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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]]
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| :<math>\varphi = {1 + \sqrt{5} \over 2}</math>
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| 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. | |
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| 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:
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| {{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]]}}
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| 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>
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| ==Derivation==
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| 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>:
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| :<math>\varphi^2 = \varphi + 1 </math>
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| into the form of the [[Pythagorean theorem]]:
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| :<math>(\varphi)^2 = (\sqrt\varphi)^2 + (1)^2. </math>
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| ==Relation to arithmetic, geometric, and harmonic mean==
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| 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>
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| ==Constructing a Kepler triangle==
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| [[File:Kepler Triangle Construction.svg|thumb|A method to construct a Kepler triangle via a [[golden rectangle]]]]
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| A Kepler triangle can be [[Compass and straightedge constructions|constructed with only straightedge and compass]] by first creating a [[golden rectangle]]:
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| # Construct a simple square
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| # Draw a line from the midpoint of one side of the square to an opposite corner
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| # Use that line as the radius to draw an arc that defines the height of the rectangle
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| # Complete the golden rectangle
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| # 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
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| 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/>
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| ==A mathematical coincidence==
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| [[File:Kepler triangle squaring the circle.gif|thumb|160px|alt=construction|The circle and the square have approximately the same perimeter]]
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| Take any Kepler triangle with sides <math>a, a \sqrt{\varphi}, a \varphi,</math> and consider:
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| * the circle that circumscribes it, and
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| * a square with side equal to the middle-sized edge of the triangle.
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| 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%.
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| 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]].
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| 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
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| | last = Markowsky
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| | first = George
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| | date =
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| | year = 1992
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| | month = January
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| | title = Misconceptions about the Golden Ratio
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| | journal = College Mathematics Journal
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| | volume = 23
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| | issue = 1
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| | doi = 10.2307/2686193
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| | url = http://www.umcs.maine.edu/~markov/GoldenRatio.pdf
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| | format = PDF
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| | accessdate =
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| | jstor = 2686193
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| | publisher = Mathematical Association of America
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| | pages = 2–19
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| | quote = It does not appear that the Egyptians even knew of the existence of φ much less incorporated it in their buildings
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| }}</ref>
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| ==See also==
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| *[[Golden triangle (mathematics)|Golden triangle]]
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| *[[Special right triangles]]
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| ==References==
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| {{reflist}}
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| [[Category:Triangles]]
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| [[Category:Golden ratio]]
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| [[Category:Elementary geometry]]
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| [[Category:Johannes Kepler]]
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