fibonacci

Find the sixth Fibonacci number by using fibonacci.

f = fibonacci(6)

f = 
8

Find the first 10 Fibonacci numbers.

n = 1:10;
f = fibonacci(n)

f = 1×10

     1     1     2     3     5     8    13    21    34    55

The ratio of successive Fibonacci numbers converges to the golden ratio $$1.61803...$$. Show this convergence by plotting this ratio against the golden ratio for the first 10 Fibonacci numbers.

n = 2:10;
ratio = fibonacci(n)./fibonacci(n-1);

plot(n,ratio,'--o')
hold on

yline(1.61803)
hold off

Use Fibonacci numbers in symbolic calculations by representing them with symbolic input. fibonacci returns the input.

Represent the ${\mathit{n}}^{\mathrm{th}}$ Fibonacci number.

syms n
f = fibonacci(n)

f = $$\mathrm{fibonacci}\left(n\right)$$

Find large Fibonacci numbers by specifying the input symbolically using sym. Symbolic input returns exact symbolic output instead of double output. Convert symbolic numbers to double by using the double function.

Find the 300th Fibonacci number.

num = sym(300);
f300sym = fibonacci(num)

f300sym = $$222232244629420445529739893461909967206666939096499764990979600$$

Convert fib300 to double. The result is a floating-point approximation.

f300double = double(f300sym)

f300double = 
2.2223e+62

For more information on symbolic and double arithmetic, see Choose Numeric or Symbolic Arithmetic.

The Fibonacci numbers are commonly visualized by plotting the Fibonacci spiral. The Fibonacci spiral approximates the golden spiral.

Approximate the golden spiral for the first 8 Fibonacci numbers. Define the four cases for the right, top, left, and bottom squares in the plot by using a switch statement. Form the spiral by defining the equations of arcs through the squares in eqnArc. Draw the squares and arcs by using rectangle and fimplicit respectively.

x = 0;
y = 1;
syms v u

axis off
hold on

for n = 1:8

    a = fibonacci(n);

    % Define squares and arcs
    switch mod(n,4)
        case 0
            y = y - fibonacci(n-2);
            x = x - a;
            eqnArc = (u-(x+a))^2 + (v-y)^2 == a^2;
        case 1
            y = y - a;
            eqnArc = (u-(x+a))^2 + (v-(y+a))^2 == a^2;
        case 2
            x = x + fibonacci(n-1);
            eqnArc = (u-x)^2 + (v-(y+a))^2 == a^2;
        case 3
            x = x - fibonacci(n-2);
            y = y + fibonacci(n-1);
            eqnArc = (u-x)^2 + (v-y)^2 == a^2;
    end

    % Draw square
    pos = [x y a a];
    rectangle('Position', pos)

    % Add Fibonacci number
    xText = (x+x+a)/2;
    yText = (y+y+a)/2;
    text(xText, yText, num2str(a))

    % Draw arc
    interval = [x x+a y y+a];
    fimplicit(eqnArc, interval, 'b')

end