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https://github.com/mcallisto/tessella

Tilings (tessellations) by regular polygons
https://github.com/mcallisto/tessella

graphs scala tessellation

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Tilings (tessellations) by regular polygons

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README 

        
⚠️⛔ This project is now deprecated, please use the new and improved [Tessella](https://github.com/scala-tessella/tessella)



# ~~Tilings by regular polygons~~



**Tessella** is a Scala library that helps working with finite unit-regular-polygon tessellations of a flat surface, a classical theme in the wider field of [tessellations](https://en.wikipedia.org/wiki/Tessellation) (or _tilings_). See a [mathematical definition of tiling](docs/tiling-definition.md) for a more accurate notion of the chosen constraints.



> ![(▲.■.⬣.■)](docs/(▲.■.⬣.■).svg)

>

> _Finite set of the (▲.■.⬣.■) Archimedean (1-uniform) tiling_



## Tessellation as graph



Each tessellation is internally described as an [undirected graph](https://en.wikipedia.org/wiki/Graph_(discrete_mathematics)#Undirected_graph), where:



*   each **node** of the graph is a _**vertex**_ of a polygon and it is represented by a unique `Int`

*   each **edge** of the graph is a _**side**_ of a polygon



The graph describing the tessellation is a `Tiling` object and can be created through algorithms.

Many of them, exploiting linear symmetries or growth strategies, are already available from the `creation` subpackage.



An undirected graph is not necessarily a valid `Tiling`, see [graph validation](docs/graph-validation.md).



Graphs are seamless Scala collections thanks to the excellent [Graph for Scala](https://scala-graph.org/) library by Peter Empen.



### Stronger constraints



A `Mono` object is a monogonal `Tiling`, with all vertices having the same gonality, see [Archimedean tilings](docs/archimedean.md).



## How to



### Use



#### From a **[Mill](http://www.lihaoyi.com/mill)** project



```scala

import mill._, scalalib._



object foo extends ScalaModule {

  def ivyDeps = Agg(

    ivy"vision.id::tessella:0.3.1"

  )

}

```



#### From an **[sbt](https://www.scala-sbt.org/)** project



```scala

libraryDependencies += "vision.id" % "tessella" % "0.3.1"

```



### Draw a tessellation



Each node of the tessellation can be `Map`ped to a cartesian point with the `Tiling.toNodesMap` method. The tessellation can then be rendered as a layered SVG image with the `Draw(…)` method of the `SVG` trait.



The lowest node is always at coordinates `(0.0, 0.0)` and the second lowest at `(1.0, 0.0)`.



On top of the edge layer, the following optional additions are possible:



*   node labels

    > ![(⬟².10)_label](docs/(⬟².10)_label.svg)



*   perimeter polygon

    > ![(⬟².10)_perimeter](docs/(⬟².10)_perimeter.svg)



*   filled polygons coloured according to number of sides

    > ![(⬟².10)_filled](docs/(⬟².10)_filled.svg)



*   full vertices coloured according to type of adjacent polygons (gonality)

    > ![(▲⁶; (⬣³)²; (▲².⬣²)²)](docs/(▲⁶;(⬣³)²;(▲².⬣²)²).svg)



### Test



The library can be built with



#### Mill



1.  [Install](http://www.lihaoyi.com/mill/#installation) Mill

2.  Open a terminal and `cd` to the repo directory

3.  Use the `mill jvm.test` command to run all tests

4.  Or use the `mill jvm.test.one [testClassName]` command to run a single test class



#### sbt



1.  Open a terminal and `cd` to the repo directory

2.  Use the `sbt` command to launch sbt

3.  Use the `test` command to run all tests

4.  Or use the `testOnly vision.id.tessella.[testClassName]` command to run a single test class



### Redraw the images in the /docs folder



Uncomment the first test in the `outputTest` class and run it.



### Draw several examples of algorithmic tessellations



Uncomment the second test in the `outputTest` class, run it and view them ordered in the temporary `out/jvm/test/myAlgos` folder.



## Vertex



For the library methods a given vertex is described by the ordered adjacent regular p-gons joined at the vertex.



Example: `(▲.■)`, alternative form `(3.4)`



_Note:_ description can be shortened if consecutive identical elements, where `(⬣.⬣.▲)` becomes `(⬣².▲)`, alternative form `(6*2.3)`



A vertex is **full** when the adjacent regular p-gons join to exactly complete a full circle.



Examples: `(▲.■.⬣.■)` and `(■.⬣.12)`



In a finite tessellation:



*   all vertices on the perimeter are NOT full

*   all other vertices are full



### 14 p-gons



There are only 21 possible combinations of regular p-gons for the full vertex and they use only 14 different p-gons:



1.  **Triangle**, **3** sides, alt symbol `▲`

2.  **Square**, **4** sides, alt symbol `■`

3.  **Pentagon**, **5** sides, alt symbol `⬟`

4.  **Hexagon**, **6** sides, alt symbol `⬣`

5.  **Eptagon**, **7** sides

6.  **Octagon**, **8** sides, alt symbol `⯃`

7.  **Ennagon**, **9** sides

8.  **Decagon**, **10** sides

9.  **Dodecagon**, **12** sides

10. **Gon15**, **15** sides

11. **Gon18**, **18** sides

12. **Icosagon**, **20** sides

13. **Gon24**, **24** sides

14. **Gon42**, **42** sides



## Infinite tiling



Can be named after the different full vertices it's composed of.



Example: `(▲.■.⬣.■)` or `(▲⁶; ▲².■.▲.■)`



[travis-badge]: https://travis-ci.org/mcallisto/tessella.svg

[travis-link]: https://travis-ci.org/mcallisto/tessella

[maven-badge]: https://maven-badges.herokuapp.com/maven-central/vision.id/tessella_2.12/badge.svg

[maven-link]: https://maven-badges.herokuapp.com/maven-central/vision.id/tessella_2.12

[codacy-badge]: https://api.codacy.com/project/badge/Grade/c9a888d7249943a3b5b82e64fdcc7a52

[codacy-link]: https://www.codacy.com/app/mcallisto/tessella?utm_source=github.com&utm_medium=referral&utm_content=mcallisto/tessella&utm_campaign=Badge_Grade

[codecov-badge]: https://codecov.io/gh/mcallisto/tessella/branch/master/graph/badge.svg

[codecov-link]: https://codecov.io/gh/mcallisto/tessella