Editing Regular polytope (section)

===Higher-dimensional polytopes===

[[Image:8-cell-simple.gif|right|thumb|A 3D projection of a rotating tesseract. This tesseract is initially oriented so that all edges are parallel to one of the four coordinate space axes. The rotation takes place in the xw plane.]]
It was not until the 19th century that a Swiss mathematician, [[Ludwig Schläfli]], examined and characterised the regular polytopes in higher dimensions. His efforts were first published in full in {{Harvtxt|Schläfli|1901}}, six years posthumously, although parts of it were published in {{Harvtxt|Schläfli|1855}} and {{Harvtxt|Schläfli|1858}}. Between 1880 and 1900, Schläfli's results were rediscovered independently by at least nine other mathematicians &mdash; see {{harvtxt|Coxeter|1973|pp=143&ndash;144}} for more details. Schläfli called such a figure a "polyschem" (in English, "polyscheme" or "polyschema"). The term "polytope" was introduced by [[Reinhold Hoppe]], one of Schläfli's rediscoverers, in 1882, and first used in English by [[Alicia Boole Stott]] some twenty years later. The term "polyhedroids" was also used in earlier literature (Hilbert, 1952).

{{Harvtxt|Coxeter|1973}} is probably the most comprehensive printed treatment of Schläfli's and similar results to date. Schläfli showed that there are six [[convex regular 4-polytope|regular convex polytopes in 4 dimensions]]. Five of them can be seen as analogous to the Platonic solids: the [[4-simplex]] (or pentachoron) to the [[tetrahedron]], the [[4-hypercube]] (or 8-cell or [[tesseract]]) to the [[cube]], the [[4-orthoplex]] (or hexadecachoron or [[16-cell]]) to the [[octahedron]], the [[120-cell]] to the [[dodecahedron]], and the [[600-cell]] to the [[icosahedron]].  The sixth, the [[24-cell]], can be seen as a transitional form between the 4-hypercube and 16-cell, analogous to the way that the [[cuboctahedron]] and the [[rhombic dodecahedron]] are transitional forms between the cube and the octahedron.

Also of interest are the star [[regular 4-polytope]]s, partially discovered by Schläfli. By the end of the 19th century, mathematicians such as [[Arthur Cayley]] and [[Ludwig Schläfli]] had developed the theory of regular polytopes in four and higher dimensions, such as the [[tesseract]] and the [[24-cell]].

The latter are difficult (though not impossible) to visualise through a process of [[Four-dimensional space#Dimensional analogy|dimensional analogy]], since they retain the familiar symmetry of their lower-dimensional analogues. The [[tesseract]] contains 8 cubical cells. It consists of two cubes in parallel hyperplanes with corresponding vertices cross-connected in such a way that the 8 cross-edges are equal in length and orthogonal to the 12+12 edges situated on each cube. The corresponding faces of the two cubes are connected to form the remaining 6 cubical faces of the tesseract. The [[24-cell]] can be derived from the tesseract by joining the 8 vertices of each of its cubical faces to an additional vertex to form the four-dimensional analogue of a pyramid. Both figures, as well as other 4-dimensional figures, can be directly visualised and depicted using 4-dimensional stereographs.<ref name="Brisson">{{cite book |editor-first=David W. |editor-last=Brisson |first=David W. |last=Brisson |chapter= Visual Comprehension in n-Dimensions |title=Hypergraphics: Visualizing Complex Relationships In Arts, Science, And Technololgy |chapter-url=https://books.google.com/books?id=zcPADwAAQBAJ&pg=PA109 |date=2019 |publisher=Taylor & Francis |isbn=978-0-429-70681-3 |pages=109–145 |series=AAAS Selected Symposium |volume=24 |orig-year=1978}}</ref>

Harder still to imagine are the more modern [[abstract regular polytope]]s such as the [[57-cell]] or the [[11-cell]]. From the mathematical point of view, however, these objects have the same aesthetic qualities as their more familiar two and three-dimensional relatives.

In five and more dimensions, there are exactly three finite regular polytopes, which correspond to the tetrahedron, cube and octahedron: these are the [[#Regular simplices|regular simplices]], [[#Measure polytopes (hypercubes)|measure polytopes]] and [[#Cross polytopes (orthoplexes)|cross polytopes]]. Descriptions of these may be found in the [[list of regular polytopes]].