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4-polytope
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{{short description|Four-dimensional geometric object with flat sides}}{| align=right class=wikitable|+ Graphs of the six convex regular 4-polytopes- the content below is remote from Wikipedia
- it has been imported raw for GetWiki
Image:4-simplex t0.svg>120px5-cellPentatope4-simplex | Image:4-cube t3.svg>121px16-cellOrthoplex4-orthoplex | Image:4-cube t0.svg>120px8-cellTesseract4-cube |
Image:24-cell t0 F4.svg>120px24-cellOctaplex | Image:600-cell graph H4.svg>120px600-cellTetraplex | Image:120-cell graph H4.svg>120px120-cellDodecaplex |
,weblink 978-3-540-85977-2, BOOK
, Capecchi, V., Contucci, P., Buscema, M., D'Amore, B., Applications of Mathematics in Models, Artificial Neural Networks and Arts
, Springer, 2010, 598,weblink 10.1007/978-90-481-8581-8
, 978-90-481-8580-1, It is a connected and closed figure, composed of lower-dimensional polytopal elements: vertices, edges, faces (polygons), and cells (polyhedra). Each face is shared by exactly two cells. The 4-polytopes were discovered by the Swiss mathematician Ludwig Schläfli before 1853.{{Sfn|Coxeter|1973|p=141|loc=§7-x. Historical remarks}}
The two-dimensional analogue of a 4-polytope is a polygon, and the three-dimensional analogue is a polyhedron.Topologically 4-polytopes are closely related to the uniform honeycombs, such as the cubic honeycomb, which tessellate 3-space; similarly the 3D cube is related to the infinite 2D square tiling. Convex 4-polytopes can be cut and unfolded as nets in 3-space., Capecchi, V., Contucci, P., Buscema, M., D'Amore, B., Applications of Mathematics in Models, Artificial Neural Networks and Arts
, Springer, 2010, 598,weblink 10.1007/978-90-481-8581-8
, 978-90-481-8580-1, It is a connected and closed figure, composed of lower-dimensional polytopal elements: vertices, edges, faces (polygons), and cells (polyhedra). Each face is shared by exactly two cells. The 4-polytopes were discovered by the Swiss mathematician Ludwig Schläfli before 1853.{{Sfn|Coxeter|1973|p=141|loc=§7-x. Historical remarks}}
Definition
A 4-polytope is a closed four-dimensional figure. It comprises vertices (corner points), edges, faces and cells. A cell is the three-dimensional analogue of a face, and is therefore a polyhedron. Each face must join exactly two cells, analogous to the way in which each edge of a polyhedron joins just two faces. Like any polytope, the elements of a 4-polytope cannot be subdivided into two or more sets which are also 4-polytopes, i.e. it is not a compound.Geometry
The convex regular 4-polytopes are the four-dimensional analogues of the Platonic solids. The most familiar 4-polytope is the tesseract or hypercube, the 4D analogue of the cube.The convex regular 4-polytopes can be ordered by size as a measure of 4-dimensional content (hypervolume) for the same radius. Each greater polytope in the sequence is rounder than its predecessor, enclosing more content{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii): The sixteen regular polytopes {p,q,r} in four dimensions|ps=: [An invaluable table providing all 20 metrics of each 4-polytope in edge length units. They must be algebraically converted to compare polytopes of unit radius.]}} within the same radius. The 4-simplex (5-cell) is the limit smallest case, and the 120-cell is the largest. Complexity (as measured by comparing configuration matrices or simply the number of vertices) follows the same ordering.{{Regular convex 4-polytopes}}Visualisation{| classwikitable alignright|+ Example presentations of a 24-cell
!colspan=2|Sectioning!Net(File:24cell section anim.gif|200px) | 150px) |
100px) | 100px) | 150px) |
- Orthogonal projection
- Perspective projection
- Sectioning
- Nets
Topological characteristics
Image:Hypercube.svg|150px|thumb|The tesseract as a Schlegel diagramSchlegel diagramThe topology of any given 4-polytope is defined by its Betti numbers and torsion coefficients.Richeson, D.; Euler's Gem: The Polyhedron Formula and the Birth of Topoplogy, Princeton, 2008.The value of the Euler characteristic used to characterise polyhedra does not generalize usefully to higher dimensions, and is zero for all 4-polytopes, whatever their underlying topology. This inadequacy of the Euler characteristic to reliably distinguish between different topologies in higher dimensions led to the discovery of the more sophisticated Betti numbers.Similarly, the notion of orientability of a polyhedron is insufficient to characterise the surface twistings of toroidal 4-polytopes, and this led to the use of torsion coefficients.Classification
Criteria
Like all polytopes, 4-polytopes may be classified based on properties like "convexity" and "symmetry".- A 4-polytope is convex if its boundary (including its cells, faces and edges) does not intersect itself and the line segment joining any two points of the 4-polytope is contained in the 4-polytope or its interior; otherwise, it is non-convex. Self-intersecting 4-polytopes are also known as star 4-polytopes, from analogy with the star-like shapes of the non-convex star polygons and KeplerâPoinsot polyhedra.
- A 4-polytope is regular if it is transitive on its flags. This means that its cells are all congruent regular polyhedra, and similarly its vertex figures are congruent and of another kind of regular polyhedron.
- A convex 4-polytope is semi-regular if it has a symmetry group under which all vertices are equivalent (vertex-transitive) and its cells are regular polyhedra. The cells may be of two or more kinds, provided that they have the same kind of face. There are only 3 cases identified by Thorold Gosset in 1900: the rectified 5-cell, rectified 600-cell, and snub 24-cell.
- A 4-polytope is uniform if it has a symmetry group under which all vertices are equivalent, and its cells are uniform polyhedra. The faces of a uniform 4-polytope must be regular.
- A 4-polytope is scaliform if it is vertex-transitive, and has all equal length edges. This allows cells which are not uniform, such as the regular-faced convex Johnson solids.
- A regular 4-polytope which is also convex is said to be a convex regular 4-polytope.
- A 4-polytope is prismatic if it is the Cartesian product of two or more lower-dimensional polytopes. A prismatic 4-polytope is uniform if its factors are uniform. The hypercube is prismatic (product of two squares, or of a cube and line segment), but is considered separately because it has symmetries other than those inherited from its factors.
- A tiling or honeycomb of 3-space is the division of three-dimensional Euclidean space into a repetitive grid of polyhedral cells. Such tilings or tessellations are infinite and do not bound a "4D" volume, and are examples of infinite 4-polytopes. A uniform tiling of 3-space is one whose vertices are congruent and related by a space group and whose cells are uniform polyhedra.
Classes
The following lists the various categories of 4-polytopes classified according to the criteria above:File:Schlegel half-solid truncated 120-cell.png|150px|thumb|The truncated 120-celltruncated 120-cellUniform 4-polytope (vertex-transitive):- Convex uniform 4-polytopes (64, plus two infinite families)
- 47 non-prismatic convex uniform 4-polytope including:
- Prismatic uniform 4-polytopes:
- {} Ã {p,q} : 18 polyhedral hyperprisms (including cubic hyperprism, the regular hypercube)
- Prisms built on antiprisms (infinite family)
- {p} Ã {q} : duoprisms (infinite family)
- Non-convex uniform 4-polytopes (10 + unknown)File:Ortho solid 016-uniform polychoron p33-t0.png|150px|thumb|The great grand stellated 120-cellgreat grand stellated 120-cell
- 10 (regular) Schläfli-Hess polytopes
- 57 hyperprisms built on nonconvex uniform polyhedra
- Unknown total number of nonconvex uniform 4-polytopes: Norman Johnson and other collaborators have identified 2189 known cases (convex and star, excluding the infinite families), all constructed by vertex figures by Stella4D software.Uniform Polychora, Norman W. Johnson (Wheaton College), 1845 cases in 2005
- 28 convex uniform honeycombs: uniform convex polyhedral tessellations, including:
- 1 regular tessellation, cubic honeycomb: {4,3,4}
- 76 Wythoffian convex uniform honeycombs in hyperbolic space, including:
- 4 regular tessellation of compact hyperbolic 3-space: {3,5,3}, {4,3,5}, {5,3,4}, {5,3,5}
- 41 unique dual convex uniform 4-polytopes
- 17 unique dual convex uniform polyhedral prisms
- infinite family of dual convex uniform duoprisms (irregular tetrahedral cells)
- 27 unique convex dual uniform honeycombs, including:
- WeaireâPhelan structure periodic space-filling honeycomb with irregular cells
See also
- Regular 4-polytope
- 3-sphere â analogue of a sphere in 4-dimensional space. This is not a 4-polytope, since it is not bounded by polyhedral cells.
- The duocylinder is a figure in 4-dimensional space related to the duoprisms. It is also not a 4-polytope because its bounding volumes are not polyhedral.
References
Notes
{{Reflist}}{{notelist}}Bibliography
- H.S.M. Coxeter:
- BOOK, Coxeter, H.S.M., Harold Scott MacDonald Coxeter, 1973, 1948, Regular Polytopes, Dover, New York, 3rd, Regular Polytopes (book),
- H.S.M. Coxeter, M.S. Longuet-Higgins and J.C.P. Miller: Uniform Polyhedra, Philosophical Transactions of the Royal Society of London, Londne, 1954
- Kaleidoscopes: Selected Writings of H.S.M. Coxeter, edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication, 1995, {{ISBN|978-0-471-01003-6}} weblink
- (Paper 22) H.S.M. Coxeter, Regular and Semi Regular Polytopes I, [Math. Zeit. 46 (1940) 380â407, MR 2,10]
- (Paper 23) H.S.M. Coxeter, Regular and Semi-Regular Polytopes II, [Math. Zeit. 188 (1985) 559â591]
- (Paper 24) H.S.M. Coxeter, Regular and Semi-Regular Polytopes III, [Math. Zeit. 200 (1988) 3â45]
- J.H. Conway and M.J.T. Guy: Four-Dimensional Archimedean Polytopes, Proceedings of the Colloquium on Convexity at Copenhagen, page 38 und 39, 1965
- N.W. Johnson: The Theory of Uniform Polytopes and Honeycombs, Ph.D. Dissertation, University of Toronto, 1966
- Four-dimensional Archimedean Polytopes (German), Marco Möller, 2004 PhD dissertation weblink {{Webarchive|url=https://web.archive.org/web/20050322235615weblink |date=2005-03-22 }}
External links
{{Commons category}}- {{Mathworld | urlname=Polychoron | title=Polychoron }}
- {{Mathworld | urlname=PolyhedralFormula | title=Polyhedral formula }}
- {{Mathworld | urlname=RegularPolychoron | title=Regular polychoron Euler characteristics}}
- Uniform Polychora, Jonathan Bowers
- weblink" title="web.archive.org/web/20110718202453weblink">Uniform polychoron Viewer - Java3D Applet with sources
- R. Klitzing, polychora
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