Призма (геометријска фигура) — разлика између измена

Set of uniform $n$ -gonal prisms
Set of uniform n-gonal prisms
	Example uniform hexagonal prism
Type	uniform in the sense of semiregular polyhedron
Стране	2 n-gonal regular polygons; n squares
Ивице	3n
Темена	2n
Конфигурација темена	4.4.n
Шлафлијев симбол	{n}×{} ; t{2, n}
Конвејова нотација	Pn
Кокстеров дијаграм
Група симетрије	Dnh, [n,2], (*n22), order 4n
Група ротације	Dn, [n,2]+, (n22), order 2n
Дуални полиедар	convex dual-uniform n-gonal bipyramid
Својства	convex, regular polygon faces, vertex-transitive, translated bases, sides ⊥ bases
Мрежа

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Верзија на датум 20. јул 2022. у 20:38

Призма је геометријски полиедар ограничен са две паралелне подударне основе (основа може бити било који многоугао) које су повезане паралелограмима (бочним странама). У зависности од многоугла у основи, призма може бити троугаона, четвороугаона, петоугаона итд. All cross-sections parallel to the bases are translations of the bases. Prisms are named after their bases, e.g. a prism with a pentagonal base is called a pentagonal prism. Prisms are a subclass of prismatoids.

Like many basic geometric terms, the word prism (од грчки πρίσμα (prisma), са значењем „something sawed”) was first used in Euclid's Elements. Euclid defined the term in Book XI as “a solid figure contained by two opposite, equal and parallel planes, while the rest are parallelograms”. However, this definition has been criticized for not being specific enough in relation to the nature of the bases, which caused confusion among later geometry writers.^[2]^[3]

Right prism, uniform prism

Right prism

A right prism is a prism in which the joining edges and faces are perpendicular to the base faces.^[4] This applies if all the joining faces are rectangular.

The dual of a right n-prism is a right n-bipyramid.

A right prism (with rectangular sides) with regular n-gon bases has Schläfli symbol { }×{n}. It approaches a cylindrical solid as n approaches infinity.

Special cases

A right rectangular prism (with a rectangular base) is also called a cuboid, or informally a rectangular box. A right rectangular prism has Schläfli symbol { }×{ }×{ }.

A right square prism (with a square base) is also called a square cuboid, or informally a square box.

Note: some texts may apply the term rectangular prism or square prism to both a right rectangular-based prism and a right square-based prism.

Uniform prism

A uniform prism or semiregular prism is a right prism with regular bases and square sides, since such prisms are in the set of uniform polyhedra.

A uniform n-gonal prism has Schläfli symbol t{2,n}.

Right prisms with regular bases and equal edge lengths form one of the two infinite series of semiregular polyhedra, the other series being antiprisms.

Шаблон:UniformPrisms

Подела

Праве и косе призме
Тростране, четворостране, петостране ...
Правилне и неправилне

Правилна призма је она призма која у основи има правилан многоугао (троугао, четвороугао, петоугао итд.)

Делови призме

Права
Основа (Basis)
Бочна страна
Основна ивица

Површина призме

Површина призме је збир површина свих страна призме. Најједноставније је израчунати површину бочних страна -М- призме и саберемо са две основе.

2B+Ph

where B is the area of the base, h the height, and P the base perimeter.

The surface area of a right prism whose base is a regular n-sided polygon with side length s and height h is therefore:

A={\frac {n}{2}}s^{2}\cot {\left({\frac {\pi }{n}}\right)}+nsh

Запремина призме

Запремину призме израчунавамо тако што површину основе призме помножимо висином призме.

V=Bh

where B is the base area and h is the height. The volume of a prism whose base is an n-sided regular polygon with side length s is therefore:

V={\frac {n}{4}}hs^{2}\cot \left({\frac {\pi }{n}}\right)

Schlegel diagrams

P3	P4	P5	P6	P7	P8

Symmetry

The symmetry group of a right n-sided prism with regular base is D_nh of order 4n, except in the case of a cube, which has the larger symmetry group O_h of order 48, which has three versions of D_4h as subgroups. The rotation group is D_n of order 2n, except in the case of a cube, which has the larger symmetry group O of order 24, which has three versions of D₄ as subgroups.

The symmetry group D_nh contains inversion iff n is even.

The hosohedra and dihedra also possess dihedral symmetry, and an n-gonal prism can be constructed via the geometrical truncation of an n-gonal hosohedron, as well as through the cantellation or expansion of an n-gonal dihedron.

Truncated prism

A truncated prism is a prism with non-parallel top and bottom faces.^[5]

Example truncated triangular prism. Its top face is truncated at an oblique angle, but it is NOT an oblique prism!

Twisted prism

A twisted prism is a nonconvex polyhedron constructed from a uniform n-prism with each side face bisected on the square diagonal, by twisting the top, usually by $π$ /n radians (180/n degrees) in the same direction, causing sides to be concave.^[6]^[7]

A twisted prism cannot be dissected into tetrahedra without adding new vertices. The smallest case: the triangular form, is called a Schönhardt polyhedron.

An n-gonal twisted prism is topologically identical to the n-gonal uniform antiprism, but has half the symmetry group: D_n, [n,2]⁺, order 2n. It can be seen as a nonconvex antiprism, with tetrahedra removed between pairs of triangles.

3-gonal	4-gonal		12-gonal
Schönhardt polyhedron	Twisted square prism	Square antiprism	Twisted dodecagonal antiprism

Frustum

A frustum is a similar construction to a prism, with trapezoid lateral faces and differently sized top and bottom polygons.

Star prism

A star prism is a nonconvex polyhedron constructed by two identical star polygon faces on the top and bottom, being parallel and offset by a distance and connected by rectangular faces. A uniform star prism will have Schläfli symbol {p/q} × { }, with p rectangle and 2 {p/q} faces. It is topologically identical to a p-gonal prism.

Examples
{ }×{ }₁₈₀×{ }	t_a{3}×{ }		{5/2}×{ }	{7/2}×{ }	{7/3}×{ }	{8/3}×{ }
D_2h, order 8	D_3h, order 12		D_5h, order 20	D_7h, order 28		D_8h, order 32

Crossed prism

A crossed prism is a nonconvex polyhedron constructed from a prism, where the vertices of one base are inverted around the center of this base (or rotated by 180°). This transforms the side rectangular faces into crossed rectangles. For a regular polygon base, the appearance is an n-gonal hour glass. All oblique edges pass through a single body center. Note: no vertex is at this body centre. A crossed prism is topologically identical to an n-gonal prism.

Examples
{ }×{ }₁₈₀×{ }₁₈₀	t_a{3}×{ }₁₈₀		{3}×{ }₁₈₀	{4}×{ }₁₈₀	{5}×{ }₁₈₀	{5/2}×{ }₁₈₀	{6}×{ }₁₈₀
D_2h, order 8	D_3d, order 12			D_4h, order 16	D_5d, order 20		D_6d, order 24

Toroidal prism

A toroidal prism is a nonconvex polyhedron like a crossed prism, but without bottom and top base faces, and with simple rectangular side faces closing the polyhedron. This can only be done for even-sided base polygons. These are topological tori, with Euler characteristic of zero. The topological polyhedral net can be cut from two rows of a square tiling (with vertex configuration 4.4.4.4): a band of n squares, each attached to a crossed rectangle. An n-gonal toroidal prism has 2n vertices, 2n faces: n squares and n crossed rectangles, and 4n edges. It is topologically self-dual.

Examples
D_4h, order 16	D_6h, order 24
v=8, e=16, f=8	v=12, e=24, f=12

Prismatic polytope

A prismatic polytope is a higher-dimensional generalization of a prism. An n-dimensional prismatic polytope is constructed from two (n − 1)-dimensional polytopes, translated into the next dimension.

The prismatic n-polytope elements are doubled from the (n − 1)-polytope elements and then creating new elements from the next lower element.

Take an n-polytope with f_i i-face elements (i = 0, ..., n). Its (n + 1)-polytope prism will have 2f_i + f_i−1 i-face elements. (With f₋₁ = 0, f_n = 1.)

By dimension:

Take a polygon with n vertices, n edges. Its prism has 2n vertices, 3n edges, and 2 + n faces.
Take a polyhedron with v vertices, e edges, and f faces. Its prism has 2v vertices, 2e + v edges, 2f + e faces, and 2 + f cells.
Take a polychoron with v vertices, e edges, f faces, and c cells. Its prism has 2v vertices, 2e + v edges, 2f + e faces, 2c + f cells, and 2 + c hypercells.

Uniform prismatic polytope

A regular n-polytope represented by Schläfli symbol {p, q, ..., t} can form a uniform prismatic (n + 1)-polytope represented by a Cartesian product of two Schläfli symbols: {p, q, ..., t}×{}.

By dimension:

A 0-polytopic prism is a line segment, represented by an empty Schläfli symbol {}.
A 1-polytopic prism is a rectangle, made from 2 translated line segments. It is represented as the product Schläfli symbol {}×{}. If it is square, symmetry can be reduced: {}×{} = {4}.
- Example: Square, {}×{}, two parallel line segments, connected by two line segment sides.
A polygonal prism is a 3-dimensional prism made from two translated polygons connected by rectangles. A regular polygon {p} can construct a uniform n-gonal prism represented by the product {p}×{}. If p = 4, with square sides symmetry it becomes a cube: {4}×{} = {4, 3}.
- Example: Pentagonal prism, {5}×{}, two parallel pentagons connected by 5 rectangular sides.
A polyhedral prism is a 4-dimensional prism made from two translated polyhedra connected by 3-dimensional prism cells. A regular polyhedron {p, q} can construct the uniform polychoric prism, represented by the product {p, q}×{}. If the polyhedron is a cube, and the sides are cubes, it becomes a tesseract: {4, 3}×{} = {4, 3, 3}.
- Example: Dodecahedral prism, {5, 3}×{}, two parallel dodecahedra connected by 12 pentagonal prism sides.

Higher order prismatic polytopes also exist as cartesian products of any two polytopes. The dimension of a product polytope is the product of the dimensions of its elements. The first examples of these exist in 4-dimensional space; they are called duoprisms as the product of two polygons. Regular duoprisms are represented as {p}×{q}.

Галерија

Права тространа призма
Коса неправила призма

Референце

^ N.W. Johnson: Geometries and Transformations, (2018) ISBN 978-1-107-10340-5 Chapter 11: Finite symmetry groups, 11.3 Pyramids, Prisms, and Antiprisms, Figure 11.3b
^ Thomas Malton (1774). A Royal Road to Geometry: Or, an Easy and Familiar Introduction to the Mathematics. ... By Thomas Malton. ... author, and sold. стр. 360—.
^ James Elliot (1845). Key to the Complete Treatise on Practical Geometry and Mensuration: Containing Full Demonstrations of the Rules ... Longman, Brown, Green, and Longmans. стр. 3—.
^ William F. Kern, James R. Bland, Solid Mensuration with proofs, 1938, p.28
^ William F. Kern, James R. Bland, Solid Mensuration with proofs, 1938, p.81
^ The facts on file: Geometry handbook, Catherine A. Gorini, 2003, ISBN 0-8160-4875-4, p.172
^ „Pictures of Twisted Prisms”.

Литература

Anthony Pugh (1976). Polyhedra: A visual approach. California: University of California Press Berkeley. ISBN 0-520-03056-7.
Љиљана Петрушевски - Полиедри
Cromwell, P.;Polyhedra, CUP hbk (1997), pbk. (1999).
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Grünbaum, B.; Are your polyhedra the same as my polyhedra? Discrete and comput. geom: the Goodman-Pollack festschrift, ed. Aronov et al. Springer (2003) pp. 461–488. (pdf Архивирано на сајту Wayback Machine (3. август 2016))
Bertrand, J. (1858). Note sur la théorie des polyèdres réguliers, Comptes rendus des séances de l'Académie des Sciences, 46, pp. 79–82.
Haeckel, E. (1904). Kunstformen der Natur. Available as Haeckel, E. Art forms in nature, Prestel USA (1998), ISBN 3-7913-1990-6, or online at https://web.archive.org/web/20090627082453/http://caliban.mpiz-koeln.mpg.de/~stueber/haeckel/kunstformen/natur.html
Smith, J. V. (1982). Geometrical And Structural Crystallography. John Wiley and Sons.
Sommerville, D. M. Y. (1930). An Introduction to the Geometry of n Dimensions. E. P. Dutton, New York. (Dover Publications edition, 1958). Chapter X: The Regular Polytopes.
Coxeter, H.S.M.; Regular Polytopes (third edition). Dover Publications Inc. ISBN 0-486-61480-8
Whitney, Hassler (1932). „Congruent graphs and the connectivity of graphs”. Amer. J. Math. 54 (1): 150—168. JSTOR 2371086. doi:10.2307/2371086. hdl:10338.dmlcz/101067.
Blind, Roswitha; Mani-Levitska, Peter (1987), „Puzzles and polytope isomorphisms”, Aequationes Mathematicae, 34 (2–3): 287—297, MR 921106, doi:10.1007/BF01830678
Kalai, Gil (1988), „A simple way to tell a simple polytope from its graph”, Journal of Combinatorial Theory, Ser. A, 49 (2): 381—383, MR 964396, doi:10.1016/0097-3165(88)90064-7
Kaibel, Volker; Schwartz, Alexander (2003). „On the Complexity of Polytope Isomorphism Problems”. Graphs and Combinatorics. 19 (2): 215—230. arXiv:math/0106093 . doi:10.1007/s00373-002-0503-y. Архивирано из оригинала 2015-07-21. г.
Büeler, B.; Enge, A.; Fukuda, K. (2000). „Exact Volume Computation for Polytopes: A Practical Study”. Polytopes — Combinatorics and Computation. стр. 131. ISBN 978-3-7643-6351-2. doi:10.1007/978-3-0348-8438-9_6.
Yao, Andrew Chi Chih (1981), „A lower bound to finding convex hulls”, Journal of the ACM, 28 (4): 780—787, MR 677089, doi:10.1145/322276.322289 ; Ben-Or, Michael (1983), „Lower Bounds for Algebraic Computation Trees”, Proceedings of the Fifteenth Annual ACM Symposium on Theory of Computing (STOC '83), стр. 80—86, doi:10.1145/800061.808735
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Спољашње везе

Weisstein, Eric W. „Prism”. MathWorld.
Paper models of prisms and antiprisms Free nets of prisms and antiprisms
Paper models of prisms and antiprisms Using nets generated by Stella

[1] N.W. Johnson: Geometries and Transformations, (2018) ISBN 978-1-107-10340-5 Chapter 11: Finite symmetry groups, 11.3 Pyramids, Prisms, and Antiprisms, Figure 11.3b

[Malton1774-2] Thomas Malton (1774). A Royal Road to Geometry: Or, an Easy and Familiar Introduction to the Mathematics. ... By Thomas Malton. ... author, and sold. стр. 360—.

[Elliot1845-3] James Elliot (1845). Key to the Complete Treatise on Practical Geometry and Mensuration: Containing Full Demonstrations of the Rules ... Longman, Brown, Green, and Longmans. стр. 3—.

[:0-4] William F. Kern, James R. Bland, Solid Mensuration with proofs, 1938, p.28

[5] William F. Kern, James R. Bland, Solid Mensuration with proofs, 1938, p.81

[6] The facts on file: Geometry handbook, Catherine A. Gorini, 2003, ISBN 0-8160-4875-4, p.172

[7] „Pictures of Twisted Prisms”.

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