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A Kepler triangle is a right triangle with edge lengths in geometric progression
Geometric progression

In mathematics, a geometric progression, also known as a geometric sequence, is a sequence of numbers where each term after the first is found by multiplying the previous one by a fixed non-zero number called the common ratio....
. The ratio of the edges of a Kepler triangle are linked to the golden ratio
Golden ratio

In mathematics and the arts, two quantities are in the golden ratio if the ratio between the sum of those quantities and the larger one is the same as the ratio between the larger one and the smaller....


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

Triangles with such ratios are named after the German mathematician
Mathematician

A mathematician is a person whose primary area of study and/or research is the field of mathematics....
 and astronomer
Astronomer

An astronomer is a scientist who studies Celestial body such as planets, stars, and Galaxy.Historically, astronomy was more concerned with the classification and description of phenomena in the sky, while astrophysics attempted to explain these phenomena and the differences between them using physical laws....
 Johannes Kepler
Johannes Kepler

Johannes Kepler was a Germans mathematician, astronomer and astrologer, and key figure in the 17th century Scientific revolution. He is best known for his eponymous Kepler's laws of planetary motion, codified by later astronomers based on his works Astronomia nova, Harmonices Mundi, and Epitome of Copernican Astrononomy....
 (1571–1630), who first demonstrated that this triangle is characterised by a ratio between short side and hypotenuse equal to the golden ratio.






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Encyclopedia


A Kepler triangle is a right triangle with edge lengths in geometric progression
Geometric progression

In mathematics, a geometric progression, also known as a geometric sequence, is a sequence of numbers where each term after the first is found by multiplying the previous one by a fixed non-zero number called the common ratio....
. The ratio of the edges of a Kepler triangle are linked to the golden ratio
Golden ratio

In mathematics and the arts, two quantities are in the golden ratio if the ratio between the sum of those quantities and the larger one is the same as the ratio between the larger one and the smaller....


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

Triangles with such ratios are named after the German mathematician
Mathematician

A mathematician is a person whose primary area of study and/or research is the field of mathematics....
 and astronomer
Astronomer

An astronomer is a scientist who studies Celestial body such as planets, stars, and Galaxy.Historically, astronomy was more concerned with the classification and description of phenomena in the sky, while astrophysics attempted to explain these phenomena and the differences between them using physical laws....
 Johannes Kepler
Johannes Kepler

Johannes Kepler was a Germans mathematician, astronomer and astrologer, and key figure in the 17th century Scientific revolution. He is best known for his eponymous Kepler's laws of planetary motion, codified by later astronomers based on his works Astronomia nova, Harmonices Mundi, and Epitome of Copernican Astrononomy....
 (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
Pythagorean theorem

In mathematics, the Pythagorean theorem or Pythagoras' theorem is a relation in Euclidean geometry among the three sides of a triangle#Types of triangles....
 and the golden ratio
Golden ratio

In mathematics and the arts, two quantities are in the golden ratio if the ratio between the sum of those quantities and the larger one is the same as the ratio between the larger one and the smaller....
—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
Great Pyramid of Giza

The Great Pyramid of Giza, also called Khufu's Pyramid or the Pyramid of Khufu, and Pyramid of Cheops, is the oldest and largest of the three Egyptian pyramidss in the Giza Necropolis bordering what is now Cairo , Egypt, and is the only remaining member of the Seven Wonders of the Ancient World....
.

Derivation

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

into Pythagorean
Pythagorean theorem

In mathematics, the Pythagorean theorem or Pythagoras' theorem is a relation in Euclidean geometry among the three sides of a triangle#Types of triangles....
 form:

Constructing a Kepler triangle


A Kepler triangle can be constructed with only straightedge and compass by first creating a golden rectangle
Golden rectangle

A golden rectangle is a rectangle whose side lengths are in the golden ratio, 1: , that is, or approximately 1:1.618.A distinctive feature of this shape is that when a square section is removed, the remainder is another golden rectangle; that is, with the same proportionality s as the first....
:

  1. Construct a simple square
  2. Draw a line from the midpoint of one side of the square to an opposite corner
  3. Use that line as the radius to draw an arc that defines the height of the rectangle
  4. Complete the golden rectangle
  5. Use the longer side of the golden rectangle to draw an arc that intersects the opposite side of the rectangle and defines the hypotenuse
    Hypotenuse

    File:Triangle Sides.svgA hypotenuse is the longest side of a right triangle, the side opposite the right angle. The length of the hypotenuse of a right triangle can be found using the Pythagorean theorem, which states that the Square of the length of the hypotenuse equals the sum of the squares of the lengths of the other two sides....
     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."

See also

  • Golden triangle
    Golden triangle (mathematics)

    A golden triangle is an isosceles triangle triangle in which the two longer sides have equal lengths and in which the ratio of this length to that of the third, smaller side is the golden ratio....