The Fibonacci Squares collection shows arrangements of Fibonacci squares whose side lengths are Fibonacci numbers.
Fibonacci squares have the fascinating property that they can be composed of the next two smaller Fibonacci squares and two Fibonacci rectangles. (The side lengths of a Fibonacci rectangle are two consecutive Fibonacci numbers.) The smaller Fibonacci squares are attached to diametrically opposite corners of the large square and touch at the diagonal.
The larger the Fibonacci numbers, the more the ratio of the side lengths of a Fibonacci rectangle is that of a golden rectangle.
Example: a square with side length 89cm can be formed from the two squares with side length 55cm and 34cm and two rectangles with side length 34x55cm. The ratio 55/34 ≈ 1.6176 is close to the golden ratio, whose first four decimals are 1.6180…
Stairway to Heaven
Stairway to Heaven uses the above property of Fibonacci squares in several ways. The image has a square format, which also represents the largest square. This is first divided into its building blocks, resulting in the two smaller squares. The smaller of the two (top right) remains as it is, and the larger one (bottom left) is in turn broken down into its building blocks. These two steps are carried out recursively until the last step, at the bottom left, is to split a 2×2 square into two 1×1 squares and two 1×1 (square) rectangles.
The color of the picture is only determined by the colors of the rectangles and the yellow 1×1 closing square. The red and blue gradients of the rectangles blend into the squares and result in their color.
The arrangement resulting from the construction produces several fascinating effects.
On the one hand, the Fibonacci squares form a staircase that extends beyond the picture, growing exponentially into the sky, resp. goes to infinity (hence the name). On the other hand, the rectangles with the squares show a herringbone pattern, which also grows exponentially.
In addition, Fibonacci effects can also be seen in the diagonal: in each square there is a small square in the center, which is held in the pincers at the top right and bottom left by two squares of the same size. The lower of the two is further divided into even smaller parts. Here you can also see the summation rule of the Fibonacci numbers: the diagonal of the small square in the middle extended by the diagonal of the next larger square at the bottom left results in the diagonal of the next but one larger square, and this in turn extended by the diagonal of the square at the top right finally results in the diagonal of the picture.
In addition, the alignment of the squares in the diagonal illustrates the well-known property of the Fibonacci numbers that the sum of the n first Fibonacci numbers is equal to the (n+2)th Fibonacci number minus 1: the sum of the diagonals of all red squares is the same the diagonal of the image minus the diagonal of the small yellow square.
Similar effects appear in the Fibonacci Hexagon image Between the Devil and the Deep Blue Sea.