Let us define a define the trigonometric tangle as the following parametric function:
where can be a rational number or an irrational number. and are any real number. If is a rational number and then we get a tangle petals defined thus:
, such that are relatively prime.
When then the envelop of the tangle converges to a figure with -fold symmetry defined thus:
such that are relatively prime.
For irrational for the above conditions we get figures coming close those of corresponding to the continued fraction approximations of the irrationals. This provides us an interesting way of visualizing the irrationals. Interestingly, visual distinction can be made between algebraic and transcendental irrationals with these curves.
Figure 1. Convergence from -lobed initial to -fold symmetry
Figure 2. Curves for small and large for integral
Figure 3. Curves for small and large for fractional rational
Figure 4. Curves for various irrational