“Geometry existed before creation”

– Plato

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The discovery of regular polyhedra is attributed to Pythagoras of Samos (6-5th c. BC.), although he probably only knew the tetrahedron, cube and dodecahedron.

Pythagoras was the first to apply himself to the concept of mathematical proof. Furthermore, he turned the numbers into abstract concepts; ‘two houses’ or ‘two people’ became simply ‘two’. All this developed from the religion that Pythagoras and his followers adhered to, which was based on the belief that everything in the world, at the deepest level, is essentially mathematical in nature. By this is meant that everything can be ‘expressed as a number or as a ratio of numbers’, an idea that stems from the observation that strings whose lengths are related to integers sound harmoniously together.

It was not until several generations later that Theaetetus of Athens (5-4th c. BC.) described the octahedron and icosahedron. This student of Socrates and Plato made important contributions to mathematics and, although no manuscripts of his have survived, we find much about him in Plato’s dialogues. Moreover, it is almost certain that books X and XIII of the Elements of Euclid describe exactly his work.

Although Theaetetus was the first to construct the five bodies, they are handed down to us through Plato’s work and are named after him.

Plato believed that observable reality is only a reflection of pure Ideas. For example, in the Timaios, Plato described the mathematical construction of the regular polyhedra, associating them with the four elements and the universe. The worldly particles earth, fire, air and water are said to be reflections of the perfect forms, the cube, tetrahedron, octahedron and icosahedron. The fifth body, the dodecahedron, is for Plato the form of the universe ‘ether’.

Euclid of Alexandria (4th-3rd c. BC.) is, like Plato, a philosopher, but his vision of mathematics is closer to science as we know it today. His most important work, The Elements, a collection of thirteen volumes, describes, among other things, the proof of the five regular polyhedra.

In this line of great Greek minds, Archimedes of Syracuse (3rd c. BC), sometimes called the greatest mathematician of antiquity, cannot be forgotten. Building on the framework laid down by Euclid, he elaborated numerous geometrical proofs and described the semi-regular polyhedra.

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*There is evidence that mankind knew of the existence of the Platonic bodies even before Plato. The Ashmolean Museum in Oxford preserves all five Platonic bodies and some semiregular forms described by Pythagoras. Chiselled in stone, these Platonic bodies are estimated to be at least a thousand years before Plato. These stones have been found in Scotland and come from a Neolithic people.*

Hundreds of carved stone spheres with a diameter of about 7cm, probably around 3000 BC., have been found at Scara Brae in Scotland. Some are carved with lines corresponding to the edges of regular polyhedra. The function of these stones is unknown, many are intricately carved with spirals or crosshatches. The material ranges from easily worked sandstone and serpentine to difficult, hard granite and quartzite.

All five platonic bodies are represented: tetrahedron, cube, octahedron, icosahedron and dodecahedron. As well as some additional compound and semi-compound forms, such as the cube octahedron and the icosidodecahedron.

Many of them, however, were not ‘perfect’ Platonic bodies, but rather very close approximations that clearly show the skills of the stonemasons.

It is a clear indication of a mathematical ability of Neolithic man. There is speculation about a possible relationship with the construction of the great astronomical stone circles of the same era in Britain. After all, the study of the heavens requires an understanding of spherical laws of three-dimensional co-ordinates. Plato, Ptolemy, Kepler and Al-Kindi all attributed cosmic significance to these polyhedra.

Photos © Ashmolean Museum in Scotland

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Tetrahedron

4 equal surfaces

4 points*Fire*

Hexahedron

6 equal surfaces

8 points*Earth*

Octahedron

8 equal surfaces

6 points*Air*

Dodecahedron

12 equal planes

20 points*Ether*

Icosahedron

20 equal surfaces

12 points*Water*

Nature provides us with a large number of symmetrical structures. The material world is made up of molecules that have a certain internal symmetrical structure. Elementary forces string the individual molecules together to form structures which we call crystals. A single crystal has the shape of a geometric polyhedron, in which the symmetries of the constituent molecules can be found. These elementary forms are the regular tetrahedron, hexahedron and octahedron.

Symmetry in the form of repetition and regularity is the basis of beauty. What is striking is that the most beautiful or rich symmetry forms are found in the simplest forms of life.

It is striking that the crystals in nature are all based on the numbers 3, 4 and 6. In crystallography, one cannot obtain a quintuple symmetry, because it is contrary to the repeatability of crystal construction on geometric grounds. The laws of crystallography make a symmetry form with a quintuple rotation impossible, so that neither dodecahedron nor ikosahedron can be present in pure form as a crystal.

In 1984, physicists created quasi-crystalline structures in a laboratory that exhibit a quasicrystalline symmetry. Quasi-crystalline patterns are similar to the Penrose mosaics, the so-called ‘aperiodic mosaics’.

It has also been found that the golden ratio is present in non-periodic rule patterns of the number 5. The diagonals of a regular pentagon intersect according to the golden ratio.

So in the world of inanimate matter there is only room for the three simple platonic bodies. A crystal that has the shape of a regular twelve plane or a twenty plane will unfortunately never be found. But what cannot be done in lifeless nature, life forms can do.

Viruses consist of a core and a spherical covering of proteins. In a number of cases, the mantle proteins show a form of symmetry that corresponds to that of dodecahedron and icosahedron.

Bibliography

– ‘Symmetrie, kunst en computers ‘, by Hans Lauwerier

– ‘Viruses and Geometry: Group, Graph and Tiling Theory Open Up Novel Avenues for Anti-Viral Therapy’, by Reidun Twarock

– https://www.geestkunde.net

– https://geometrymatters.tumblr.com

– https://www.cosmic-core.org

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Radiolaria illustration from the Challenger Expedition 1873–1876. Including in the center a *Circogonia icosahedra*, a type of Radiolaria, in the shape of an ordinary icosahedron.

* Braarudosphaera bigelowii *is a unicellular phytoplankton alga in the shape of a docahedron from the geological Cretaceous Period.

* Dictyochophyceae – Silicoflagellates *is a unicellular heterokontalga from the geological Cretaceous Period, 145 to 66 million years ago.

* Actinoptychus senarius *is a unicellular phytoplankton alga, currently living all over the world.

* Braarudosphaera perampla, *unicellular phytoplankton alga from the Pleistoon geological period.

* Azpeitia nodulifera* is a diatomaceous seaweed, a unicellular alga with a silica exoskeleton. Here with a Fibonacci spiral. From the Quaternary Geological Age 2.58 million years ago.

*Azpeitia nodulifera*

Microscopic view of three sten towers, a type of single-celled freshwater protozoa.

(Photo © Dr. Igor Siwanowicz)

Microscopic photo of the unicellular alga *Triceratium morlandii*.

(Photo © Larry G. Gouliard)

* Penicillium vulpinum *(fungus).

*(Photo © Tracy Debenport)*

*Pyrocystis fusiformis* (alga).

(Photo © Gerd Günther)

Diatom art by Klaus Kemp.

Discover how artist Klaus Kemp makes beautiful art from algae.

Selections from the movie Proteus. The film tells of Ernst Haeckel, a nineteenth-century naturalist, and his detailed engravings of Radiolaria, single-celled marine organisms.

3D image of a Bacillus phage Basilisk (a), herpes simplex virus 1 (b) and bacteriophage lambda (c) with an icosahedral pattern.

(Photo © Nature Communications)

A large majority of viruses exhibit complete icosahedral symmetry.

Adenovirus (*Adenoviridae*) in the form of an icosahedron. Creates respiratory infections.

Bacteriophage consisting of an icosahedral capsid and a long non-contractile tail that allows for host recognition and genome delivery.

Microscopic image of a housefly’s eye pattern.

(Photo © Dr Razvan Cornel Constantin)

Butterfly egg

Although common dodecahedrons do not occur in crystals, the shape occurs in the crystals of the mineral pyrite.

Snow crystals.

(Photo © Kenneth Libbrecht)

(Photo’s © Alexey Kljatov)

The Bubbleologist – The Code

Discover how platonic shapes are created in soap bubbles.

Tetrahedron in a soap bubble collection

(Photo © Tom Noddy)

Octahedron in a soap bubble collection

(Photo © Tom Noddy)

Dodecahedron in a soap bubble collection.

(Photo © Tom Noddy)

‘Muscovite’ trapiche.

A special stone is the ‘cherry-blossom-stone’ from Japan, consisting of a variant of muscovite, namely pinite. The base stone is indialite, converted to cordierite with a hexagonal shape (pseudomorphosis).

(Photo © unknown)

Emerald trapiche crystal.

During the formation of the emerald gemstone mixture, inclusions of albite, quartz, or other carbonaceous materials can penetrate. The hexagonal crystal structure of emerald creates a six-point radial pattern.

(Photo © Jeffery Bergman, Primagem)

Diffraction patterns of quasicrystals ‘Icosohedral quasicrystal’.

X-ray diffraction of Halite (rock salt).

X-ray diffraction of iridium.

(Photo © Omikron)

Water sound image, Chladnic sound figure, pitch 28.6 Hz.

(Photo © Alexander Lauterwasser)

Chladnic sound figure, pitch 102.528 Hz.

(Photo © Alexander Lauterwasser)

Laue diffraction pattern via X-ray of Beryl crystal.

(Photo © ‘Structures in Art and in Science’ Gyorgy Kepes)

Laue diffraction pattern via X-rays of quasicrystals (zinc-magnesium-holmium).

Microscopic image of chemical bonds between atoms.

(Photo © IBM research Zurich)

These images are derived from simulations of light in the cavities of nanolasers and provide a variety of standing wave patterns.

The first image is a whispering gallery mode. They take their name from the whispering gallery phenomenon that was observed with sound waves in domes. Whispering gallery modes appear not only for light and sound, but also for other types of waves, such as matter waves and gravity waves.

Hologram of a single photon: reconstructed from raw measurements (left) and theoretically predicted (right).

Photons are pulses of electromagnetic energy. When atoms absorb or give off energy, the energy is transferred in the form of photons. A photon is a pulse travelling through the ether/zero energy field.

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