Lineare Algebra II SS 2000

http://www.mathematik.uni-bielefeld.de/birep/linalg/

8. Noch einmal: Bewegungen. Und reguläre Polyeder.

29. (a) Zeige: Die Zuordnung η: B(n) → GL(n + 1, ℝ)

η(t_{a} ∘ f_{A}) = [ \begin{matrix}1 & 0 \\ a & A \end{matrix} ] für a ∈ ℝ^{n} und A ∈ O(n)

ist ein injektiver Gruppen-Homomorphismus; ist ϕ ∈ B(n) und x ∈ ℝ^{n}, so gilt:

η(ϕ) [ \begin{matrix}1 \\ x \end{matrix} ] = [ \begin{matrix}1 \\ ϕ(x) \end{matrix} ] .

(b) Sei nun n = 2. Wie sieht η(ϕ) aus, wenn ϕ eine Translation, eine Drehung mit Drehwinkel 0 < α < 2π, eine Spiegelung, bzw. eine Gleitspiegelung ist?

30. Sei n_{1}, …, n_{t} eine Folge natürlicher Zahlen n_{i} ≥ 3 mit ∑_{i} \frac{n_{i}−2}{n_{i}} = 2.

(a) Zeige: Bis auf zyklische Vertauschungen und Umkehr der Reihenfolge gibt es genau 11 solche Folgen (zum Beispiel (3, 3, 3, 3, 3, 3), (3, 3, 4, 3, 4), (3, 3, 3, 4, 4), (3, 3, 3, 3, 6), …).

(b) Zu jeder solchen Folge kann man eine “Pflasterung” der Ebene aus regulären n_{i} -Ecken mit Kantenlänge 1 konstruieren, so daß in jedem Eckpunkt t n-Ecke aneinanderstoßen, und zwar ein n_{1} -Eck, ein n_{2} -Eck, usw., entsprechend der Folge (n_{1}, n_{2}, …, n_{t}). Skizziere alle diese Pflasterungen. Hier als Beispiel ein Ausschnitt der Pflasterung zur Folge (3, 3, 3, 3, 6):

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Zeige: die jeweilige Symmetriegruppe ist eine ebene Kristallgruppe. Bestimme jeweils den Typ.

31. Seien k, n ≥ 3 natürliche Zahlen mit \frac{k(n−2)}{n} < 2. Bestimme alle Möglichkeiten für k, n und zeige, daß zu jedem dieser Fälle in natürlicher Weise ein reguläres Polyeder gehört (Tetraeder, Würfel, Oktaeder, Dodekaeder, Isokaeder).

32. Betrachte die Punkte A = [ \begin{matrix}−a \\ 0 \\ 1 \end{matrix} ] , B = [ \begin{matrix}a \\ 0 \\ 1 \end{matrix} ] , C = [ \begin{matrix}0 \\ −a \\ −1 \end{matrix} ] , D = [ \begin{matrix}0 \\ a \\ −1 \end{matrix} ] ,

mit a = \sqrt{2}. Bestimme alle Drehungen des ℝ^{3} , die die Menge \{A, B, C, D\} in sich überführen (gesucht sind die entsprechenden (3 × 3)-Matrizen bezüglich der kanonischen Basis des ℝ^{3}). Bestimme jeweils auch die Drehachse.

Abgabe: 08.05.2000 Die 17 ebenen Kristallgruppen nach A.Schoenfließ (1891) und E.S.Fedorow (1892).

Es gibt 17 verschiedene ebene Kristallgruppen. Hier ein Entscheidungsverfahren zur Bestimmung des jeweiligen Typs. Sei G eine ebene Kristallgruppe, sei d die höchste Ordnung einer Drehung in G. Es sei \overline{G} die zugehörige Punktgruppe.

(Gleitspiegelachsen bezeichnen wir einfach als Achsen. Unter einem Viererzentrum verstehen wir das Drehzentrum einer Drehung der Ordnung 4; analog ist ein Dreierzentrum definiert.)

d  Typ  \overline{G}

Fall 1. Alle Symmetrien in G sind eigentliche Bewegungen.

1  p1  C_{1}
2  p2  C_{2}
3  p3  C_{3}
4  p4  C_{4}
6  p6  C_{6}

Fall 2. Es gibt Symmetrien in G, die uneigentliche Bewegungen sind.

1 Jede Achse ist Spiegelachse.  pm  D_{1}
Es gibt keine Spiegelachse.  pg  D_{1}

Es gibt Spiegelachsen,

aber auch Achsen, die keine Spiegelachsen sind.  cm  D_{1}
2 Jedes Drehzentrum liegt auf zwei orthogonalen Spiegelachsen. pmm  D_{2}
Ein Drehzentrum liegt auf genau einer Achse.  pmg  D_{2}
Ein Drehzentrum liegt auf keiner Achse.  pgg  D_{2}

Es gibt ein Drehzentrum, das auf zwei orthogonalen Achsen,

die keine Spiegelachsen sind, liegt.  cmm  D_{2}
3 Jedes Dreierzentrum liegt auf einer Spiegelachse.  p3m1  D_{3}
Es gibt ein Dreierzentrum, das auf keiner Spiegelachse liegt. p31m  D_{3}
4 Ein Viererzentrum liegt auf einer Spiegelachse.  p4m  D_{4}
Ein Viererzentrum liegt auf keiner Spiegelachse.  p4g  D_{4}
6  p6m  D_{6}

Es gibt genau 230 räumliche Kristallgruppen. Auch sie wurden von A.Schoenfließ und E.S.Fedorow klassifiziert.

Die regulären Polyeder:

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Würfel  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Oktaeder  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