Computer Generated Islamic Star Patterns

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Computer Generated Islamic Star Patterns

• Mustafa Shabib

Based on paperKaplan, Computer Generated
Islamic Star Patterns, Bridges
2000http//www.cs.washington.edu/homes/csk/tile/p
apers/kaplan_bridges2000.pdf
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What Are Islamic Star Patterns?

• Over 1000 years ago, Muslim artists used these
  patterns as architectural decorations
• Typically found on mosques throughout the Islamic
  world
• They are called star patterns since they are
  most often seen as the division of a plane into
  star-shaped regions
• Geometrically intriguing, as the artists who
  created them never revealed their techniques
• Lastly, they look badass as youre about to see

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Mind Boggling Examples
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Technique Used

• Start with a regular n-gon
• For n gt 3, let the unit circle be parameterized
  viag(t) (cos (2pt/n), sin (2pt/n))
• Using a notation (n/d) where d is some number, we
  can create an n-pointed star based on the
  original regular n-gon
• When d is an integer, we do this by connecting
  for 0 lt i lt n the line segment si formed from
  g(i) and g(id)
• Certain restrictions apply on the variables
  mentioned d lt n/2 and when d 1 it creates the
  original n-gon
• For some values k ¹ i, si intersects with sk and
  so a new term, s, is introduced changing the
  notation from (n/d) to (n/d)s where we now choose
  to draw only the first s sub-segments of si

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So That Probably Made Very Little Sense
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Good Thing We Have Eyes

• (n/d)s examples where d is an integer (and in
  this case n8)

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What if d isnt an integer?

• Kaplan is way better at math than Ill ever be
  and so hes generalized this implementation to
  where d can take on any value, as long as d is
  between 1, n/2)
• Here, a point P is computed as the intersection
  of line segments g(i)g(id) and g(i d - d )g(i
  d )
• Then the segment si is replaced with two line
  segments g(i)P and Pg(i d )
• Hopefully, this figure clarifies everything

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Rosettes

• When six-pointed stars are arranged in a certain
  way, a new pattern emerges each star is
  surrounded by a ring of regular hexagons, called
  rosettes
• Tiling these rosettes in the plane leave behind
  gaps, which in the example Im about to show
  happen to be more six-pointed stars

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Rosette Example
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More On Rosettes

• The rosette, a central star surrounded by
  hexagons, appears frequently in Islamic art but
  is not limited to only six-pointed stars
• This implementation has allowed for the
  construction of the rosette to handle n-pointed
  stars

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Rosette Madness!

• Steps to create these n-fold rosettes
• Inscribe a regular n-gon in the unit circle and
  then another n-gon tilted so that its vertices
  intersect the first n-gons edges
• Let A and B be adjacent vertices of the outer
  n-gon and C and D be adjacent vertices of the
  inner one, where D bisects AB
• We then identify point E, which is the
  intersection of segment CD with the angle
  bisector of ÐOAB
• Next we get point F which is the intersection of
  segment OA with the line segment which passes
  through E and is parallel to OD
• These steps are repeated around the n-gon

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Eh?
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Filling in the Plane

• Using a periodic tiling of regular polygons
  across the plane
• Then irregular polygons are thrown in as needed
  to fill gaps
• For each regular n-gon where ngt4, we make an
  n-fold star, rosette or extended rosette to place
  in it and replicate this motif everywhere that
  n-gon appears in the tiling. It is placed such
  that its points bisect the edges of the n-gon

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Final Steps

• Where the motif isnt placed, little gaps are
  left behind
• To fill these gaps, the lines that terminate at
  the edges are extended until the meet again in
  the void, and then the original n-gon lines are
  removed, finishing the design

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Rendering Styles

• The output of this construction is a planar
  graph, which can be rendered in a variety of ways
• Plain line representation of graph edges
• Outline lines are thickened and outlines
  darkened
• Emboss 3D effect is added to the lines where
  center of each edge is raised and one side is
  darkened to simulate a light source
• Interlace line segments are added at each
  intersection to suggest an over-under
  relationship between the crossing edges, where
  crossings are chosen so that the over and
  under relationship is maintained
• Checkerboard renders only the faces of the graph
  and not the edges, and where all vertices have an
  even numbered degree, a 2-coloring of the graph
  can always be achieved
• Since Checkerboard doesnt render edges, any of
  the edge-rendering techniques can be used to
  overlap the checkerboard pattern, approximating a
  style called Zellij

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Demo

• http//www.cs.washington.edu/homes/csk/taprats/app
  let.html