POLYHEDRA
Polyhedra
· 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)
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
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.
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
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