Tilings by regular polygons
Tessella is a Scala library that helps working with finite unitregularpolygon tessellations of a flat surface, a classical theme in the wider field of tessellations (or tilings). See a mathematical definition of tiling for a more accurate notion of the chosen constraints.
Finite set of the (▲.■.⬣.■) Archimedean (1uniform) tiling
Tessellation as graph
Each tessellation is internally described as an 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.
Graphs are seamless Scala collections thanks to the excellent Graph for Scala library by Peter Empen.
Stronger constraints
A Mono
object is a monogonal Tiling
, with all vertices having the same gonality, see Archimedean tilings.
How to
Use
Mill project
From aimport mill._, scalalib._
object foo extends ScalaModule {
def ivyDeps = Agg(
ivy"vision.id::tessella:0.3.1"
)
}
sbt project
From anlibraryDependencies += "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

perimeter polygon

filled polygons coloured according to number of sides

full vertices coloured according to type of adjacent polygons (gonality)
Test
The library can be built with
Mill
 Install Mill
 Open a terminal and
cd
to the repo directory  Use the
mill jvm.test
command to run all tests  Or use the
mill jvm.test.one [testClassName]
command to run a single test class
sbt
 Open a terminal and
cd
to the repo directory  Use the
sbt
command to launch sbt  Use the
test
command to run all tests  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 pgons 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 pgons 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 pgons
There are only 21 possible combinations of regular pgons for the full vertex and they use only 14 different pgons:
 Triangle, 3 sides, alt symbol
▲
 Square, 4 sides, alt symbol
■
 Pentagon, 5 sides, alt symbol
⬟
 Hexagon, 6 sides, alt symbol
⬣
 Eptagon, 7 sides
 Octagon, 8 sides, alt symbol
⯃
 Ennagon, 9 sides
 Decagon, 10 sides
 Dodecagon, 12 sides
 Gon15, 15 sides
 Gon18, 18 sides
 Icosagon, 20 sides
 Gon24, 24 sides
 Gon42, 42 sides
Infinite tiling
Can be named after the different full vertices it's composed of.
Example: (▲.■.⬣.■)
or (▲⁶; ▲².■.▲.■)