Fibonacci Circles
The Fibonacci Circles collection shows arrangements of circles whose diameter corresponds to the Fibonacci numbers. The diameter of a circle is therefore the sum of the diameters of the next two smaller circles. The colored areas grow as the square of the Fibonacci numbers.
Encircling Circles
In the Encircling Circles series, one circle is surrounded by the next larger circle so that they touch on the inside. The larger circle therefore encloses all the smaller Fibonacci circles. The only exceptions to this rule are the two innermost circles with a diameter of 1 cm, which touch each other on the outside as the origin of the image.
More pictures from the Encircling Circles series:
Encircling Circles Triplets
The picture Encircling Circles Triplets shows three instances of the picture Encircling Circles I, whereby the outermost circles with a diameter of 55 cm are not white but colored.
Each of the three instances shows two series of Fibonacci circles alternately in black and white. Both series therefore contain identical circles.
The first series of circles spreads out like a wave front, starting in the center at the left edge with the red circle. Smaller circles are placed over the larger circles and cover them. Circles with odd Fibonacci numbers are painted black, those with even Fibonacci numbers are painted white, with two exceptions: the first circle with a diameter of 1 cm (the origin) is painted red, the last one is colored. All the circles in this first series touch on the inside of the left edge.
The second series is a linear sequence of circles starting to the right of the red circle. Here, circles with odd Fibonacci numbers are painted white and those with even Fibonacci numbers are painted black.
Because the sum of the diameters of two consecutive Fibonacci circles equals the diameter of the next larger one, the circles of the second series always fit into the gap of the first series and are visible there due to their complementary color.
With the second series of circles, the picture illustrates the well-known formula that the sum of the n first Fibonacci numbers is equal to the (n+2)th Fibonacci number minus 1: The sum of all circle diameters of the second series is the diameter of the large circle comprising the series minus the diameter of the small red circle.
If you look at the image from the side at a small angle (see detailed image 1), the circles become ellipses and the image takes on a spatial shape. The first series of circles becomes a bowl and the second series becomes a tower.
Each of the three instances shows two series of Fibonacci circles alternately in black and white. Both series therefore contain identical circles.
The first series of circles spreads out like a wave front, starting in the center at the left edge with the red circle. Smaller circles are placed over the larger circles and cover them. Circles with odd Fibonacci numbers are painted black, those with even Fibonacci numbers are painted white, with two exceptions: the first circle with a diameter of 1 cm (the origin) is painted red, the last one is colored. All the circles in this first series touch on the inside of the left edge.
The second series is a linear sequence of circles starting to the right of the red circle. Here, circles with odd Fibonacci numbers are painted white and those with even Fibonacci numbers are painted black.
Because the sum of the diameters of two consecutive Fibonacci circles equals the diameter of the next larger one, the circles of the second series always fit into the gap of the first series and are visible there due to their complementary color.
With the second series of circles, the picture illustrates the well-known formula that the sum of the n first Fibonacci numbers is equal to the (n+2)th Fibonacci number minus 1: The sum of all circle diameters of the second series is the diameter of the large circle comprising the series minus the diameter of the small red circle.
If you look at the image from the side at a small angle (see detailed image 1), the circles become ellipses and the image takes on a spatial shape. The first series of circles becomes a bowl and the second series becomes a tower.