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snowflakes and fractalstsp water project 2004
Pictures from http://www.its.caltech.edu/~atomic/snowcrystals/photos/photos.htm, http://www.lsbu.ac.uk/water/molecule.html, Presentation by Enoch Lau
team leader
David Curtin
…team members
Thomas ClementJulian GibbonsJeff GordonEnoch LauOzan OnayJohn Sun
…
Snowflakes and FractalsTSP Water Project 2004
Snowflakes and Fractals
• Mathematics Focus Group
• Computing Focus Group
Picture from http://www.fractaldomains.com/ronda/joe/ronda.shtml
Snowflakes and FractalsTSP Water Project 2004
L-Systems
• Iterative approach to fractal construction.• Repeatedly replaces line segments with
pre-determined constructions.• Fractal is the result of infinitely many
iterations.
BeforeF
AfterF + F – – F + F
Snowflakes and FractalsTSP Water Project 2004
Koch Snowflake
Snowflakes and FractalsTSP Water Project 2004
L-System Snowflake
Snowflakes and FractalsTSP Water Project 2004
Metric Spaces
• Different definitions of distance.• Use the new definition to prove desired
results.
Snowflakes and FractalsTSP Water Project 2004
Koch Snowflake
• The Hausdorff metric is complete – all Cauchy sequences in it converge.
• For Koch curve, Hausdorff distance translated into the height of the triangles added by each iteration (red).
Snowflakes and FractalsTSP Water Project 2004
Implications
• The Koch curve exists – it does not spread out infinitely far.
• The image we produce is unique –this method of generating fractals is not chaotic, but thoroughly systematic.
• Snowflake formation follows a very different model!
END OF MATHEMATICS SECTION
Snowflakes and FractalsTSP Water Project 2004
Snowflakes on Computer
Picture from http://www.alunw.freeuk.com/fractaltour.html
Snowflakes and FractalsTSP Water Project 2004
Physical Principles
• Snowflakes are not merely frozen water, as they exhibit crystalline structures.
• Variations in snowflake formation is due to dust particles, temperature and humidity.
Dust Picture from http://www.math.niu.edu/~rusin/known-math/index/tour_alg.html
Snowflakes and FractalsTSP Water Project 2004
Physical Principles
Picture from http://library.tedankara.k12.tr/chemistry/vol2/hydrogen%20bonding/z20.htm
Snowflakes and FractalsTSP Water Project 2004
Diffusion Limited Aggregation
• DLA approximates real-life snowflake construction.
• Begin with a seed cluster in the middle.• Free particles move until they collide with
the main cluster.
Snowflakes and FractalsTSP Water Project 2004
Grid-based Implementation
• Matrix elements store different numbers to represent “on” and “off” positions.
• By treating each element as two points, it behaves like a triangular/hexagonal grid.
Snowflakes and FractalsTSP Water Project 2004
Grid-based Implementation
• Triangular grid had no effect on symmetry.
Snowflakes and FractalsTSP Water Project 2004
Particle-based Implementation
• Virtual particles move in a simulated environment.
Simulated Environment
Snowflakes and FractalsTSP Water Project 2004
Particle-based Implementation
QuickTime version QuickTime version
Snowflakes and FractalsTSP Water Project 2004
Particle-based Implementation
• Fits the dimension equation very well:
N(r) = k.rd
y = 1.48x + 0.99R2 = 1.00
0
1
2
3
4
0 0.5 1 1.5 2
Log(Radius in pixels)
Log(
Part
icle
s)
Snowflakes and FractalsTSP Water Project 2004
Conclusion
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