This Demonstration shows two figures. On the left is a torus knot (coil or rosette) pattern, winding a thickened curve in a spiral around a torus plotted using parametric equations.

The right graphic shows a Lotus Kolam. This variety of Kolam pattern comes mainly from Tamil Nadu, South India, and is hand drawn using a star-polygon-like framework but with different definitions (in form and process) and with more degrees of freedom. The drawings connect a dot on each adjacent radial line in sequence, and finally create a loop or multi-loop pattern.

A torus knot (without the torus) has the same topology as the Lotus Kolam. The number of components of the knot link and the loop number of the Lotus Kolam are the same as (in Mathematica given by KnotData["TorusKnot",{l,n}]), where is the number of leaves in the Lotus Kolam or turns in the torus knot, and is the number of periods in the Lotus Kolam or the torus knot.

When the number of loops (components) is greater than 1, up to three loops are drawn, in the order: purple, green, and blue. The number of crossings is . The crossings alternate if selected by the checkbox, changing the modulation frequency of to .

An odd period makes a symmetrical form in both the torus knot and the Lotus Kolam and has additional radial axes of half-interval angles and extra inner dots for each axis. The asymmetric form of the Lotus Kolam removes half-interval radial axes and alternating crossings and shows only one loop of odd periods.

For more information on these patterns, see [1, 2]. For colored animations of some Lotus Kolams, see [3, 4].

References

[1] S. Nagata, "Loop Patterns of Lotus Kolam, Phyllotaxis, Designs in Renaissance and Knot Patterns", Forma, Special Issue 2016 (forthcoming)