Sierpinski Curves

1
Sierpinski Curves

• Russell Centanni

2
Overview

• Waclaw Sierpinski
• Sierpinski Curves
• Generating a Sierpinski Curve
• Other Fractals from Sierpinski
• Uses of Sierpinski Fractals
• Summary

3
Waclaw Sierpinski

• Born 14, March 1882 Warsaw, Russian Empire.
• Russians forced the Poles to learn the Russian
  language and to adopt the Russian culture.
• Resulted in extremely high illiteracy and a drop
  in the number of students.

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Sierpinski the Student

• 1899 Entered the Department of Mathematics and
  Physics at the Czars University (University of
  Warsaw).
• Staff was entirely Russian.
• 1903 Sierpinski received a gold medal for the
  best essay on Voronoys contribution to number
  theory.
• His essay was not published until 1907, due to
  strikes against Russian schools and because I
  did not want to have my first work printed in the
  Russian language.
• we had to attend a yearly lecture on the Russian
  language. ... Each of the students made it a
  point of honour to have the worst results in that
  subject. ... I did not answer a single question
  ...
• Almost resulted in Sierpinski receiving a lesser
  degree.

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Career

• 1920 Sierpinski studied the Sierpinski Curve.
• 1921 He became Dean of faculty at the University
  of Warsaw.
• 1928 Vice Chairman of the Warsaw Scientific
  Society and chairman of the Polish Mathematical
  Society.

6
WWII

• Was able to make publications by sending them to
  Italy.
• House was burned after an uprising against the
  Nazis in 1944.
• In a lecture at Jagiellonian University in
  Krakóv, he mourned the loss of more than half of
  all Polish mathematicians.

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Contributions and Death

• Made his greatest contributions in set
  theory(axiom of choice and to the continuum
  hypothesis), point set topology and number
  theory.
• Died 21 October, 1969.
• During his life he authored 724 papers and 50
  books.

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Sierpinski Curves

• Closed Path containing every interior point of a
  square.
• Fills two dimensional space. AKA space-filling
  curve.
• It is the limiting curve of an infinite sequence
  of curves numbered by an index of n 1,2,3
• By scaling the curve down ½ at each n, it comes
  closer and closer to every point in the region.
• The length of the curve is infinity (hence a
  fractal)
• Area enclosed is 5/12 that of the square

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What does it look like?
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Sierpinski Curve
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Sierpinski Arrowhead
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Generating Sierpinski Curves

• Sierpinski Curves can be generated through a
  recursive algorithm.
• ZIG
• ZAG
• Base case of n 1
• Can also be unrolled into string of a Sierpinski
  Curve Grammar
• Grammar generation may still be done recursively.

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Recursive Algorithm

• ZIG(n)
• if n 1 then
• turn left
• advance 1
• turn left
• advance 1
• else
• ZIG(n/2)
• ZAG(n/2)
• ZIG(n/2)
• ZAG(n/2)

• ZAG(n)
• if n 1 then
• turn right
• advance 1
• turn right
• advance 1
• turn left
• advance 1
• else
• ZAG(n/2)
• ZAG(n/2)
• ZIG(n/2)
• ZAG(n/2)

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Recursive Algorithm

• MAIN
• ZIG(2)
• ZIG(2)

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Example
CALLS ZIG(2) ZIG(1) turn left advance 1 turn
left advance 1
ZIG(n) if n 1 then turn left advance 1 turn
left advance 1 else ZIG(n/2) ZAG(n/2) ZIG(n/
2) ZAG(n/2)
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Example
CALLS ZAG(1) turn right advance 1 turn
right advance 1 turn left advance 1
ZAG(n) if n 1 then turn right advance
1 turn right advance 1 turn left advance
1 else ZAG(n/2) ZAG(n/2) ZIG(n/2) ZAG(n/2)
ZIG(n) if n 1 then turn left advance 1 turn
left advance 1 else ZIG(n/2) ZAG(n/2) ZIG(n/
2) ZAG(n/2)
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Example
CALLS ZIG(1) turn left advance 1 turn
left advance 1
ZIG(n) if n 1 then turn left advance 1 turn
left advance 1 else ZIG(n/2) ZAG(n/2) ZIG(n/
2) ZAG(n/2)
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Example
CALLS ZAG(1) turn right advance 1 turn
right advance 1 turn left advance 1
ZAG(n) if n 1 then turn right advance
1 turn right advance 1 turn left advance
1 else ZAG(n/2) ZAG(n/2) ZIG(n/2) ZAG(n/2)
ZIG(n) if n 1 then turn left advance 1 turn
left advance 1 else ZIG(n/2) ZAG(n/2) ZIG(n/
2) ZAG(n/2)
The next call is ZIG(2), which will complete the
Sierpinski Curve (n2).
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Grammar Algorithm

• aa
• a ? abab
• b ? bbab

n 1 aa n 2 abababab n 3
ababbbabababbbabababbbabababbbab n 4
ababbbabababbbabbbabbbabababbb
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Grammar Algorithm

• Generated strings can then be iterated over
• a ? ZIG
• b ? ZAG
• Recursion is unrolled into the Sierpinski Curve
  String.

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Other Fractals associated with Sierpinski

• Sierpinski Triangle
• Sierpinski Carpet

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Sierpinski Fractal Uses

• Antenna Research.
• Work equally well at all requencies.
• Higher frequencies, they are naturally broadband.
• Multiband performance.
• Good performance at smaller sizes.
• Simplifies circuit design.

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Summary

• Sierpinski curves are space-filling curves.
• Easily generated using recursion.
• Also easily implemented using a grammar.
• Sierpinski triangle researched for use as an
  antenna design.

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References

• Dewdney, A. K. The New Turing Omnibus. Computer
  Science Press/Freeman, New York, 1993.
• McLauchlan and C. Phillips. Sixth Form
  Conference - Sierpinski Curve. Taster Conference
  2001 Practical Session P6. Interactive Java
  Programming December 12, 1995. March 20, 2005.
• OConnor, J.J. and E F Robertson. Waclaw
  Sierpinski. The MacTutor History of Mathematics
  archive. March 21, 2005.
• Eric W. Weisstein. "Sierpinski Arrowhead Curve."
  From MathWorld--A Wolfram Web Resource.
  http//mathworld.wolfram.com/SierpinskiArrowheadCu
  rve.html
• Eric W. Weisstein. "Sierpinski Curve." From
  MathWorld--A Wolfram Web Resource.
  http//mathworld.wolfram.com/SierpinskiCurve.html