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The Entropy Reduction Laboratory

From Platonic bodies to geodesic spheres, fullerenes, and virus

this page.

Platonic bodies serve as the basis of essentially three types of extended structures:
  • The Buckminster Fuller (1,0) frequency geodesic spheres are identical to Platonic bodies. Since there are 5 Platonic bodies, there are 5 principal Buckminster Fuller (1,0) frequency geodesic spheres. Higher frequency geodesic spheres are produced by finer and finer triangulation of the Platonic bodies's faces. In the limit of infinite frequency, geodesic spheres of infinitely fine tessellation look identical to the Platonic spheres, even though they are of different genealogy.
  • By truncation of the icosahedron, there results the simplest of the fullerenes, a stable molecule consisting of carbon only atoms only. Its surface is structured into 12 pentagons and 20 hexagons. Further stable fullerenes result from insertion of further hexagons, while the number of pentagons stays constant.
  • Geodesic domes according to the Buckminster Fuller procedure result in structures which are made up by quite a number of different geometric components. A new class of structures which might be called the Ditchfield structures, are built up by arcs of just one size. Only at the vertices of the native Platonic bodies, a few arcs of a second size are required for compliance.
  • Nature extends the principal Platonic bodies in the direction of higher complexity. In the virus, a highly complex molecular structure still exhibits the symmetry of the icosahedron.
The dimension of these structures span a range of 1012, from 7 angstroms to about 70 meters. This demonstrates the universal relevance of symmetry principles inherent in the Platonic bodies.
I.Nobel Price Chemistry 96
Comment Most famous became the geodesic dome designed by the architect R.Buckminster Fuller for the 1967 World Exhibition in Montreal.

For the discovery of carbon atoms bound in the form of closed shells the

was awarded to In 1985 they had found high stability in C60 , which suggested a molecular structure of great symmetry. In 1990, macroscopic samples of fullerenes were produced by Wolfgang Kraetschmer and Donald E.Huffman.
The newly discovered structure was named buckminsterfullerene because of the similarity of the structure to geodesic domes as invented by Buckminster Fuller.
The following links of this page will elaborate on this structural similarity in a variety of different disciplines.
For the official website of the Nobel Foundation see
Nobel Foundation
II.Geodesic Spheres
Kirby Urner discusses basic concepts and modes of construction of Buckminster Fuller geodesic spheres and domes. For the icosahedron, an applet demonstrates the process of triangulation, i.e. the formation of higher frequency geodesic spheres. - Geodesic domes are fractional parts of complete geodesic spheres. The Spaceship Earth Pavilion, a 17-storey high structure, is one of the rare constructions of a full sphere.
For detailed information, please see the
Article by Kirby Urner
III.The Sudbury
Neutrino Detector
The Sudbury Neutrino Observatory (SNO) is taking data that provides insight into the properties of neutrinos and the core of the sun. The detector was built 6800 feet under ground, in INCO's Creighton mine near Sudbury, Ontario. SNO is designed to detect neutrinos produced by fusion reactions in the sun. It is a heavy-water Cherenkov detector and uses 1000 tonnes of heavy water, contained in a 12 meter diameter acrylic vessel. Neutrinos react with the heavy water (D2O) to produce flashes of light (Cherenkov radiation) which is then detected by an array of 9600 photomultiplier tubes mounted on a geodesic support structure surrounding the heavy water vessel. The detector is immersed in light (normal) water within a 30 meter (that's 10 stories!) barrel-shaped cavity excavated from Norite rock.The overburden of rock shields the detector from cosmic rays. The detector laboratory is extremely clean to reduce background signals from radioactive elements present in the mine dust which would otherwise hide the very weak signal from neutrinos.

First Results from the Sudbury Neutrino Observatory reveal New Neutrino Properties Physicists from Canada, the UK and the US announced that their results provide a solution to a 30-year old mystery - the puzzle of the missing solar neutrinos. The Sudbury Neutrino Observatory (SNO) finds that the solution lies not with the Sun, but with the neutrinos, which change as they travel from the core of the Sun to the Earth.As a special aspect experiments suggest, that the mass of the neutrino is not zero, even though extremely small indeed.

For mores see the site of the
Sudbury Neutrino Observatory, Canada
IV.Descartes' Angular Deficit
It has been stated already by Rene Descartes (1596-1650) that the amount of curvature , expressed as angular deficit in closed figures, such as the polyhedra or the sphere, is always 720°.
Why just 720°?
As the applet demonstrates, there are 360° necessary to built up the curvature of halve a sphere from the south pole. Another 360° are necessary to bring the curvature back to zero at the north pole of the sphere( resp. the octahedron). Descartes' law is of importance for obtaining a detailed specification of the properties of Platonic bodies.
For a more detailed discussion of Descartes' law, see the
page DESCarte of this site

A complementary discussion, including a reference to Euler's Law, is given in Kirby Urmer 1
, a nice demonstration of higher frequency in Kirby Urner 2
V.Platonic spheres
Comment The icosahedral sphere is one of the five Platonic spheres.
Platonic spheres are the central projection of Platonic bodies onto the surface of the circumscribing sphere. From the point of view of geodesic spheres, Platonic bodies are geodesic spheres of lowest frequency, i.e. f=1 and Platonic spheres appear as Buckminster Fuller geodesic spheres in the limit of infinite frequency. Also, from this point of view, there exist 5 different hidden types of ideal round spheres.
For a derivation of the properties of Platonic spheres see
the page PLASpher of this site.
VI.Paper model
This is a (8,0) frequency geodesic sphere. It consists of 1280 arc-shaped strips glued back-to-back in triangles. - It may be noted, that any of the higher frequency geodesic spheres will be tiled with triangles only. Almost all of them are arranged as hexagons with six triangles in the center. It will be difficult to find the critical vertices, from which the type of the original Platonic body can be inferred. As an example, there are exactly six points with four triangles arranged around a fourfold rotation axis on geodesic spheres of any frequency as long as they are derived from the octahedron.
This model has been constructed and put onto the web by
Peter T. Wang
VII.Big BuckyBall
Move the BuckyBall by dragging the mouse. - After proper orientation of the ball, you may verify by stepwise rotation around the vertical axis that the surface of this ball shows 12 pentagons as required by the symmetry of the dodecahedron. You may also notice that each pentagon is surrounded by 5 hexagons. Thus, this structure is not a geodesic sphere as obtained by triangulation. It may be visualized alternatively to be a truncated icosahedron as obtained from an icosahedron by truncating of the twelve vertices: each vertex is replaced by a pentagon and the twenty former triangular faces are replaced by twenty hexagons. On the microscopic level, this structure is the pioneer fullerene C60 , a model of which is shown under the next link.
This ball results from the fullerene research of
John C. Huffman
Indiana University Molecular Structure Center, Bloomington
VIII.The fullerene C60
The shape of the fullerene C60 originally was not observed directly. The key proof of the structure as shown in this ball and stick model was obtained from mass spectrometry in combination with NMR (nuclear magnetic resonance) spectrometry. As in the BuckyBall model, this structure consists of twelve pentagons and twenty hexagons and can be considered to be a truncated icosahedron. Each vertex of the truncated icosahedron is here occupied by a carbon atom. Each carbon is connected to three other carbon atoms by one double bond and two single bonds.
This model has been put onto the web by
Chemistry Department, Muskingum College
IX. More general fullerenes
Here ball and stick models of fullerenes other than the molecule C60 are shown. All of these structures consist of twelve pentagons and a variable, even number of hexagons. Both pentagons and hexagons would contribute to the curvature, if the the hexagons would not be exactly regular. However, it turns out that the hexagons are quite regular and thus the curvature is carried exclusively by the pentagons. This results in nearly cylindrical geometries, the nano tubes, or in structures resembling a polyhedron, if the principal symmetry of the C60 fullerene is preserved as in C240 and C540 .
Graphite - which consists of regular hexagons - cannot form a cage and is essentially planar. - The completely truncated icosahedron, the dodecahedron, cannot be realized as a fullerene because of the large resulting strain of the carbon bonds.
The Fullerene Structure Gallery has been assembled by
Mitsuko Yoshido
X. The SoccerBall
Technical requirements have been set up by the
FIFA Laws, Qualities and Measurements:

By law #2 the ball is:
  • spherical
  • made of leather or other suitable material
  • of a circumference of not less than 62 cm and not more than 64 cm
  • not less than 400 grams nor more than 440 grams in weight at the start of the match
  • of a pressure equal to 0.4-0.6 atmosphere (400-600g/cm²) at sea level
There are further requirements on FIFA approved logos on the ball. Also, the logo of the ball's manufacture may be applied.
Apparently, there is no indication on the way, how a "spherical" ball may be manufactured and what the allowed deviations from the perfect sphere may be. Without going to an even finer tessellation, the truncated icosahedron is the optimal solution. But football players would value a ball that allowed more predictable shots. To this purpose a finer tessellation like that of the proposed Brazilian Football would render the surface of the ball closer to a perfect sphere than the familiar pentagon-hexagon structured ball in use today.
For a large collection of different quality and logo types of soccerballs, all being truncated icosahedra, see
and click there on "Hartwarenkatalog" etc
XI. The Honeycomb Spherical Figure
While the geodesic dome structures of Buckminster Fuller are based on the icosahedron, the honeycomb figure of Alan Neil Ditchfield is based on the truncated icosahedron which consists already of 20 hexagons in addition to the native 12 pentagons. The ideal honeycomb consists of regular hexagons with equal angles and equal sides and does not exhibit any curvature. In the honeycomb spherical figure of Ditchfield, the sides, resp. arcs are of equal length, and the curvature is produced by 260 hexagons of type a,b,c,and d and 12 pentagons with dihedral angles not corresponding to those required by a regular polygon.

          Specification of the Honeycomb Spherical Figure

type of   # on a   dihedral angles [radians]    spherical excess*    
hexagon   sphere    1   2   3   4   5   6       each       total
  a         20      i   i   i   i   i   i     0.050622   1.012431
  b        120      j   j   k   j   j   k     0.050625   6.075002
  c         60      l   m   n   i   m   n     0.050053   3.003153 
  d         60      o   p   q   q   p   o     0.040054   2.403214
pentagon e  12      r   r   r   r   r         0.006048   0.072571
                     total spherical excess of figure = 12.566371  
                                                            = 4pi

   i=2.102832   j=2.090177   l=1.830767   o=2.198510   r=1.886165 
                k=2.128145   m=2.314516   p=2.226209
                             n=2.026896   q=1.878493
      Sides, resp. arcs of the hexagons are of equal length 
      except those, which are identical to one of the sides
         of the 12 pentagons with arcs of equal length 
* While the term "angular deficit" refers to angles around a vertex, the term "spherical excess" refers to the internal angles of a polygon: (sum of the "n" dihedral angles) - (n-2)*180°. For both points of view, the sum over all constituents of a closed figure is by Descartes' proposition equal to 4pi.

The relevance of the Honeycomb Spherical Figure (HTML short version) or (WORD 2000 DOC long version) extends to many areas:

  • since the number of different construction components is greatly reduced in the honeycomb geometry, it may be recommended for application in all kinds of dome- and sphere- mechanical structures and buildings where besides stability there is the requirement of cost efficiency.
  • the determination of the detailed configuration of the C540 fullerene is a topic of current experimental research. Electron microscopy images suggest that there are indeed "protuberances" at the positions of the pentagons at an otherwise spherical molecule, as is predicted by the geometry of the Honeycomb Spherical Figure.
  • as a possible new international standard for a global referencing grid.
The detailed specification and use of the Honeycomb Spherical Figure in coordinate systems, discrete global grids, global referencing systems and telecommunications was presented at
Internat.Conf. on Discrete Global Grids, 3/2000, Santa Barbara
in a
Paper by Alan Neil Ditchfield ( Word 6 doc )
where the actual global grid is given in
Global Grid by Alan Neil Ditchfield ( Word 2000 doc )

XII. The C540- and C840- Fullerenes - Groundstate Calculations

While the Ditchfield approach aims at minimizing the geometrical complexity of a fullerene structure, the research of Heggie and Breuer and relates to the key questions in carbon chemistry: What is the ground state of a given number of carbon atoms? Are there for certain numbers of carbon atoms new structures arising with unexpected properties? What determines whether one or another structure is more stable. The key problem lies in the nature of bonding in carbon: its ease of rehybridisation and p-bond formation and the unique nature of the interaction between p-bonds in different planes. Recent quantitative total energy calculations starting from first principles have shown that that the interlayer interaction in graphite is not simply of van der Waals type.

With respect to the fullerenes, computer calculations, requiring powerful parallel computing facilities, show that in the ground state of a cluster of n carbon atoms, the configuration changes from a single shell fullerene at low n to a multi-shell onion structure as n increases. The energy difference between the two types of structure is a delicate balance between the benefits of a larger radius of curvature and greater p delocalisation in the single shell, and the benefit of interlayer attraction in the onion.

The optimized ground state structure for a possible C540 shell has D2h symmetry and is almost spherical as in the geometrical model of Ditchfield. The theoretical C840 struction appears to be an multi-shell onion with the outer shape of an icosahedron. The onion being favoured for certain high values of n because of the higher interlayer energy, which seems to correspond to the theoretical energy for AA stacked graphite. The authors conclude that apparently many of the onions observed in experiment, which are most often spherical, are not in fact ground state structures.

The C540-, C840-Fullerene computer modeling is reported on in the
Article of Malcolm Heggie & Stephen Breuer
XIII. The CVB3 Virus
Now we proceed from complex molecular structures to even more complex biological structures. Here the structure of coxsackievirus B3 (CVB3) is shown in 3.5 angstrom resolution. Coxsackie B viruses (CVBs) are etiologic agents of a number of human diseases that range in severity from asym*ptomatic to lethal infections. They are small, single stranded RNA icosahedral viruses that belong to the enterovirus genus of the picornavirus family. - Two of the twelve prominent five-fold axes of symmetry of the icosahedron are seen in yellow. Surrounding the 5 fold axes of symmetry can be seen the canyons, regions that have been demonstrated in other enteroviruses to be binding sites for the cell membrane receptor. - As compared to the ideal icosahedron, here the triangular faces appear to be compressed by wide interstitial areas, which in turn present a two-fold symmetry. Despite minor distortions, the symmetry of the icosahedron is preserved.
These results of research on the coxsackievirus CVB3 in 3.5 angstrom - resolution were obtained by
Prof. Michael Rossmann,
Department of Biological Sciences, Purdue University, West Lafayette
XIV. The sand dollar
The sand dollar is one of numerous circular sea urchins. Its structure corresponds to just one vertex of an icosahedron. As discussed under the topic of Platonic spheres, only planar structures may have a hexagonal symmetry - a slightly curved structure will form a vertex of fivefold symmetry (or less). Since the sand dollar is not entirely flat, by its slight curvature it is able to adjust to curved shapes of sandy bottoms in shallow water, where it mainly lives. - Similar observations hold for starfish and other some other primitive animals. Also flowers and fruits (cut an apple perpendicular to its axis! ) often exhibit the fivefold symmetry of the icosahedral vertex.
In contrast, a typical symmetry element of crystals is the sixfold symmetry axis.
This picture is part of the collection of
Jill Britton
XV. Droplet Surface-Structure
In size between virus and sanddollar here spherical water droplets, covered with polystyrene beads are shown. In a plane, the beads would arrange in a hexagonal lattice - each bead being surrounded by 6 neigbouring beads. To allow for the curvature inherent to the sphere, at 12 critical points, the vertices, a bead is expected to be surrounded by only 5 neighbouring beads.
In Fig.A, a phase contrast light microscope image of about 13% of the surface of a water droplet with radius R= 12 micrometer and a bead distance of a=2.9 micrometer is shown. The corresponding surface structure, Fig.B, exhibits indeed a fivefold symmetry around one of the beads, extrapolation to the total surface area results in 12 such "defects"..
In Fig.C and Fig.D are shown corresponding images for the case R=43.9 microns and a=3.1 micrometer. Thus, the distance a between beads is about the same as in the first, example, but the radius of the water droplet about 4 times as large. Consequently, in units of a, the curvature is much smaller and the surface is flatter. This leads to an increase of strain energy around the defect-vertices. Here an entirely new mechanism for reducing the elastic strain energy is observed: the little red dots in Fig.D stand for beads with 5 neighbours, the yellow dots stand for beads with 7 neigbours
In case of a three-dimensinal crystal, the strain energy of the lattice around a perfect dislocation my be reduced by dissociation into two partial Shockley dislocations, resulting in an extended dislocation enclosing a stacking fault area. .
Here, in the case of a slighly curved twodimensional lattice, the strain energy around the zero dimensional defect is reduced by formation of one-dimensional "scars", which ares chains consisting of beads with 5-7-5-7-5-7-5 coordination.. Evidently, as to be expected from an Generalized view on Platonic Spheres, the 7-coordination defects counteract the strain produced by the 5-coordination defects. The length of these chains increases linear with the size of the droplets.
The droplet structures are always in a state of rearrangement and therefeor difficult to observe. The images on the droplet structure are the results of experiments and theoretical considerations of
A.R.Bausch, Physics Department, Technische Universitaet Muenchen,
and groups of scientist from other well known scientific institutions as published in the article:
A. R. Bausch, M. J. Bowick, A. Cacciuto, A. D. Dinsmore, M. F. Hsu, D. R. Nelson, M. G.Nikolaides, A. Travesset, and D. A. Weitz
Grain Boundary Scars and Spherical Crystallography
SCIENCE, Vol.299, 2003, p.1716
The theoretical interpretation of the droplet structure by the authors of the images is much more profound than our discussion given above, also it differs in certain details.
XVI. Bucky living
Buckminster Fuller's geodesic structures are realized also as wooden structures to live in. If you wish, you may start right away to build one with
the assistance of
Einar Thorstein
Construction Lab & Kingdome
XVII. Biomemoa -
the futuristic Biosphere

Objective of Biomemoa - the futuristic Biosphere:

  • create a spectacular theatre in which to tell the story of human dependence on plants
  • to build a series of giant conservatories to stand high, wide and handsome
  • be high enough for the towering trees of the rainforests
  • be wide enough for the sun baked landscapes of the Mediterranean
  • be handsome; an international symbol, world class architecture, the 8th wonder of the world:

The "hex-tri-hex": The final design comprised a two-layer steel curved space frame with an outer layer of hexagons, (with the occasional pentagon thrown in to make it all fit together like a football) and an inner layer of hexagons and triangles (resembling huge steel stars). Diagonals connected the node points of the layers together to make the structure rigid. This structure - the first of its kind on this scale in the world - is called the 'hex-tri-hex'.

Why this shape?

  • efficient: maximum size and strength with minimum steel
  • energy saving: maximum volume with minimum surface area (minimising heat loss)
  • adaptable: moulds to the shape of the site
  • transparent: it lets through more light than glass, even including UV light
  • lightweight: the weight of the steel is almost the same as the weight of air it encloses
  • roomy: spans 110 metres at its widest point with no internal supports - allowing complete freedom for the plants and the landscape architects
Visit the
Eden Project in Cornwall, Great Britain

Especially, look at the design of
The Biomemoa Building
XVIII. The Jena






Here we close our mini anthology by reference to the most impressive technical innovation based on the icosahedron structure. Before Buckminster Fuller popularized the idea of Geodesic Domes, already in 1913 the Carl Zeiss optical works of Germany recognized the problem of designing a huge sphere that would both hold a large number of people and show the motions of the planets as well as the stars. After much work, no satisfactory solution was found.
Then in 1919, Dr. Walter Bauersfeld of Zeiss:

...caught an entirely different idea: reversing the plan of a mechanically rotatable hollow sphere with illuminated images of the stars, he transferred the entire mechanism for the movements to a collection of projectors which would project luminous images of the stars on to a stationary white hemispherical dome of much larger dimensions than those originally conceived. Within the dome, the centre of which would be occupied by the projectors, all would be in darkness. By means of suitable mechanisms the projectors would be moved and guided so that their illuminated images of the heavenly bodies would conform on the dome to the motions which actually occur in nature...1

For five years a large staff of scientists, engineers and mechanics worked with Bauersfeld at the huge Zeiss plant in Jena, Germany to design the projector and the projection dome.

The projection of the starry sky required a certain number of projectors, arranged in the center of the dome. Each projector should illuminate an area of the same size as the dome. If the vertices of an icosahedron are cut in such a way that the new surface consists of 12 pentagons and 20 hexagons the area within each is nearly of the same size. The projectors are arranged in the centers of the pentagons and hexagons and produce 32 starfields on the same dome. (Actually only 31, since one area is used for the support.)...2

To test the projector Bauersfeld needed a hemispheric dome as a replica of the sky. It had to be lightweight, as it was to be placed on the roof of the Zeiss factory. He built a light iron rod framework, the design a highly sub-divided icosahedron, with great circle arcs.

Thus both the dome frame and the projection pattern
were derived from the icosahedron.
Not until the complex skeleton (3,480 struts accurate in length to 2/1,000 of an inch) was complete did Bauersfeld seek professional advice to construct the permanent dome. This was accomplished by engineers of Dyckerhoff and Widmann, a company engaged with factory buildings of ferro concrete.
For present activities see the Zeiss Planetarium in Jena

The idea of Symbiosis has been envisaged by Kisho Kurokawa . He discusses Symbiosis in architecture and philosophy of life in his book

EACH ONE A HERO, the Philosophy of Symbiosis,
whith reference to many of his architectural works, including the The Ehime Museum of Science, Planetarium.

Thus the planetarium, both as a fine instrument and as an institution, has come a long way since 1923 when astronomer Elis Stromgren wrote:
"Never before was an instrument created which is so instructive as this; never before one so bewitching; and never before did an instrument speak so directly to the beholder. The machine itself is precious and aristocratic... The planetarium is school, theater, and cinema in onea classroom under the eternal dome of the sky."

Read more of this from the original
Shelter Publications, Inc., 1973
Bolina, Cal.
XIX. Global-Model GME
Weather Prediction Model
Based on the work of John Baumgardner the Deutsche Wetterdienst, Offenbach, develops a worldwide weather model. The parameters of this model are properties such as local temperature, humidity, windvelocity,... The coordinate system is based on the icosaeder, which is subdivided into triangles, which are of nearly equal size and therefore of essential advantage compared to the conventional coordinate systems based on longitude and lattitude. The length of a triangle side of the icosaeder is 7054 km, by the subdivision a mesh size of about 60 km is achieved. There are 163830 hexagonal areas apart from the 12 special cases, the 12 pentagons at the original vertices of the icosaeder. The mesh size of 60 km allows prediction of a rather high resolution.
For further information on this model visit the
Deutscher Wetterdienst
( The German Weather Service )
See also other pagesof this site

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Last update of the page: May 12, 2003