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. A sphere can be inscribed touching the centers  of the polygons.

  

The 13 semiregular polyhedra (1, 2, 3,) 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. 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 (alt) 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

.

 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 (1, 2, 3,) were described by Archimedes (c. 257 - 212 B.C). 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.

In Mouseion in Alexandria (from c. 300 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 by Pappos in his Collectiones.

During the period of Arabic high culture (c. 700 - 1 200) the Greek geometry literature was translated to Arabic.

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. 

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”.  He also designed his own polyhedra (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)
Truncated octahedra and rhombidodecahedra have the smallest ratio surface / volume and
can be closely packed,thus saving energy.
The polygons and polyhedra are ralely perfect in nature.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 C-60 fullerene(1 )  (Nobel Prize in chemistry in 1996). This Archimedean polyhedron can also be recognised in the European football from about 1965. Before that time an other type was used.

Zeolites ,aluminiasilicates, of polyhedral structure

Art

The Renaissance artists e.g. Pierro della Fransesca ,Lorenz Stör (1, 2, 3 )Ucello, Hans Hayden ,,Lorenz Zicken, Jean Cousin (id.) Wenzeln Jamnitzer, Luca Pacioli , Leonardo da Vinci used the Greek geometry in their paintings and sculptures. Battista Alberti (1404-1472) described in Della Pittura  for the first time the mathematical construction of the perspective with a “centric point”. The first three-diemensional figures to appear were the regular polyhedra , often made in the form of intarsia (wood inlay) by intarsiatori . Giovanni da Verona  is one of the most famous.

   Dürer designed his own polyhedron in his etching Melancholia.( http://www.artglobe.se/Ghist/ghist_06.htm

 In modern art polyhedra occur in pictures and sculptures, e.g. in Dali´s The Sacrament of theLast Supper (takes place in a dodecahedron (alt), http://ellensplace.net/dali.html  Dodecahedra also occur in Dali´s Searching for the Fourth Dimension ( 1979), in The Sacrament of the Last Supper,  in Roch and Infuriated Horse Sleeping under the See (1947) ,  in Pentagonale Sardana (1979) and in some illustrations in Esseys of Michel de Montaigne.

 Several Russian artists, Malevitch and Rodtjenko were familiar with polyhedron and polygon structures (c. 1920 -1930).

 M. C. Escher often used polyhedra in his etchings, e.g. the dodecahedron and the small stellated dodecahedron. Stars, wood engraving 1948, stellated polyhedron.

.
Goethe brought an ancient sculpture from Italy
Islamic art is often geometric.
A Belgian note with platonic solids
The Golden triangle in a pentagon.

Even today several artists construct polygons and platonic bodies (alt))as well as Archimedean polyhedra ,e.g.
Collection of Dodecahedra
K G Nilson : Red Score
Pål Svensson Platonic solids
Per Svensson: : Platonic solids and Truncated icosahedron (id.)
Lennart Mörk: Platonic solids
Legotype of Bauhaus

In architecture, particularly in the USA, polyhedra occur in constructions.

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

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 2001-11-11 25 apr 2001