正圖形

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一些正圖形與正多胞形的例子
Regular pentagon.svg
正五邊形是一個多邊形,是一個正圖形,由5個邊組成的二維正多胞形,其施萊夫利符號為{5}.
POV-Ray-Dodecahedron.svg
正十二面體是一個多面體,是一個正圖形,由12個正五邊形面組成的三維正多胞形,其施萊夫利符號為{5,3}.
Schlegel wireframe 120-cell.png
正一百二十胞體是一個四維多胞體,是一個正圖形,由120個正十二面體胞組成的四維正多胞形,其施萊夫利符號為{5,3,3}。(这里展示的是施莱格尔图像英语Schlegel diagram
Cubic honeycomb.png
正方體堆砌是一個三维空間堆砌,可被看作是四维的无穷胞体,施萊夫利符號為{4,3,4}.
Octeract Petrie polygon.svg
八维超正方体的256个顶点和1024条棱可以用正交投影来展示。(皮特里多边形英语Petrie polygon

幾何學中,正圖形又稱正多胞形英语Regular polytope),是一種對稱性对于英语Flag (geometry)可递的幾何體,且具有高度對稱性,對於該幾何體內所有同維度的元素(如:點、線、面)都完全具有相同的性質,並且每一個元素皆為一個正圖形,例如,正方體所有的面的面積及形狀皆相同,且皆為正方形,是一個二維正多胞形、所有邊的長度也相同,所有角的角度及形式也相同,因此正方體是一個正圖形或正多胞形。對於所有元素,或叫j維面(對所有的 0 ≤ j ≤ n,其中n是該幾何體所在的維度) — 胞、面等等 — 也都对于多胞形的对称性可递,也是≤ n维的正圖形。

正图形是正多边形(例如,正方形或者正五边形)和正多面体(例如立方体)的向任意维度的推广类比。正图形极强的对称性使它们拥有极强的审美价值,吸引着数学家和数学爱好者。

一般地,n维正图形被定义为有正维面英语Facet (geometry)[(n − 1)-表面]和正顶点图英语Vertex Figure。这两个条件已经能充分地保证所有面、所有顶点都是相似的。但要注意的是,这一定义并不适用于抽象多胞形英语abstract polytope

一个正图形能用形式为{a, b, c, ...., y, z}的施莱夫利符号代表,其正的面为{a, b, c, ..., y},顶点图为{b, c, ..., y, z}。

分類和描述[编辑]

正图形最基础的分类是按其维度。

它们能够按照对称性进一步分类。例如,正方体正八面体有着相同的对称性,同样,正十二面体正二十面体也是。事实上,对称群大多依照正图形命名,例如正四面体对称群和正二十面体对称群。

3种特殊类型的正图形存在于所有维度:

在二维,这里有无穷多个正多边形。在三维和四维这里有许多上述三种之外的正多面体正多胞体。在五维及以上维,只存在这三种类型的正图形。另见正图形列表英语list of regular polytopes

正图形的概念有时被扩展,使其包括了另外一些相关的几何对象。其中一些有正的例子,下面“历史发现”一章将会详细说明。

施萊夫利符號[编辑]

施萊夫利符號是一個簡潔有力的多面體表示法,是19世紀由路德維希·施萊夫利所發明的,一个改进了的版本随后成为了标准。这种记号可通过维度依次增加一获得最好的解释。

  • 一个有n条边的正多边形可以标记为{n}。所以一个等边三角形是{3},一个正方形是{4}……一个绕其中心旋转m圈的正星形多边形被标记为分式{n/m},这里nm互质的,例如正五角星是{5/2}。
  • 一个有着面{n},并且一个顶点处有p个面相交的正多面体标记为{n, p}。九个正多面体是:{3, 3}、{3, 4}、{4, 3}、{3, 5}、{5, 3}、{3, 5/2}、{5/2, 3}、{5, 5/2}和{5/2, 5}。{p}就是这个正多面体的顶点图
  • 一个有着胞{n, p},并且每一条棱处有q个胞相交的正多胞体标记为{n, p, q}。其顶点图为{p, q}。
  • 一个五维正多胞体是{n, p, q, r},等等。

正图形的对偶性[编辑]

正图形的对偶形也是正图形。对偶图形的施莱夫利符号就是将原来的符号倒过来写:{3, 3}为自身对偶,{3, 4}与{4, 3}对偶,{4, 3, 3}与{3, 3, 4}对偶,以此类推。

正图形的顶点图英语Vertex Figure的对偶即是其对偶图形的维面。例如{3, 3, 4}的顶点图是{3, 4},其对偶即是{4, 3} — {4, 3, 3}的一个胞。

任何维的超方形正轴形都是互相对偶的。

如果其施莱夫利符号是回文,即正反读都一样,那么这个正图形就是自身对偶的。自身对偶正图形包括:

正单纯形[编辑]

1-正单纯形 到 4-正单纯形 的图像
1-simplex t0.svg 2-simplex t0.svg 3-simplex t0.svg 4-simplex t0.svg
线段 正三角形 正四面体 正五胞体
  Regular triangle.svg Tetrahedron.svg Schlegel wireframe 5-cell.png

我们从点A开始。标下与A相距r的点B,并连接它们,形成线段。在垂直与它的第二维度标下与AB都相距r的第三点C,并连接ACBC,形成正三角形。在垂直与它的第三维度标下与三点都相距r的第四点D,连接四点,便形成正四面体。用同样的方法,我们可以得到更高维的类似正图形。

这些就是正单纯形。以维度来排序,它们是:

0.
1. 线段
2. 正三角形(正三边形)
3. 正四面体
4. 正五胞体 4-单纯形
5. 五维正六胞体 5-单纯形
... n-单纯形有n+1个顶点。

超方形[编辑]

2-超方形 到 4-超方形 的图像
Cross graph 2.svg Cube graph ortho vcenter.png Hypercubestar.svg
正方形 立方体 超正方体
Kvadrato.svg Hexahedron.svg Schlegel wireframe 8-cell.png

从一个点A开始。连一条线到距离为rB,形成一条线段。延伸第二条长为r的线,垂直于AB,将B连接到C,同样链接AD,形成一个正方形ABCD。从每个顶点同样延伸出长为r的线,同时垂直于ABBC,标记点EFGH形成立方体ABCD-EFGH。用同样的方法,我们可以得到更高维的类似正图形。

它们就是超方形或称正测形。以维度来排序,它们是:

0. 点
1. 线段
2. 正方形(正四边形)
3. 立方体(正六面体)
4. 四维超正方体(正八胞体)4-超方体
5. 五维超正方体(正十超胞体)5-超方体
...一个n-超方体有2n个顶点。

正轴形[编辑]

2-正轴形 到 4-正轴形 的图像
2-orthoplex.svg 3-orthoplex.svg 4-orthoplex.svg
正方形 正八面体 正十六胞体
Kvadrato.svg Octahedron.svg Schlegel wireframe 16-cell.png

从一个点O开始。从O向两个相反的方向延出两条线到距O点距离为rAB,互相之间距离为2r,形成一条线段。同样再画线段COD,长度为2r,以O为中点而垂直于AB。连接4个顶点形成正方形ACBD。再画线段EOF,同样长度为2r,中点为O,同时垂直于ABCD(即上下方向)。将其顶点与正方形顶点一一相连得到正八面体。用同样的方法,我们可以得到更高维的类似正图形。

这样得到的图形称为正轴形交叉形。以维度来排序,它们是:

0. 点
1. 线段
2. 正方形(正四边形)
3. 正八面体
4. 正十六胞体4-正轴形
5. 正三十二超胞体(五维正三十二胞体)5-正轴形
...n-正轴形有2n个顶点。

History of discovery[编辑]

Convex polygons and polyhedra[编辑]

The earliest surviving mathematical treatment of regular polygons and polyhedra comes to us from ancient Greek mathematicians. The five Platonic solids were known to them. Pythagoras knew of at least three of them and Theaetetus (ca. 417 B.C. – 369 B.C.) described all five. Later, Euclid wrote a systematic study of mathematics, publishing it under the title Elements, which built up a logical theory of geometry and number theory. His work concluded with mathematical descriptions of the five Platonic solids.

Platonic solids
Tetrahedron.jpg Hexahedron.jpg Octahedron.svg POV-Ray-Dodecahedron.svg Icosahedron.jpg
Tetrahedron Cube Octahedron Dodecahedron Icosahedron

Star polygons and polyhedra[编辑]

Our understanding remained static for many centuries after Euclid. The subsequent history of the regular polytopes can be characterised by a gradual broadening of the basic concept, allowing more and more objects to be considered among their number. Thomas Bradwardine (Bradwardinus) was the first to record a serious study of star polygons. Various star polyhedra appear in Renaissance art, but it was not until Johannes Kepler studied the small stellated dodecahedron and the great stellated dodecahedron in 1619 that he realised these two were regular. Louis Poinsot discovered the great dodecahedron and great icosahedron in 1809, and Augustin Cauchy proved the list complete in 1812. These polyhedra are known as collectively as the Kepler-Poinsot polyhedra.

Main article Regular polyhedron - History.
Kepler-Poinsot polyhedra
SmallStellatedDodecahedron.jpg GreatStellatedDodecahedron.jpg GreatDodecahedron.jpg GreatIcosahedron.jpg
Small stellated
dodecahedron
Great stellated
dodecahedron
Great dodecahedron Great icosahedron

Higher-dimensional polytopes[编辑]

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 (Schläfli, 1901), six years posthumously, although parts of it were published in 1855 and 1858 (Schläfli, 1855), (Schläfli, 1858). Interestingly, between 1880 and 1900, Schläfli's results were rediscovered independently by at least nine other mathematicians — see (Coxeter, 1948, pp143–144) for more details. Schläfli called such a figure a "polyschem" (in English, "polyscheme" or "polyschema"). The term "polytope" was introduced by Hoppe in 1882, and first used in English by Mrs. Stott some twenty years later. The term "polyhedroids" was also used in earlier literature (Hilbert, 1952).

Coxeter (1948) is probably the most comprehensive printed treatment of Schläfli's and similar results to date. Schläfli showed that there are six 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 hypercube (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 hypercube and 16-cell, analogous to the way that the cuboctahedron and the rhombic dodecahedron are transitional forms between the cube and the octahedron.

In five and more dimensions, there are exactly three regular polytopes, which correspond to the tetrahedron, cube and octahedron: these are the regular simplices, measure polytopes and cross polytopes. Descriptions of these may be found in the List of regular polytopes. Also of interest are the nonconvex regular 4-polytopes, 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, but still retain the aesthetically pleasing symmetry of their lower dimensional cousins. 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.[1]

Harder still to imagine are the more modern abstract regular polytopes 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.

At the start of the 20th century, the definition of a regular polytope was as follows.

  • A regular polygon is a polygon whose edges are all equal and whose angles are all equal.
  • A regular polyhedron is a polyhedron whose faces are all congruent regular polygons, and whose vertex figures are all congruent and regular.
  • And so on, a regular n-polytope is an n-dimensional polytope whose (n − 1)-dimensional faces are all regular and congruent, and whose vertex figures are all regular and congruent.

This is a "recursive" definition. It defines regularity of higher dimensional figures in terms of regular figures of a lower dimension. There is an equivalent (non-recursive) definition, which states that a polytope is regular if it has a sufficient degree of symmetry.

  • An n-polytope is regular if any set consisting of a vertex, an edge containing it, a 2-dimensional face containing the edge, and so on up to n−1 dimensions, can be mapped to any other such set by a symmetry of the polytope.

So for example, the cube is regular because if we choose a vertex of the cube, and one of the three edges it is on, and one of the two faces containing the edge, then this triplet, or flag, (vertex, edge, face) can be mapped to any other such flag by a suitable symmetry of the cube. Thus we can define a regular polytope very succinctly:

  • A regular polytope is one which is transitive on its flags.

In the 20th century, some important developments were made. The symmetry groups of the classical regular polytopes were generalised into what are now called Coxeter groups. Coxeter groups also include the symmetry groups of regular tessellations of space or of the plane. For example, the symmetry group of an infinite chessboard would be the Coxeter group [4,4].

正无穷胞体 — 无穷多胞形[编辑]

In the first part of the 20th century, Coxeter and Petrie discovered three infinite structures {4, 6}, {6, 4} and {6, 6}. They called them regular skew polyhedra, because they seemed to satisfy the definition of a regular polyhedron — all the vertices, edges and faces are alike, all the angles are the same, and the figure has no free edges. Nowadays, they are called infinite polyhedra or apeirohedra. The regular tilings of the plane {4, 4}, {3, 6} and {6, 3} can also be regarded as infinite polyhedra.

In the 1960s Branko Grünbaum issued a call to the geometric community to consider more abstract types of regular polytopes that he called polystromata. He developed the theory of polystromata, showing examples of new objects he called regular apeirotopes, that is, regular polytopes with infinitely many faces. A simple example of an apeirogon {∞} would be a zig-zag. It seems to satisfy the definition of a regular polygon — all the edges are the same length, all the angles are the same, and the figure has no loose ends (because they can never be reached). More importantly, perhaps, there are symmetries of the zig-zag that can map any pair of a vertex and attached edge to any other. Since then, other regular apeirogons and higher apeirotopes have continued to be discovered.

Regular complex polytopes[编辑]

A complex number has a real part, which is the bit we are all familiar with, and an imaginary part, which is a multiple of the square root of minus one. A complex Hilbert space has its x, y, z, etc. coordinates as complex numbers. This effectively doubles the number of dimensions. A polytope constructed in such a unitary space is called a complex polytope.

Abstract polytopes[编辑]

The Hemicube is derived from a cube by equating opposite vertices, edges, and faces. It has 4 vertices, 6 edges, and 3 faces.

Grünbaum also discovered the 11-cell, a four-dimensional self-dual object whose facets are not icosahedra, but are "hemi-icosahedra" — that is, they are the shape one gets if one considers opposite faces of the icosahedra to be actually the same face (Grünbaum, 1977). The hemi-icosahedron has only 10 triangular faces, and 6 vertices, unlike the icosahedron, which has 20 and 12.

This concept may be easier for the reader to grasp if one considers the relationship of the cube and the hemicube. An ordinary cube has 8 corners, they could be labeled A to H, with A opposite H, B opposite G, and so on. In a hemicube, A and H would be treated as the same corner. So would B and G, and so on. The edge AB would become the same edge as GH, and the face ABEF would become the same face as CDGH. The new shape has only three faces, 6 edges and 4 corners.

The 11-cell cannot be formed with regular geometry in flat (Euclidean) hyperspace, but only in positively-curved (elliptic) hyperspace.

A few years after Grünbaum's discovery of the 11-cell, H. S. M. Coxeter independently discovered the same shape. He had earlier discovered a similar polytope, the 57-cell (Coxeter 1982, 1984).

By 1994 Grünbaum was considering polytopes abstractly as combinatorial sets of points or vertices, and was unconcerned whether faces were planar. As he and others refined these ideas, such sets came to be called abstract polytopes. An abstract polytope is defined as a partially ordered set (poset), whose elements are the polytope's faces (vertices, edges, faces etc.) ordered by containment. Certain restrictions are imposed on the set that are similar to properties satisfied by the classical regular polytopes (including the Platonic solids). The restrictions, however, are loose enough that regular tessellations, hemicubes, and even objects as strange as the 11-cell or stranger, are all examples of regular polytopes.

A geometric polytope is understood to be a realization of the abstract polytope, such that there is a one-to-one mapping from the abstract elements to the geometric. Thus, any geometric polytope may be described by the appropriate abstract poset, though not all abstract polytopes have proper geometric realizations.

The theory has since been further developed, largely by Egon Schulte and Peter McMullen (McMullen, 2002), but other researchers have also made contributions.

Regularity of abstract polytopes[编辑]

Regularity has a related, though different meaning for abstract polytopes, since angles and lengths of edges have no meaning.

The definition of regularity in terms of the transitivity of flags as given in the introduction applies to abstract polytopes.

Any classical regular polytope has an abstract equivalent which is regular, obtained by taking the set of faces. But non-regular classical polytopes can have regular abstract equivalents, since abstract polytopes don't care about angles and edge lengths, for example. And a regular abstract polytope may not be realisable as a classical polytope.

All polygons are regular in the abstract world, for example, whereas only those having equal angles and edges of equal length are regular in the classical world.

Vertex figure of abstract polytopes[编辑]

The concept of vertex figure is also defined differently for an abstract polytope. The vertex figure of a given abstract n-polytope at a given vertex V is the set of all abstract faces which contain V, including V itself. More formally, it is the abstract section

Fn / V = {F | VFFn}

where Fn is the maximal face, i.e. the notional n-face which contains all other faces. Note that each i-face, i ≥ 0 of the original polytope becomes an (i − 1)-face of the vertex figure.

Unlike the case for Euclidean polytopes, an abstract polytope with regular facets and vertex figures may or may not be regular itself – for example, the square pyramid, all of whose facets and vertex figures are regular abstract polygons.

The classical vertex figure will, however, be a realisation of the abstract one.

Constructions[编辑]

Polygons[编辑]

The traditional way to construct a regular polygon, or indeed any other figure on the plane, is by compass and straightedge. Constructing some regular polygons in this way is very simple (the easiest is perhaps the equilateral triangle), some are more complex, and some are impossible ("not constructible"). The simplest few regular polygons that are impossible to construct are the n-sided polygons with n equal to 7, 9, 11, 13, 14, 18, 19, 21,...

Constructibility in this sense refers only to ideal constructions with ideal tools. Of course reasonably accurate approximations can be constructed by a range of methods; while theoretically possible constructions may be impractical.

Polyhedra[编辑]

Euclid's Elements gave what amount to ruler-and-compass constructions for the five Platonic solids. (See, for example, Euclid's Elements.) However, the merely practical question of how one might draw a straight line in space, even with a ruler, might lead one to question what exactly it means to "construct" a regular polyhedron. (One could ask the same question about the polygons, of course.)

The English word "construct" has the connotation of systematically building the thing constructed. The most common way presented to construct a regular polyhedron is via a fold-out net. To obtain a fold-out net of a polyhedron, one takes the surface of the polyhedron and cuts it along just enough edges so that the surface may be laid out flat. This gives a plan for the net of the unfolded polyhedron. Since the Platonic solids have only triangles, squares and pentagons for faces, and these are all constructible with a ruler and compass, there exist ruler-and-compass methods for drawing these fold-out nets. The same applies to star polyhedra, although here we must be careful to make the net for only the visible outer surface.

If this net is drawn on cardboard, or similar foldable material (for example, sheet metal), the net may be cut out, folded along the uncut edges, joined along the appropriate cut edges, and so forming the polyhedron for which the net was designed. For a given polyhedron there may be many fold-out nets. For example, there are 11 for the cube, and over 900000 for the dodecahedron. Some interesting fold-out nets of the cube, octahedron, dodecahedron and icosahedron are available here.

Numerous children's toys, generally aimed at the teen or pre-teen age bracket, allow experimentation with regular polygons and polyhedra. For example, klikko provides sets of plastic triangles, squares, pentagons and hexagons that can be joined edge-to-edge in a large number of different ways. A child playing with such a toy could re-discover the Platonic solids (or the Archimedean solids), especially if given a little guidance from a knowledgeable adult.

In theory, almost any material may be used to construct regular polyhedra. Instructions for building origami models may be found here, for example. They may be carved out of wood, modeled out of wire, formed from stained glass. The imagination is the limit.

Higher dimensions[编辑]

A perspective projection (Schlegel diagram) for tesseract
An animated cut-away cross-section of the 24-cell.

In higher dimensions, it becomes harder to say what one means by "constructing" the objects. Clearly, in a 3-dimensional universe, it is impossible to build a physical model of an object having 4 or more dimensions. There are several approaches normally taken to overcome this matter.

The first approach, suitable for four dimensions, uses four-dimensional stereography.[1] Depth in a third dimension is represented with horizontal relative displacement, depth in a fourth dimension with vertical relative displacement between the left and right images of the stereograph.

The second approach is to embed the higher-dimensional objects in three-dimensional space, using methods analogous to the ways in which three-dimensional objects are drawn on the plane. For example, the fold out nets mentioned in the previous section have higher-dimensional equivalents. Some of these may be viewed at [1]. One might even imagine building a model of this fold-out net, as one draws a polyhedron's fold-out net on a piece of paper. Sadly, we could never do the necessary folding of the 3-dimensional structure to obtain the 4-dimensional polytope, or polychoron, because of the constraints of the physical universe. Another way to "draw" the higher-dimensional shapes in 3 dimensions is via some kind of projection, for example, the analogue of either orthographic or perspective projection. Coxeter's famous book on polytopes (Coxeter, 1948) has some examples of such orthographic projections. Other examples may be found on the web (see for example [2]). Note that immersing even 4-dimensional polychora directly into two dimensions is quite confusing. Easier to understand are 3-d models of the projections. Such models are occasionally found in science museums or mathematics departments of universities (such as that of the Université Libre de Bruxelles).

The intersection of a four (or higher) dimensional regular polytope with a three-dimensional hyperplane will be a polytope (not necessarily regular). If the hyperplane is moved through the shape, the three-dimensional slices can be combined, animated into a kind of four dimensional object, where the fourth dimension is taken to be time. In this way, we can see (if not fully grasp) the full four-dimensional structure of the four-dimensional regular polytopes, via such cutaway cross sections. This is analogous to the way a CAT scan reassembles two-dimensional images to form a 3-dimensional representation of the organs being scanned. The ideal would be an animated hologram of some sort, however, even a simple animation such as the one shown can already give some limited insight into the structure of the polytope.

Another way a three-dimensional viewer can comprehend the structure of a four-dimensional polychoron is through being "immersed" in the object, perhaps via some form of virtual reality technology. To understand how this might work, imagine what one would see if space were filled with cubes. The viewer would be inside one of the cubes, and would be able to see cubes in front of, behind, above, below, to the left and right of himself. If one could travel in these directions, one could explore the array of cubes, and gain an understanding of its geometrical structure. An infinite array of cubes is not a polytope in the traditional sense. In fact, it is a tessellation of 3-dimensional (Euclidean) space. However, a 4-dimensional polychoron can be considered a tessellation of a 3-dimensional non-Euclidean space, namely, a tessellation of the surface of a four-dimensional sphere (a 4-dimensional spherical tiling).

A regular dodecahedral honeycomb, {5,3,4}, of hyperbolic space projected into 3-space.

Locally, this space seems like the one we are familiar with, and therefore, a virtual-reality system could, in principle, be programmed to allow exploration of these "tessellations", that is, of the 4-dimensional regular polytopes. The mathematics department at UIUC has a number of pictures of what one would see if embedded in a tessellation of hyperbolic space with dodecahedra. Such a tessellation forms an example of an infinite abstract regular polytope.

Normally, for abstract regular polytopes, a mathematician considers that the object is "constructed" if the structure of its symmetry group is known. This is because of an important theorem in the study of abstract regular polytopes, providing a technique that allows the abstract regular polytope to be constructed from its symmetry group in a standard and straightforward manner.

Regular polytopes in nature[编辑]

For examples of polygons in nature, see:

Each of the Platonic solids occurs naturally in one form or another:

Higher polytopes can obviously not exist in a three-dimensional world. However this might not rule them out altogether. In cosmology and in string theory, physicists commonly model the Universe as having many more dimensions. It is possible that the Universe itself has the form of some higher polytope, regular or otherwise. Astronomers have even searched the sky in the last few years, for tell-tale signs of a few regular candidates, so far without definite results.

See also[编辑]

參考文獻[编辑]

  1. ^ 1.0 1.1 Brisson, David W., Visual Comprehension in n-Dimensions//Brisson, David W., Hypergraphics: Visualizing Complex Relationships in Art, Science and Technology, AAAS Selected Symposium, 24, Washington, D.C.: AAAS. 1978:  109–145 
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  • (Schläfli, 1858), Schläfli, L.; On The Multiple Integralndx dy ... dz, Whose Limits Are p_1 =a_1 x+b_1y+ \cdots +h_1z\ge 0, p_2 > 0, \ldots , p_n > 0 and x^2+y^2+\cdots+z^2<1 Quarterly Journal Of Pure And Applied Mathematics 2 (1858) pp269–301, 3 (1860) pp54–68, 97–108.
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  • Olshevsky, George, Regular polytope at Glossary for Hyperspace.
  • Stella: Polyhedron Navigator Tool for exploring 3D polyhedra, 4D polytopes, and printing nets
  • Ernst Haeckel's Kunstformen der Natur online (German)
  • Interesting fold-out nets of the cube, octahedron, dodecahedron and icosahedron