Higher Dimensions - Versionsgeschichte

Rudolf.Grossmann: version="4.5"

2016-05-17T13:14:29Z

version="4.5"

M. Ruppert am 12. Juni 2012 um 13:46 Uhr

2012-06-12T13:46:45Z

M. Ruppert am 12. Juni 2012 um 13:44 Uhr

2012-06-12T13:44:19Z

M. Ruppert am 12. Juni 2012 um 13:42 Uhr

2012-06-12T13:42:50Z

M. Ruppert am 12. Juni 2012 um 13:37 Uhr

2012-06-12T13:37:26Z

M. Ruppert am 12. Juni 2012 um 13:36 Uhr

2012-06-12T13:36:29Z

Änderungen zeigen

M. Ruppert am 4. Juni 2012 um 20:46 Uhr

2012-06-04T20:46:59Z

M. Ruppert am 4. Juni 2012 um 20:45 Uhr

2012-06-04T20:45:27Z

Änderungen zeigen

M. Ruppert am 4. Juni 2012 um 20:42 Uhr

2012-06-04T20:42:45Z

Änderungen zeigen

M. Ruppert am 4. Juni 2012 um 20:35 Uhr

2012-06-04T20:35:57Z

Änderungen zeigen

@@ Zeile 24: / Zeile 24: @@
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 <div align="center">''Fig. 2.1: Projection of a square (Author: Sebastian Hammer, University of Würzburg)''</div>
@@ Zeile 30: / Zeile 30: @@
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 <div align="center">''Fig. 2.2:Projection of a cube (Author: Sebastian Hammer, University of Würzburg)''</div>
 <br/><br/>

@@ Zeile 96: / Zeile 96: @@
 == Intersections of cubes ==
 A dynamic representation of a four-dimensional hypercube is given by considering the different shapes of intersection while intersecting it with  a three-dimensional hyper plane. Initially, a cube (in a three-dimensional space), which intersects a plane will be regarded. It is assumed, that the objects move with (relative) velocity v.
-It is especially easy to describe the situation while considering the cube (edge length a) with its edges  fixed  on the axes of a coordinate system. The plane <math>E(t):\ x_{1}+x_{2}+x_{3}-\sqrt{3} \cdot v \cdot t = 0</math> moves along a cube diagonal with velocity v through this cube. Since the cube is a convex body itself, it is sufficient to determine the points of intersection of the edges at any time. We get the cross section as the convex hull of these intersection points.
+It is especially easy to describe the situation while considering the cube (edge length a) with its edges  fixed  on the axes of a coordinate system. The plane
 <br/>
@@ Zeile 103: / Zeile 109: @@
 A four-dimensional hypercube with an edge length <math>a</math> will be intersected with a space <math>R(t)</math>, which moves with a velocity of <math>v</math> along the body diagonal of the hypercube. Therefore we define analogously:
-<math>R(t):\ x_{1}+x_{2}+x_{3}+x_{4}-2 \cdot v \cdot t = 0</math>.
+<br/>
 <br/>

@@ Zeile 39: / Zeile 39: @@
 <br/>
 <br/>
-<div align="center"><math>g_{i}: \ \vec{X}= \vec{A_{i}} + k \cdot \begin{pmatrix}1 \\ 1 \end{pmatrix} , \ \ (1\le i \le 4)</math> <div/>
+<div align="center"><math>g_{i}: \ \vec{X}= \vec{A_{i}} + k \cdot \begin{pmatrix}1 \\ 1 \end{pmatrix} , \ \ (1\le i \le 4)</math> </div>
 <br/>
 <br/>
@@ Zeile 45: / Zeile 45: @@
 <br/>
 <br/>
-<div align="center"><math>h:\ x_{1}+x_{2}=0.</math><div/>
+<div align="center"><math>h:\ x_{1}+x_{2}=0.</math></div>
 <br/>
 <br/>
@@ Zeile 51: / Zeile 51: @@
 <br/>
 <br/>
-<div align="center"><math>g_{i}: \ \vec{X}= \vec{A_{i}} + k \cdot \begin{pmatrix}1 \\ \vdots  \\1 \end{pmatrix} , \ \ (1\le i \le 2^{n})</math><div/>
+<div align="center"><math>g_{i}: \ \vec{X}= \vec{A_{i}} + k \cdot \begin{pmatrix}1 \\ \vdots  \\1 \end{pmatrix} , \ \ (1\le i \le 2^{n})</math></div>
 <br/>
 <br/>
@@ Zeile 57: / Zeile 57: @@
 <br/>
 <br/>
-<div align="center"><math>R:\ x_{1}+ \cdots +x_{2}=0.</math><div/>
+<div align="center"><math>R:\ x_{1}+ \cdots +x_{2}=0.</math></div>
 <br/>
 <br/>

@@ Zeile 39: / Zeile 39: @@
 <br/>
 <br/>
-<math>g_{i}: \ \vec{X}= \vec{A_{i}} + k \cdot \begin{pmatrix}1 \\ 1 \end{pmatrix} , \ \ (1\le i \le 4)</math> with the line <math>h:\ x_{1}+x_{2}=0.</math>
+<div align="center"><math>g_{i}: \ \vec{X}= \vec{A_{i}} + k \cdot \begin{pmatrix}1 \\ 1 \end{pmatrix} , \ \ (1\le i \le 4)</math> <div/>
 <br/>
 <br/>
@@ Zeile 45: / Zeile 51: @@
 <br/>
 <br/>
-<math>g_{i}: \ \vec{X}= \vec{A_{i}} + k \cdot \begin{pmatrix}1 \\ \vdots  \\1 \end{pmatrix} , \ \ (1\le i \le 2^{n})</math> and the ''(n-1)''-hyper-plane <math>R:\ x_{1}+ \cdots +x_{2}=0.</math>
+<div align="center"><math>g_{i}: \ \vec{X}= \vec{A_{i}} + k \cdot \begin{pmatrix}1 \\ \vdots  \\1 \end{pmatrix} , \ \ (1\le i \le 2^{n})</math><div/>
 <br/>
 <br/>

@@ Zeile 1: / Zeile 1: @@
-== 1. Looking for the next dimension ==
+== Looking for the next dimension ==
 [[Datei:Calabi-Yau.png|miniatur|rechts|300px|Fig. 1: Illustration of a Calabi-Yau-manifold (Important for the description of higher dimensional models in superstring-theory)]]
 Does our world really have more than three dimensions? If so, do objects in higher dimension have a relation to the world around us? Is it possible to get a perception of these objects or do they withdraw any representation? Questions like these will be posed by students if you talk about space dimensions in school. Students want to understand the meaning of a four-, five or even n-dimensional space. The relativity theory uses four dimensions to explain the concept of space-time, six dimensions are necessary to describe the bending of space-time and different string theories  even use representations in up to 26 dimensions (e.g. L. Botelho, R. Botelho, 1999).  Another current domain of application for higher dimensional objects and their three-dimensional representations is the study of non-periodic structures in modern crystallography. Within the concept of quasicrystals projections of higher dimensional point-sets (such as the integer-lattice in dimension 5) to three dimensional space are supposed to be good models for non-periodic crystalline structures (see section 5 below).
@@ Zeile 15: / Zeile 15: @@
-== 2. Projections ==
+== Projections ==
 In this section, the basic idea of describing higher dimensional objects by means of projection will be generalized in higher dimensions. Especially the orthogonal projection along a body diagonal of an ''n''-dimensional hyper cube in an ''(n-1)''-dimensional space can easily be generalized.
@@ Zeile 82: / Zeile 82: @@
 <br />
-== 3. Intersections of cubes ==
+== Intersections of cubes ==
 A dynamic representation of a four-dimensional hypercube is given by considering the different shapes of intersection while intersecting it with  a three-dimensional hyper plane. Initially, a cube (in a three-dimensional space), which intersects a plane will be regarded. It is assumed, that the objects move with (relative) velocity v.
 It is especially easy to describe the situation while considering the cube (edge length a) with its edges  fixed  on the axes of a coordinate system. The plane <math>E(t):\ x_{1}+x_{2}+x_{3}-\sqrt{3} \cdot v \cdot t = 0</math> moves along a cube diagonal with velocity v through this cube. Since the cube is a convex body itself, it is sufficient to determine the points of intersection of the edges at any time. We get the cross section as the convex hull of these intersection points.
@@ Zeile 103: / Zeile 103: @@
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-== 4. Geometry of coordinates ==
+== Geometry of coordinates ==
 The unit line segment and the unit square can be considered as a one‐ or two‐dimensional analog of the
 unit cube. Looking at the coordinates of the vertices in a coordinate system, we get
@@ Zeile 228: / Zeile 228: @@
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-== 5. Quasicrystals – Projections from higher dimensions ==
+== Quasicrystals – Projections from higher dimensions ==
 Starting with the assumption of the classical crystallography, that the characteristic of real crystallographic structures is their translational symmetry (i.e. invariance under three independent translations), mathematical representations of these structures lead to the well-known “crystallographic restriction”, which allows non-trivial rotational symmetries only of orders 2, 3, 4 and 6. This corresponds to physical observations, until Shechtman et al. (1984) found a non-periodic structure of crystals (without translational symmetry) inside an Al-Mn-alloy, which has fivefold rotational symmetry. Crystallographers call these structures quasicrystals. To be more  precise:

@@ Zeile 163: / Zeile 163: @@
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 <big>''' Example 3: The three‐dimensional cube (''n'' = 3):'''</big>
-[[Datei:Klein_Bild_1_table1.PNG|miniatur|rechts|600px|'''Table 1: Number of the k‐dimensional boundaries of an n‐dimensional cube''']]
 {|