4-polytope




























Graphs of six convex regular 4-polytopes
{3,3,3}
{3,3,4}
{4,3,3}

4-simplex t0.svg
5-cell
Pentatope
4-simplex

4-cube t3.svg
16-cell
Orthoplex
4-orthoplex

4-cube t0.svg
8-cell
Tesseract
4-cube
{3,4,3}
{5,3,3}
{3,3,5}

24-cell t0 F4.svg
Octaplex
24-cell

120-cell graph H4.svg
Dodecaplex
120-cell

600-cell graph H4.svg
Tetraplex
600-cell

In geometry, a 4-polytope (sometimes also called a polychoron[1], polycell, or polyhedroid) is a four-dimensional polytope.[2][3] 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 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.




Contents






  • 1 Definition


  • 2 Visualisation


  • 3 Topological characteristics


  • 4 Classification


    • 4.1 Criteria


    • 4.2 Classes




  • 5 See also


  • 6 References


    • 6.1 Notes


    • 6.2 Bibliography




  • 7 External links





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.


The most familiar 4-polytope is the tesseract or hypercube, the 4D analogue of the cube.




Visualisation

























Example presentations of a 24-cell
Sectioning

Net

24cell section anim.gif

Polychoron 24-cell net.png
Projections

Schlegel
2D orthogonal
3D orthogonal

Schlegel wireframe 24-cell.png

24-cell t0 F4.svg

Orthogonal projection envelopes 24-cell.png

4-polytopes cannot be seen in three-dimensional space due to their extra dimension. Several techniques are used to help visualise them.


Orthogonal projection

Orthogonal projections can be used to show various symmetry orientations of a 4-polytope. They can be drawn in 2D as vertex-edge graphs, and can be shown in 3D with solid faces as visible projective envelopes.


Perspective projection

Just as a 3D shape can be projected onto a flat sheet, so a 4-D shape can be projected onto 3-space or even onto a flat sheet. One common projection is a Schlegel diagram which uses stereographic projection of points on the surface of a 3-sphere into three dimensions, connected by straight edges, faces, and cells drawn in 3-space.


Sectioning

Just as a slice through a polyhedron reveals a cut surface, so a slice through a 4-polytope reveals a cut "hypersurface" in three dimensions. A sequence of such sections can be used to build up an understanding of the overall shape. The extra dimension can be equated with time to produce a smooth animation of these cross sections.


Nets

A net of a 4-polytope is composed of polyhedral cells that are connected by their faces and all occupy the same three-dimensional space, just as the polygon faces of a net of a polyhedron are connected by their edges and all occupy the same plane.



Topological characteristics




The tesseract as a Schlegel diagram


The topology of any given 4-polytope is defined by its Betti numbers and torsion coefficients.[4]


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.[4]


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.[4]



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 Kepler–Poinsot 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:




The truncated 120-cell is one of 47 convex non-prismatic uniform 4-polytopes


Uniform 4-polytope (vertex-transitive):




  • Convex uniform 4-polytopes (64, plus two infinite families)

    • 47 non-prismatic convex uniform 4-polytope including:
      • 6 Convex regular 4-polytope



    • Prismatic uniform 4-polytopes:

      • {} × {p,q} : 18 polyhedral hyperprisms (including cubic hyperprism, the regular hypercube)

      • Prisms built on antiprisms (infinite family)

      • {p} × {q} : duoprisms (infinite family)






  • Non-convex uniform 4-polytopes (10 + unknown)


    The great grand stellated 120-cell is the largest of 10 regular star 4-polytopes, having 600 vertices.



    • 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 1849 known cases (convex and star), all constructed by vertex figures by Stella4D software.[5]




Other convex 4-polytopes:



  • Polyhedral pyramid

  • Polyhedral prism




The regular cubic honeycomb is the only infinite regular 4-polytope in Euclidean 3-dimensional space.


Infinite uniform 4-polytopes of Euclidean 3-space (uniform tessellations of convex uniform cells)


  • 28 convex uniform honeycombs: uniform convex polyhedral tessellations, including:
    • 1 regular tessellation, cubic honeycomb: {4,3,4}


Infinite uniform 4-polytopes of hyperbolic 3-space (uniform tessellations of convex uniform cells)


  • 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}


Dual uniform 4-polytope (cell-transitive):



  • 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:

    • Rhombic dodecahedral honeycomb

    • Disphenoid tetrahedral honeycomb




Others:



  • Weaire–Phelan structure periodic space-filling honeycomb with irregular cells



The 11-cell is an abstract regular 4-polytope, existing in the real projective plane, it can be seen by presenting its 11 hemi-icosahedral vertices and cells by index and color.


Abstract regular 4-polytopes:



  • 11-cell

  • 57-cell


These categories include only the 4-polytopes that exhibit a high degree of symmetry. Many other 4-polytopes are possible, but they have not been studied as extensively as the ones included in these categories.



See also



  • Regular 4-polytope

  • The 3-sphere (or glome) is another commonly discussed figure that resides 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





  1. ^ N.W. Johnson: Geometries and Transformations, (2018) .mw-parser-output cite.citation{font-style:inherit}.mw-parser-output q{quotes:"""""""'""'"}.mw-parser-output code.cs1-code{color:inherit;background:inherit;border:inherit;padding:inherit}.mw-parser-output .cs1-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/thumb/6/65/Lock-green.svg/9px-Lock-green.svg.png")no-repeat;background-position:right .1em center}.mw-parser-output .cs1-lock-limited a,.mw-parser-output .cs1-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/thumb/d/d6/Lock-gray-alt-2.svg/9px-Lock-gray-alt-2.svg.png")no-repeat;background-position:right .1em center}.mw-parser-output .cs1-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/thumb/a/aa/Lock-red-alt-2.svg/9px-Lock-red-alt-2.svg.png")no-repeat;background-position:right .1em center}.mw-parser-output .cs1-subscription,.mw-parser-output .cs1-registration{color:#555}.mw-parser-output .cs1-subscription span,.mw-parser-output .cs1-registration span{border-bottom:1px dotted;cursor:help}.mw-parser-output .cs1-hidden-error{display:none;font-size:100%}.mw-parser-output .cs1-visible-error{font-size:100%}.mw-parser-output .cs1-subscription,.mw-parser-output .cs1-registration,.mw-parser-output .cs1-format{font-size:95%}.mw-parser-output .cs1-kern-left,.mw-parser-output .cs1-kern-wl-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right,.mw-parser-output .cs1-kern-wl-right{padding-right:0.2em}
    ISBN 978-1-107-10340-5 Chapter 11: Finite Symmetry Groups, 11.1 Polytopes and Honeycombs, p.224



  2. ^ Vialar, T. (2009). Complex and Chaotic Nonlinear Dynamics: Advances in Economics and Finance. Springer. p. 674. ISBN 978-3-540-85977-2.


  3. ^ Capecchi, V.; Contucci, P.; Buscema, M.; D'Amore, B. (2010). Applications of Mathematics in Models, Artificial Neural Networks and Arts. Springer. p. 598. doi:10.1007/978-90-481-8581-8. ISBN 978-90-481-8580-1.


  4. ^ abc Richeson, D.; Euler's Gem: The Polyhedron Formula and the Birth of Topoplogy, Princeton, 2008.


  5. ^ Uniform Polychora, Norman W. Johnson (Wheaton College), 1845 cases in 2005




Bibliography




  • H.S.M. Coxeter:

    • 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

    • H.S.M. Coxeter, Regular Polytopes, 3rd Edition, Dover New York, 1973




  • 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 [1]

    • (Paper 22) H.S.M. Coxeter, Regular and Semi Regular Polytopes I, [Math. Zeit. 46 (1940) 380–407, MR 2,10]

    • (Paper 23) H.S.M. Coxeter, Regular and Semi-Regular Polytopes II, [Math. Zeit. 188 (1985) 559–591]

    • (Paper 24) H.S.M. Coxeter, Regular and Semi-Regular Polytopes III, [Math. Zeit. 200 (1988) 3–45]




  • 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 [2]



External links







  • Weisstein, Eric W. "Polychoron". MathWorld.

  • Weisstein, Eric W. "Polyhedral formula". MathWorld.

  • Weisstein, Eric W. "Regular polychoron Euler characteristics". MathWorld.


  • Four dimensional figures page, George Olshevsky.


  • Olshevsky, George. "Polychoron". Glossary for Hyperspace. Archived from the original on 4 February 2007.


  • Uniform Polychora, Jonathan Bowers

  • Uniform polychoron Viewer - Java3D Applet with sources

  • Dr. R. Klitzing, polychora































































































Fundamental convex regular and uniform polytopes in dimensions 2–10


Family

An

Bn

I2(p) / Dn

E6 / E7 / E8 / F4 / G2

Hn

Regular polygon

Triangle

Square

p-gon

Hexagon

Pentagon

Uniform polyhedron

Tetrahedron

Octahedron • Cube

Demicube


Dodecahedron • Icosahedron

Uniform 4-polytope

5-cell

16-cell • Tesseract

Demitesseract

24-cell

120-cell • 600-cell

Uniform 5-polytope

5-simplex

5-orthoplex • 5-cube

5-demicube



Uniform 6-polytope

6-simplex

6-orthoplex • 6-cube

6-demicube

122 • 221


Uniform 7-polytope

7-simplex

7-orthoplex • 7-cube

7-demicube

132 • 231 • 321


Uniform 8-polytope

8-simplex

8-orthoplex • 8-cube

8-demicube

142 • 241 • 421


Uniform 9-polytope

9-simplex

9-orthoplex • 9-cube

9-demicube



Uniform 10-polytope

10-simplex

10-orthoplex • 10-cube

10-demicube


Uniform n-polytope

n-simplex

n-orthoplex • n-cube

n-demicube

1k2 • 2k1 • k21

n-pentagonal polytope
Topics: Polytope families • Regular polytope • List of regular polytopes and compounds



這個網誌中的熱門文章

12.7 cm/40 Type 89 naval gun

Rikitea

University of Vienna