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The Astroid, the Deltoid and the Fish Within the Fish

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The Astroid, the Deltoid and the Fish Within the Fish The astroid, the deltoid and the fish within the fish 1 Prologue oṃ hrīṃ bhuvaneśvaryai namaḥ | We start by worshiping Devī Bhuvaneśvarī the mistress of the cakra in the form of the astroid. As a kid the light from the street lamp used to pass through the corrugated glass of the win- dow in our room and produce some interesting curves. The most prominent of these was one with four cusps, which we learned from a book of our father to be an astroid. We soon discovered for ourselves that it could be constructed using the spirograph principle: when rotated inside another circle, with radius 4 times its radius, a point a point on the circumference of a circle traces out an astroid (Figure 1). Likewise, when we did the same with circles that had radii in the ratio of 1:3 we got a tricuspid curve that we learned was called the deltoid (Figure 2). Finally, when we did this with circles having radii in the ratio of 1:2 we got the diameter of the larger circle (Figure 3). In doing so we had recapitulated what the Hashishin marūnmatta scientist Nasir al-din al-Tusi had done in the days of the Il-Khan Hulegu. This also corresponded to what we could learn about the construction of this family of curves, namely the hypocycloids, from our father’s book. However, we were keen to have other means for constructing these curves because our spirograph could not draw them properly as the hole for the pen was not ideally positioned. Figure 1: The astroid as a Figure 2: The deltoid as a Figure 3: The al-Tusi diameter hypocycloid with radial ratio of hypocycloid with radial ratio of as a hypocycloid with radial ra- 1:4 1:3 tio of 1:2 1 Around the 10th year of our life the means for drawing these curves and many others came to us from an unexpected quarter as we wandered free from cares in the Karṇāṭa country. We often went with our maternal grandfather to a shrine built around a millennium ago by kālāmukha-s patronized by Cālukya-s. In the vicinity of that shrine we saw a traditional Indian chair-maker who was making wicker chairs (known as auśīra in Sanskrit; frequently called a moḍā in Southern deśī tongues) by tying bamboo sticks. We saw him deftly craft hyperboloid surfaces of different curvatures using linear bamboos. This immediately gave us the inspiration that we might be able use a similar technique to draw curves on a plane by placing lines which smoothly vary in their inclinations. We applied ourselves to it right away upon reaching home. While our initial attempts ended in failure, by the end of that vacation we had discovered for ourselves many constructions for curves using such arrays of lines. Some time after that we learned that the yantra of the great Mahāvidyā Bhuvaneśvarī was based on the astroid. The inspiration from that Devī further impelled these constructions and by our fifteenth year we achieved several such constructions of the astroid, deltoid and other curves: re-discovering by ourselves what had been done by many in the past and learning others from some sources like Lockwood’s famous “A book of Curves”. We had by then realized that they were all envelope constructions. This exercise had given us some understanding of these entities, which greatly helped us with our visualization of calculus. We re-experience several of these constructions below. • Construction of an astroid from rectangles with diagonals of constant length Figure 4: The right side shows the rectan- gles forming the bounding circle while the Figure 5: The complete astroid formed as an left side shows the half rectangles with diag- envelope of the diagonals of the rectangles as onals forming astroid constructed in the adjacent figure 2 2 Construction of astroids, deltoids and related curves Our first experiment which yielded an easy construction of the astroid wentthus: 1) Draw all rectangles starting from “tall and narrow” to “short and wide” with two adjacent sides defined by common vertical (y-axis) and horizontal (x-axis) lines such that their diagonals havea constant length (Figure 4). 2) The common corner of these rectangles is the center while the opposite corner traces out a circle (Figure 4). 3) The envelope of the second diagonal of these rectangles i.e. the one other than that connecting the above points is the astroid. Thus, they are the tangents to the astroid (Figure 5). This construction can be seen as a simulation of a ladder sliding along a wall, which is the famous description of an astroid in literature. • Construction of an astroid as the locus of point defined using another point on the bounding circle Figure 6: Astroid Figure 7: Astroid and Quadrifolium The above experiment led to a second experiment that allowed us to obtain the astroid as the locus of a point rather than as an envelope (Figure 6). 1) Draw the bounding circle of radius a with center O within which we would construct the astroid. 2) Take any point C on the above circle and draw perpendiculars from it to the x-axis and y-axis. These perpendiculars ! intersect the axes respectively at points A and B. 3) Draw AB connecting points A and B. ! 4) Drop a perpendicular to AB from point C to intersect it a point D. 5) As point C moves along the circle point D traces out an astroid. ! As one can see, the above construction uses a similar principle as the previous one. The AB is the! equivalent of the diagonal of the rectangle in Figures 4 and 5. Here, the point of tangency of AB 3 and the astroid is directly identified using the perpendicular from point C. This construction also allowed us to construct another notable curve related to the astroid thus (Figure 7): ! ! 1) Drop a perpendicular from the center of the circle, point O, to AB. It intersects AB at point E. 2) The locus of point E as point C move around the bounding circle is a four-petalled rose also known as the quadrifolium. From the construction it can be seen that the quadrifolium is the pedal curve of the astroid for normals from the origin. Moreover the quadrifolium’s circumcircle is the incircle of the astroid and these three curves are concurrently tangent to each other (Figure 7). Using the bhujā-koṭi-karṇa nyāya one can show that this incircle of the astroid is half the radius of the astroid’s circumcircle. We shall consider some further details of this circle and the quadrifolium further ahead. Figure 8: Pedal point on diameter Figure 9: Pedal point on circumference Moving the pedal point from origin O to a point H on the diameter of the bounding circle we can perform a similar construction of the pedal curve (Figure 8). The resulting curve is a quadrifolium with bilateral symmetry where one pair of petals is larger than the other pair. When we move the pedal point to where the diameter meets the circumference of the bounding circle we see that two of the petals vanish and the curve crosses itself at point of contact between the circumcircle and the astroid (Figure 9). Visual examination of this family of quadrifolia allowed us, much to our pleasure, to derive their general polar equation: r = a sin 2θ − b cos (θ) (2.1) When b = 0 in 1 we get the classic quadrifolium. When a = 1 and b = 2 we get the version in 4 Figure 9. The other values give other versions of this curve. • Construction of astroid as the envelope of a family of ellipses with a constant sum of axes Acknowledging our fascination for the conics, our father obtained for us a most beautiful set of devices - templates for ellipses and circles. Using this we were able to achieve another easy construction of the astroid: 1) Draw ellipses around a common central point such that the sum of their semi-major and semi- minor axes is a constant. 2) The envelope of such ellipses is a astroid (Figure 10). As with the rectangles with a constant diagonal here too we find the narrow and tall or equivalent long and short ellipses converging at the limit to the cusps of the astroid, while the rest contribute to the four arcs of it. The central ellipse where the major and minor axes become equal is the in- circle of the astroid (Figure 10), which we showed above in the context of the quadrifolium (Figure 7). Figure 10: Astroid as envelope of 100 ellipses Figure 11: Astroid as envelope of normals or of given form. Central ellipse is also incircle evolute of an ellipse • An elongated astroid as the evolute of an ellipse In our early youth we received with interest our father’s impatient but clear lesson on evolutes of curves: An evolute is a curve formed by the envelope of the normals of that curve. We quickly followed up the lesson with the construction of the evolute of an ellipse. To our surprise as a kid it turned out to be another type of astroid (Figure 11). Notably this astroid was longer along the minor axis and shorter along the major axis; its cusps defined a rhombus, which was not a square. 5 As the major and minor axes become more equal in length the astroid also becomes more equal but collapses to a point when the ellipse becomes a circle. We later learned that this construction was first achieved by the French scientist Lamé whowas considered by Karl Gauss as one of the great mathematicians of the age.
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