Kepler triangle

A Kepler triangle is a right triangle with edge lengths in geometric progression. The ratio of the edges of a Kepler triangle are linked to the golden ratio

:$varphi\; =\; \{1\; +\; sqrt\{5\}\; over\; 2\}$

and can be written: $1\; :\; sqrtvarphi\; :\; varphi$, or approximately 1 : 1.2720196 : 1.6180339. [cite book | title = The Shape of the Great Pyramid | author = Roger Herz-Fischler | publisher = Wilfrid Laurier University Press | year = 2000 | isbn = 0889203245 | url = http://books.google.com/books?id=066T3YLuhA0C&pg=PA81&dq=kepler-triangle+geometric&ei=ux77Ro6sGKjA7gLzrdjlDQ&sig=bngzcQrK9nHOkfZTo5O0ieNdtUs ]

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.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|id=ISBN 0-7679-0815-5|pages=149] Kepler triangles combine two key mathematical concepts—the Pythagorean theorem and the golden ratio—that fascinated Kepler deeply, as he expressed in this quotation:

Some sources claim that a triangle with dimensions closely approximating a Kepler triangle can be recognized in the Great Pyramid of Giza. [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 = 1425970400 | year = 2006 ] [ [http://www.dartmouth.edu/~matc/math5.geometry/unit2/unit2.html Squaring the circle, Paul Calter] ]

Derivation

The fact that a triangle with edges $1$, $sqrtvarphi$ and $varphi$, forms a right triangle follows directly from rewriting the defining quadratic polynomial for the golden ratio $varphi$:

:$varphi^2\; =\; varphi\; +\; 1$

into Pythagorean form:

:$(varphi)^2\; =\; (sqrtvarphi)^2\; +\; (1)^2.$

Constructing a Kepler triangle

A Kepler triangle can be 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 longer rectangular edge 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."

ee also

*Golden triangle

References