General matrix notation of a VAR(p): Difference between revisions

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[[File:Kepler triangle.svg|right|thumb|A '''Kepler triangle''' is a right triangle formed by three squares with areas in geometric progression according to the '''[[golden ratio]]'''.]]
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A '''Kepler triangle''' is a [[Special right triangle|right triangle]] with edge lengths in [[geometric progression]]. The ratio of the edges of a Kepler triangle are linked to the [[golden ratio]]
 
:<math>\varphi = {1 + \sqrt{5} \over 2}</math>
 
and can be written: <math> 1 : \sqrt\varphi : \varphi</math>, or approximately '''1 : 1.272 : 1.618'''.<ref>{{cite book | title = The Shape of the Great Pyramid | author = Roger Herz-Fischler | publisher = Wilfrid Laurier University Press | year = 2000 | isbn = 0-88920-324-5 | url =  http://books.google.com/books?id=066T3YLuhA0C&pg=PA81&dq=kepler-triangle+geometric&ei=ux77Ro6sGKjA7gLzrdjlDQ&sig=bngzcQrK9nHOkfZTo5O0ieNdtUs }}</ref> The squares of the edges of this triangle (see figure) are in [[geometric progression]] according to the golden ratio.
 
Triangles with such ratios are named after the German [[mathematician]] and [[astronomer]] [[Johannes Kepler]] (1571–1630), who first demonstrated that this triangle is characterised by a ratio between short side and [[hypotenuse]] equal to the golden ratio.<ref name="livio">{{cite book|last=Livio|first=Mario|year=2002|title=The Golden Ratio: The Story of Phi, The World's Most Astonishing Number|publisher=Broadway Books|location=New York|isbn=0-7679-0815-5|pages=149}}</ref> Kepler triangles combine two key mathematical concepts—the [[Pythagorean theorem]] and the golden ratio—that fascinated Kepler deeply, as he expressed in this quotation:
 
{{quote|Geometry has two great treasures: one is the theorem of Pythagoras, the other the division of a line into mean and extreme ratio. The first we may compare to a mass of gold, the second we may call a precious jewel.<ref>{{cite book | title = A Brief History of Mathematics: An Authorized Translation of Dr. Karl Fink's Geschichte der Elementar-Mathematik | author = Karl Fink, Wooster Woodruff Beman, and David Eugene Smith | publisher = Chicago: Open Court Publishing Co | year = 1903 | edition = 2nd ed. | url = http://books.google.com/books?id=3hkPAAAAIAAJ&pg=PA223&dq=%22Geometry+has+two+great+treasures%22&lr=&as_brr=1&ei=sQ1GSI_KH4fstgO_rvCpDQ }}</ref> | [[Johannes Kepler]]}}
 
Some sources claim that a triangle with dimensions closely approximating a Kepler triangle can be recognized in the [[Great Pyramid of Giza]].<ref>{{cite book | title = The Best of Astraea: 17 Articles on Science, History and Philosophy | url = http://books.google.com/books?id=LDTPvbXLxgQC&pg=PA93&dq=kepler-triangle&ei=vCH7RuG7O4H87gLJ56XlDQ&sig=6n43Hhu5pE3TN5BW18tbQJGRHTQ | publisher = Astrea Web Radio | isbn = 1-4259-7040-0 | year = 2006 }}</ref><ref name="Squaring the circle, Paul Calter">[http://www.dartmouth.edu/~matc/math5.geometry/unit2/unit2.html Squaring the circle, Paul Calter]</ref>
 
==Derivation==
The fact that a triangle with edges <math>1</math>, <math>\sqrt\varphi</math> and <math>\varphi</math>, forms a right triangle follows directly from rewriting the defining quadratic polynomial for the golden ratio <math>\varphi</math>:
 
:<math>\varphi^2 = \varphi + 1 </math>
 
into the form of the [[Pythagorean theorem]]:
 
:<math>(\varphi)^2 = (\sqrt\varphi)^2 + (1)^2. </math>
 
==Relation to arithmetic, geometric, and harmonic mean==
 
For positive real numbers ''a'' and ''b'', their [[arithmetic mean]], [[geometric mean]], and [[harmonic mean]] are the lengths of the sides of a right triangle if and only if that triangle is a Kepler triangle.<ref>Di Domenico, Angelo, "The golden ratio—the right triangle—and the arithmetic, geometric, and harmonic means," ''[[The Mathematical Gazette]]'' 89, 2005.</ref>
 
==Constructing a Kepler triangle==
 
[[File:Kepler Triangle Construction.svg|thumb|A method to construct a Kepler triangle via a [[golden rectangle]]]]
 
A Kepler triangle can be [[Compass and straightedge constructions|constructed with only straightedge and compass]] by first creating a [[golden rectangle]]:
 
# Construct a simple square
# Draw a line from the midpoint of one side of the square to an opposite corner
# Use that line as the radius to draw an arc that defines the height of the rectangle
# Complete the golden rectangle
# Use the longer side of the golden rectangle to draw an arc that intersects the opposite side of the rectangle and defines the [[hypotenuse]] of the Kepler triangle
 
Kepler constructed it differently. In a letter to his former professor [[Michael Mästlin]], he wrote, "If on a line which is divided in extreme and mean ratio one constructs a right angled triangle, such that the right angle is on the perpendicular put at the section point, then the smaller leg will equal the larger segment of the divided line."<ref name=livio/>
 
==A mathematical coincidence==
 
[[File:Kepler triangle squaring the circle.gif|thumb|160px|alt=construction|The circle and the square have approximately the same perimeter]]
 
Take any Kepler triangle with sides <math>a, a \sqrt{\varphi}, a \varphi,</math> and consider:
* the circle that circumscribes it, and
* a square with side equal to the middle-sized edge of the triangle.
Then the [[perimeter]]s of  the square (<math>4a \sqrt{\varphi}</math>) and the circle (<math>a \pi \varphi</math>) coincide up to an error less than 0.1%.
 
This is the [[mathematical coincidence]] <math>\pi \approx 4/\sqrt\varphi</math>. The square and the circle cannot have exactly the same perimeter, because in that case one would be able to solve the classical (impossible) problem of the [[Squaring the circle|quadrature of the circle]]. In other words, <math>\pi \neq 4/\sqrt\varphi</math> because <math>\pi</math> is a [[transcendental number]].
 
According to some sources, Kepler triangles appear in the design of Egyptian pyramids.<ref name="Squaring the circle, Paul Calter"/><ref>[http://www.petrospec-technologies.com/Herkommer/pyramid/pyramid.htm The Great Pyramid, The Great Discovery, and The Great Coincidence, Mark Herkommer]</ref>  However, the ancient Egyptians probably did not know the mathematical coincidence involving the number <math>\pi</math> and the golden ratio <math>\phi</math>.<ref>{{Cite journal
| last = Markowsky
| first = George
| date =
| year = 1992
| month = January
| title = Misconceptions about the Golden Ratio
| journal = College Mathematics Journal
| volume = 23
| issue = 1
| doi = 10.2307/2686193
| url = http://www.umcs.maine.edu/~markov/GoldenRatio.pdf
| format = PDF
| accessdate =
| jstor = 2686193
| publisher = Mathematical Association of America
| pages = 2–19
| quote = It does not appear that the Egyptians even knew of the existence of &phi; much less incorporated it in their buildings
}}</ref>
 
==See also==
*[[Golden triangle (mathematics)|Golden triangle]]
*[[Special right triangles]]
 
==References==
{{reflist}}
 
[[Category:Triangles]]
[[Category:Golden ratio]]
[[Category:Elementary geometry]]
[[Category:Johannes Kepler]]

Latest revision as of 15:46, 18 June 2014

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