Fibonacci Pyramid - Gauthier Cerf Art

Details - Overview

The pictures all show the same sculpture, how it changes like a chameleon depending on the viewing angle under the same lighting.

The Fibonacci pyramid consists of 32 Fibonacci cubes and Fibonacci cuboids, which together form a pyramid 132.8 cm high. The sides of the cubes and cuboids are made of Plexiglas with different semi-transparent colours. This makes light and colour an important characteristic of the sculpture.

The complex interior is made visible through the transparent Plexiglas shell of the cubes, which create a fascinating kaleidoscopic image. As a result, the sculpture changes its appearance like a chameleon depending on the lighting, which sets changing accents both directly and indirectly, from inside and outside. All the photos on this page, which show one and the same sculpture, bear witness to this.

The Fibonacci Pyramid radiates simplicity, clarity and lightness, yet still displays finesse and sophistication. The haptics of the cubes, the interior views and many mirror effects make the sculpture tangible for people.

Fibonacci Cubes

The Fibonacci Cubes collection shows arrangements of Fibonacci cubes and cuboids.

A Fibonacci cube is a cube whose side length corresponds to a Fibonacci number (e.g. 55 x 55 x 55 cm). A Fibonacci cuboid is a cuboid whose side lengths (length, width, height) correspond to three consecutive Fibonacci numbers (e.g. 55 x 34 x 21 cm).

Fibonacci cubes have a wonderful property: each Fibonacci cube can be built from two smaller Fibonacci cubes positioned in diametrically opposite corners and three identical Fibonacci cuboids that encompass the two smaller cubes.

Mathematically, this can be shown using the cubic binomial formula (a + b)³, applied to the Fibonacci numbers. (a + b)³ = a³ + 3a²b + 3ab² + b³ Since each Fibonacci number F_n is the sum of the two preceding Fibonacci numbers F_n-1 und F_n-2 , i.e., F_n = F_n-1 + F_n-2, we get: F_n³ = (F_n-1 + F_n-2)³ = F_n-1³ + 3F_n-1²F_n-2 + 3F_n-1F_n-2² + F_n-2³ = = F_n-1³ + 3F_n-1F_n-2(F_n-1 + F_n-2) + F_n-2³ = F_n-1³ + 3F_nF_n-1F_n-2 + F_n-2³ Because a Fibonacci number to the power of three, i.e. F_n³ , corresponds to a cube with side length F_n, we have shown that: Cube_n = Cube_n-1 + 3 Cuboid_n + Cube_n-2 !

Arrangement

The Fibonacci pyramid is created by stacking several Fibonacci cubes on top of each other. The base is formed by the largest cube, Cuben. This is set up so that its diagonal points vertically upwards through the two inner cubes. To ensure stability, the larger of the two sub-cubes (Cube_n-1) is removed so that the remaining cube stands on three corners of the Fibonacci cuboids. These three corners form an equilateral triangle.

The next smaller Fibonacci cube (Cube_n-1) is now placed on top of the base cube. Because Cube_n-1 is also broken down again (into Cube_n-2 and Cube_n-3 and the corresponding cuboids), the lower part (Cube_n-2) can be taken from the base. This process is continued with smaller and smaller Fibonacci cubes until there is only one cube of the size 1 × 1 × 1 cm at the top of the pyramid.

In the Fibonacci pyramid shown here, with the base cube measuring 55 × 55 × 55 cm (Cube₁₀), a total of 8 such nested cubes are stacked on top of each other.

Cubes in Nature

Alongside the sphere, the cube is one of the purest spatial forms in nature. Cubes occur, for example, in metals with a cubic crystal lattice and show wonderful natural sculptures with often intergrown cubes (see picture on the right: cube-shaped, intergrown pyrite crystals from the Ampliación a Victoria de Navajún mine, La Rioja, Spain).

Foto by JJ Harrison (https://www.jjharrison.com.au/) – Own work, provided under the CC BY-SA 3.0 Licence.

Pyramids

In geometry, the best-known shape of a pyramid consists of a square as the base and a point that lies vertically above the centre of the square. The apex is connected to all corners of the square by a straight line. In the more general case, the base can be any polygon and the apex can also be off-centre.

Man-made pyramids have existed all over the world for thousands of years. Most of them are built on a square base. With sides measuring 450 metres, the pyramid of Cholula in Mexico has the largest footprint and also the largest volume.

Probably the most famous and the highest pyramid is the pyramid of Khufu in Giza, which was built over 4,500 years ago. The picture below shows the pyramid of Kukulcán in Chichén Itzá, Mexico.

Photography by: Fcb981, Rotstich entfernt: Wladyslaw – http://de.wikipedia.org/wiki/Bild:El_Castillo_Stitch_2008.jpg, CC BY-SA 3.0,
https://commons.wikimedia.org/w/index.php?curid=12185740

Pyramids have been built again and again right up to modern times. A fascinating modern pyramid was built in front of the Louvre Museum in Paris, commissioned by François Mitterand to the architect I.M. Pei (see picture below).

The Louvre Pyramid by scarletgreen from Japan – LouvrePyramid03f, CC BY 2.0, https://de.wikipedia.org/w/index.php?curid=3749517

Symbolism

For a long time, pyramids were the highest buildings of mankind and thus symbolised the closeness of people to the sky, the sun, light and the gods. Very often, pyramids served as tombs and as a starting point for a new life through rebirth.

Pyramids and their orientation also showed the builders’ knowledge of the course of the sun (e.g. solstices, equinoxes), planets and stars.