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 φ=(1+√5)/2 and can be written: 1:√φ:φ , or approximately 1 : 1.2720196 : 1.6180339. 解説 †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 : φ^2=φ+1 into Pythagorean form: (φ)^2 = (√φ)^2 + (1)^2
作図方法 †A Kepler triangle can be constructed with only straightedge and compass by first creating a golden rectangle:
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 † |