Descartes' theorem

René Descartes

In geometry , Descartes 'Theorem ( Descartes' Four Circles Theorem ), named after René Descartes , describes a relationship between four circles that touch each other . The sentence can be used to find, for three given circles that touch each other, a fourth one that touches the other three. It is a special case of the Apollonian problem .

history

Geometric problems related to circles touching each other were contemplated more than 2000 years ago. In ancient Greece of the 3rd century BC Apollonios von Perge dedicated an entire book to this topic. Unfortunately, this work, entitled On Touch, has not survived.

René Descartes briefly mentioned the problem (in accordance with the customs of the time) in a letter to Princess Elisabeth of Bohemia in 1643 . He essentially came to the solution described in equation (1) below, even if his proof was incorrect. Therefore the four-circle theorem is named after Descartes today.

The phrase was rediscovered several times independently, including in a special case in Japanese temple problems , by Jakob Steiner (1826), by the British amateur mathematician Philip Beecroft (1842) and by Frederick Soddy (1936). The Soddy circles are sometimes referred to , perhaps because Soddy published his version of the sentence in the form of a poem entitled The Kiss Precise , which was published in Nature (June 20, 1936). Soddy also generalized Descartes' theorem to a theorem about spheres in 3-dimensional space and Thorold Gosset to n-dimensions.

Allan Wilks and Colin Mallows of Bell Laboratories discovered in the late 1990s that a complex version of Descartes' Theorem also defines the locations of the circles.

If you continue the construction, you get a fractal structure with smaller and smaller touching circles. While the first four curvatures are connected by a quadratic equation according to Descartes' theorem, a linear equation holds for the following circles. If you start with four integer curvatures, then the following curvatures of the circles in the construction also have integer values. The number theoretic aspects of the problem were followed up in particular by Wilks, Jeffrey Lagarias , Ronald Graham , Peter Sarnak , Alex Kontorovich and Hee Oh .

Definition of the signed curvature

Descartes' theorem is most easily expressed by the term curvature . The signed curvature of a circle is defined by , where r denotes the radius. The larger the circle, the smaller the amount of its curvature and vice versa. ${\ displaystyle k = \ pm 1 / r}$

The minus sign in is for a county, the other three circuits including touches. Otherwise the plus sign is to be used. ${\ displaystyle k = \ pm 1 / r}$

If a straight line is viewed as a degenerate circle with curvature , Descartes' theorem can also be applied when a straight line and two circles are given that touch each other, and a third circle is sought that touches the straight line and the given circles. ${\ displaystyle k = 0}$

Descartes' theorem

Given four circles touching each other with the radii , , and . If one defines the signed curvature (for ) for each of these circles as above , the following equation is fulfilled: ${\ displaystyle r_ {1}}$ ${\ displaystyle r_ {2}}$ ${\ displaystyle r_ {3}}$ ${\ displaystyle r_ {4}}$ ${\ displaystyle k_ {i}}$ ${\ displaystyle i = 1, \ ldots, 4}$

{\ displaystyle (1)}

{\ displaystyle (k_ {1} + k_ {2} + k_ {3} + k_ {4}) ^ {2} = 2 \, (k_ {1} ^ {2} + k_ {2} ^ {2} + k_ {3} ^ {2} + k_ {4} ^ {2})}

Solving this equation allows you to determine the radius of the fourth circle: ${\ displaystyle k_ {4}}$

{\ displaystyle (2)}

{\ displaystyle k_ {4} = k_ {1} + k_ {2} + k_ {3} \ pm 2 {\ sqrt {k_ {1} k_ {2} + k_ {2} k_ {3} + k_ {3 } k_ {1}}}.}

The plus-minus symbol expresses that there are generally two solutions .

example

Given are three circles with radii , and . Accordingly, the signed curvature has the values , and . The two solutions and result from equation (2) . The tiny circle (red) between the given circles therefore has the radius . The large circle (also red) that encloses the given circles has the radius . ${\ displaystyle r_ {1} = 1/2}$ ${\ displaystyle r_ {2} = 1/6}$ ${\ displaystyle r_ {3} = 1/3}$ ${\ displaystyle k_ {1} = 2}$ ${\ displaystyle k_ {2} = 6}$ ${\ displaystyle k_ {3} = 3}$ ${\ displaystyle k_ {4} = 23}$ ${\ displaystyle {k_ {4}} '= - 1}$ ${\ displaystyle r_ {4} = 1/23}$ ${\ displaystyle {r_ {4}} '= 1}$

Special cases

If, for example, the third of the three given circles is replaced by a straight line, then this equals 0 and falls out of equation (1). Equation (2) becomes much simpler in this case: ${\ displaystyle k_ {3}}$

{\ displaystyle (3)}

{\ displaystyle k_ {4} = k_ {1} + k_ {2} \ pm 2 {\ sqrt {k_ {1} k_ {2}}}.}

example

Given are two circles with the radii and as well as a straight line that is understood as a circle with an infinite radius. The corresponding signed curvature values are , and . Using equation (3) one again obtains two possible values, namely and . For the radii of the two circles drawn in red there is consequently or . ${\ displaystyle r_ {1} = 1/4}$ ${\ displaystyle r_ {2} = 1/9}$ ${\ displaystyle k_ {1} = 4}$ ${\ displaystyle k_ {2} = 9}$ ${\ displaystyle k_ {3} = 0}$ ${\ displaystyle k_ {4} = 25}$ ${\ displaystyle {k_ {4}} '= 1}$ ${\ displaystyle r_ {4} = 1/25}$ ${\ displaystyle {r_ {4}} '= 1}$

Descartes' theorem cannot be applied if two or even all three given circles are replaced by straight lines. The sentence also does not apply if there is more than one encompassing touching circle, i.e. in the case of three circles lying one inside the other with a common point of contact.

Complex theorem of Descartes

To fully determine a circle, not just its radius (or curvature), one must also know its center point. The easiest way to express the equation for this is to interpret the coordinates of the center point ( x , y ) as a complex number . The equation for is very similar to Descartes' Theorem and is therefore called the Descartes Complex Theorem . ${\ displaystyle z = x + iy}$ ${\ displaystyle z_ {1}, z_ {2}, z_ {3}, z_ {4}}$

Given are four circles with the centers and the signed curvatures (see above) that touch each other. Then the relationship applies in addition to (1) ${\ displaystyle z_ {1}, z_ {2}, z_ {3}, z_ {4}}$ ${\ displaystyle k_ {1}, k_ {2}, k_ {3}, k_ {4}}$

{\ displaystyle (4)}

{\ displaystyle (k_ {1} z_ {1} + k_ {2} z_ {2} + k_ {3} z_ {3} + k_ {4} z_ {4}) ^ {2} = 2 \, (k_ {1} ^ {2} z_ {1} ^ {2} + k_ {2} ^ {2} z_ {2} ^ {2} + k_ {3} ^ {2} z_ {3} ^ {2} + k_ {4} ^ {2} z_ {4} ^ {2}).}

The substitution results in: ${\ displaystyle q_ {i} = k_ {i} \ cdot z_ {i}}$

{\ displaystyle (5)}

{\ displaystyle (q_ {1} + q_ {2} + q_ {3} + q_ {4}) ^ {2} = 2 \, (q_ {1} ^ {2} + q_ {2} ^ {2} + q_ {3} ^ {2} + q_ {4} ^ {2})}

This equation is analogous to and has the solution: ${\ displaystyle (1)}$

{\ displaystyle (6)}

{\ displaystyle q_ {4} = q_ {1} + q_ {2} + q_ {3} \ pm 2 {\ sqrt {q_ {1} q_ {2} + q_ {2} q_ {3} + q_ {3 } q_ {1}}}.}

Here, too, there are generally two solutions.

If one has determined from equation (2), one obtains through ${\ displaystyle k_ {4}}$ ${\ displaystyle z_ {4}}$ ${\ displaystyle z_ {4} = q_ {4} / k_ {4}}$

various

The primitive integer solutions of the four radii are exactly the diagonal products and line products of the two two-parametric representations of the primitive Pythagorean triples , e.g. the primitive Pythagorean triplet with the parameter representations (written as columns) and the diagonal products and the line products , which as Radii, interpreted, satisfy Descartes' theorem. ${\ displaystyle (5,12,13)}$ ${\ displaystyle (2,3) ^ {T}}$ ${\ displaystyle (1 = 3-2.5 = 3 + 2) ^ {T}}$ ${\ displaystyle 3.10}$ ${\ displaystyle 2.15}$