Hyperanimator

hypertiling offers a simple API to animate hyperbolic tilings using the FuncAnimation feature of matplotlib. The hyperanimator class is a wrapper which simplifies the associated syntax and let’s you realize simple animation with very little effort

import numpy as np
from matplotlib import pyplot as plt
import matplotlib.cm as cmap

from hypertiling import HyperbolicTiling
from hypertiling.graphics.plot import convert_polygons_to_patches

enable interactive plot in notebook

%matplotlib notebook

define a function which takes one configuration to the next. In this case we implement a simple Ising model

J = 1 # coupling constant (positiv: ferromagnet, negativ: antiferromagnet)

def step(state, beta):

    for i in range(len(state)):

        site = np.random.choice(len(state))

        nbrhood = 0
        for n in nbrs[site]:
            nbrhood += state[n]

        phi_old = state[site]
        phi_new = -state[site]
        E1 = -J*phi_old*nbrhood
        E2 = -J*phi_new*nbrhood
        dE = E2-E1

        if dE < 0:
            state[site] = phi_new
        elif np.random.rand() < np.exp(-beta*dE):
            state[site] = phi_new

    return state

construct the tiling and generate an initial state for the model

T = HyperbolicTiling(3, 7, 6)
nbrs = T.get_nbrs_list()

colors = 2*np.random.choice(2, len(T))-1

Live Animation

# import hyperanimator class
from hypertiling.graphics.animation import AnimatorLive

# create figure
fig, axs = plt.subplots(1,1,figsize=(7,7))

# plot tiling using built in function "poly2patch"
patchkwargs = {"cmap": cmap.RdYlGn, "edgecolor": "#f1f2f0ff", "lw": 0.7, "clim":[-1.5,1.5]}
pgons = convert_polygons_to_patches(T, colors, **patchkwargs)
axs.add_collection(pgons)

# ------- this is the animation part -------
# note: the argument "frames" is only relevant for saving the animation, in the
# live plot it will run forever (i.e. until manually stopped)

# extra arguments to the step functions; leave away if there are none
stepargs = {"beta": 2}
# arguments to be passed through to the matplotlib FuncAnimator
animargs = {"frames": 300, "interval": 5, "blit": True}
# the actual animator
ani = AnimatorLive(colors, fig, pgons, step, stepargs, animargs)
# -------


# some usual plot adjustments
axs.add_patch(plt.Circle((0.0, 0.0), radius=1, edgecolor='k', facecolor='None', zorder=7))
axs.set_xlim(-1.02,1.02)
axs.set_ylim(-1.02,1.02)
axs.axis("off"); axs.set_aspect('equal'); plt.show()

hyperanimations can be easily saved like this

ani.save("/home/user/anim.mp4", fps=10, codec="libx264")

List Animation

from hypertiling.graphics.animation import AnimatorList

Pre-compute some data and store it in 2D array-like container. Here we use our Ising model, but of course data can come from anywhere

# first dimension: max number of frames, second dimension: len(tiling)
colors_list = np.zeros((100,len(T)))

for i in range(100):
    colors_list[i] = step(colors, beta=0.1)

# create figure
fig, axs = plt.subplots(1,1,figsize=(7,7))

# plot tiling using built in function "poly2patch"
patchkwargs = {"cmap": cmap.RdYlGn, "edgecolor": "#f1f2f0ff", "lw": 0.7, "clim":[-1.5,1.5]}
pgons = convert_polygons_to_patches(T, colors, **patchkwargs)
axs.add_collection(pgons)

# ------- this is the animation part -------

# arguments to be passed through to the matplotlib FuncAnimator
animargs = {"interval": 200, "blit": True}

# the actual animator
ani = AnimatorList(colors_list, fig, pgons, animargs)

# -------

# some usual plot adjustments
axs.add_patch(plt.Circle((0.0, 0.0), radius=1, edgecolor='k', facecolor='None', zorder=7))
axs.set_xlim(-1.02,1.02)
axs.set_ylim(-1.02,1.02)
axs.axis("off"); axs.set_aspect('equal'); plt.show()

ani.save("/home/user/anim.mp4", fps=10, codec="libx264")

Path Animation

With AnimatorPath you can parse a predefined path which the camera will track throughout the animation. Here we will demonstrate how walking in a circle on a hyperbolic disk we accumulate rotation.

from hypertiling.graphics.plot import plot_tiling
from hypertiling.graphics.animation import AnimatorPath

T = HyperbolicTiling(4, 5, 3)
T.rotate(-np.pi/4)

plot_tiling(T)

for poly in T.polygons:
    x, y = poly.get_center().real, poly.get_center().imag
    plt.text(x, y, f"{poly.idx}", ha="center", va="center", fontsize=17*(1-0.7*np.abs(x*x+y*y)))

Our path will be: 0 → 1 → 2 → 48 → 46 → 0

We paint the cells with random colors in order to make it easier to follow the movement

path = [0, 1, 2, 48, 46, 0]
colors = np.random.rand(len(T))

fig, ax = plt.subplots()
ani = AnimatorPath(colors, fig, ax, T, path)

Note that while our path returns back to central polygon, the lattice is now rotated by \(\frac{\pi}{2}\) in clock-wise direction. This is due to the curvature of the space and does not happen in flat space (try it out on a piece of grid paper yourself!).

ani.save("/home/user/anim.mp4", fps=60)