## Geometry

The Tusi Couple

The Tusi Couple, named for Nasir al-Din al-Tusi (1201 – 1274), is a planar geometric construct consisting of two circles, the smaller of which is half the diameter of the larger and inside it, touching at a single point. The smaller circle rolls around the circumference of the larger without sliding in a fixed direction, anticlockwise say.

The singular feature of the Tusi Couple is that each point on the circumference of the smaller circle exhibits a periodic motion along a fixed diameter of the larger. The pdf explains the geometry.

A far more comprehensive discussion of the Tusi couple, and the family of hypocycloids of which it is a member, can be found here.

Pappus Chains

Pappus Chains are a geometric construct formed by dividing the diameter of a semi-circle into two unequal parts, the left hand part forming the diameter of a second semi-circle within the first one. A Pappus Chain is an infinite sequence of circles tangent (externally) to the first two semi-circles and to each other. The two Pappus Chains of particular interest are shown below.

Pappus Chains have been studied since antiquity and were popularised by Martin Gardner in his Mathematical Games column in the January 1979 edition of the Scientific American.

Salient properties of Pappus chains are described in a Mathematical Association of America Monthly article of March 2006 by Harold P Boas, a preprint of which is available here.

Interesting properties of Pappus chains are (1) a simple relationship between the diameter of a given circle, the height of its centre above the diameter of the outer circle, and its index number within the sequence and (2) the fact that the tangent points of the circles in the chain lie on a circle. This is invariably explained using inversion (or inversive) geometry. The purpose of the pdf is to give a relatively straightforward explanation using Cartesian coordinates.

Descartes’ Circle Theorem

Suppose there exist three circles (C1, C2, C3, in black, below) that are mutually tangent externally and have radii r1, r2, r3, and a fourth circle (C4 in red, below – there are two possiblities) having radius r4 that is tangent to the first three.

Descartes’ Circle Theorem states that the radii are related by

$\displaystyle \left( \frac{1}{r_{1}} + \frac{1}{r_{2}} + \frac{1}{r_{3}} \pm \frac{1}{r_{4}} \right) ^2 = 2 \left( \frac{1}{{r_{1}}^{2}} + \frac{1}{{r_{2}}^2} + \frac{1}{{r_{3}}{^2}} + \frac{1}{{r_{4}}^2} \right)$

The minus sign is taken if the fourth circle is external to the first three and the plus sign if it is internal.

The pdf reproduces two proofs of the theorem.

Steiner Chains

Steiner Chains were briefly discussed in Martin Gardner’s 1979 Scientific American article referred to above.

A distorted annulus is formed by two circles, one within the other. The Chain is an infinite sequence of circles tangent (externally) to the circles forming the distorted annulus and to each other, using a particular inscribed circle as the first in the sequence.

In certain circumstances the chain reproduces itself after one ‘circuit’ of the distorted annulus. In these cases the chain will close itself regardless of where the first circle is placed (see diagram below reproduced from Gardner’s article which illustrates a chain of six circles).

Three properties of Steiner Chains are:

(1)  The centres of the circles forming the chain lie on an ellipse

(2)  The points at which the circles forming the chain are tangent to each other lie on a  circle

(3)  The chain reproduces itself after one ‘circuit’ of the distorted annulus if and only if

$\displaystyle \sin^2 \frac{\pi}{n} = \frac{(a')^2 - d^2}{a^2 - d^2}$

where r and r’ are the radii of the exterior and interior circles respectively, 2d is the distance between their centres and n is the number of circles in the chain, a = (r + r’)/2 and a’ = (r – r’)/2.

It follows that the existence of a closed chain of any particular length and configuration d, r and r’ is independent of the position of the starting circle in the chain.

The pdf uses Cartesian coordinate geometry to verify the first two properties and a mapping of the plane to itself which is adapted from the invertive geometry method to verify the third property.