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Regular and semi-regular convex polytopes a short historical overview:

Phytagoras born about 569 BC in Samos, Ionia Greece, died about 475 BC.

Although early findings acknowledged by mathematicians and historians date back before the time of Phytagoras like the Babylonians who were aquainted with the famous Pythagoras's theorem c^2=a^2+b^2 as early as 3750 BC, this was not discoverd until 1962. Some of the first basic geometric theorems are credited to Phytagoras. Phytagoras is often called the first pure mathematician; he founded a school "the semicircle" and many pupils elaborated on his findings and thoughts.
Besides his famous theorem some basic polygon theorems are credited to Phytagoras and his pupils:
A polygon with n sides has sum of interior angles 2*n - 4 right angles (90 degrees) and sum of exterior angles equal to four right angles (360 degrees). This was later described in more detail by Euclid.

Although it is unknown who first brought up the idea, followers of Phytagoras, who were familiar with rough estimates of pi, later calculated in more detail by Archimedes, must have acknowledged that there are an INFINITE number of regular polygons, and a regular polygon with an infinite number of vertices equals a perfect circle. These polygons are now also called regular convex polytopes in 2D having Schläfli notation {p}.

Plato, lived from 427 BC-347 BC in Athens, Greece.

In about 387 BC Plato founded a school in Athens "the Academy". In his work "the Timaeus" there is a mathematical construction of the elements, in which the cube, tetrahedron, octahedron, and icosahedron are given as the shapes of the atoms of earth, fire, air, and water. The fifth Platonic solid, the dodecahedron, is Plato's model for the whole universe. The original work was lost but was described later by Euclid.

Euclid, lived about 325 BC-265 BC died in Alexandria, Egypt.

In 300 B.C. Euclid, student of the school of Plato, proved in his manuscript "The Elements XII" that in 3 dimensions there are just 5 regular bodies called polyhedra with the properties of having faces made out of regular polygons all being convex.
Euclid also mentions that although the cube, the tetrahedron (called pyramid) and the dodecahedron are credited to the Pytagoreans (scholars of phytagoras), the octahedron and icosahedron are due to Theaetetus of Athens, a friend of Plato.
Since Plato first described the complete group of the 5 cosmic figures, Euclid attached Plato's name to the solids. In his eleborate work the concept of 3 dimensional rectangular space, which is now called Euclidian space, was introduced for the first time as well. It is disputed whether Euclid wrote "The Elements" or his students or a group of mathematicians calling themselves Euclid did.

Main findings Plato/Euclid:
(In 3 dimensions), there are exactly FIVE regular convex polytopes called polyhedra:

• The tetrahedron, with 4 triangular faces, 6 edges and 4 vertices, (Schläfli notation {3,3}), which has, proven later, a so called a "self-dual" property which means when assuming an imaginary vertex on the centre of each face, these vertices will form the same body, also the number of faces, edges and vertices and Schläfli notation can be mirrored.
• The cube or hexahedron, with 6 square faces, 12 edges and 8 vertices, (Schläfli notation {4,3}), which is, proven later, a dual of the octahedron i.e. when assuming an imaginary vertex on the centre of each face, these vertices will form an octahedron, also the number of faces, edges and vertices and Schläfli notation can be mirrored to satisfy the respective notation of an octahedron.
• The octahedron, with 8 triangular faces, 12 edges and 6 vertices, (Schläfli notation {3,4}), which is, proven later, a dual of the cube.
• The dodecahedron, with 12 pentagonal faces, 30 edges and 20 vertices, (Schläfli notation {5,3}), which is, proven later, a dual of the icosahedron.)
• The icosahedron, with 20 triangular faces, 30 edges and 12 vertices, (Schläfli notation {3,5}), which is, proven later ,a dual of the dodecahedron.
  

The final proposition of the Elements states: "No other figure, besides the said five figures, can be constructed which is contained by equilateral and equiangular figures equal to one another".
The above proposition is false however. A so-called group of 8 deltahedra exist all being constructed from equilateral triangles.
Furthermore a number of star polyhedra and certain honeycomb structures found by J.F. Petrie which are not exluded by the proposition exist as well.
Strictly taken, regular figures beyond 3 dimensions are not excluded either, but when the proposition is seen in the context of it's time this may seem far fetched.
Modern definitions restrict the vertices of the 5 Platonic solids to lay on a three dimesional sphere, excluding all the others.

Besides the famous legendary cry of "Eureka", when introducing the theory of buoyancy, some important findings on mathematic s and geometry are contributed to Archimedes.
He calculated the first accurate estimation of pi being 223/71 < pi < 22/7 acquired through the method of exhaustion, which obtains 3.1418, an error of about +/-0.0002 * pi, far more precise than the ancient Egyptians who used a value of 3 1/8 or 3.125.
By introducing calculus for the very first time, later 'reïnvented' by Newton, Archimedes proved in his book "the method" that the area and volume of a sphere are in the same ratio (2/3) to the area and volume of a circumscribed straight cylinder, a result he was so proud of that he instructed to have it inscripted on his tombstone. The original manuscript and the tombstone were lost, but a copy of the manuscript reemerged in a suprising way at a Christie's auction in 1998, and is still being decyphered today because of it's poor condition.
The original manuscripts dealing with polyhedra were also lost, but Pappus of Alexandria writes in the 4th century AD in Book V, about Archimedes' finding that (in three dimensions), there are exactly THIRTEEN semi-regular convex polytopes:

The first 5 are obtained when truncating the Platonic solids:

• The truncated tetrahedron, with 8 faces of 4 triangles and 4 hexagons, 12 vertices and 18 edges. (3,6,6)
• The truncated octahedron, with 14 faces of 6 squares and 8 hexagons, 24 vertices and 36 edges. (4,6,6)
  
• The truncated cube, with 14 faces of 8 triangles and 6 squares, 24 vertices and 36 edges. (3,8,8)
  
• The truncated icosahedron, with 32 faces of 12 pentagons and 20 hexagons, 60 vertices and 90 edges. (5,6,6)
• The truncated dodecahedron, with 32 faces of 20 triangles and 12 decagons, 60 vertices and 90 edges. (3,10,10)

If you divide the edges of the cube or octahedron in half and truncate and likewise for the dodecahedron or icosahedron the following solids are formed:

• The cuboctahedron, with 14 faces of 8 triangles and 6 squares, 12 vertices and 24 edges. (3,4,3,4)
• The icosidodecahedron, with 32 faces of 20 triangles and 12 pentagons, 30 vertices and 60 edges. (3,5,3,5)

Truncating the edges and vertices of the above two will give:

• The great rhombicuboctahedron, with 26 faces of 18 triangles and 8 square faces, 24 vertices and 48 edges. (4,6,8)
  
• The great rhombicosidodecahedron, with 62 faces of 20 triangles, 30 squares and 12 pentagons, 60 vertices and 120 edges (4,6,10)

Truncating the edges and vertices of the cube or octahedron and icosahedron or dodecahedron will give:

• The small rhombicuboctahedron, with 26 faces of 12 squares, 8 hexagons and 6 octagons, 48 vertices and 72 edges. (3,4,4,4)
• The small icosidodecahedron, with 62 faces of 30 squares, 20 hexagons and 12 octagons, 120 vertices and 180 edges. (3,4,5,4)

The last 2 solids are obtained by moving the faces of a cube and dodecahedron outward while giving each face a twist. The resulting spaces are then filled with ribbons of equilateral triangles. These two are not mirror symmetric and are so-called chiral:

• The snub cube, with 38 faces of 32 triangles and 6 squares, 24 vertices and 60 edges. (3,3,3,3,4)
• The snub dodecahedron, with 92 faces of 80 triangles and 12 pentagons, 60 vertices and 150 edges. (3,3,3,3,5)

In book V Pappus writes: "Although many solid figures having all kinds of faces can be conceived, those which appear to be regularly formed are most deserving of attention. These not only include the five figures found in the godlike Plato...but also the solids thirteen in number, which were discovered by Archimedes and are contained by equilateral and equiangular, but not similar polygons."

Again the above definition is incomplete; there are about 75 convex polyhedra whose faces are regular polygons of more than one kind. Furthermore in three-dimensions there are an INFINITE number of prisms and anti-prisms which also satisfy the definition.
This has lead to diffferent schools of thought as to what are Archimedean solids and what not.

Analogue to the two-dimensional theorhem of Pythagoras i.e. "A polygon with n sides has sum of interior angles 2*n - 4 right angles (of 90 degrees) and sum of exterior angles equal to four right angles (360 degrees), in his "De Solidorum Elementis" Descartes theorem states:
"The sum of deficiences of the solid angles in a polyhedron is eight right angles",
which means (2*pi -ó)*V=4*pi, where ó is the sum of face-angles at a vertex and 2*pi-ó is the deficiency or external steradial angle of each vertex and V is the number of vertices.
Descartes gave no prove for his theorem, proof was given later by Steinitz and Rademacher in 1934 and Ball and Coxeter in 1987.
Another theorem states that for regular convex polyhedra (2*V-4)/F and (2*F-4)/V must be integers where V is the no. of vertices and F the no. of faces, this only possible for V=4, 6, 8, 12 or 20 and F= 4, 8, 6, 20 or 12, which is the respective vertex and face count of the 5 Platonic solids.

Leonhard Euler lived from 15 April 1707 in Basel, Switzerland, died 18 Sept 1783 in St Petersburg, Russia.
Euler discovered in 1750 what is now called the Euler polyhedral theorem, which states that for all polyhedra the number of Faces-Edges+Vertices = 2.
Euler himself was surprised that this obvious characteristics had gone unnoticed for two millenea and writes in a letter revealing the formula to his friend Christian Goldback in November of 1750; "I find it surpising that these general result in solid geometry have not been noticed by anyone, so far as i am aware.."
He was not able to give satisfactory proof for his formula, the first generally accepted proof was given by Adrien-Marie Legendre in 1794.

Ludwig Schläfli lived from 15 Jan 1814 till 20 March 1895 in Grasswil, Bern, Switzerland.
Although independently English recreational mathematician Alicia Boole Stott daughter of George Boole experimentally found similar results which were published with help of the Dutch mathematician Pieter Hendrik Schoute in 1900 and 1910, Swiss mathematician Ludwig Schläfli proved in 1852 in his manuscript "Theorie der vielfachen Kontinuität" that besides the 5 Platonic solids there are excactly 6 regular bodies with Platonic properties in 4 dimensions and only 3 in 5 dimensions and 3 in all other higher dimensions.
His work which contained no illustrations remained practically unknown for a considerable time. In 1858 and 1860 his paper was partly published in English by Arthur Cayley. Other mathematicians like W.I. Stringham discovered similar results independently in 1880. Schläfli's work was published in it's entirety in 1901, six years after his dead.
Schläfli also devised a notation still in use today; a regular convex n-gon has Schläfli symbol {n}, a polytope like the 4 dimensional hypercube is denoted as {4,3,3} i.e. it consist of 8 cubes {4,3} 3 of which are meeting at each vertex, a cube {4,3} consists out of 6 squares {4} 3 of which are meeting at each vertex, a square {4} has 4 vertices.
Furthermore a polyhedron with notation {p,q} has 4*p/z vertices, 2*p*q/z edges and 4*q/z faces, where z=4-(p-2)*(q-2).

Harold Scott MacDonald Coxeter born 9 Feb 1907 in London, England died 31 March 2003 in Toronto, Canada.
Coxeter has made contributions of major importance in the theory of polytopes, non-Euclidean geometry, group theory and combinatorics. In 1936 he joined the Faculty of the University of Toronto and remained active there till his death.
In his famous books Regular polytopes, 3d ed. New York, Dover Publications, 1973 and Regular and Semi-Regular Polytopes Math. Z. 46, 1940 Coxeter has made a solid foundation for geometry in hyperspace. Furthermore he popularised the term "polytope", which was used by his friend Alicia Boole Stott.
There is still an outdated homepage of Coxeter available at the department of mathematics of the university of Toronto. A more modern website including a short video is available at science.ca.

Main findings Schläfli generalised by Coxeter:
Regular convex polytopes {p,q,r} should satisfy cos(pi/q)<sin(pi/p)*sin(pi/r) which leads to the conclusion that in 4 dimensions, there are exactly SIX regular convex polytopes:

• The hypertetrahedron, 'pentatope', '4-simplex', '5-cell' or 'pentachoron' - with 5 tetrahedral cells, 10 triangular faces, 10 edges and 5 vertices, Schläfli notation {3,3,3}, which is self-dual.
• The hypercube, 'tesseract' or 'octachoron' - with 8 cubical cells, 24 square faces, 32 edges and 16 vertices, Schläfli notation {4,3,3}, which is a dual of the hyperoctahedron.
• The hyperoctahedron or '4D-dimensional cross-polytope', '16-cell' or 'hexadecachoron' - with 16 tetrahedral cells, 32 triangular faces, 24 edges and 8 vertices, Schläfli notation {3,3,4}, which is a dual of the hypercube.
• The hyperdodecahedron, '4D-dodecahedron', '120-cell' or 'hecatonicosachoron' - with 120 dodecahedral cells, 720 pentagonal faces, 1200 edges and 600 vertices, Schläfli notation {5,3,3} which is a dual of the hypericosahedron.
• The hypericosahedron, '4D-icosahedron','600-cell' or 'hexacosichoron' - with 600 tetrahedral cells, 1200 triangular faces, 720 edges and 120 vertices, Schläfli notation{3,3,5}, which is a dual of the hyperdodecahedron.
• The '24-cell' or 'icositetrachoron' - with 24 octahedral cells, 96 triangular faces, 96 edges and 24 vertices, Schläfli notation {3,4,3}, which is self-dual and is unique in the sence that it has no direct three-dimensional analog.

In 5 or more dimensions, there exist only THREE regular convex polytopes:

• The hyper-n-tetrahedron, called the 'n-simplex', with n+1 cells each of which is an (n-1) simplex, n*(n + 1) / 2 edges and n+1 vertices, Schläfli notation {3,3,3,...,3}, all are self-dual.
• The hyper-n-cube, called the 'n-cube', with 2*n cells all of which are (n-1)-cubes with 2*n*(n - 1) square faces, n*2^(n-1) edges and 2^n vertices, Schläfli notation {4,3,3,..,3}, being a dual of the n-dimensional cross-polytope.
• The hyper-n-octahedron, called the 'n-dimensional cross-polytope', with 2^ⁿ cells, all of which are (n-1)-simplices with n*2^(n - 1) triangular faces, 2*n*(n - 1) edges and 2*n vertices, Schläfli notation {3,3,3,..,4}, being a dual of the n-cube.

John Horton Conway, born 26 Dec 1937 in Liverpool, England
John Conway and Michael Guy discovered in 1965, that one there is one anomalous, entirely non-Wythoffian antiprismatic convex uniform 4 dimensional polytope which also needed to be added to complete the collection of semi-regular convex 4D polytopes.
Using a computer search, they proved that the set is indeed complete, and published the result in a short note titled “Four-dimensional Archimedean Polytopes” in the proceedings of a colloquium on convexity at Copenhagen.
The full set of 64 polytopes or polychora (including the 6 4D Platonic solids and the ones built from prisms) can be found at the website of George Olshevsky.
The complete list of Archimedean polytopes in 5D or higher is to this date not fully known although various have been described or calculated or can be easily derived from it's Wythoff symbol.

References:
The MacTutor History of Mathematics archive http://turnbull.dcs.st-and.ac.uk/~history/
Eric W. Weisstein http://mathworld.wolfram.com/PlatonicSolid.html © 1999-2002 Wolfram Research, Inc.
Pi history http://www-groups.dcs.st-and.ac.uk/~history/HistTopics/Pi_through_the_ages.html
Polyhedra, Peter R. Cromwel, Cambridge university press, 1997.
The geometry junkyard, Seventeen Proofs of Euler's Formula: V-E+F=2 http://www.ics.uci.edu/~eppstein/junkyard/euler/
Coxeter, H. S. M. Regular Polytopes, 3rd ed. New York: Dover, 1973.
The story of the 120-cell by John Stillwell http://www.ams.org/notices/200101/fea-stillwell.pdf
Archimedes - Wikipedia http://en.wikipedia.org/wiki/Archimedes
ARCHIMEDES ANCIENT MANUSCRIPT http://www.omogenia.com/arch.htm

2004, Symen H. Hovinga

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