Kepler triangle

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A Kepler triangle is a right triangle formed by three squares with areas in geometric progression according to the golden ratio.

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: $1 : \sqrt\varphi : \varphi$ , or approximately 1 : 1.2720196 : 1.6180339.^[1]

Triangles with such ratios are named after the German mathematician and astronomer Johannes Kepler (1571–1630), who first demonstrated that a right triangle with edges in geometric progression are characterised by a geometric factor linked to the golden ratio.^{[citation needed]}

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

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.^[2]

— Johannes Kepler

A triangle with dimensions closely approximating a Kepler triangle can be recognised in the Great Pyramid of Giza.^[3]^[4] Whether the relationship to the golden ratio in this pyramid occurs by design or by accident remains a topic of controversy.^[5]

1 Derivation
2 Constructing a Kepler triangle
3 See also
4 References

[edit] Derivation

The fact that a triangle with edges $1$ , $\sqrt\varphi$ 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:

$(\varphi)^2 = (\sqrt\varphi)^2 + (1)^2.$

[edit] Constructing a Kepler triangle

A method to construct a golden rectangle. The resulting dimensions are in the ratio 1: $\varphi$ , the golden ratio.

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