Please enable JavaScript.
Coggle requires JavaScript to display documents.
Self-similarity (How may shapes fit in a big copy (2) (LIne S=1; MF = 3,…
Self-similarity
How may shapes fit in a big copy (2)
LIne S=1; MF = 3, BCopy =3
1 Dimension
Square S=1, MF = 3, BCopy = 9
2 Dimension
Cube S=1, MF = 3, BCopy = 27
3 Dimension
Raising to the power
Square = (MF)^D = (MF)^2 = 9
Cube = (MF)^D = (3)^3 = 27
Line = (MF)^D = (3)^1 = 3
How many small shapes fit in a big copy (1)
Shape (S)
Magnification Factor (MF)
Difference in BCopy - Dimension of starting shape
Number of small copies = (Magnification Factor)^D
D = self-similarity dimension
Scale-free
self-similarity repeats over scales
Mathematical continues for ever
Often said to be scale-free
No clue to size - e.g. shrunk in a Sierpinski triangle - never know as always the same
Ideal world of maths (not unique to fractals e.g. circles)
In science & everyday life
Shrunk but know immediately that small - as big as cat
Cut of scale - self-similarity does not go on forever
Geometric idea of fractals - useful although no perfect fractal in real world
What if D is between 1 & 2?
E.g. 'Snowflake' fractal
Example only had 4 stems.
But Using MF^D when D= 1 or 2 gave answer above or below 5 (MF=3)
No. of small copies = 5
3^1 =1 & 3^2 = 9
Need to use Logs to solve (for D)
Log(5) = log (3^D)
becomes log(5) = D log(3)
Divide both sides by log(3)
log(5)/log(3) = D log(3)/log(3)
Log(5)/log(3) = D
D = approx 1.465
