# Riley Slice animations

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## Deformations of limit sets down parabolic pleating rays

A journey down the 1/2-pleating ray (i.e. the positive imaginary axis).

A journey down the 1/2-pleating ray (Markov algorithm).

A journey down the 1/3-pleating ray.

A journey down a curve given by a pertubation of a low-order term of the 1/3-slope Farey polynomial, showing what a trip down a ray which is a deformation of a pleating ray looks like.

A journey down the 4/5-pleating ray.

## The branches of the hyperbolic locus of $$\Phi_{1/4}$$

The inverse images of $$t$$ under the Farey polynomial of slope $$1/4$$, with $$t$$ now running forwards from $$-100$$ to $$0$$. This exhibits the pleating ray as the branch of the hyperbolic locus of the Farey polynomial which leaves the Riley slice last.

A journey down the 1/4-pleating ray (i.e. the branch of the locus shown in the previous video in the top-right quadrant).

A journey down the branch of the hyperbolic locus of $$\Phi_{1/4}$$ with asymptotic slope $$3\pi/4$$ (i.e. the branch of the locus in the top-left quadrant).

## Deformations of limit sets down elliptic pleating rays

A journey down the 1/2-pleating ray for the group with cone angles $$2\pi/3$$ and $$2\pi/4$$ (Markov algorithm).

A journey down the 1/2-pleating ray for the group with cone angles $$2\pi/4$$ and $$2\pi/4$$ (Markov algorithm).

A journey down the 1/3-pleating ray for the group with cone angles $$2\pi/3$$ and $$2\pi/4$$.

A journey down the 1/3-pleating ray for the group with cone angles $$2\pi/3$$ and $$2\pi/4$$ (Markov algorithm).

A journey down the 1/3-pleating ray for the group with cone angles $$2\pi/4$$ and $$2\pi/4$$ (Markov algorithm).

## Miscellaneous

The inverse images of $$t$$ under the Farey polynomials of slope denominator $$\leq 32$$, as $$t$$ decreases from 0 to $$-30$$. Observe that the pleating rays converge to the boundary at a rate corresponding to the slope denominator.

The roots of $$\Phi_\alpha$$ as $$\alpha$$ runs from $$0$$ to $$1$$ over the rational numbers of denominator $$\leq 32$$.