From the Golden Ratio to Polytopes

Golden Ratio and Fibonacci numbers:

Consider the line ABC in the following diagram:

Golden rectangle

Point B divides the line in such a way that the ratio between the longer segment (AB) and the shorter segment (BC) is the same as that between the whole line (AC) and the longer segment (AB).

This proportion is known by various names: the golden section, the golden mean, the golden ratio, the extreme and mean ratio, the divine proportion, or phi (φ). If the distance AB equals 1 unit, then BC = 0.6180339887… and AC = 1.6180339887… . The second of these two numbers is the golden section, or phi (sometimes this name is also given to the first number). Many designs in nature are related to the golden section, and it has been widely used in sacred architecture and artwork throughout the ages.

The golden section is part of an endless series of numbers in which any number multiplied by 1.618 gives the next higher number, and any number multiplied by 0.618 gives the next lower number.

The large rectangle below is a golden rectangle, meaning that its sides are in the proportion 1.000:1.618. If a square is removed from this rectangle, the remaining rectangle is also a golden rectangle. Continuing this process produces a series of nested golden rectangles. Connecting the successive points where the ‘whirling squares’ divide the sides of the rectangles in golden ratios produces a logarithmic spiral, which is found in many natural forms (see next section). A similar spiral can be generated from a golden triangle (an isosceles triangle whose sides are in the golden ratio), by repeatedly bisecting one of the angles to generate a smaller golden triangle.

Spiral

Spiral 2

A number series closely related to the golden section is the Fibonacci sequence: it begins with 0 and 1, and each subsequent number is generated by adding the two preceding numbers:

0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987 …

If we take these numbers two at a time, and divide the larger one by the smaller one, the value will oscillate alternately above and below the golden section, while gradually converging on it.

Many of nature’s patterns are related to the golden section and the Fibonacci numbers. For instance, the golden spiral is a logarithmic or equiangular spiral – a type of spiral found in unicellular foraminifera, sunflowers, seashells, animal horns and tusks, beaks and claws, whirlpools, hurricanes, and spiral galaxies. An equiangular spiral does not alter its shape as its size increases. Because of this remarkable property (known as self-similarity), it was known in earlier times as the ‘miraculous spiral’.

Spiral galaxy

Whirlpool Galaxy

Nautilus

Nautilus shell

Geometrical forms: polygons and polyhedra

While pentagonal patterns abound in living forms, the mineral world favours twofold, threefold, fourfold, and sixfold symmetry. The hexagon is a ‘close-packing’ shape that allows for maximum structural efficiency. It is very common in the realm of molecules and crystals, in which pentagonal forms are almost never found. Steroids, cholesterol, benzene, TNT, vitamins C and D, aspirin, sugar, graphite – all show sixfold symmetry. The most famous hexagonal architecture is built by bees, wasps, and hornets.

Honeycomb

Honeycomb

Fly eye

Fly’s eye

Snow

Six water molecules form the core of each snow crystal.

Snow

The great majority of viruses are icosahedral, including the polio virus and the 200 kinds of viruses responsible for the common cold. Icosahedral symmetry is believed to allow for the lowest-energy configuration of particles interacting on the surface of a sphere. The five platonic solids are also found in radiolarian skeletons.

Platonic solids in sea

The platonic solids have been found living in the sea. The tetrahedron, somewhat rounded as if from internal pressure, is embodied in a protozoan called Callimitra agnesae, the cube is Lithocubus geometricus, the octahedron Circoporus octahedrus, the dodecahedron Circorrhegma dodecahedrus, and the icosahedron Circognia icosahedrus.

From: http://ourworld.compuserve.com/homepages/dp5/pattern1.htm

Polyhedra in four dimensions

From: http://math.ucr.edu/home/baez/platonic.html

While there are 5 Platonic Solids in 3 dimensional space, in 4 dimensions, there are exactly six regular polytopes.

How can we visualize these? Well, a Platonic solid looks a lot like a sphere in ordinary 3-dimensional space, with its surface chopped up into polygons. So, a 4d regular polytope looks a lot like a sphere in 4-dimensional space with its surface chopped up into polyhedra! A sphere in 4-dimensional space is called a ‘3-sphere’, since people living on its surface would experience it as a 3-dimensional universe with the curious feature that if you hop aboard a rocket and shoot off straight in any direction, you eventually wind up back where you started. (This is just like what happens when you start walking in a straight line in any direction on an ordinary sphere.)

So, we can visualize the regular polytopes in 4 dimensions by taking a 3-sphere and drawing it chopped up into polyhedra. A 3-sphere is hard to draw until you realize it looks just like ordinary 3d space except that it ‘wraps around’… very far away from here. But if we ignore that, and just draw a nearby portion of the 3-sphere chopped up into polyhedra, with everything outside this portion being one big polyhedron, we’ll do okay. And this is what we get:

5 cell

Hypertetrahedron 5 tetrahedral faces

8 cell

Hypercube (tesseract) 8 cubical faces

16 cell

Hyperoctahedron: 16 tetrahedral faces

120 cell

Hyperdodecahedron: 120 dodecahedral faces

600 cell

Hypericosahedron: 600 tetrahedral faces

24 cell

24 cell: 24 octahedral faces

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