Reptiles

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Rep-tiles

• Melissa Brooks

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Some Background Info
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Tiling

• collection of plane figures that fills the plane
  with no overlaps and no gaps
• examples jigsaw puzzles, floor tiles, mosaics

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• The term rep-tiles was termed by Solomon Golomb
  in 1964
• Inventor of polyominoesfigures constructed by
  placing a number of identical squares next to one
  another in such a way that at least one edge of
  each square coincides with an edge of one of the
  other squares.
• A certain form of polyominoestetrominoesis used
  in the popular game Tetris

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Rep-tiles
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• Tiles that can be fit together to form replicas
  of themselves
• Tiles that can be subdivided into a finite number
  of congruent tiles that are each similar to the
  original tile

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Examples of Rep-tiles
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• Classified as Rep-ntiles
• Tile in which n copies can be fitted together to
  create a larger similar figure
• Exist for any natural numbers n1
• Simplest example is four small squares put
  together to make one large square. This is
  Rep-4.

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Examples of Rep-ntiles

(a) Rep-2 tile
(b) Rep-3 tile
(c) Rep-4 tile
(d) Rep-4 tile
(e) Rep-5 tile
(f) Rep-6 tile
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• Some rep-ntiles can be have more than one
  classification

Rep-4 tile
Rep-4 tile
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• The set of rep-tiles is a subset of the set of
  irreptiles
• Any shape that tiles a larger version of itself
  using differently sized or identical copies of
  itself
• Example below trapezoid

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Rep-tiles and Science
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• Some tilings of rep-tiles form a non-lattice
  structure (like the example on page 159 in On the
  Shoulders of Giants).
• These tilings are of great interest to scientists
  and mathematicians because they share some of the
  same properties of crystalline materials called
  quasicrystals.
• Quasicrystals, very simply stated, are aperiodic
  structures which produce diffraction. They have
  apereodic tiling (tiling which never repeats
  itself). They lack translational symmetry, which
  means that a shifted copy will never match
  exactly with its original.

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• An example of a quasicrystal is the Penrose tile.
• Being aperiodic, it has no translational symmetry
  (it never repeats itself exactly), but
  nevertheless it has a fivefold rotational
  symmetry.

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• Standards
• NCTM geometry, problem solving,
  representations, connections
• Program of Studies problem solving,
  mathematical connections, multiple
  representations
• Sources
• "Covering the Plane with Rep-tiles." NCTM
  Illuminations. 2007. National Council of
  Teachers of Mathematics. 18 Feb 2007
  IDL251.
• Friedman, Erich. "Problem of the Month (October
  2002)." Math Magic. 05 Aug 2003. 18 Feb 2007
  html.
• Penrose Tiling. Wikipedia. 2007. 25 Feb 2007
  
• Steen, Lynn Arthur. On The Shoulders of Giants.
  Washington, D.C. National Academy Press,
  1990.
• "Quasicrystals." Wikipedia. 2007. 25 Feb 2007
  .
• Weisstein, Eric W.. "Rep-tiles." 2007. Wolfram
  Mathworld. 18 Feb 2007 am.com/Rep-Tile.html.