Also the great 120-cell {5,5/2,5} and grand stellated 120-cell {5/2,5,5/2}. ==History== Polygons and polyhedra have been known since ancient times. An early hint of higher dimensions came in 1827 when August Ferdinand Möbius discovered that two mirror-image solids can be superimposed by rotating one of them through a fourth mathematical dimension.

By the 1850s, a handful of other mathematicians such as Arthur Cayley and Hermann Grassmann had also considered higher dimensions. Ludwig Schläfli was the first to consider analogues of polygons and polyhedra in these higher spaces.

He described the six convex regular 4-polytopes in 1852 but his work was not published until 1901, six years after his death.

By 1854, Bernhard Riemann's Habilitationsschrift had firmly established the geometry of higher dimensions, and thus the concept of n-dimensional polytopes was made acceptable.

Schläfli's polytopes were rediscovered many times in the following decades, even during his lifetime. In 1882 Reinhold Hoppe, writing in German, coined the word polytop to refer to this more general concept of polygons and polyhedra.

In due course Alicia Boole Stott, daughter of logician George Boole, introduced the anglicised polytope into the English language. In 1895, Thorold Gosset not only rediscovered Schläfli's regular polytopes but also investigated the ideas of semiregular polytopes and space-filling tessellations in higher dimensions.

He described the six convex regular 4-polytopes in 1852 but his work was not published until 1901, six years after his death.

Polytopes also began to be studied in non-Euclidean spaces such as hyperbolic space. An important milestone was reached in 1948 with H.

Branko Grünbaum published his influential work on Convex Polytopes in 1967. In 1952 Geoffrey Colin Shephard generalised the idea as complex polytopes in complex space, where each real dimension has an imaginary one associated with it.

Branko Grünbaum published his influential work on Convex Polytopes in 1967. In 1952 Geoffrey Colin Shephard generalised the idea as complex polytopes in complex space, where each real dimension has an imaginary one associated with it.

Peter McMullen and Egon Schulte published their book Abstract Regular Polytopes in 2002. Enumerating the uniform polytopes, convex and nonconvex, in four or more dimensions remains an outstanding problem. In modern times, polytopes and related concepts have found many important applications in fields as diverse as computer graphics, optimization, search engines, cosmology, quantum mechanics and numerous other fields.

In 2013 the amplituhedron was discovered as a simplifying construct in certain calculations of theoretical physics. ==Applications== In the field of optimization, linear programming studies the maxima and minima of linear functions; these maxima and minima occur on the boundary of an n-dimensional polytope.

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