Symmetry

rotationalSymmetry

Computes the order of rotational symmetry, the number >= 1 of rotations in which a circular sequence looks exactly the same.

Example

Seq(0, 1, 2, 0, 1, 2).rotationalSymmetry // 2

symmetryIndices

Finds the indices at which each axis of reflectional symmetry intersects the circular sequence — a line of symmetry splits the sequence into two identical halves.

Example

Seq(2, 1, 2, 2, 1, 2, 2, 1, 2, 2, 1, 2).symmetryIndices // List(0, 3, 6, 9)

reflectionalSymmetryAxes

Each axis of reflectional symmetry crosses the ring at two points, which can be vertices (through an element) or edges (between two consecutive elements). reflectionalSymmetryAxes returns those pairs as (AxisLocation, AxisLocation), where AxisLocation is Vertex(i) or Edge(i, n) (n is the ring size; the edge is between index i and (i + 1) mod n).

Example

import io.github.scala_tessella.ring_seq.RingSeq._ // brings AxisLocation, Vertex and Edge into scope

Seq(1, 1, 1).reflectionalSymmetryAxes
// List(
//   (Vertex(1), Edge(2, 3)),
//   (Vertex(2), Edge(0, 3)),
//   (Vertex(0), Edge(1, 3))
// )

A sequence with no reflectional symmetry returns Nil; a sequence with mirror symmetry through two vertices (e.g. a doubled triangle) returns vertex–vertex pairs.

Seq(1, 2, 1, 2, 1, 2).reflectionalSymmetryAxes
// List((Vertex(2), Vertex(5)), (Vertex(1), Vertex(4)), (Vertex(0), Vertex(3)))

symmetry

Computes the order of reflectional (mirror) symmetry, the number >= 0 of reflections in which a circular sequence looks exactly the same.

Example

Seq(2, 1, 2, 2, 1, 2, 2, 1, 2, 2, 1, 2).symmetry // 4