.. DO NOT EDIT. .. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY. .. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE: .. "auto_examples/04_demos/viz_fractals.py" .. LINE NUMBERS ARE GIVEN BELOW. .. only:: html .. note:: :class: sphx-glr-download-link-note :ref:Go to the end  to download the full example code .. rst-class:: sphx-glr-example-title .. _sphx_glr_auto_examples_04_demos_viz_fractals.py: ======== Fractals ======== Fractals are geometric structures that are self-similar at any scale. These structures are easy to generate using recursion. In this demo, we'll be implementing the following fractals: - Sierpinski Tetrahedron or Tetrix - Menger Sponge - Moseley Snowflake Let's begin by importing some necessary modules. We need fury.primitive to avoid having to hardcode the geometry of a tetrahedron and a cube. fury.utils also contains a repeat_primitive function which we will use for this demo. .. GENERATED FROM PYTHON SOURCE LINES 19-26 .. code-block:: Python import math import numpy as np from fury import primitive, ui, utils, window .. GENERATED FROM PYTHON SOURCE LINES 27-51 Before we create our first fractal, let's set some ground rules for us to work with. 1. Instead of creating a new actor to represent each primitive of the fractal, we will compute the centers of each primitive and draw them at once using repeat_primitive(). 2. How many primitives do we need? For each fractal, we define a depth which will prevent infinite recursion. Assuming we have a depth of :math:N, and at each level the shape is divided into :math:k smaller parts, we will need :math:k^{N} primitives to represent the fractal. 3. Ideally, we want to allocate the array of centers upfront. To achieve this, we can use the method of representing a binary tree in an array, and extend it to work with k-ary trees (formulas for the same can be found here_). In this scheme of representation, we represent every primitive as a node, and each sub-primitive as a child node. We can also skip storing the first :math:\frac{k^{N} - 1}{k - 1} + 1 entries as we only need to render the leaf nodes. This allows us to create an array of exactly the required size at the start, without any additional overhead. .. _here: https://book.huihoo.com/data-structures-and-algorithms-with-object-oriented-design-patterns-in-c++/html/page356.html # noqa ----------------------------------------------------------------------------- .. GENERATED FROM PYTHON SOURCE LINES 53-58 The tetrix is a classic 3d fractal, a natural three-dimensional extension of the Sierpinski Triangle. At each level, we need to calculate the new centers for the next level. We can use the vertices of a tetrahedron as the offsets for the new centers, provided that the tetrahedron is centered at the origin (which is the case here). .. GENERATED FROM PYTHON SOURCE LINES 58-96 .. code-block:: Python def tetrix(N): centers = np.zeros((4**N, 3)) # skipping non-leaf nodes (see above) offset = (4**N - 1) // 3 + 1 # just need the vertices U, _ = primitive.prim_tetrahedron() def gen_centers(depth, pos, center, dist): if depth == N: centers[pos - offset] = center else: idx = 4 * (pos - 1) + 2 for i in range(4): # distance gets halved at each level gen_centers(depth + 1, idx + i, center + dist * U[i], dist / 2) # the division by sqrt(6) is to ensure correct scale gen_centers(0, 1, np.zeros(3), 2 / (6**0.5)) vertices, faces = primitive.prim_tetrahedron() # primitive is scaled down depending on level vertices /= 2 ** (N - 1) # compute some pretty colors bounds_min, bounds_max = np.min(centers, axis=0), np.max(centers, axis=0) colors = (centers - bounds_min) / (bounds_max - bounds_min) vertices, triangles, colors, _ = primitive.repeat_primitive( centers=centers, colors=colors, vertices=vertices, faces=faces ) return utils.get_actor_from_primitive(vertices, triangles, colors) .. GENERATED FROM PYTHON SOURCE LINES 97-104 For a Menger Sponge, each cube is divided into 27 smaller cubes, and we skip some of them (face centers, and the center of the cube). This means that on every level we get 20 new cubes. Here, to compute the points of each new center, we start at a corner cube's center and add the offsets to each smaller cube, scaled according to the level. .. GENERATED FROM PYTHON SOURCE LINES 104-164 .. code-block:: Python def sponge(N): centers = np.zeros((20**N, 3)) offset = (20**N - 1) // 19 + 1 # these are the offsets of the new centers at the next level of recursion # each cube is divided into 20 smaller cubes for a snowflake V = np.array( [ [0, 0, 0], [0, 0, 1], [0, 0, 2], [0, 1, 0], [0, 1, 2], [0, 2, 0], [0, 2, 1], [0, 2, 2], [1, 0, 0], [1, 0, 2], [1, 2, 0], [1, 2, 2], [2, 0, 0], [2, 0, 1], [2, 0, 2], [2, 1, 0], [2, 1, 2], [2, 2, 0], [2, 2, 1], [2, 2, 2], ] ) def gen_centers(depth, pos, center, dist): if depth == N: centers[pos - offset] = center else: # we consider a corner cube as our starting point start = center - np.array([1, 1, 1]) * dist**0.5 idx = 20 * (pos - 1) + 2 # this moves from the corner cube to each new cube's center for i in range(20): # each cube is divided into 27 cubes so side gets divided by 3 gen_centers(depth + 1, idx + i, start + V[i] * dist, dist / 3) gen_centers(0, 1, np.zeros(3), 1 / 3) vertices, faces = primitive.prim_box() vertices /= 3**N bounds_min, bounds_max = np.min(centers, axis=0), np.max(centers, axis=0) colors = (centers - bounds_min) / (bounds_max - bounds_min) vertices, triangles, colors, _ = primitive.repeat_primitive( centers=centers, colors=colors, vertices=vertices, faces=faces ) return utils.get_actor_from_primitive(vertices, triangles, colors) .. GENERATED FROM PYTHON SOURCE LINES 165-168 A snowflake is exactly the same as above, but we skip different cubes (corners and center). I think this looks quite interesting, and it is possible to see the Koch snowflake if you position the camera just right. .. GENERATED FROM PYTHON SOURCE LINES 168-219 .. code-block:: Python def snowflake(N): centers = np.zeros((18**N, 3)) offset = (18**N - 1) // 17 + 1 V = np.array( [ [0, 0, 1], [0, 1, 0], [0, 1, 1], [0, 1, 2], [0, 2, 1], [1, 0, 0], [1, 0, 1], [1, 0, 2], [1, 1, 0], [1, 1, 2], [1, 2, 0], [1, 2, 1], [1, 2, 2], [2, 0, 1], [2, 1, 0], [2, 1, 1], [2, 1, 2], [2, 2, 1], ] ) def gen_centers(depth, pos, center, side): if depth == N: centers[pos - offset] = center else: start = center - np.array([1, 1, 1]) * side**0.5 idx = 18 * (pos - 1) + 2 for i in range(18): gen_centers(depth + 1, idx + i, start + V[i] * side, side / 3) gen_centers(0, 1, np.zeros(3), 1 / 3) vertices, faces = primitive.prim_box() vertices /= 3**N bounds_min, bounds_max = np.min(centers, axis=0), np.max(centers, axis=0) colors = (centers - bounds_min) / (bounds_max - bounds_min) vertices, triangles, colors, _ = primitive.repeat_primitive( centers=centers, colors=colors, vertices=vertices, faces=faces ) return utils.get_actor_from_primitive(vertices, triangles, colors) .. GENERATED FROM PYTHON SOURCE LINES 220-222 Now that we have the functions to generate fractals, we can start setting up the Scene and ShowManager. .. GENERATED FROM PYTHON SOURCE LINES 222-226 .. code-block:: Python scene = window.Scene() showmgr = window.ShowManager(scene, 'Fractals', (800, 800), reset_camera=True) .. GENERATED FROM PYTHON SOURCE LINES 227-229 These values are what work nicely on my machine without lagging. If you have a powerful machine, you could bump these up by around 2-3. .. GENERATED FROM PYTHON SOURCE LINES 229-232 .. code-block:: Python fractals = [tetrix(6), sponge(3), snowflake(3)] .. GENERATED FROM PYTHON SOURCE LINES 233-236 We want to be able to switch between the three fractals. To achieve this we'll create a RadioButton and register a callback which will remove existing fractals and add the selected one. This also resets the camera. .. GENERATED FROM PYTHON SOURCE LINES 236-264 .. code-block:: Python options = { 'Tetrix': 0, 'Sponge': 1, 'Snowflake': 2, } shape_chooser = ui.RadioButton( options.keys(), padding=10, font_size=16, checked_labels=['Tetrix'], position=(10, 10), ) def choose_shape(radio): showmgr.scene.rm(*fractals) showmgr.scene.add(fractals[options[radio.checked_labels[0]]]) showmgr.scene.reset_camera() shape_chooser.on_change = choose_shape # selected at start showmgr.scene.add(fractals[0]) showmgr.scene.add(shape_chooser) .. GENERATED FROM PYTHON SOURCE LINES 265-269 Let's add some basic camera movement to make it look a little interesting. We can use a callback here to update a counter and calculate the camera positions using the counter. sin and cos are used here to make smooth looping movements. .. GENERATED FROM PYTHON SOURCE LINES 269-283 .. code-block:: Python counter = 0 def timer_callback(_obj, _event): global counter counter += 1 showmgr.scene.azimuth(math.sin(counter * 0.01)) showmgr.scene.elevation(math.cos(counter * 0.01) / 4) showmgr.render() showmgr.add_timer_callback(True, 20, timer_callback) .. rst-class:: sphx-glr-script-out .. code-block:: none 4 .. GENERATED FROM PYTHON SOURCE LINES 284-287 Finally, show the window if running in interactive mode or render to an image otherwise. This is needed for generating the documentation that you are reading. .. GENERATED FROM PYTHON SOURCE LINES 287-293 .. code-block:: Python interactive = False if interactive: showmgr.start() else: window.record(showmgr.scene, out_path='fractals.png', size=(800, 800)) .. image-sg:: /auto_examples/04_demos/images/sphx_glr_viz_fractals_001.png :alt: viz fractals :srcset: /auto_examples/04_demos/images/sphx_glr_viz_fractals_001.png :class: sphx-glr-single-img .. rst-class:: sphx-glr-timing **Total running time of the script:** (0 minutes 0.193 seconds) .. _sphx_glr_download_auto_examples_04_demos_viz_fractals.py: .. only:: html .. container:: sphx-glr-footer sphx-glr-footer-example .. container:: sphx-glr-download sphx-glr-download-jupyter :download:Download Jupyter notebook: viz_fractals.ipynb  .. container:: sphx-glr-download sphx-glr-download-python :download:Download Python source code: viz_fractals.py  .. only:: html .. rst-class:: sphx-glr-signature Gallery generated by Sphinx-Gallery _