Fibonacci Rectangles Galore

Fibonacci Squares

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…

Fibonacci Rectangles Galore uses the above property of Fibonacci squares several times. The picture has a square format of 144 x 144 cm, 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) thus remains, and the larger (bottom left) is again divided into its building blocks. These two steps are continued recursively until, at the bottom left, a 2×2 square is split into two 1×1 squares and two (square) 1×1 rectangles.

The image is coloured using the horizontal and vertical strips of 144 cm length resulting from the above construction. The height or width of the strips decreases from top to bottom and from right to left. The heights and widths start at 34 cm and follow the Fibonacci numbers downwards until they reach 1 cm. This creates all combinations of squares and rectangles and their side lengths up to 34 cm.

The colour of the picture is determined solely by the colours of the lanes. The vertical tracks are coloured in a blue gradient: dark to light from left to right. The horizontal strips are coloured in a red gradient: dark to light from top to bottom. The transparent colouring allows all mixtures of red and blue to be seen. The colour within each rectangle is homogeneous, although the human eye perceives a colour gradient from light to dark, which is created by the different brightness levels of the neighbouring rectangles (an optical illusion).

The yellow 1×1 end square at the bottom left is special because it represents the origin or the end.

Arrangement

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 to infinity. On the other hand, the rectangles with the squares show a herringbone pattern, which also grows exponentially.

The picture is related to the picture Stairway to Heaven.