The Ara and the Arbelos

When I was young, I was fascinated by the blade held by the clobber and the way in which he plied it in fixing shoes and slippers. This blade (Ara in saMskR^ita) had a special shape and I realized a weapon like that was held by some of our deva-s. There was an Indian arrow that had a head much like this tool. Much later, I learnt of the great yavanAchArya Archimedes having studied a curve derived of 3 circles that he named the Arbelos or the clobber’s knife. Obviously he was meaning the same object apparently still in use in leather-work. I wondered if the proto-Indoeuropeans or at least the common ancestor of the Hindus and Greeks had a common device like that (Arbelos and Ara) used in leather work. The Arbelos fascinates mathematicians and laymen alike, and I was no exception to it. I simply wished to relive my journey through the Arbelos – perhaps relive a fantasy of the halcyon days of my childhood. Many fascinating properties of the Arbelos are apparent even at sight:

The most obvious being that the perimeter of the upper boundary of the curve = perimeter of the lower boundary because: d=d1+d2; pi*d/2=pi*(d1+d2)/2;

The hemichord formed by the shared tangent of the two smaller semi-circles is indicated by T. It is the geometric mean of their respective diameters: T²=d1*d2

Further, the area of the Arbelos (A) can be expressed independently of the 3 radii of the semi-circles using just the value T: A=pi*T²/4; thus, the area of the Arbelos is the same as that of a circle with diameter T. this is obvious from baudhAyana’s (Pythagoras) theorem and that any angle inscribed in a semicircle is a right angle.

There are many other interesting properties of the arbelos that have made it a fertile object of investigation in Euclidean geometry over the ages. Even the great geometer Jakob Steiner worked on its properties, like the issue of Pythagorean triplets in the chain of circles. Even in the past 100 years it has been a source of rich new results. One of the most famous features of the arbelos from ancient Greek mathematics concerned the chain of circles inscribed within it. The first circle (K1) in this chain is a tangent to all the 3 semicircles of the arbelos. The next K2 is a tangent to K1 and two of the other semicircles. In this chain of circles the height of the of the center of Kn from the base line is equal to n*dn, where dn is the diameter of Kn. The centers of K1-Kn lie on an ellipse whose foci are the centers of the two semicircles that bound the chain. If the ratio of the diameters of the two smaller semicircles of the arbelos (d1/d2) is rational then the following is observed:
The right triangle formed by the center of Kn, the center of the largest semicircle of the arbelos and the foot of the perpendicular dropped from the center of Kn to the baseline has sides forming a Pythagorean triplet. This chain of circles can be constructed using the method of reflection of points on a circle. Here the reflecting circle is of radius=d and it reflects the right line as the largest semicircle, the left parallel line as the second largest semicircle and the first semicircle bounded by these two lines as the 3rd semicircle of the arbelos. All other inscribed tangential circles are reflected to form the chain.

The family of circles – Archimedean twin, the Bankoff circle and the quadruplet are shown.

Finally there are the famous Archimedean twin circles of the arbelos, which are the tangential to the largest semicircle, either one of the two other semicircles and the segment T. Archimedes showed that these two circles are congruent. Strikingly, numerous other congruent circles come up all over the arbelos, and their discovery and properties has formed a major area of modern studies on this figure. One of these discovered by Bankoff, a rich dentist who was an amateur mathematician: One takes the following 3 points- 1) & 2) the two touches by the first circle of the chain of inscribed circles on the smaller two semicircles of the arbelos. 3) the point where the two smaller semicircles of the arbelos touch. Then one draws the circumcircle of these points. It is congruent to the Archimedean twin circles.Another more recently discovered quadruplet of the family is described thus: Take the perpendiculars from the centers of the smaller semicircles of the arbelos and let them intersect the semicircles at A and B. Draw the lines from the center of the biggest semicircle of the arbelos to the A and B. Then there are two circles passing through each of these points which are tangential to each other and the lines from the former sentence, which are also tangential to the biggest circle to the arbelos. These two pairs of circles form a quadruplet which is congruent to the Archimedean twin circles.

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