Polyhedra
 
POLYHEDRA

Definition

·  Polyhedra are solids consisting of polygons (faces). These are joined together by edges and vertices (corners).

The polygons in the five regular polyhedra (Platonic bodies)(alt)) are triangles, squares and pentagons. Only one kind of polygon is present in each regular polyhedron. These are the tetrahedron, cube, octahedron, dodecahedron, and icosahedron. Four other polyhedra are considered to be of regular type: the small stellated dodecahedron (with 12 pentagrams), the great stellated dodecahedron (with 12 pentagrams), the great dodecahedron  (with 12 pentagons), and the great icosahedron with 12 triangles.
The five Platonic bodies can be circumscribed by a sphere  ) touching  the vertices  (ibidem) A sphere can be inscribed touching the centers  of the polygons.

  

The 13 semiregular polyhedra , Archimedes 13 semi-regular polyhedra contain furthermore three polygons : hexagon, octagon and decagon. Two or three different polygons are present in each semiregular polyhedron.  The 13 semiregular polyhedra can be circumscribed by a sphere touching the vertices. A sphere can not be inscribed touching the centers of the polygons.

The circle and the sphere were considered to be the most harmonic figures; in ancient Greece but also even until the Renaissance, eg by Johannes  Kepler.

Duals of polyhedra (reciprocals) are constructed by connecting the centers of the polygons. Then the surfaces and the vertices change places. The number of edges remains the same. All polygons are identical. The duals to regular polyhedra (1 , 2) are still regular polyhedra. The duals to semiregular polyhedra are new polyhedra; e.g. the dual to the cuboctahedron is the rhombic dodecahedron  and the dual to icosidodecahedron is the rhombic tricontahedron.  Duals were first decribed by M.E.Calatan in 1865 and were called polyèdres conjugués.  A sphere can be inscribed in the duals touching the centers of the polygons .A sphere can not be circumscribed  the duals .

Prisms, according to Kepler, are formed from two polygons in parallel planes connected by a ring of squares or rectangles.

Antiprisms, according to Poinsot, are similar but the ring is composed of isosceles triangles.

               Non-regular polyhedra. There are 92 possible.

 

History
The oldest known regular polyhedra are carved stones from the Neolethic time ,about 1 000 years before Plato.

·  Several of the regular polyhedra and the semiregular cuboctahedron were known in Babylon, Egypt, India and China (c. 3 000 - 2 000 B.C.). The geometry was developed during the golden period of ancient Greek culture (c. 700 - 100 B.C.)

 

Plato /427 - 347 B.C.) described the five regular polyhedra in his dialogue Timaeus These polyhedra are later named the five platonic bodies or solids. Plato believed that the tetrahedron, octahedron, icosahedron and the cube corresponded to the elements of which the Greeks thought the material world was composed; fire, air, water and earth. The dodecahedron corresponded to quintessence, the elemnt of the heavens. God used this solid for the whole universe. ( In 2003 an American satellite examined the microwave rasiation generated shortly after  the universe began. The scientists believe that  the universe is dodecahedron-shaped) Plato lived in the Athens where he established the philosophy school  Academeia, probably with an inscription on the door:Let no one enter who does not know geometry.All philosophical schools were closed by Justinian in 563  in order to eliminate paganism.

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 Euclid of Alexandria(c.325 – c.265  B.C.) described in Elementa (in Greek : Stochein) the regular polyhedra and polygons and their relation to spheres (Book XIII) and circles. He described combinations of regular polyhedra. However, he did not describe the surface or the circumference of the circle. The Elementa was based on previous works by eg. Plato, Eudoxus, Theaetetus. It is divided in 13 books. More than 1 000 editions of the Elementa has been published since it was first translated to Latin from Arabic in 1482. The first edition translated to Latin from Greek was published in 1505.

The 13 semiregular polyhedra  were described by Archimedes (c. 257 - 212 B.C)(Arabic: Arsamides or E rsemidess ; Greek: ARCIMHDOUS He lived in Syracuse on Sicily.  Eleven of the 13 can be constructed by truncation of the regular polyhedra.  In his work On the sphere and cylinder Archimedes proved that the ratio of the volume of a sphere to the volume of the cylinder, that contains it, is 2 : 3. He also proved the same ratio of the surfaces of the sphere and the cylinder.Archimedes requested his friends that they would place over his tomb a cylinder containing a sphere (Plutarch AD 45-120). Cicero saw (75 BC) the tomb, at theAgrigentine gate,  with a column surmounted by a sphere and a cylinder. It was partly damaged. The likely remainder of the tomb is today present in a hotel garden

In Mouseion in Alexandria (from c. 280 B.C), with a library with c. 700 000 scrolls, the geometry was commented by e.g. Pappi Alexandrini  (Pappos) and Heron Alexandrini. Archimedes' work on the semiregular polyhedra was lost but previously commented byPappos in his Collectiones. It continued until the 5th century, but even in the 7th century Alexandria was a cultural center. There was also a library connected to the temple Serapeum of Alexandria. This was destroyed in 391.
In Pergamon  there was also a famous library.

The Great Greece (Greece, parts of Minor Asia, Southern Italy with Sicily and Egypt) was conquered by the Roman  Empire. This was in 395 divided into the West Roman Empire and Byzantium with Constantinople as the capital. Hereby the Greek sciences and literature became unknown in Western Europe for about 1 000 years.

An Arabic expansion started about in 630. (Mohammed died in 632) and the Saracenic Empire (c.700-1 200)covered territories from Syria via Nothern Africa to Spain. There was a period of Arabic high culture for several hundred years.  Bagdad was a culture center with a “house of wisdom”.Euclides Elementa was translated to Arabic about   in 700 and most of the Greek sciences was translated  about in  900.The Grrek literature was translated to Arabic via the Syrian language.

In the Renaissance (c. 1300 - 1600) the Greek literature was translated to Latin from Arabic and later from Greek.

Pierro della Francesca  (1412 - 1492) studied the five regular platonic solids and six of the semi-regular Archimedean polyhedra ; truncated cube, truncated tetrahedron, truncated octahedron, cuboctahedron, truncated icosahedron, truncated dodecahedron. He has several references to Euclid´s  Elementa.  Battista Alberti (1404-72) and Pierro della Francesca  Introduced the theory of the perspective according to Vitruvius (c.50 B.C).

Luca Pacioli (1445 - 1518) described the Platonic bodies and six of Archimedes´ 13 polyhedra. Leonardo da Vinci drew the polyhedra in Pacioli´s book De Divina Proportione . He did not know of the Collection by Pappus and thought that there were an unlimited number of semi-regular polyhedra .

Johannes Kepler (1571 - 1630) studied the polyhedra  in several ways in his works Harmonices Mundi and Mysterium Geographicum. The illustrations of the polyhedra were drawn by W.Schickard , a professor in mathematics.  He tried to find a relation for the distances between the five known planets and the five regular polyhedra but had at last to give up. He studied the rhombic dodecahedron and the rhombic tricontahedron (1, 2,)  as well as the small stellted dodecahedron and the great stellated dodecahedron.   Kepler gave the names to the 13 semi-regular polyhedra. He was the first to describe all the 13 polyhedra and described the formulas for volume, surface  etc.

Albrecht Dürer (1471 - 1518) designed in Underweisung,  in 1525,  the polygons of the regular polyhedra and nine of the semiregular polyhedra onto a plane,  in the form of “nets” (flat patterns).  He also designed his own polyhedra (1 ,2) and a polyhedron in the Melancolia  ( 1 , 2 ).

Nature and science

Several polyhedra are present in Nature, e.g. the rhombic dodecahedron (1, 2, 3) in bee´s cells, in pomegranate seeds and in the mineral garnet.

 Mineral crystals have the form of several polyhedra. For example:Tetrahedron:silicate (Si/O) and chalcopyrite (Cu/Fe/S) ; Octahedron: diamond (C), gold Au) and cuprite (Cu/O); Dodecahedron: pyrite (Fe/S) and cuprite (Cu/O) ; Icosahedron: pyrite (Fe/S) ; Rhombdodecahedron: garnet (silicates) ; magnetite (Fe/O) ; Truncated cube : galena (Pb/S) ; Truncated octahedron : cuprite Cu/O) ; Rhombhexahedron : calcite , graphite ; Octahedra / tetrahedra (closely packed ) : pendiandite (Fe/Ni/S).However, the forms are commonly imperfect.


Truncated octahedra and rhombidodecahedra have the smallest ratio surface / volume and
can be closely packed, thus saving energy. This polyhedron was studied by Lord Kelvin and called tetrakaidekahedron (polyhedron with 14 faces). It is also called Kelvin´s polyhedron .
The polygons and polyhedra are ralely perfect in nature(e.g. crystals and cells).There is, however, a tendency to make energy and closepacking forms. Kepler called this Facultas formatrix.

The surface protein ( capsid) of virus has often the form of icosahedron (alt) and sometimes the form of rhombdodecahedron.
Pomegranate seeds with rhombdodecal form.

An enzyme with the structure of rhombdodecahedron has been found.

Radiolaria are silicous plankton of polyhedral structure.

Hexagonal cells in a wasp´s nest

Closest packing (1, 2, )  means that polyhedra fit together without intermediate space. .The truncated octahedron and rhombic dodecahedron have the smallest surface in relation to the volume among polyhedra with closest packing possibility, thus saving space and energy. Several other polyedra have the ability to to be closely packed.These forms are present in animal and plant cells, as well as in beer foam.

Spheres can be closely packed . There are 13 spheres in an icosahedron and 12 spheres in a cuboctahedron.

In the beginning of the 20th century polyhedron forms were found in atomic, molecular, cluster and crystal structures.

Molecules consisting only of carbon atoms were discovered; the first were that with 60 carbon atoms.This has the form of a truncated icosahedron with 12 pentagons and 20 hexagons, i.e the C60   fullerene(1 )  (Nobel Prize in chemistry in 1996). This fullerene is present in the outer space since more than 10 billion years. It is formed in soot  and has been found in meteorites on  earth.

     

 This Archimedean polyhedron can also be recognised in the European football (soccerball) from 1962.  Previously an other type was used. The Icosidodecahedron football is also on the market.

Zeolites ,aluminiasilicates, of polyhedral structure

 

Archaeology
In about 90 excavations in Celtic areas dodecahedra were found. They have 12 holes in the pentagones and 20 spheres on the corners. What they have been used for is unknown.
Cubes used in game of dice have been found in several countries. Icosahedra, octahedra and dodecahedra were also used.
The twelve zodiacal signs ,known in ancient Babylonia, have been found on dodecahedron.
Also truncated cuboctahedra , rhombcuboctahedra and rhombtricontahedra occur in archaeologic excavations.
Cuboctahedral weights were common in The Middle Ages. 

 

EXHIBITIONS OF CERAMIC MODELS

1990 Arts Center, University of Warwick, UK
1992 Forum,Trelleborg, Sweden
1992 Sheraton Hotel, Malmö, Sweden
1993 Tivoli, Copenhagen,Sweden
1995 University Library,Lund, Sweden
1996 Technical Museum,Malmö, Sweden
1996 Royal Society of Sciences, Stockholm,Sweden
Group exhibition (Ingmar Bergström).(Nobel prize exhibition, chemistry)
1997 International Festival of Sciences, City Museum,Gothenbourg, Sweden
Group exhibition (IB)
1998 Steno museum, Århus,Denmark, Group exhibition (IB)
1999 University Hospital, Malmö, Sweden
2000 Slide Show , Scantic Hotel Slussen , Stockholm

SELECTED REFERENCES

 

Brune´s, Tons, The Secrets of Ancient Geometry - and its use, Rhodos Copenhagen, 2 Vol. 1967

Ching,Francis D.K., Architecture, Form, Space, and order, John Wiley & Son,Inc., 1996

Cundy, H.M. & Rollet, A.P., Mathematical Models. Talquin Publ. 1951, reprint 1997.

Cromwell, Peter R. , Polyhedra, Cambridge Press, 1957

Bergström,I. & Fregert,S., On the origin of the Platonic bodies and some of their relatives, In Symmetry 2000 Part 1,

Edited by I. Hargittai and T.C. Laurent ,  Wennergren International Series, Vol. 80, p.247.

Dijksterhuis,E.J., Archimedes. Princeton University Press, 1938, reprint1987.

Gabriel, J.Francois, (ed), The CUBE. The Architecture of Space, Frames and Polyhedra. John Wiley & Sons, 1997.

Field.J.V.,Rediscovering the Archimedean Polyhedra: Pierro della Francesca, Luca Pacioli, Leonardo da Vinci, Albrecht Dürer, Daniele Barbaro, and Johannes Kepler. Archives for History of Exact Sciences,vol.49, 1996, p.241-289.

Frawley, David., The Five Elements , East and West. Chinese Culture, vol XXII,No 1, p.57. 1981

Ghyca, Matila, The Geometry of Art and Life. Dover Publ. Inc. NY. 1946 reprint 1977.

Hargittai, István , Hargittai, Magdolna, Symmetry, A Unifying Concept. Shelter Publ.,inc.,Cal. 1994.

Heath, Thomas, A History of Greek Mathematics,2 Vol. Dover Sciences Books, Dover Publ. 1921, reprint 1981.

----- Euclid. The Thirteen Books of the Elements. 1-3 Vol.,Ibid.publ., 1956, reprint

Holden, Alan, Shapes, Space and Symmetry. Dover Publ. 1991.

Lindemann,F., Zur Geschichte der Polyeder und der Zahlzeichen. Sitzungsberichte der mathematisch -
physicalischen Classe der k.b. Akademie der Wissenschaften zu München . Band XXVI, Jahrgang 1896.s 625,1897.

Kappraff, Jay, Connections. The Geometry Bridge between Art and Sciences. Mc Graw-Hill,1991

Kepler, Johannes, The Six-cornered Snowflake. Oxford at the Clarendon Press. 1966

----- Mysterium Cosmographicum, Tübingen 1595 in Caspar M. Ed. Johannes Kepler Gesammelte Werke, Beck, Munich 1938

------ Harmonices Mundi, Linz 1619. in Caspar M. Ed. Johannes Kepler Gesammelte Werke.

Lawlor, Robert, Sacred Geometry, Thames & Hudson, 1982 reprint 1994.

Plato, Timaeus and Critias, Penguin Books, 1965 reprint 1977.

Sachs, E., Die Fünf Platonischen Körper, Weidmann, Berlin, 1917, reprint Arno Press 1976, NY.

Steck, Max, Dürer´ s Gestaltlehre. Der Mathematik und der bildende Künste. Max Niemayer Verlag, Halle 1948.

Thompson, D´Arcy.W., On Groth and Form , Cambridge University Press,1942.

Waterhouse, William C., The Discovery of the Regular Solids. Arch for Hist of Exact Sci. vol 9 p.212-21, 1972/73

Wenninger, Magnus,J., Polyhedron Models, 1971

Williams, Robert,The Geometrical Foundation of Natural Structure . A Source Book of Design. 1979

 

Last modified on 2005-11-15