The cardioid and the arbelos: the scimitar and the axe
The arbelos of Archimedes,
an object most wondrous;
it brought pleasure to us,
when stalked by enemies,
as the old yavana by Romans,
who ended for good his days.
What is the mystery of the scimitar and the axe which are worth the same?
In the 10th year of our life we became fascinated by the curves generated by the reflections of various utensils (and the water in them) in the kitchen. Being no Maxwell, we did not get far enough to develop a good theoretical understanding of the caustic curves as he had done around that age. But we did get some empirical feel for how most of these curves have a point (i.e. a cusp) where they become undifferentiatable. One of these was the cardioid. Playing with a device known as the spirograph, which immensely captivated us as a kid, we quickly discovered for ourselves that the cardioid could be constructed as the first integer epicycloid. About four years later we were interested in all manner of classic curves, beyond the obvious circle, appearing in biological forms. We observed appearances of the limacon in very disparate biological entities: 1) It appeared as the inner dark blue pattern on the peacock’s feather. 2) It was the outline of the cells of certain algae belonging to the pyramimonad clade. This got us interested in how a cardioid might emerge in biological systems, naively reasoning that it might form at the intersection of two signals moving along linear paths. It was then that we discovered for ourselves that the cardioid was the locus of the foot of the tangents to circle c from a point P on it.
This then lead to our realization of an interesting relationship between the area of the cardioid and the generating circle c. Moreover, at that point we were enamored by the empirical recovery of the interesting relationship between the excess area of the cardioid with respect to the generating circle and the symmetric arbelos of that circle.
All this can be geometrically proved with minimal fuss by means of what has been termed a visual calculus discovered by the Armenian physicist named Mamikon. The foundation of this method pertains to forming a tangent cluster, namely the collation of all the tangent segments by bringing them together to a common point of origin by the translation transform (This is illustrated below).
Generalizing from the above, the basic theorem of Mamikon states that the area under the tangent segment sweep of a curve is equal to the area of the equivalent tangent cluster formed by collating the tangent segments to a common point of origin. Intuitively, since the tangents define the derivatives to the curve at a given point, the geometric construction of their collation to form the tangent cluster is effectively the visual equivalent of integration. Mamikon apparently first discovered this by studying the well known problem of the area of the circular ring (illustrated below). The area of the ring can be expressed using just the length of the tangent segment of the inner circle bounded by the outer one, independently of the radii of the inner and outer circles. The sweep of this tangent segment, which forms the ring, results in a tangent cluster which defines a circle with radius equal to the tangent segment. Thus, the area of the ring is equal to the area of that circle which has radius equal to the tangent segment. Thus, the Mamikon theorem also implies the bhujā-koṭi-karṇa nyāya (known in the west as the Pythagoras theorem).
This method of Mamikon can now be used to determine the area of the cardioid. As can be seen from the below construction:
1) The sweep of the tangent segments of the generating circle c as formed by the intersection with their feet (i.e. perpendiculars dropped to them from the point P) defines the cardioid.
2) Now by transforming the tangent segments into a tangent cluster we get locus .
3) By definition of foot of tangent ; .
4) should be a semicircle and locus a circle.
5) Thus by Mamikon’s theorem the area of the claw of the cardioid PCP’B’= area of circle
6) We can see from the construction that the longest tangent segment, i.e. diameter of circle , is equal in length to the radius of circle c. Thus , we get
7) Thus area of circle one fourth area of circle c: .
8) Likewise, the mirror image claw of the cardioid has the same area. We can construct that area as an equivalent circle by reflecting circle on the X-axis.
9) Thus, the arbelos PBP’A has the same area as circle , circle and the claw of the cardioid PCP’B’
10) Thus, the area of the cardioid
Hence, the cardioid has 50% excess area over the generating circle.
If the claw of the cardioid were a scimitar and and the arbelos an axe-head we get a scimitar with same surface are as an axe-head.