Perfect-precision 2D polygonal tiles over cyclotomic integer rings - no floats, no coordinates, no grids.
This repository provides the following main features:
- practical implementation of various cyclotomic rings and fields, which are subsets of complex numbers that admit exact representation
- exact representation of a rich family of polygonal tiles based on these rings, using a minimalistic and discrete form of turtle semantics
- some useful string-based algorithms to work with these objects and visualize them
These features taken together grant us multiple freedoms:
- freedom of floating-point numbers and subtle bugs due to numerical issues
- freedom of coordinates, as polygons are described by relative angles and movements to trace out their boundary
- freedom of a (typical) grid, when using the polygons in the context of tiles and tilings
The target audience are primarily math enthusiasts (like myself) or researchers working on (a)periodic tiles or other areas where exact representation of polygons is crucial and where numerical approximation and the use of floating point arithmetic are not acceptable for the given purpose, yet the use of more generic solutions, such as computer algebra systems is not desirable, too inconvenient or too inefficient.
The concepts this work is based on are described in the following blog posts:
- https://pirogov.de/blog/perfect-precision-2d-geometry-complex-integers/
- https://pirogov.de/blog/intersecting-segments-without-tears/
- (more posts will probably be added over time)
Note that due to time constraints, this is a work-(not-so-fast-)in-progress. The API is not stable, and maybe never will be, so usage as a library is at your own risk.
To be able to execute the demos on your computer, make sure to build the crate with the cli feature enabled.
Left: BFS from the origin, Right: BFS in the unit square, starting in the corners and normalizing discovered point modulo unit square. Each image shows the new points discovered in the corresponding round.
To generate images like these, check out the cyc_explore binary.
Left: All 965 distinct polyominos with boundary length up to 16 over Z[ζ_4] (computation time: ~25 ms),
Right: All 933 distinct matchstick polygons with boundary length up to 8 over Z[ζ_12] (computation time: ~0.7 s).
The polygon sets are computed by a single-threaded DFS over angle sequences with a lex-min rotation prune
that collapses each polygon's n cyclic walks down to one - see rat_enum.
To generate images like these, check out the rat_enum binary.
The Rat Explorer is an interactive and mobile-friendly database of simple cyclotomic matchstick polygons.
This crate provides the abstract geometric API for using concrete representations of constructible cyclotomic rings for some fixed root of unity, and polygonal tiles build on top of these rings.
Instead of representing polygons by segments and coordinates, they are described using a form of turtle semantics, i.e. interpreting a sequence of discrete external angles as instructions for tracing out a polygon or segment chain from a given starting point by performing unit-length steps in some direction. This, together with the fact that we use cyclotomics for actual coordinates, helps avoiding dependence on floating point numbers and explicit coordinates.
Here is a conceptual mapping for the relevant geometrical objects:
- a point corresponds to a turtle, which is an "oriented point" (it has an angle, defining its facing direction)
- a polygonal chain corresponds to a snake, which consists of instructions for a turtle
- a simple polygon corresponds to a closed snake, which I call a rat (for rational tile)
- a tileset is a collection of distinct rats you can take as building blocks
- a patch is a collection of rats from a tileset glued along matching boundaries edge-to-edge
The reason why I call these segment chains and polygons rational is because all the side lengths can be expressed as integer multiples of a common length, which then can be interpreted as normalized unit steps. So contrary to classical polygon representation, the smallest meaningful unit is not a point (you cannot connect arbitrary points), but a unit step.
By using cyclotomic integers for coordinates and expressing all geometric objects in terms of unit steps into some direction, each simple polygon allows for a natural representation as a sequence of external angles along its boundary. As the sequence is cyclic, there is one cyclically shifted sequence for each starting vertex.
The canonical representation is then simply the lexicographically minimal such sequence. Note that this gives us a simple and efficient equivalence check on polygons: two polygons are equal iff they have the same canonical angle sequence. Treating (rational) polygons as strings of angles also allows us to use other efficient string-based algorithms, e.g. to compute combinations of tiles.
This is a library and an experimentation sandbox of data structures trying to push the basic ideas outlined above as far as possible.
The library also provides a 2D rendering pipeline that helps visualizing the
provided data structures by rendering them into various output formats, such as
SVG, PNG (raster feature) or GIF (animation feature).
Add the crate to a Rust project:
[dependencies]
tilezz = "0.1.2"Full API reference (auto-built by docs.rs against the current published version): https://docs.rs/tilezz
Default features are intentionally empty so the dependency stays lean - just the core cyclotomic types + geometry + string algorithms, no clap, threading, or rendering deps unless you opt in. Enable what you need:
raster: PNG output viaresvg+tiny-skiaanimation: multi-frame GIF (impliesraster)cli: bundled CLI binaries (rat_enum,cyc_explore,patch_enum,polyomino,tileset_collect); impliesanimation
To install the binaries above directly without cloning the repo:
cargo install tilezz --features cliThe binaries land under ~/.cargo/bin/. Each accepts --help for its
specific options. See the Demonstration section above for
what they produce.
The library integrates with Jupyter notebooks
through the evcxr Rust kernel. A
Scene exposes an evcxr_display-aware wrapper via scene.display(&vp)
that renders inline as SVG; scene.display_png(&vp) does the same as
PNG (requires the raster feature).
- Jupyter Notebook (Arch Linux users see here)
- evcxr
- evcxr_jupyter
If you have all the dependencies installed correctly and can create/open Jupyter Notebooks using a Rust kernel (check out the evcxr documentation), then you are already set up for using this crate interactively.
Here is how you can quickly construct and render the spectre tile:
Step 1: (Recommended) This step is only needed if you want to use the most recent development version from this repository.
Clone this repository and change to its directory, i.e. in a terminal run:
git clone https://github.com/apirogov/tilezz
cd tilezzStep 2: Run jupyter notebook
Step 3: Open the minimal example notebook OR
Create a new Rust notebook (which is powered by evcxr) and add the following code into a cell:
// Build and load the crate. The `raster` feature enables PNG output;
// SVG works without it.
:dep tilezz = { path = "..", features = ["raster"] }
use tilezz::cyclotomic::*;
use tilezz::geom::rat::Rat;
use tilezz::geom::snake::Turtle;
use tilezz::vis::draw::{MarkerStyle, TileStyle};
use tilezz::vis::scene::{Color, Fill, Scene, Stroke, TextStyle, Viewport};
// Define a sequence of external angles. All segments have unit length,
// so this fully determines a polygon.
let external_angles: &[i8] = &[3, 2, 0, 2, -3, 2, 3, 2, -3, 2, 3, -2, 3, -2];
// Instantiate an abstract polygon over the cyclotomic ring Z[ζ_12].
let r: Rat<ZZ12> = external_angles.try_into().unwrap();
// Trace it out in the cartesian plane.
let pts: Vec<(f64, f64)> = r.to_polyline_f64(Turtle::default());
// Build a scene with one filled tile + red vertex markers + index labels.
let mut scene: Scene = Scene::new().with_background(Color::WHITE);
scene.draw_tile(
&pts,
&TileStyle::filled(
Fill::solid(Color::YELLOW.with_alpha(96)),
Stroke::solid(Color::BLACK, 0.04),
)
.with_vertex_marker(MarkerStyle::filled_circle(0.2, Color::RED))
.with_vertex_labels(TextStyle::new(0.15, Color::WHITE).bold()),
);
// Render inline as SVG.
let vp: Viewport = Viewport::square_for(500, scene.auto_bounds().unwrap(), 16);
scene.display(&vp)After waiting for some seconds (the required dependencies have to be built first, after that it is faster), you should see a plot showing the spectre tile.
It seems that people who work on/with tiles use software like:
- computer algebra systems, such as SageMath
- SAT or SMT solvers, such as Z3
- PolyForm Puzzle Solver
SageMath or similar systems are of course much more heavy and require deeper algebraic understanding to even ask what you want. This crate provides much simpler (and I hope more efficient) solutions to much more specific problems.
Similarly, SAT or SMT solvers are excellent tools for certain NP-complete problems (I have some experience using them), but encoding tiling problems into suitable formulas is far from trivial and requires some thought and work (even though it sometimes is possible). It can really pay off and work well if you have enough knowledge about SAT/SMT encoding tricks and also have a deeper grasp on the structure of the problem at hand. But it is not something I'd pull out to just quickly come up with a polygon and check its behavior as a tile.
The PolyForm Solver seems to have some adoption by the tiling community and looks like a mature and feature-rich package that I eventually want to try out myself. From a cursory look, it seems like the biggest conceptual difference is that the PolyForm Solver is grid-based - it requires selecting some underlying grid of primitive cells from which the polyform tiles can be built from. My approach is grid-free, so it is more general. Technically, the cyclotomic integers do provide another kind of grid, but it is not a typical periodic cell grid as typically used to describe tiles and tilings. This means that more exotic and irregular tiles can be expressed, but also that there is less fixed structure to work with and exploit, and I don't provide any sophisticated solvers (yet).
I have no clue about abstract algebra and number theory (I just stumbled into this topic trying to represent some tiles exactly), but it seems like there are a few very general implementations of cyclotomic fields.
Compared to these packages, I do not try to implement the full set of cyclotomic integers (i.e. one type that includes and works with all roots of unity at once). This crate provides a separate ring/field for each supported root of unity instead, which then serves as the backbone for representing geometry of suitable polygons.
The corresponding underlying representations are optimized for and limited to the respective ring, there is no overhead due to management of symbolic representations or anything like that, because for each ring, the data type implements an efficient implementation using a minimal basis, together with ring-specific implementations of multiplication and some other operations.
I have not tried to compare this crate to the other approaches or benchmark anything yet, because the implementations of the complex integers were not the intended main feature of this crate. If someone is mainly interested in this crate for the implementation of the cyclotomic integer rings, I would be happy to get some feedback on how this compares to the more generic implementations with similar features.
Licensed under the MIT License.