snowflakes and fractals tsp water project 2004 team leader David - - PowerPoint PPT Presentation

snowflakes and fractals

tsp 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 Clement Julian Gibbons Jeff Gordon Enoch Lau Ozan Onay John Sun

…

Snowflakes and Fractals

TS P Water Proj ect 2004

S nowflakes and Fractals

• Mathematics Focus Group
• Computing Focus Group

Picture from http: / / www.fractaldomains.com/ ronda/ joe/ ronda.shtml

Snowflakes and Fractals

TS P Water Proj ect 2004

L-S ystems

• Iterative approach to fractal construction.
• Repeatedly replaces line segments with

pre-determined constructions.

• Fractal is the result of infinitely many

iterations.

Before F After F + F – – F + F

Snowflakes and Fractals

TS P Water Proj ect 2004

Koch S nowflake

Snowflakes and Fractals

TS P Water Proj ect 2004

L-S ystem S nowflake

Snowflakes and Fractals

TS P Water Proj ect 2004

Metric S paces

• Different definitions of distance.
• Use the new definition to prove desired

results.

Snowflakes and Fractals

TS P Water Proj ect 2004

Koch S nowflake

• 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 Fractals

TS P Water Proj ect 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 Fractals

TS P Water Proj ect 2004

S nowflakes on Computer

Picture from http: / / www.alunw.freeuk.com/ fractaltour.html

Snowflakes and Fractals

TS P Water Proj ect 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 Fractals

TS P Water Proj ect 2004

Physical Principles

Picture from http: / / library.tedankara.k12.tr/ chemistry/ vol2/ hydrogen% 20bonding/ z20.htm

Snowflakes and Fractals

TS P Water Proj ect 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 Fractals

TS P Water Proj ect 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 Fractals

TS P Water Proj ect 2004

Grid-based Implementation

• Triangular grid had no effect on

symmetry.

Snowflakes and Fractals

TS P Water Proj ect 2004

Particle-based Implementation

• Virtual particles move in a simulated

environment.

Sim ulated Environm ent

Snowflakes and Fractals

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Particle-based Implementation

QuickTim e version QuickTim e version

Snowflakes and Fractals

TS P Water Proj ect 2004

Particle-based Implementation

• Fits the dimension equation very well:

N(r) = k.rd

y = 1.48x + 0.99 R2 = 1.00

1 2 3 4

0.5 1 1.5 2

Log(Radius in pixels) Log(Particles)

Snowflakes and Fractals

TS P Water Proj ect 2004

Conclusion