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


and can be written: 1:√φ:φ , or approximately 1 : 1.2720196 : 1.6180339.



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. Kepler triangles combine two key mathematical concepts—the Pythagorean theorem and the golden ratio—that fascinated Kepler deeply, as he expressed in this quotation:

導出 Derivation

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


into Pythagorean form:

(φ)^2 =  (√φ)^2 + (1)^2
  • 直角三角形(right triangle)において,hypotenuseは「斜辺」,adjacentは「底辺」, oppositeは「対辺」。
  • アルキメデスの定理 "The length of the hypotenuse of the right triangle is equal to the square-root of the sum of the squares of the other two sides (the adjacent sides). Assuming that the lengths of the other two sides are 8 [cm] and 5 [cm], the length of the hypotenuse is about 9.43398 [cm]."


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 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."


Golden Rectangle with Golden Spiral

Golden Rectangle with Golden Spiral.JPG

添付ファイル: fileGolden Rectangle with Golden Spiral.JPG 415件 [詳細] filekepler2.JPG 378件 [詳細] filekepler1.JPG 387件 [詳細]

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Last-modified: 2009-10-02 (金) 13:51:00 (4010d)