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