maandag 14 maart 2016

Fascinating things about regularity – in Antiquity and now

A mathematical form –the regular pentagon – puts us on a journey of discovery along a personal memento of my childhood, the ancient Greek, and we find a remarkable piece of craftsmanship at Brightlands Chemelot Campus.

Self-made dodecahedrons

It must have been about 1977, when I was in the last year of my high school, when I took six sheets of DIN A4 100 gram paper and fabricated three dodecahedrons, with a ruler, a drafting compass, a pencil, a pair of scissors and a Pritt stick. I stacked them together and assembled the object on a pentagonal paper pedestal. Above, a recent photo; my craftwork has survived decades of relocations and other discomfort.

In my childhood I was fascinated by regular mathematical forms, a fascination that I not only share with lovers of diamonds and crystals (Swarovski), but also with philosophers of antiquity.

And thus we arrive at specific belief systems. Whatever you think of that, it's quite intriguing and worth exploring.

Timaeus” is a dialogue by Plato from 360 B.C., in which the astronomer and physicist Timaeus of Locri gives a description of nature, which has been qualified by the English philosopher Bertrand Russell as ‘silly’.

Timaeus states that God created the Universe, shaped in a perfect form: a sphere, where the ends are everywhere equidistant from the center – a suitable form for the Living, which in itself should include all living beings.

By analogy, the human head is spherical: according to Timaeus the gods gave the head, the most divine and sacred residence in us, the body as a vehicle to move.

Four elements defined
According to Timaeus the visible and tangible world is composed of fire, earth, water, and air. These four elements are enclosed by triangles, or more accurately three-dimensional combinations thereof.

4: tetrahedron – FIRE

The element fire is represented as a tetrahedron (with four surfaces).

8: octahedron – AIR

Air is represented as a octahedron (with eight surfaces).

20: icosahedron – WATER

Water is depicted as an icosahedron (with twenty surfaces).

6: cube – EARTH

The fourth element, earth is represented as a hexahedron (cube, with six surfaces).

According to Timaeus these are forms that possess the property that they divide the entire sphere (which is a model for the Universe) in equal and uniform parts.

Fire is represented as the smallest of these bodies (tetrahedron), the icosahedron (water) rolls easily, and the cube (earth) is the least susceptible to movement.

Four elements pictured
We find the concept of the four elements at various places, for instance in the thriller "The Bernini Mystery” by Dan Brown (2000). There and elsewhere alien forces are assigned to the elements.

Glass screen at Brightlands Chemelot Campus

We also encounter the four elements at Brightlands Chemelot Campus. In the hall of one of the buildings we find a freestanding stained glass screen with unclear signature, on which an element is pictured on each quadrant.


Bottom right fire is displayed, with burning torches, a flaming fagot, lightning, and the sun as distinctive attributes.

And God said: “Let there be light.” (Genesis 1 : 3).


Top left air is depicted with birds, clouds, and a kite.

And God said:" …let birds fly above the earth across the vault of the sky.” (Genesis 1 : 20).


Bottom left water is displayed, using marine animals, rain, and a sailing ship.

And God said: “Let the water teem with living creatures…” (Genesis 1 : 20).


Finally, top right we find earth, presented by plants, a cornucopia, and a globe.

And God said: “Let the land produce vegetation: seed-bearing plants and trees on the land that bear fruit with seed in it, according to their various kinds.” (Genesis 1 : 11).

12: dodecahedron

The fifth element
In Timaeus another body is mentioned, with which Plato posits a fifth element and which completes the classic collection of Platonic bodies: the dodecahedron (with twelve surfaces), the object of the craftwork from my youth.
The dodecahedron is supposed to be the classical symbol of health.

Incidentally, Aristotle also introduces a fifth element: ether, the first element, which allows the circular motion of celestial bodies.

Dodecahedron derivates
From Wenzel Jamnitzer, Perspectiva corporum regurarium,
Nurnberg 1568

Complex mathematics
All Platonic bodies meet three requirements:
  1. The surfaces are regular polygons.
  2. All surfaces, edges, and vertices are equal.
  3. The complete sphere is ‘convex’, i.e. the vertices form a sphere (which is the perfect form of the Universe, according to the Timaeus).
So there are five Platonic bodies? Surely not, because many variations are possible – there is even a separate discipline of mathematics that deals with this. Variations are possible by playing with the requirements of the bodies. For example, this results in the object below.

The Timaeus is available in English online: It’s about much more than the elements, for example about the mythical Atlantis.
This blog post is a repost of my (Dutch) February 1, 2016 post.
Read my May 20, 2013 blog post about the reason why of my English reposts.