tessagon: tessellation / tiling with python

Tessellate your favorite 3D surfaces (technically, 2D manifolds) with triangles, hexagons, or a number of other curated tiling types!

News

A nascent Inkscape extension to create tiling patterns: https://github.com/cwant/inkscape-tiling-extension

You can find a Blender (3.6) pull request to add support for Tessagon in the "XYZ function" addon here: https://projects.blender.org/blender/blender-addons/pulls/104726

Check out my presentation on the creation of Tessagon as an open source Python project here: https://cwant.github.io/python-os-presentation (or watch on YouTube)

A demo of using Tessagon with Inkscape Simple Scripting can be found here.

TL;DR

Check out the repository and look in the demo directory.

• Blender: you'll find a blender file and tessagon_blender_demo.py which creates the meshes in the demo. The demo has examples of each tessagon class, and an example that uses tessagon with one of my other projects, wire_skin.
• VTK: Take a look at tessagon_vtk_demo.py for a script that creates all of the current tessagon classes.

This software may also be installed via pip:

python3 -m pip install tessagon

or

pip3 install tessagon

How it works

Three things are needed to use tessagon to tessellate the surface of a 2D-manifold (or more accurately, a patch on a 2D-manifold in 3-space):

• Tessagon provides a bunch of classes (subclasses of a class called Tessagon) that will tessellate a portion of UV-space with mesh patterns. Parameters provide the details of the bounds in UV-space, the resolution of the tiling, whether the tiling is cyclic, whether a cyclic domain "twists" (think a topological identification space, like a Mobius strip or a Klein bottle), whether it is rotated, etc. These classes are in the tessagon.types module.
• The programmer must provide a formula that describes the shape of surface of the 3-dimensional object to create. This function maps two-dimensional space (also known as UV-space) into 3-dimensional space. The tiling happens in the input two dimensional space, and the function maps the tiling onto the surface of the 3-dimensional shape. There are some demo functions in tessagon.misc.shapes, such as torii (a.k.a. donuts), spheres, cylinders, etc.

• Finally, an adaptor is chosen to create a mesh in a supported 3D software package. Currently Blender and VTK are supported applications. There is an adaptor that writes an SVG group to a string, with various options (root SVG node, style options). There is also a generic ListAdaptor that does not depend on any external package, and can aid in the creation of importers (and is useful for testing/debugging):

  • adaptor BlenderAdaptor from the module tessagon.adaptors.blender_adaptor. The output from the adaptor's get_mesh method is of type BMesh.
  • adaptor VtkAdaptor from the module tessagon.adaptors.vtk_adaptor. The output from the adaptor's get_mesh method is of type VtkPolydata.
  • adaptor SvgAdaptor from the module tessagon.adaptors.svg_adaptor. The output from the adaptor's get_mesh method is a string (by default, an SVG group of polygons).
  • adaptor ListAdaptor from the module tessagon.adaptors.list_adaptor. The output from the adaptor's get_mesh method is a dict with keys vert_list, face_list and color_list, which point to lists of vertices, faces (as indices into the vertex list), and color indices for each face.

  (Note that the get_mesh methods mentioned above are usually called indirectly through the Tessagon method create_mesh.)

The reader should check out the demos, but here is some very basic usage using blender:

from tessagon.types.hex_tessagon import HexTessagon
from tessagon.adaptors.blender_adaptor import BlenderAdaptor

def my_func(u,v):
  return [u, v, u**2 + v**2]

options = {
    'function': my_func,
    'u_range': [0.0, 1.0],
    'v_range': [0.0, 1.0],
    'u_num': 8,
    'v_num': 20,
    'u_cyclic': False,
    'v_cyclic': False,
    'adaptor_class' : BlenderAdaptor
  }
tessagon = HexTessagon(**options)

bmesh = tessagon.create_mesh()

# Do something with the bmesh ...

Tessagon classes

Additional tessagon classes can be added by deconstructing how a tessellation fits within a rectangular patch in the plane (check out the ASCII art in each source file in tessagon.types). The current Tessagon subclasses include:

Regular tilings

• SquareTessagon (8 color patterns)
  
• HexTessagon (2 color patterns)
  
• TriTessagon (3 color patterns)
  

Archimedean tilings

• OctoTessagon (1 color pattern)
  
• HexTriTessagon (1 color pattern)
  
• HexSquareTriTessagon (1 color pattern)
  
• DodecaTessagon (1 color pattern)
  
• SquareTriTessagon (2 color patterns)
  
• SquareTri2Tessagon (1 color pattern)
  
• DodecaTriTessagon (1 color pattern)
  
• BigHexTriTessagon (1 color pattern, 1 extra parameter)
  

Laves tilings

• RhombusTessagon (2 color patterns)
  
• FloretTessagon (3 color patterns)
  
• DissectedSquareTessagon (2 color patterns)
  
• DissectedTriangleTessagon (1 color pattern)
  
• DissectedHexQuadTessagon (2 color patterns)
  
• DissectedHexTriTessagon (1 color pattern)
  
• PentaTessagon (1 color pattern)
  
• Penta2Tessagon (1 color pattern)
  

Non-edge-to-edge tilings

• PythagoreanTessagon (1 color pattern)
  
• BrickTessagon (1 color pattern)
  
• WeaveTessagon (1 color pattern, 1 extra parameter)
  
• HexBigTriTessagon (2 color patterns)
  
• ZigZagTessagon (1 color pattern)
  
• ValemountTessagon (1 color pattern)
  
• CloverdaleTessagon (1 color pattern)
  

Non-convex tilings

• StanleyParkTessagon (2 color patterns)
  
• IslamicHexStarsTessagon (1 color pattern)
  
• IslamicStarsCrossesTessagon (1 color pattern)
  

Usage and Options

Each tessagon class is initialized with number of keyword options, e.g.:

from tessagon.types.dodeca_tessagon import DodecaTessagon
from tessagon.adaptors.vtk_adaptor import VtkAdaptor
tessagon = DodecaTessagon(function=my_func,
                          u_range=[0.0, 1.0],
                          v_range=[0.0, 1.0],
                          u_num=8,
                          v_num=20,
                          u_cyclic=True,
                          v_cyclic=False,
                          adaptor_class=VtkAdaptor)
poly_data = tessagon.create_mesh()

The create_mesh() method creates a tessellated mesh using your provided function and the tile type corresponding to the tessagon class used. The chosen adaptor dictates the 3D data type the mesh will be (for the BlenderAdaptor the output is BMesh, for the VTKAdaptor the output is vtkPolyData).

Each Tessagon class may supply options that are specific to that class (they will be listed on the documentation page for each class).

Here are the options that are common to all Tessagon classes:

• function: the function to be used to generate the geometry. This is a function that takes two arguments u, v and returns a list of three items [x, y, z]
• u_range: a list with two items indicating the minimum and maximum values for u (the first argument to the function passed);
• v_range: a list with two items indicating the minimum and maximum values for v (the second argument to the function passed);
• u_num: the number of tiles to be created in the u-direction;
• v_num: the number of tiles to be created in the v-direction;

As you work with the software, keep in mind the difference between a face and a tile. A tile is not a face! A tile is a four-sided region (often a rectangle) that holds one or more faces that form a repeated pattern. A tile often shares faces with neighboring tiles:

The source code files for each tessagon class usually contain some ASCII art that illustrates how the pattern of faces is arranged on a tile.

• corners: instead of using u_range and v_range, you can also specify your domain in the uv plain as a quadrilateral by specifying the corners of the region you would like to map. This is a list of four tuples in the following order: bottom-left, bottom-right, top-left, top-right.

• u_cyclic: a boolean indicating whether the u-direction is cyclic (wraps around to the beginning again). You're function needs to be periodic in the u-direction for this to look nice. Note: the default for this is True, so set it to False if you don't want things to be cyclic
• v_cyclic: a boolean indicating whether the v-direction is cyclic. Default: True

• rot_factor: this is an integer greater than zero that allows you to rotate the tiles in the UV-domain is such a way that the tiling can still be cyclic.

  

  The rot_factor specifies how many tiles you go across before you go up one unit (essentially the reciprocal of the slope of the grid lines. The image depicts a rot_factor of three, which generates 45 tiles (the purple interior and the blue boundary squares, each of which is blasted with the tessellation pattern when the function is applied). Here the meaning of u_num and v_num are interpreted differently: whereas in the non-rotated case, u_num = 3 and v_num = 2 would yeild 6 tiles, here we have 45 tiles. Niether U nor V are cyclic in the picture; had they both been cyclic, 60 tiles whould have been generated. The interior tiles form groups of (rot_factor-1)**2 tiles (here each group is 2 x 2, for 24 total interior tiles), and each boundary shares rot_factor x 1 tiles with it's neighbors (there are 7 such boundaries, so 21 boundary tiles).

  It hurt my brain developing this feature, so don't feel bad if it does't make any sense for you. Play around with it, and keep in mind that the number of tiles generated is somewhere between u_num * v_num * (rot_factor - 1)**2 and u_num * v_num * (rot_factor**2 + 1) (depending on which tiles have neighbors, due to periodicity), so you typically want to set u_num and v_num to be a lot lower than you would in the non-rotated case.

• parallelogram_vectors: two vectors (e.g., [[5, 1], [-1, 6]]) that form a parallelogram to warp the tiles in a way that cyclic tiles can occur (this is more general than rot_factor, in fact the rot_factor implementation uses parallelogram_vectors). This one will require it's own page to describe.
• u_twist: this is used with v_cyclic (note that says v_cyclic not u_cyclic). As the v values wrap around, the u values reconnect in the opposite direction: the low u values connect to the high u values, and vice versa. This allows you to make things like Möbius strips and Klein bottles.
• v_twist: works with u_cyclic, analogous to how u_twist works.
• color_pattern: some Tessagon types support some curated patterns (applying more than one material/color to the output shape). These are identified by a number (e.g., color_pattern=2). The number has no meaning other than as an identifier. Checkout the description of the Tessagon types above to see what color patterns exist. The values of u_num and v_num may need to be tweaked to make a specific color pattern wrap correctly for cyclic tilings.
• simple_2d (default: False), multiplier_2d, translate_2d: these parameters (when simple_2d is True) help with creating tilings in the XY-plane (2D). The "natural" undistorted aspect ratio of the tiling is preserved. multiplier_2d (a number) allows you to scale such tilings, and translate_2d (a tuple) places a corner of the tiling (either top-left or bottom-left, depending on how you handle the up direction). With simple_2d, no function should be supplied to the Tessagon class.

Writing your own tessellation classes

All tesselations are found in the types module, so check out the source code there for numerous examples. The source code documentation in hex_tessagon is more verbose than the others, which hopefully will aid understanding.

Each tessellation involve two classes: a tile class (a child of Tile) and a child of class Tessagon. The Tessagon subclass is easy (it just declares which tile class with be used), so writing the tile class will take most of your time. (The FloretTessagon is an exception because the math is more complex.)

There are five methods that you will want to write:

• __init__: typically you will want to call the constructor of the super-class. You can declare your tile class to be symmetric in either the u-direction or the v-direction, which will help you greatly if you have structured your vertices and faces well.

• init_verts: here you define the structure of your verts in a nested hash. You can use any combination of keys you want as pathways to addressing your vertices. If you declared your class to be u-symmetric, then the words left and right will have significance in your pathways. Creating a vertex with the word left in it's pathway will automagically also create the symmetric right vertex, halfing the amount of work you need to do. For v-symmetric tiles, the words top and bottom have meaning, in an analogous way. If the tile is both u-symmetric and v-symmetric, then each vert created that has a combination of both left/right and top/bottom in it's pathway will actually create four symmetric vertices.

• init_faces: here you define the structure of your faces in a nested hash. As with vertices, you can reduce your work by exploiting symmetry and using left, right, top and bottom when addressing your faces.

• calculate_verts: here you define the locations of the vertices in the tiling, via the add_vert method. The method takes an array of keys to indicate which vertex is being defined, and a location expressed as a u_ratio and a v_ratio, both between 0 and 1. A u_ratio of 0 is left and a u_ratio of 1 is right. A v_ratio of 0 is bottom and a v_ratio of 1 is top. The symmetry of the tile may cause additional reflected vertices to be defined. Using the boolean keywords u_boundary, v_boundary and corner, you can tell the system whether the vert is shared with neighboring tiles (this reduces the work that needs to be done, and can also keep the model topologically sound and water-tight).

• calculate_faces: here you define the faces for the tiling using the previously defined vertices, via the add_face method. The method takes an array of face to indicate which face is being defined, and an array of vertex keys that indicate which vertices are in the face. These vertices are either on the current tile, or some of them can being on neighboring tiles (in which case the keys for the neighboring tile are also included). The symmetry of the tile may cause other reflected faces to be defined. Using the boolean keywords u_boundary, v_boundary and corner, you can tell the system whether the face is shared with neighboring tiles.

Metadata and Discovery

Each Tessagon subclass has some metadata attached to it (class TessagonMetadata) that describes the properties of the tiling. This metadata will be expanded as needs require, but currently consists of information about how many color patterns the tiling has, what type (regular, archimedean, laves, non_edge, non_convex), and what sorts of shapes are produced by instances of the class.

A nascent helper class tessagon.TessagonDiscovery exists that can help you search for tilings with certain properties (for example, this could be used to create a menu for an external application that categorizes the tilings). The methods in the class are intended to support chaining of operations (e.g., they results returned are also of type TessagonDiscovery, that can then be reified using the to_list method). Some examples:

find_all = TessagonDiscovery()
find_all.to_list() # a list of all of the tessagons
regular = find_all.with_classification('regular')
regular.to_list() # The three regular tilings (HexTessagon, SquareTessagon, TriTessagon)
regular.inverse().to_list() # All tilings except for the regular ones

Checkout the test suite for more example usage.

The TessagonDiscovery class also has a class method get_class for fetching tessagon classes without having to deal with importing that looks like this:

tessagon_cls = TessagonDiscovery.get_class('TriTessagon')
tessagon = tessagon_cls(function=my_func, ...)

wire_skin

Check out my other project wire_skin to add some interesting effects to the tessellated manifolds you create:

https://github.com/cwant/wire_skin

3D Printing on Shapeways

I primarily use Tessagon to create objects for 3D printing (on Shapeways). I tag my models with the keyword tessagon. If you also use Shapeways to print models made with Tessagon, feel free to use this tag -- I'd be very interested to see what you make!

https://www.shapeways.com/marketplace?q=tessagon