Kepler triangle

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

${\displaystyle \varphi ^{2}=\varphi +1}$

into the form of the Pythagorean theorem:

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

For positive real numbers a and b, their arithmetic mean, geometric mean, and harmonic mean are the lengths of the sides of a right triangle if and only if that triangle is a Kepler triangle.[6]

A method to construct a Kepler triangle via a golden rectangle

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

1. Construct a unit 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 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."[2]

The circle and the square have approximately the same perimeter

In the Kepler triangle with sides ${\displaystyle 1,{\sqrt {\varphi }},\varphi ,}$ consider:

• the circle that circumscribes it, and
• a square with side equal to the middle-sized edge of the triangle.

Then the perimeters of the square (${\displaystyle 4{\sqrt {\varphi }}}$) and the circle (${\displaystyle \pi \varphi }$) coincide up to an error less than 0.1%.

This is the mathematical coincidence ${\displaystyle \pi \approx 4/{\sqrt {\varphi }}}$. The square and the circle cannot have exactly the same perimeter, because in that case one would be able to solve the classical (impossible) problem of the quadrature of the circle. In other words, ${\displaystyle \pi \neq 4/{\sqrt {\varphi }}}$ because ${\displaystyle \pi }$ is a transcendental number.

According to some sources, Kepler triangles appear in the design of Egyptian pyramids. The diagonal of the floor of the King's Chamber, plus the width of the chamber, divided by the length of the chamber is very close to the golden ratio.[5][7] However, according to various scholars who have investigated this relationship, the ancient Egyptians probably did not know the mathematical coincidence involving the number ${\displaystyle \pi }$ and the golden ratio ${\displaystyle \varphi }$.[8]

• Golden triangle
• Special right triangles

Footnotes

Citations