## Trigonometric tangles-2

We had earlier described our exploration of the spirograph, hypocycloids, epicycloids and related curves. In course of our study of the śaiva tantra-s of the kaula tradition we started thinking about a remarkable piece of imagery mentioned in them. Tantra-s like the Virūpākṣa-pañcākśika and Nityāṣoḍaṣikārṇava talk of the waves and spinning cakra-s of Śakti-s. The Virūpākṣa-pañcākśika describes these innumerable Śakti-waves and wheels of Śakti-s as emanating the universe:

sambheda-vibhedābhyāṃ sṛjati dhvaṃsayati cai(e)ṣa jagat || 2.13

In his (Śiva’s) own body, in the void of the conscious-self, resembling the milk-ocean, through the constructive and destructive interference of the waves of his own śakti-s this universe is generated and destroyed.

rūpādi-pañca-viṣayātmani bhogya-hṛṣīka-bhoktṛ-rūpe ‘asmin |

In the contents of the five sense-streams starting with form, in the objects being sensed, in the sense organs and the first-person experiencer, in the world generated by the innumerable wheels of his Śakti-s consciousness should be conceived.

The Nityāṣoḍaṣikārṇava states:
śakti-cakrocchala-cchaktivalanā-kavalīkṛtām | 2.24-25

The goddess Tripurā is worshipped as: The gushing intoxicant waves setting in vibration the vulval receptacle, setting in motion the śakti wheels, spinning the Śakti wheels and devouring all.

This account of the waves of Śakti-s and their wheels led us to conceive the imagery as related to the more general epicycloid problem: Imagine a point on the circumference of a wheel rotating at some speed either clockwise or anticlockwise. That point is the center of another wheel, which is likewise rotating at a distinct speed. On the circumference of the second wheel is a point which in turn is the center of yet another wheel rotating at yet another speed and so on. Now what curve would a point on the rim of the terminal wheel of this system would trace out? [See link for animation

More than 20 years ago mathematician Frank Farris, who has a great eye for symmetry and beauty, showed that this can be described rather simply by a function in the complex plane thus:
$z(t)=a_1e^{k_1it}+a_2e^{k_2it}...+a_ne^{k_nit} ; \; z(t) \in \mathbb{C}$
$a_j=a_1,a_2...a_n$ represent the relative radii and $k_j=k_1, k_2...k_n$ represent the relative speeds of rotation of the $n$ wheels in the system. If the wheel is rotating clockwise then $k_j<0$ and $a_j=i\cdot a_j$; where $i=\sqrt{-1}$

Figure 1: The curve generated by a system of 9 wheels of relative radii: 1.0000000, 0.6180340, 0.3197333, 0.1227690, 0.2256262, 0.2233212, 0.4724658 and rotation speeds: -1, 5, 11, 17, 23, 29, 35.

Thus more generally we can define a complex function which would generate “trigonometric tangles” related to those generated by a n-wheel system:
$z(t)=\displaystyle \sum _{j=1}^{n}a_je^{k_jit}$
where $k_j$ is real number and $a_j$ might be real or imaginary. Now these functions are symmetric and closed as $t$ takes all values $[0,2\pi]$ when the following conditions are satisfied:
$k_j=l\times n+m; \; l \in \mathbb{Z}; \; n,m \in \mathbb{N}_1\; and \; \textrm{gcd}(n,m)=1$
i.e if $l$ is any integer and $n,m$ are relatively prime, positive integers then the resultant curve is closed and has n-fold symmetry. Thus, keeping to symmetric closed curves this equation can literally produce nava-nava-chamatkAra.

Figure 2: A 12 term system with 8-fold symmetry allowing only real $a_j$

Figure 3: The svastika-system with 4-fold symmetry, random number of terms up to 10, and allowing both real and imaginary $a_j$

Figure 4: The 17 term system with 24-fold symmetry allowing both real and imaginary $a_j$. It somewhat resembles the guilloche security patterns used on bank notes.

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