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Transformations

Transformation Log

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Instructions

How to Use

• Two necklaces are equivalent if one can be transformed into the other through rotations and/or reflections

• Use transformation buttons to explore

• Answer challenges to test understanding

• Choose Easy mode for direct colors or Hard mode for a different color-to-color mapping

Difficulty Modes

• Easy: Colors displayed directly on beads

• Hard: Each Color is mapped to a different color.

• In Hard mode, use the color mapping guide to decode

Controls

• Rotate: Rotate the necklace one position

• Reflect: Mirror the necklace

• Reset: Reset statistics and pattern

Settings

Difficulty Mode

Direct color representation

Different Color Matching

Parameters

Equivalence Relations

An equivalence relation ~ on a set S must satisfy:

  1. Reflexive: a ~ a for all a ∈ S
  2. Symmetric: If a ~ b, then b ~ a
  3. Transitive: If a ~ b and b ~ c, then a ~ c

Necklace Equivalence:

Two necklaces are equivalent if one can be obtained from the other by:

  • Rotation (cyclic permutation)
  • Reflection (mirror image)
  • Combination of both

Burnside's Lemma

Number of equivalence classes:

|X/G| = (1/|G|) Σ |Fix(g)|

For n-bead, k-color necklaces:

  • Rotations: n elements
  • Reflections: n elements
  • Total group size: 2n

Number of distinct necklaces involves counting fixed points under each symmetry.