tilezz

This repository provides two main features:

1. practical implementation of various cyclotomic integer rings (I simply call them complex integers)
2. grid-free (but still exact) representation of a rich family of polygonal tiles based on these rings

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/
• TODO: write followup post

Note that due to time constraints, this is a work-(not-so-fast-)in-progress.

Usage

This crate provides the abstract geometric API for using concrete representations of complex integer rings (i.e. cyclotomic fields for some fixed root of unity without division) and polygonal tiles build on top of these rings.

It also provides some plotting functionality based on plotterrs, which means that you can easily render tiles built with this crate into various backends, including PNG, SVG, web pages (HTML5/WASM), including interactive usage in Jupyter.

Interactive (Jupyter Notebook)

Thanks to the capabilities of the plotters library, it is easy to use Jupyter notebooks with this crate to visualize polygonal tiles.

Requirements

• 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 official evcxr documentation), then you are already set up for using this crate interactively.

Rendering the Spectre Tile Over the Cyclotomic Integer Ring ZZ12

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 tilezz

Step 2: Run jupyter notebook

Step 3: Open the example notebook OR Create a new Rust notebook (which is powered by evcxr) and add the following code into a cell:

:dep plotters = { version = "^0.3.7", features = ["evcxr", "all_series"] }
// 1(a) if you want to use a published version of this crate:
// :dep tilezz = "*"
// 1(b) (RECOMMENDED) if you cloned this repository and want to use the current development version:
:dep tilezz = { path = "." }

use plotters::prelude::*;
use tilezz::snake::constants::spectre;
use tilezz::snake::{Snake, Turtle};
use tilezz::plotters::plot_tile;
use tilezz::zz::ZZ12;

evcxr_figure((500,500), |root| {
    let s: Snake<ZZ12> = spectre();
    let tile = s.to_polyline_f64(&Turtle::default());

    let _ = root.fill(&WHITE);
    let root = root.margin(10, 10, 10, 10);

    plot_tile(
        &mut ChartBuilder::on(&root)
            .caption("Spectre tile", ("sans-serif", 40).into_font())
            .x_label_area_size(20)
            .y_label_area_size(40),
        &tile,
    );
    root.present()?;
    Ok(())
})

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.

TODO: improve/finalize API, update the example and add a screenshot.

See Also

Tiles and Tilings

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 something similar is of course much more heavy and requires 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.

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 tiling pieces from which the polyform tiles can be built from. My approach is completely grid-free, so it is more general. However, it also means less fixed structure to work with and exploit, and I don't provide any sophisticated solvers (yet).

Cyclotomic Integer Rings (i.e. Complex Integers)

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.

• https://github.com/CyclotomicFields/cyclotomic (Rust)
• https://github.com/walck/cyclotomic (Haskell)

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 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 the respective ring, there is no overhead due to management of symbolic representations or anything like that, because for each ring, the provided data type encodes the values of each ring as vectors over a linearly independent set base of units (and all their distinct symbolic products), together with a ring-specific implementation of multiplication, which hard-codes the symbolic simplifications of expressions that appear during the evaluation of multiplication.

I have 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.

Roadmap

• implement complex integer rings generically
• implement simple polygonal chains over complex integers
• implement simple polygons with useful operations
• implement (interactive) visualization utilities (i.e. render images and/or use WebGL)