would be the game's final one.<br><br>Fashion and luxury. Women care about fashion and beauty than warmth and comfy on the boots, which is women's style and design. But UGG Boots combine every one of these features, which give a fabulous choice for females.<br><br>If you cherished this article and you simply would like to collect more info regarding [http://horizonafrica.com/img/ cheap uggs outlet] please visit our web site. |
| | |
| In [[mathematics]], a [[Pythagorean triple]] is a set of three positive [[integer]]s ''a, b,'' and ''c'' having the property that they can be respectively the two legs and the [[hypotenuse]] of a [[right triangle]], thus satisfying the equation <math>a^2+b^2=c^2</math>; the triple is said to be '''primitive''' [[if and only if]] ''a, b,'' and ''c'' share no common [[divisor]]. The set of all primitive Pythagorean triples has the structure of a rooted [[Tree structure|tree]], specifically a [[ternary tree]], in a natural way. This was first discovered by B. Berggren in 1934.<ref>B. Berggren, "Pytagoreiska trianglar" (in Swedish), ''Tidskrift för elementär matematik, fysik och kemi'' 17 (1934), 129–139. See page 6 for the rooted tree.</ref>
| |
| | |
| F. J. M. Barning showed<ref>Barning, F. J. M. (1963), "Over pythagorese en bijna-pythagorese driehoeken en een generatieproces met behulp van unimodulaire matrices" (in Dutch), Math. Centrum Amsterdam Afd. Zuivere Wisk. ZW-011: 37, http://oai.cwi.nl/oai/asset/7151/7151A.pdf
| |
| </ref> that when any of the three [[Matrix (mathematics)|matrices]] | |
| : <math>
| |
| \begin{array}{lcr}
| |
| A = \begin{bmatrix} 1 & -2 & 2 \\ 2 & -1 & 2 \\ 2 & -2 & 3 \end{bmatrix} &
| |
| B = \begin{bmatrix} 1 & 2 & 2 \\ 2 & 1 & 2 \\ 2 & 2 & 3 \end{bmatrix} &
| |
| C = \begin{bmatrix} -1 & 2 & 2 \\ -2 & 1 & 2 \\ -2 & 2 & 3 \end{bmatrix}
| |
| \end{array}
| |
| </math>
| |
| is [[matrix multiplication|multiplied]] on the right by a [[column vector]] whose components form a Pythagorean triple, then the result is another column vector whose components are a different Pythagorean triple. If the initial triple is primitive, then so is the one that results. Thus each primitive Pythagorean triple has three "children". All primitive Pythagorean triples are descended in this way from the triple (3 ,4, 5), and no primitive triple appears more than once. The result may be graphically represented as an infinite ternary tree with (3, 4, 5) at the root node (see classic tree at right). This tree also appeared in papers of A. Hall in 1970<ref>A. Hall, "Genealogy of Pythagorean Triads", ''[[The Mathematical Gazette]]'', volume 54, number 390, December, 1970, pages 377–9.</ref> and A. R. Kanga in 1990.<ref>Kanga, A. R., "The family tree of Pythagorean triples," ''Bulletin of the [[Institute of Mathematics and its Applications]]'' 26, January/February 1990, 15–17.</ref> | |
| | |
| | |
| | |
| ==Proofs==
| |
| | |
| | |
| | |
| | |
| ===Presence of exclusively primitive Pythagorean triples===
| |
| | |
| It can be shown [[Induction (mathematics)|inductively]] that the tree contains primitive Pythagorean triples and nothing else by showing that starting from a primitive Pythagorean triple, such as is present at the initial node with (3, 4, 5), each generated triple is both Pythagorean and primitive.
| |
| | |
| ====Preservation of the Pythagorean property====
| |
| If any of the above matrices, say ''A'', is applied to a triple (''a'', ''b'', ''c'')<sup>T</sup> having the Pythagorean property ''a''<sup>2</sup>+''b''<sup>2</sup>=''c''<sup>2</sup> to obtain a new triple (''d'', ''e'', ''f'')<sup>T</sup> = ''A''(''a'', ''b'', ''c'')<sup>T</sup>, this new triple is also Pythagorean. This can be seen by writing out each of ''d'', ''e'', and ''f'' as the sum of three terms in ''a'', ''b'', and ''c'', squaring each of them, and substituting ''c''<sup>2</sup>=''a''<sup>2</sup>+''b''<sup>2</sup> to obtain ''f''<sup>2</sup>=''d''<sup>2</sup>+''e''<sup>2</sup>. This holds for ''B'' and ''C'' as well as for ''A''.
| |
| | |
| ====Preservation of primitivity====
| |
| The matricies ''A'', ''B'', and ''C'' are all [[unimodular matrix|unimodular]]—that is, they have only integer entries and their determinants are ±1. Thus their inverses are also unimodular and in particular have only integer entries. So if any one of them, for example ''A'', is applied to a primitive Pythagorean triple (''a'', ''b'', ''c'')<sup>T</sup> to obtain another triple (''d'', ''e'', ''f'')<sup>T</sup>, we have (''d'', ''e'', ''f'')<sup>T</sup> = ''A''(''a'', ''b'', ''c'')<sup>T</sup> and hence (''a'', ''b'', ''c'')<sup>T</sup> = ''A''<sup>−1</sup>(''d'', ''e'', ''f'')<sup>T</sup>. If any prime factor were shared by any two of (and hence all three of) ''d'', ''e'', and ''f'' then by this last equation that prime would also divide each of ''a'', ''b'', and ''c''. So if ''a'', ''b'', and ''c'' are in fact pairwise coprime, then ''d'', ''e'', and ''f'' must be pairwise coprime as well. This holds for ''B'' and ''C'' as well as for ''A''.
| |
| | |
| ===Presence of every primitive Pythagorean triple exactly once===
| |
| To show that the tree contains every primitive Pythagorean triple, but no more than once, it suffices to show that for any such triple there is exactly one path back through the tree to the starting node (3, 4, 5). This can be seen by applying in turn each of the unimodular inverse matrices ''A''<sup>−1</sup>, ''B''<sup>−1</sup>, and ''C''<sup>−1</sup> to an arbitrary primitive Pythagorean triple (''d'', ''e'', ''f''), noting that by the above reasoning primitivity and the Pythagorean property are retained, and noting that for any triple larger than (3, 4, 5) exactly one of the inverse transition matrices yields a new triple with all positive entries (and a smaller hypotenuse). By induction, this new valid triple itself leads to exactly one smaller valid triple, and so forth. By the finiteness of the number of smaller and smaller potential hypotenuses, eventually (3, 4, 5) is reached. This proves that (''d'', ''e'', ''f'') does in fact occur in the tree, since it can be reached from (3, 4, 5) by reversing the steps; and it occurs uniquely because there was only one path from (''d'', ''e'', ''f'') to (3, 4, 5).
| |
| | |
| ==Properties==
| |
| The transformation using matrix ''A'', if performed repeatedly from (''a'', ''b'', ''c'') = (3, 4, 5), preserves the feature ''b'' + 1 = ''c''; matrix ''B'' preserves ''a'' – ''b'' = ±1 starting from (3, 4, 5); and matrix ''C'' preserves the feature ''a'' + 2 = ''c'' starting from (3, 4, 5).
| |
| | |
| A geometric interpretation for this tree involves the [[excircle]]s present at each node. The three children of any parent triangle “inherit” their [[inradius|inradii]] from the parent: the parent’s excircle radii become the inradii for the next generation.<ref name="Price">{{cite arXiv |last=Price |first=H. Lee |eprint=0809.4324 |title=The Pythagorean Tree: A New Species |year=2008}}</ref>{{rp|p.7}} For example, parent ( 3, 4, 5) has excircle radii equal to 2, 3, and 6. These are precisely the inradii of the three children (5, 12, 13), (15, 8, 17) and {{nowrap|(21, 20, 29)}} respectively.
| |
| | |
| If either of ''A'' or ''C'' is applied repeatedly from any Pythagorean triple used as an initial condition, then the dynamics of any of ''a'', ''b'', and ''c'' can be expressed as the dynamics of ''x'' in
| |
| | |
| :<math> x_{n+3} - 3x_{n+2} + 3x_{n+1} - x_n = 0 \, </math>
| |
| | |
| which is patterned on the matrices' shared [[Characteristic polynomial#Characteristic equation|characteristic equation]]
| |
| | |
| :<math>\lambda ^3 -3 \lambda ^2 + 3 \lambda -1 = 0. \, </math>
| |
| | |
| If ''B'' is applied repeatedly, then the dynamics of any of ''a'', ''b'', and ''c'' can be expressed as the dynamics of ''x'' in
| |
| | |
| :<math> x_{n+3} - 5x_{n+2} - 5x_{n+1} + x_n = 0, \, </math>
| |
| | |
| which is patterned on the characteristic equation of ''B''.<ref>Mitchell, Douglas W., "Feedback on 92.60", ''Mathematical Gazette'' 93, July 2009, 358–9.</ref>
| |
| | |
| Moreover, an infinitude of other third-order univariate [[difference equations]] can be found by multiplying any of the three matrices together an arbitrary number of times in an arbitrary sequence. For instance, the matrix ''D'' = ''CB'' moves one out the tree by two nodes (across, then down) in a single step; the characteristic equation of ''D'' provides the pattern for the third-order dynamics of any of ''a'', ''b,'' or ''c'' in the [[Collectively exhaustive events|non-exhaustive]] tree formed by ''D''.
| |
| | |
| ==Alternative methods of generating the tree==
| |
| Another approach to the dynamics of this tree<ref>{{citation|last1=Saunders|first1=Robert A.|last2=Randall|first2=Trevor|title=The family tree of the Pythagorean triplets revisited|journal=Mathematical Gazette|volume=78|date=July 1994|pages=190–193|jstor= 3618576}}.</ref> relies on the standard formula for generating all primitive Pythagorean triples:
| |
| | |
| :<math>a = m^2 - n^2, \, </math>
| |
| :<math>b = 2mn, \, </math>
| |
| :<math>c = m^2 + n^2, \, </math>
| |
| | |
| with ''m'' > ''n'' > 0 and ''m'' and ''n'' coprime and of opposite parity. Pairs (''m'', ''n'') can be iterated by pre-multiplying them (expressed as a column vector) by any of
| |
| | |
| :<math>
| |
| \begin{array}{lcr}
| |
| \begin{bmatrix} 2 & -1 \\ 1 & 0 \end{bmatrix}, &
| |
| \begin{bmatrix} 2 & 1 \\ 1 & 0 \end{bmatrix}, &
| |
| \begin{bmatrix} 1 & 2 \\ 0 & 1 \end{bmatrix},
| |
| \end{array}
| |
| </math>
| |
| | |
| each of which preserves the inequalities, coprimeness, and opposite parity. The resulting ternary tree contains every such (''m'', ''n'') pair exactly once, and when converted into (''a'', ''b'', ''c'') triples it becomes identical to the tree described above.
| |
| | |
| Another way of using two underlying parameters to generate the tree of triples<ref>Mitchell, Douglas W., "An alternative characterisation of all primitive Pythagorean triples", ''Mathematical Gazette'' 85, July 2001, 273–275.</ref> uses an alternative formula for all primitive triples:
| |
| | |
| :<math>a = uv, \, </math>
| |
| :<math>b = \frac{u^2-v^2}{2},</math>
| |
| :<math>c = \frac{u^2+v^2}{2},</math>
| |
| | |
| with ''u'' > ''v'' > 0 and ''u'' and ''v'' coprime and ''both odd''. Pairs (''u'', ''v'') can be iterated by pre-multiplying them (expressed as a column vector) by any of the above 2 × 2 matrices, all three of which preserve the inequalities, coprimeness, and the odd parity of both elements. When this process is begun at (3, 1), the resulting ternary tree contains every such (''u'', ''v'') pair exactly once, and when converted into (''a'', ''b'', ''c'') triples it becomes identical to the tree described above.
| |
| | |
| ==A different tree==
| |
| A different tree found by Price<ref name="Price" /> may be produced in a similar way using matrices A',B',C' shown below. (See [[Formulas for generating Pythagorean triples#Pythagorean triples by use of matrices and linear transformations|Pythagorean triples by use of matrices and linear transformations]].)
| |
| | |
| : <math>
| |
| \begin{array}{lcr}
| |
| A' = \begin{bmatrix} 2 & 1 & -1 \\ -2 & 2 & 2 \\ -2 & 1 & 3 \end{bmatrix} &
| |
| B' = \begin{bmatrix} 2 & 1 & 1 \\ 2 & -2 & 2 \\ 2 & -1 & 3 \end{bmatrix} &
| |
| C' = \begin{bmatrix} 2 & -1 & 1 \\ 2 & 2 & 2 \\ 2 & 1 & 3 \end{bmatrix}
| |
| \end{array}
| |
| </math> | |
| | |
| == Notes and references ==
| |
| {{reflist}}
| |
| | |
| == External links ==
| |
| * [http://www.cut-the-knot.org/pythagoras/PT_matrix.html The Trinary Tree(s) underlying Primitive Pythagorean Triples] at [[cut-the-knot]]
| |
| * Frank R. Bernhart, and H. Lee Price, "Pythagoras' garden, revisited", Australian Senior Mathematics Journal 01/2012; 26(1):29-40.[http://www.eric.ed.gov/PDFS/EJ992372.pdf]
| |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| [[Category:Number theory]]
| |
| [[Category:Graph theory]]
| |
| [[Category:Diophantine equations]]
| |
He hangs way up. The first person to call him in two days is a telemarketer. He grips his aluminum cane challenging he can feel the calluses bite into his palms. The damned thing. He shouldn't even be on it anymore, but he is not doing his tools. No way in Hell should he have come back from a ten-day vacation with Diabetes type 2 Decompression Sickness. Off of a lousy 35-foot dive.
First, absolutely set up car dvd player to concentrate to pieces. Music is a really good companion components are operating. the car dvd player supports many mainstream formats such as: MP4, DVD, SVCD, CD, MP3, etc. Merchandise in your articles are upset with uploding the music, you may use the high sensitive FM/AM tuner, FM/AM auto search, frequency computer screen. With this function, you can catch develop the ugg news.
May be you hope to buy your pair of ugg boots of trainers at a reduced store to obtain lesser amount. But it is not easy various other an UGGS Order at these the minimum price as these quality shoes will not come at low value. Nonetheless there are occasions that damages to lead it to in a procurement. But the difficulty is that there're lots of counterfeits relevant to. As a result if they tell you to send your UGGS Order and they can provide in a reduced price you will need consider care of that retail store. Probably they are talking approximately a counterfeit.
A associated with nice shoes or boots: in places where rain a lot, the dogs' paws suffer when they walk their mud or maybe dirt. It is the same in too hot places, while the roads can burn them. Target sells very cute and cheap boots for doggies.
Redemption Of the islands. As we flash retrace to RI, we see Matt at ease with the isle he has called home for nearly a four weeks. He's endured 2 blindsides and 29 days on RI while only being in the actual game for 7 days and nights. Tree Mail comes to them, and Mike hopes it's a pizza. Yet it will help pizza, about the did let them know that their next Duel would be the game's final one.
Fashion and luxury. Women care about fashion and beauty than warmth and comfy on the boots, which is women's style and design. But UGG Boots combine every one of these features, which give a fabulous choice for females.
If you cherished this article and you simply would like to collect more info regarding cheap uggs outlet please visit our web site.