Fractals

Flashcards about Fractals

Define a fractal.

A rough or fragmented geometric shape that can be split into parts, each of which is at least approximately a reduced-size copy of the whole, exhibiting self-similarity.

Who is Lewis Fry Richardson?

A Quaker pacifist who wondered how the length of the border between two countries affected the chance of war; also contributed to work on Chaos.

What equation is used to find the fractal dimension of a rough line?

L(λ) = n^(1-D) where λ is the length of the ruler, L(λ) is the total measured length, n is the number of segments, and D is the fractal dimension.

What is the formula for calculating the fractal dimension (D) using the slope (m) of a log-log plot?

D = 1 - m

What is the Complex Iterating Equation for the Mandelbrot Set?

Znew = Zold^2 + C, where Z and C are complex numbers.

How is the Mandelbrot Set plotted?

Count the number of steps until the distance from (0,0) is greater than or equal to 2, plotting a color corresponding to the number of steps taken, up to a maximum of 15 steps, using each location on the x,y plane as the starting value (C).

What is the logistic map equation for population growth?

rnew = rold * b * (1 - r_old), where r is the rabbit population [0,1] and b is the starvation factor [0,1]

What phenomenon is associated with period doubling in the logistic map?

Bifurcation

What does the Feigenbaum constant (δ) describe?

The ratio of successive distances between bifurcation points in the logistic map, approximately equal to 4.669.

Describe the process of generating the Logistic map

Run through a few iterations until system settles down. Plot points for at all (r,b) values for a large number of iterations. Sequence through each b value looking for range of values produced by r.

Describe the equations for rabbit and fox populations.

rnew = a*rold * (1 - rold) - fold and fnew = bfoldr_old, where a and b are success rates for rabbits and foxes, respectively.

Describe the algorithm for simulating rabbit and fox populations

Select values for (a,b), say a=3, b=4. Set r=0.5 and f=0.5. Repeat 40,000 times. Scale r and f value to fit on screen. Plot point (sr,sf).

Describe the steps to generate a Sierpinski triangle.

1. Choose three points on the plane to form a triangle, A, B, and C. 2. Select a point P at random inside the triangle. 3. Choose one of the vertices of the triangle at random (e.g., B). 4. Locate the midpoint of the line between the vertex, B, and point chosen P. 5. Set the pixel at this midpoint, Pm.

Describe the Koch curve Generation.

Take a line, split it into three equal lengths, replace the middle segment with two sides of an equilateral triangle, and repeat the process.

What happens to the perimeter and area of a Koch snowflake as the number of iterations approaches infinity?

The perimeter length is infinite, but the area is finite (approaching 1.6).

Define a space-filling curve.

A curve whose range contains the entire 2-dimensional unit square (or more generally an N-dimensional hypercube); also known as Peano curves.

What are some mathematical constants related to fractals and chaos?

Pi (π), the Golden Number (φ), and the Feigenbaum constant (δ).

What are Cellular Automata?

Cellular automata (CA) were invented by Stanislaw Ulam and von Neumann in the 1940s to explain complex systems.

Describe Conway's Game of Life.

A simulation on an infinite, two-dimensional grid where cells are either live or dead, and the state of each cell evolves based on the states of its eight neighbors according to a set of rules for survival, death, and birth.

Describe the rules for deriving a generation from the previous one in Conway's Game of Life

Death: If an occupied cell has not got 2 or 3 occupied neighbours. Survival: If an occupied cell has two or three neighbours, the organism survives to the next generation. Birth: If an unoccupied cell has three occupied neighbours, it becomes occupied.