## 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

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

**A****rchaeology****
**In about 90 excavations in

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

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

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Last modified on 2001-11-11 25 apr 2001