Intro to Recursion in Programming with Fractals

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What is recursion?

Recursion simply is applying a procedure, breaking it into smaller parts, and applying the same to the smaaller parts.

Example: Fibonacci series: F(n) = F(n-1)+F(n-2)

Usually one has to get some experience before getting a good feel of the concept.

But if you could see what happens under the hood you could get a better understanding more quickly.

Fractals are perfect for this purpose besides being beautiful.

What are fractals?

A Fractal, basically, is a repeating pattern.

Each smaller part of the fractal is identical to the bigger part.

If you zoom into an ideal fractal, you couldnt tell how much you have zoomed in.

An Example of fractals

The mandelbrot set is a famous fractal

Let's make some fractals!

The Koch Line

We will begin with the simplest fractal : The Koch Line

The rule to make Koch line:

Draw (an imaginary) straight line.

Split it into three equal parts.

Replace the middle part with an equilateral triangle without a base

Apply the same procedure to all the new straight lines(recursion).

Finally draw the lines.

Defining iterations

Usually it is not feasible Step 4, in the last slide, a lot of times.

We need to stop after a finite number of times.

Each execution of step 1-3 is an iteration.

The number of times we apply the rule is called the number of iterations.

The Koch Line

Next we see the result of executing the steps.

I used simple C graphics to make the fractals.

The code is available on GitHub.

The Koch Line

One Iteration:

The Koch Line

Two Iterations:

The Koch Line

Three Iterations:

The Koch Line

Four Iterations:

The Koch Snowflake

The Koch Line procedure could be used to make a Koch Snowflake

To make a Koch Snowflake:

Make an equlateral trinangle

Apply Koch Line procedure to all the sides of the triangle

Let's see what the resulting fractal looks like.

The Koch Snowflake

The Koch Snowflake

In different colours

One More Example:
The Sierpinski Arrowhead Curve

The Sierpinski Arrowhead Curve

Following are the steps to make a sierpinski arrowhead curve

Draw an imaginary line. Think of it as a base of an equilateral triangle.

Move up (while drawing) one side. Stop halfway.

Move parallel to base till you reach the other side.

Move down the side to the base line.

Apply same procedure to all lines( Recursion ). The triangles on slanting lines must point inwards. The triangle on the horizontal line must point outwards.

The Sierpinski Arrowhead Curve

One Iteraton:

The Sierpinski Arrowhead Curve

Two Iteratons:

The Sierpinski Arrowhead Curve

Three Iteratons:

The Sierpinski Arrowhead Curve

Four and Five iterations skipped..

The Sierpinski Arrowhead Curve

Six iterations:

The Sierpinski Arrowhead Curve

In different colours:

Conclusions

Things to Note About Fractals

The final structure is completely different form what we started with.

Each smaller part is completely identical to the bigger part.

Recursion in Simple Language

Recursion mainly consists of following two steps:

Apply some procedure to a thing.

Start from Step 1 for all smaller parts.

In practice, we stop when we meet a base case.

Fractals are everywhere!

Romanesco Broccoli

Copper Crystals

Applications of recursion in Programming

Quick Sort

algorithm quicksort(A, lo, hi) is

if lo < hi then

p := partition(A, lo, hi)

quicksort(A, lo, p - 1)

quicksort(A, p + 1, hi)

RecursiveCalls

Merge Sort

function merge_sort(list m)

// Base case. A list of zero or one elements is sorted, by definition.

if m is empty then

return m

// Recursive case. First, divide the list into equal-sized sublists

// consisting of the even and odd-indexed elements.

//omitted

// RECURSIVELY sort both sublists.

left := merge_sort(left)

right := merge_sort(right)

// Then merge the now-sorted sublists.

return merge(left, right)

Tower of Hanoi

function hanoi is:

input: integer n, such that n >= 1

1. if n is 1 then return 1

2. return [2 * [call hanoi(n-1)] + 1]

end hanoi

Interestingly, graphical representation of tower of hanoi is similar to that of sierpinski arrowhead curve

Recursive Call

Useful Resources

Original blog post

Code Used to make fractals

Snowflakes

Fractals in Nature

Wolfram Fractals Reference App

Think Python book

Thank You

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