Approaches to Infinity
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Transcript of Approaches to Infinity
Introduction
Computers are particularly good at repetition.
The high precesion with which the modern
computers can do calculations allows an alogorithm
to get closer and closer looks.
Now we are jumping to infinitely, however it has the
finite resolution and finite size and it must be made
in finite amount of time.
Thus we make the approximation to the creatures
being studied.
Fractals and self Similarity
We want methods that takes us to infinity.
This will feature Recursion.
Self similar here means the same at every scale
some curves are Exactly self similar and others are
Statistically similar.
Self similar curves were called “fractal”.
Successive Refinement in
Curves
Successive genration of KOCH curve are denoteds
as Ko, K1, K2 ….
The zeroth is just a horizontal line.
To create the K1 divide the line K0 in to three equal
parts and replace the middle section with a
triangular bump. Sides of length 1/3.
The total length of one line is 4/3.
Successive Refinement in
Curves
To form Kn+1 from Kn
Sub divide each segment of kn into three equal
parts and replace the middle part with a bump in the
shape of an equilateral Triangle.
Ki has total length of (4/3)I
As I tend to infinity, the length of the curve become
infinite
String Production and Peano
Curves
A large number of curves can be generated by
Refining the line Segments. (to generate these
curves, L-systems)
Turtle Graphics Reads the String and interprets
each character a command to perform some
operation.
„F‟ means forward(1,1) (Go forward 1 m the current direction)
„+‟ means turn(A) (turn right through angle A degrees)
„-‟ means turn(A) (turn left through angle A degrees)
String Production and Peano
Curves
“F-F++F-F “ with an angle of 60 the turtle would
draw an the first genration Koch curve.
String Production and Peano
Curves
How to generate a simple String in to the Longer
one that will generate Rich curve?
This is based on the String Production Rules .
„F‟->”F-F++F-F”
In the first stage the initial string is called the atom.
S1=“F-F++F-F”(1st curve)
S2=“F-F++F-F-F-F++F-F++F-F++F-F-F-F++F-F”(2nd
curve)
String Production and Peano
Curves
Call Produce String Function and read a string from
the Input file which contains the atoms and finally
writes it in the Output file the Desired Curve.
String Production and Peano
Curves
Where in each string is repeatedly fed back into same function and
produce the next higher order object.
Extending the Language
A richer set of curves can be generated by adding more rules to
string production process
The rule is that any X and Y characters are ignored where as F , +
and – are interpreted as before.
The atom is FX here.
For Example X and Y in FX+YF is interpreted as F+F+
Extending the Language
For the second order Dragon the turtle Responds to F+F++-F-F+
instead of S2=FX+YF++-FX-YF+.
The Resultant Curve is shown Below.
Producing String Recursively
and Drawing in a program.
To do this we store the variable called Order, the number of times to
apply the rule to the String.
For + and – the turtle turn accordingly but if the order is zero the turtle
moves forward for F. else it calls the Function it self to produce the
desired string
Curves based on String
Production
The five key intergradient's for each curve (atom, F-string, X-String,
Y-String, angle in degrees)
Allowing Branching
We need a character that commands the turtle to save its
current state
State of turtle = [cp,cd].
A turtle Stack is maintained.
Fractal Trees
The bush is based on the atom F an angle of 22 and the F
string.
F-> “FF-[-F+F+F]+[+F-F-F]”
Tiling the Plane
Another way to move towards the infinity is to repeat a shape
again and again, covering the Entire Plane
The notion is to take many copy of some shape .
Tiling the Plane
Another way to move towards the infinity is to repeat a shape
again and again, covering the Entire Plane
The notion is to take many copy of some shape .
Monohedral Tiling
Those based on a single Polygon
The polygon in question is called the prototile of the
tesselation
Only three possible regular tiling can be shown. A square ,
Hexagon and triangle.
Monohedral Tiling
Famous polyominoes formed by connecting unit squares Edge
to edge
Polyimanoids are formed by connecting congruent triangles
together..
Reptiles
Reptiles are class of non periodic tilings that are most Easily
described Recursively.
Different replicas of a reptile fit together to form a large reptile
of the same shapes