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Diffstat (limited to '')
| -rw-r--r-- | buch/papers/punktgruppen/Makefile.inc | 12 | ||||
| -rw-r--r-- | buch/papers/punktgruppen/main.tex | 58 | ||||
| -rw-r--r-- | buch/papers/punktgruppen/packages.tex | 5 | ||||
| -rw-r--r-- | buch/papers/punktgruppen/teil0.tex | 22 | ||||
| -rw-r--r-- | buch/papers/punktgruppen/teil1.tex | 55 | ||||
| -rw-r--r-- | buch/papers/punktgruppen/teil2.tex | 40 | ||||
| -rw-r--r-- | buch/papers/punktgruppen/teil3.tex | 40 | ||||
| -rw-r--r-- | vorlesungen/punktgruppen/.gitignore | 8 | ||||
| -rw-r--r-- | vorlesungen/punktgruppen/Makefile | 18 | ||||
| -rw-r--r-- | vorlesungen/punktgruppen/crystals.py | 472 | ||||
| -rw-r--r-- | vorlesungen/punktgruppen/poetry.lock | 591 | ||||
| -rw-r--r-- | vorlesungen/punktgruppen/pyproject.toml | 15 | ||||
| -rw-r--r-- | vorlesungen/punktgruppen/script.pdf | bin | 0 -> 25412 bytes | |||
| -rw-r--r-- | vorlesungen/punktgruppen/script.tex | 63 |
14 files changed, 1203 insertions, 196 deletions
diff --git a/buch/papers/punktgruppen/Makefile.inc b/buch/papers/punktgruppen/Makefile.inc index 7c6e70d..629abca 100644 --- a/buch/papers/punktgruppen/Makefile.inc +++ b/buch/papers/punktgruppen/Makefile.inc @@ -3,12 +3,8 @@ # # (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule # -dependencies-punktgruppen = \ - papers/punktgruppen/packages.tex \ - papers/punktgruppen/main.tex \ - papers/punktgruppen/references.bib \ - papers/punktgruppen/teil0.tex \ - papers/punktgruppen/teil1.tex \ - papers/punktgruppen/teil2.tex \ - papers/punktgruppen/teil3.tex +dependencies-punktgruppen = \ + papers/punktgruppen/packages.tex \ + papers/punktgruppen/main.tex \ + papers/punktgruppen/references.bib diff --git a/buch/papers/punktgruppen/main.tex b/buch/papers/punktgruppen/main.tex index fc91913..603f293 100644 --- a/buch/papers/punktgruppen/main.tex +++ b/buch/papers/punktgruppen/main.tex @@ -3,34 +3,38 @@ % % (c) 2020 Hochschule Rapperswil % -\chapter{Thema\label{chapter:punktgruppen}} -\lhead{Thema} +\chapter{Crystal M\rotatebox[origin=c]{180}{a}th\label{chapter:punktgruppen}} +\lhead{Crystal M\rotatebox[origin=c]{180}{a}th} \begin{refsection} -\chapterauthor{Hans Muster} - -Ein paar Hinweise für die korrekte Formatierung des Textes -\begin{itemize} -\item -Absätze werden gebildet, indem man eine Leerzeile einfügt. -Die Verwendung von \verb+\\+ ist nur in Tabellen und Arrays gestattet. -\item -Die explizite Platzierung von Bildern ist nicht erlaubt, entsprechende -Optionen werden gelöscht. -Verwenden Sie Labels und Verweise, um auf Bilder hinzuweisen. -\item -Beginnen Sie jeden Satz auf einer neuen Zeile. -Damit ermöglichen Sie dem Versionsverwaltungssysteme, Änderungen -in verschiedenen Sätzen von verschiedenen Autoren ohne Konflikt -anzuwenden. -\item -Bilden Sie auch für Formeln kurze Zeilen, einerseits der besseren -Übersicht wegen, aber auch um GIT die Arbeit zu erleichtern. -\end{itemize} - -\input{papers/punktgruppen/teil0.tex} -\input{papers/punktgruppen/teil1.tex} -\input{papers/punktgruppen/teil2.tex} -\input{papers/punktgruppen/teil3.tex} +\chapterauthor{Tim T\"onz, Naoki Pross} + +%% TODO: remove +%% Some ideas to motivate the topic: +%% - Physics in a crystal lattice structure +%% - Birifrencenge and scattering of light / Xray in Crystals +%% - Electron density function in a lattice +%% - Heat diffusion with lattice model +%% - Ising model for ferromagnetism (?? => H.D. Lang) +%% +%% - Homomorphic encryption (or lattice based cryptography) +%% + Q: Is it possible to edit encrypted data without decrypting it first? + +%% TODO: translated and move into a file {{{ + +\section{Motivation} +% birifrengence + +\section{Math} +% lattice group +% symmetry +% space group + +\section{Physics} +\subsection{Electromagnetic Waves} +\subsection{Crystal Lattice} + + +%% }}} \printbibliography[heading=subbibliography] \end{refsection} diff --git a/buch/papers/punktgruppen/packages.tex b/buch/papers/punktgruppen/packages.tex index 971bcfe..9953339 100644 --- a/buch/papers/punktgruppen/packages.tex +++ b/buch/papers/punktgruppen/packages.tex @@ -4,7 +4,4 @@ % (c) 2019 Prof Dr Andreas Müller, Hochschule Rapperswil % -% if your paper needs special packages, add package commands as in the -% following example -%\usepackage{packagename} - +\usepackage{tikz-3dplot} diff --git a/buch/papers/punktgruppen/teil0.tex b/buch/papers/punktgruppen/teil0.tex deleted file mode 100644 index 5a8278e..0000000 --- a/buch/papers/punktgruppen/teil0.tex +++ /dev/null @@ -1,22 +0,0 @@ -% -% einleitung.tex -- Beispiel-File für die Einleitung -% -% (c) 2020 Prof Dr Andreas Müller, Hochschule Rapperswil -% -\section{Teil 0\label{punktgruppen:section:teil0}} -\rhead{Teil 0} -Lorem ipsum dolor sit amet, consetetur sadipscing elitr, sed diam -nonumy eirmod tempor invidunt ut labore et dolore magna aliquyam -erat, sed diam voluptua \cite{punktgruppen:bibtex}. -At vero eos et accusam et justo duo dolores et ea rebum. -Stet clita kasd gubergren, no sea takimata sanctus est Lorem ipsum -dolor sit amet. - -Lorem ipsum dolor sit amet, consetetur sadipscing elitr, sed diam -nonumy eirmod tempor invidunt ut labore et dolore magna aliquyam -erat, sed diam voluptua. -At vero eos et accusam et justo duo dolores et ea rebum. Stet clita -kasd gubergren, no sea takimata sanctus est Lorem ipsum dolor sit -amet. - - diff --git a/buch/papers/punktgruppen/teil1.tex b/buch/papers/punktgruppen/teil1.tex deleted file mode 100644 index 228af33..0000000 --- a/buch/papers/punktgruppen/teil1.tex +++ /dev/null @@ -1,55 +0,0 @@ -% -% teil1.tex -- Beispiel-File für das Paper -% -% (c) 2020 Prof Dr Andreas Müller, Hochschule Rapperswil -% -\section{Teil 1 -\label{punktgruppen:section:teil1}} -\rhead{Problemstellung} -Sed ut perspiciatis unde omnis iste natus error sit voluptatem -accusantium doloremque laudantium, totam rem aperiam, eaque ipsa -quae ab illo inventore veritatis et quasi architecto beatae vitae -dicta sunt explicabo. -Nemo enim ipsam voluptatem quia voluptas sit aspernatur aut odit -aut fugit, sed quia consequuntur magni dolores eos qui ratione -voluptatem sequi nesciunt -\begin{equation} -\int_a^b x^2\, dx -= -\left[ \frac13 x^3 \right]_a^b -= -\frac{b^3-a^3}3. -\label{punktgruppen:equation1} -\end{equation} -Neque porro quisquam est, qui dolorem ipsum quia dolor sit amet, -consectetur, adipisci velit, sed quia non numquam eius modi tempora -incidunt ut labore et dolore magnam aliquam quaerat voluptatem. - -Ut enim ad minima veniam, quis nostrum exercitationem ullam corporis -suscipit laboriosam, nisi ut aliquid ex ea commodi consequatur? -Quis autem vel eum iure reprehenderit qui in ea voluptate velit -esse quam nihil molestiae consequatur, vel illum qui dolorem eum -fugiat quo voluptas nulla pariatur? - -\subsection{De finibus bonorum et malorum -\label{punktgruppen:subsection:finibus}} -At vero eos et accusamus et iusto odio dignissimos ducimus qui -blanditiis praesentium voluptatum deleniti atque corrupti quos -dolores et quas molestias excepturi sint occaecati cupiditate non -provident, similique sunt in culpa qui officia deserunt mollitia -animi, id est laborum et dolorum fuga \eqref{000tempmlate:equation1}. - -Et harum quidem rerum facilis est et expedita distinctio -\ref{punktgruppen:section:loesung}. -Nam libero tempore, cum soluta nobis est eligendi optio cumque nihil -impedit quo minus id quod maxime placeat facere possimus, omnis -voluptas assumenda est, omnis dolor repellendus -\ref{punktgruppen:section:folgerung}. -Temporibus autem quibusdam et aut officiis debitis aut rerum -necessitatibus saepe eveniet ut et voluptates repudiandae sint et -molestiae non recusandae. -Itaque earum rerum hic tenetur a sapiente delectus, ut aut reiciendis -voluptatibus maiores alias consequatur aut perferendis doloribus -asperiores repellat. - - diff --git a/buch/papers/punktgruppen/teil2.tex b/buch/papers/punktgruppen/teil2.tex deleted file mode 100644 index b48e785..0000000 --- a/buch/papers/punktgruppen/teil2.tex +++ /dev/null @@ -1,40 +0,0 @@ -% -% teil2.tex -- Beispiel-File für teil2 -% -% (c) 2020 Prof Dr Andreas Müller, Hochschule Rapperswil -% -\section{Teil 2 -\label{punktgruppen:section:teil2}} -\rhead{Teil 2} -Sed ut perspiciatis unde omnis iste natus error sit voluptatem -accusantium doloremque laudantium, totam rem aperiam, eaque ipsa -quae ab illo inventore veritatis et quasi architecto beatae vitae -dicta sunt explicabo. Nemo enim ipsam voluptatem quia voluptas sit -aspernatur aut odit aut fugit, sed quia consequuntur magni dolores -eos qui ratione voluptatem sequi nesciunt. Neque porro quisquam -est, qui dolorem ipsum quia dolor sit amet, consectetur, adipisci -velit, sed quia non numquam eius modi tempora incidunt ut labore -et dolore magnam aliquam quaerat voluptatem. Ut enim ad minima -veniam, quis nostrum exercitationem ullam corporis suscipit laboriosam, -nisi ut aliquid ex ea commodi consequatur? Quis autem vel eum iure -reprehenderit qui in ea voluptate velit esse quam nihil molestiae -consequatur, vel illum qui dolorem eum fugiat quo voluptas nulla -pariatur? - -\subsection{De finibus bonorum et malorum -\label{punktgruppen:subsection:bonorum}} -At vero eos et accusamus et iusto odio dignissimos ducimus qui -blanditiis praesentium voluptatum deleniti atque corrupti quos -dolores et quas molestias excepturi sint occaecati cupiditate non -provident, similique sunt in culpa qui officia deserunt mollitia -animi, id est laborum et dolorum fuga. Et harum quidem rerum facilis -est et expedita distinctio. Nam libero tempore, cum soluta nobis -est eligendi optio cumque nihil impedit quo minus id quod maxime -placeat facere possimus, omnis voluptas assumenda est, omnis dolor -repellendus. Temporibus autem quibusdam et aut officiis debitis aut -rerum necessitatibus saepe eveniet ut et voluptates repudiandae -sint et molestiae non recusandae. Itaque earum rerum hic tenetur a -sapiente delectus, ut aut reiciendis voluptatibus maiores alias -consequatur aut perferendis doloribus asperiores repellat. - - diff --git a/buch/papers/punktgruppen/teil3.tex b/buch/papers/punktgruppen/teil3.tex deleted file mode 100644 index 94abd74..0000000 --- a/buch/papers/punktgruppen/teil3.tex +++ /dev/null @@ -1,40 +0,0 @@ -% -% teil3.tex -- Beispiel-File für Teil 3 -% -% (c) 2020 Prof Dr Andreas Müller, Hochschule Rapperswil -% -\section{Teil 3 -\label{punktgruppen:section:teil3}} -\rhead{Teil 3} -Sed ut perspiciatis unde omnis iste natus error sit voluptatem -accusantium doloremque laudantium, totam rem aperiam, eaque ipsa -quae ab illo inventore veritatis et quasi architecto beatae vitae -dicta sunt explicabo. Nemo enim ipsam voluptatem quia voluptas sit -aspernatur aut odit aut fugit, sed quia consequuntur magni dolores -eos qui ratione voluptatem sequi nesciunt. Neque porro quisquam -est, qui dolorem ipsum quia dolor sit amet, consectetur, adipisci -velit, sed quia non numquam eius modi tempora incidunt ut labore -et dolore magnam aliquam quaerat voluptatem. Ut enim ad minima -veniam, quis nostrum exercitationem ullam corporis suscipit laboriosam, -nisi ut aliquid ex ea commodi consequatur? Quis autem vel eum iure -reprehenderit qui in ea voluptate velit esse quam nihil molestiae -consequatur, vel illum qui dolorem eum fugiat quo voluptas nulla -pariatur? - -\subsection{De finibus bonorum et malorum -\label{punktgruppen:subsection:malorum}} -At vero eos et accusamus et iusto odio dignissimos ducimus qui -blanditiis praesentium voluptatum deleniti atque corrupti quos -dolores et quas molestias excepturi sint occaecati cupiditate non -provident, similique sunt in culpa qui officia deserunt mollitia -animi, id est laborum et dolorum fuga. Et harum quidem rerum facilis -est et expedita distinctio. Nam libero tempore, cum soluta nobis -est eligendi optio cumque nihil impedit quo minus id quod maxime -placeat facere possimus, omnis voluptas assumenda est, omnis dolor -repellendus. Temporibus autem quibusdam et aut officiis debitis aut -rerum necessitatibus saepe eveniet ut et voluptates repudiandae -sint et molestiae non recusandae. Itaque earum rerum hic tenetur a -sapiente delectus, ut aut reiciendis voluptatibus maiores alias -consequatur aut perferendis doloribus asperiores repellat. - - diff --git a/vorlesungen/punktgruppen/.gitignore b/vorlesungen/punktgruppen/.gitignore new file mode 100644 index 0000000..068a456 --- /dev/null +++ b/vorlesungen/punktgruppen/.gitignore @@ -0,0 +1,8 @@ +# directories +__pycache__ +media +build + +# files +script.log +missfont.log diff --git a/vorlesungen/punktgruppen/Makefile b/vorlesungen/punktgruppen/Makefile new file mode 100644 index 0000000..eb4eed6 --- /dev/null +++ b/vorlesungen/punktgruppen/Makefile @@ -0,0 +1,18 @@ +TEX=xelatex +TEXARGS=--output-directory=build --halt-on-error + +all: script.pdf media + +.PHONY: clean +clean: + @rm -rfv build + # @rm -rfv media + +script.pdf: script.tex + mkdir -p build + $(TEX) $(TEXARGS) script.tex + cp build/script.pdf . + +media: + poetry install + poetry run manim -ql crystals.py diff --git a/vorlesungen/punktgruppen/crystals.py b/vorlesungen/punktgruppen/crystals.py new file mode 100644 index 0000000..76dee1f --- /dev/null +++ b/vorlesungen/punktgruppen/crystals.py @@ -0,0 +1,472 @@ +from manim import * + +import math as m +import numpy as np +import itertools as it + +# configure style +config.background_color = '#202020' +config.tex_template.add_to_preamble( + r"\usepackage[p,osf]{scholax}" + r"\usepackage{amsmath}" + r"\usepackage[scaled=1.075,ncf,vvarbb]{newtxmath}" +) + +# scenes +class Geometric2DSymmetries(Scene): + def construct(self): + # title + title = Tex(r"Geometrische \\ Symmetrien") + title.scale(1.5) + self.play(Write(title)) + self.wait() + self.play(FadeOut(title)) + self.wait() + + self.intro() + self.cyclic() + self.dihedral() + + def intro(self): + # create square + square = Square() + square.set_fill(PINK, opacity=.5) + self.play(SpinInFromNothing(square)) + self.wait() + + # the action of doing nothing + action = MathTex(r"\mathbb{1}") + self.play(Write(action)) + self.play(ApplyMethod(square.scale, 1.2)) + self.play(ApplyMethod(square.scale, 1/1.2)) + self.play(FadeOut(action)) + + # show some reflections + axis = DashedLine(2 * LEFT, 2 * RIGHT) + sigma = MathTex(r"\sigma") + sigma.next_to(axis, RIGHT) + + self.play(Create(axis)) + self.play(Write(sigma)) + + self.play(ApplyMethod(square.flip, RIGHT)) + self.wait() + + for d in [UP + RIGHT, UP]: + self.play( + Rotate(axis, PI/4), + Rotate(sigma, PI/4, about_point=ORIGIN)) + + self.play(Rotate(sigma, -PI/4), run_time=.5) + self.play(ApplyMethod(square.flip, d)) + + self.play( + FadeOutAndShift(sigma), + Uncreate(axis)) + + # show some rotations + dot = Dot(UP + RIGHT) + figure = VGroup(square, dot) + + rot = MathTex(r"r") + self.play(Write(rot), Create(dot)) + + last = rot + for newrot in map(MathTex, [r"r", r"r^2", r"r^3"]): + self.play( + ReplacementTransform(last, newrot), + Rotate(figure, PI/2, about_point=ORIGIN)) + self.wait() + last = newrot + + self.play(Uncreate(dot), FadeOut(square), FadeOut(last)) + + + def cyclic(self): + # create symmetric figure + figure = VGroup() + prev = [1.5, 0, 0] + for i in range(1,6): + pos = [ + 1.5*m.cos(2 * PI/5 * i), + 1.5*m.sin(2 * PI/5 * i), + 0 + ] + + if prev: + line = Line(prev, pos) + figure.add(line) + + dot = Dot(pos, radius=.1) + if i == 5: + dot.set_fill(RED) + + prev = pos + figure.add(dot) + + group = MathTex(r"G = \langle r \rangle") + self.play(Write(group), run_time = 2) + self.wait() + self.play(ApplyMethod(group.to_edge, UP)) + + actions = map(MathTex, [ + r"\mathbb{1}", r"r", r"r^2", + r"r^3", r"r^4", r"\mathbb{1}"]) + + action = next(actions, MathTex(r"r")) + + self.play(Create(figure)) + self.play(Write(action)) + self.wait() + + for i in range(5): + newaction = next(actions, MathTex(r"r")) + self.play( + ReplacementTransform(action, newaction), + Rotate(figure, 2*PI/5, about_point=ORIGIN)) + action = newaction + + self.play(Uncreate(figure), FadeOut(action)) + + whole_group = MathTex( + r"G = \langle r \rangle" + r"= \left\{\mathbb{1}, r, r^2, r^3, r^4 \right\}") + + self.play(ApplyMethod(group.move_to, ORIGIN)) + self.play(ReplacementTransform(group, whole_group)) + self.wait() + + cyclic = MathTex( + r"Z_n = \langle r \rangle" + r"= \left\{\mathbb{1}, r, r^2, \dots, r^{n-1} \right\}") + + cyclic_title = Tex(r"Zyklische Gruppe") + cyclic_title.next_to(cyclic, UP * 2) + + cyclic.scale(1.2) + cyclic_title.scale(1.2) + + self.play(ReplacementTransform(whole_group, cyclic)) + self.play(FadeInFrom(cyclic_title, UP)) + + self.wait(5) + self.play(FadeOut(cyclic), FadeOut(cyclic_title)) + + def dihedral(self): + # create square + square = Square() + square.set_fill(PINK, opacity=.5) + + # generator equation + group = MathTex( + r"G = \langle \sigma, r \,|\,", + r"\sigma^2 = \mathbb{1},", + r"r^4 = \mathbb{1},", + r"(\sigma r)^2 = \mathbb{1} \rangle") + + self.play(Write(group), run_time = 2) + self.wait() + self.play(ApplyMethod(group.to_edge, UP)) + self.play(FadeIn(square)) + + axis = DashedLine(2 * LEFT, 2 * RIGHT) + sigma = MathTex(r"\sigma^2 = \mathbb{1}") + sigma.next_to(axis, RIGHT) + self.play(Create(axis), Write(sigma)) + self.play(ApplyMethod(square.flip, RIGHT)) + self.play(ApplyMethod(square.flip, RIGHT)) + self.play(Uncreate(axis), FadeOut(sigma)) + + # rotations + dot = Dot(UP + RIGHT) + rot = MathTex(r"r^4 = \mathbb{1}") + rot.next_to(square, DOWN * 3) + + figure = VGroup(dot, square) + + self.play(Write(rot), Create(dot)) + for i in range(4): + self.play(Rotate(figure, PI/2)) + self.play(FadeOut(rot), Uncreate(dot)) + + # rotation and flip + action = MathTex(r"(\sigma r)^2 = \mathbb{1}") + action.next_to(square, DOWN * 5) + + dot = Dot(UP + RIGHT) + axis = DashedLine(2 * LEFT, 2 * RIGHT) + self.play(Create(dot), Create(axis), Write(action)) + + figure = VGroup(dot, square) + + for i in range(2): + self.play(Rotate(figure, PI/2)) + self.play(ApplyMethod(figure.flip, RIGHT)) + self.wait() + + self.play(Uncreate(dot), Uncreate(axis), FadeOut(action)) + self.play(FadeOut(square)) + + # equation for the whole + whole_group = MathTex( + r"G &= \langle \sigma, r \,|\," + r"\sigma^2 = r^4 = (\sigma r)^2 = \mathbb{1} \rangle \\" + r"&= \left\{" + r"\mathbb{1}, r, r^2, r^3, \sigma, \sigma r, \sigma r^2, \sigma r^3" + r"\right\}") + + self.play(ApplyMethod(group.move_to, ORIGIN)) + self.play(ReplacementTransform(group, whole_group)) + self.wait(2) + + dihedral = MathTex( + r"D_n &= \langle \sigma, r \,|\," + r"\sigma^2 = r^n = (\sigma r)^2 = \mathbb{1} \rangle \\" + r"&= \left\{" + r"\mathbb{1}, r, r^2, \dots, \sigma, \sigma r, \sigma r^2, \dots" + r"\right\}") + + dihedral_title = Tex(r"Diedergruppe: Symmetrien eines \(n\)-gons") + dihedral_title.next_to(dihedral, UP * 2) + + dihedral.scale(1.2) + dihedral_title.scale(1.2) + + self.play(ReplacementTransform(whole_group, dihedral)) + self.play(FadeInFrom(dihedral_title, UP)) + + self.wait(5) + self.play(FadeOut(dihedral), FadeOut(dihedral_title)) + + +class Geometric3DSymmetries(ThreeDScene): + def construct(self): + self.symmetric() + + + @staticmethod + def get_cube(): + verts = np.array(list(it.product(*3 * [[-1, 1]]))) + edges = [ + (v1, v2) + for v1, v2 in it.combinations(verts, 2) + if sum(v1 == v2) == 2 + ] + corner_dots = Group(*[ + Sphere().set_height(0.25).move_to(vert) + for vert in verts + ]) + corner_dots.set_color(GREY_B) + edge_rods = Group(*[ + Line3D(v1, v2) + for v1, v2 in edges + ]) + + faces = Cube(square_resolution=(10, 10)) + faces.set_height(2) + faces.set_color(BLUE_E, 0.3) + # faces.add_updater(lambda m: m.sort(lambda p: np.dot(p, [np.sign(self.euler_angles[0]) * 0.2, -1, 0.2]))) + + cube = Group(corner_dots, edge_rods, faces) + cube.corner_dots = corner_dots + cube.edge_rods = edge_rods + cube.faces = faces + return cube + + def symmetric(self): + self.renderer.camera.light_source.move_to(3*IN) # changes the source of the light + self.set_camera_orientation(phi=60 * DEGREES, theta=5 * DEGREES) + + cube = Cube() + self.play(GrowFromCenter(cube)) + + axes = list( + map(lambda v: v / np.linalg.norm(v), + map(np.array, [ + [0, 0, 1], + [0, 1, 1], + [1, 1, 1], + ]) + )) + angles = [ PI, PI, PI * 2/3 ] + lines = list(map(lambda x: Line(-2 * x, 2 * x), axes)) + + camera_thetas = list(map(lambda x: x * DEGREES, [10, 100, 110])) + for axis, line, angle, camera_angle in zip(axes, lines, angles, camera_thetas): + self.move_camera(theta=camera_angle) + self.play(Create(line)) + self.play(Rotate(cube, angle, axis=axis, run_time=3)) + + self.wait(7) + + +class AlgebraicSymmetries(Scene): + def construct(self): + # title + title = Tex(r"Algebraische \\ Symmetrien") + title.scale(1.5) + self.play(Write(title)) + self.wait() + self.play(FadeOut(title)) + self.wait() + + self.cyclic() + self.matrices() + + def cyclic(self): + # show the i product + product = MathTex( + r"1", r"\cdot i &= i \\", + r"i \cdot i &= -1 \\", + r"-1 \cdot i &= -i \\", + r"-i \cdot i &= 1") + product.scale(1.5) + + for part in product: + self.play(Write(part)) + + self.wait() + self.play(ApplyMethod(product.scale, 1/1.5)) + + # gather in group + group = MathTex(r"G = \left\{ 1, i, -1, -i \right\}") + self.play(ReplacementTransform(product, group)) + self.wait() + + # show Z4 + grouppow = MathTex( + r"G &= \left\{ 1, i, i^2, i^3 \right\} \\", + r"Z_4 &= \left\{ \mathbb{1}, r, r^2, r^3 \right\}") + self.play(ReplacementTransform(group, grouppow[0])) + self.wait() + + self.play(Write(grouppow[1])) + self.wait() + self.play(ApplyMethod(grouppow.to_edge, UP)) + + # define morphisms + morphism = MathTex(r"\phi: Z_4 \to G \\") + morphism.shift(UP) + self.play(Write(morphism)) + + # show an example + mappings = MathTex( + r"\phi(\mathbb{1}) &= 1 \\", + r"\phi(r) &= i \\", + r"\phi(r^2) &= i^2 \\", + r"\phi(r^3) &= i^3 \\") + mappings.next_to(morphism, DOWN) + + self.play(Write(mappings)) + self.wait() + self.play(FadeOutAndShift(mappings, DOWN)) + + # more general definition + homomorphism = MathTex( + r"\phi(r\circ \mathbb{1}) &= i\cdot 1 \\", + r"&= \phi(r)\cdot\phi(\mathbb{1})") + homomorphism.next_to(morphism, DOWN).align_to(morphism, LEFT) + for part in homomorphism: + self.play(Write(part)) + + hom_bracegrp = VGroup(morphism, homomorphism) + + self.play( + ApplyMethod(grouppow.shift, 2.5 * LEFT), + ApplyMethod(hom_bracegrp.shift, 2.5 * LEFT)) + + hom_brace = Brace(hom_bracegrp, direction=RIGHT) + hom_text = Tex("Homomorphismus").next_to(hom_brace.get_tip(), RIGHT) + hom_text_short = MathTex(r"G \simeq Z_4").next_to(hom_brace.get_tip(), RIGHT) + + self.play(Create(hom_brace)) + self.play(Write(hom_text)) + self.play(ReplacementTransform(hom_text, hom_text_short)) + self.wait() + + self.play(FadeOut(hom_brace), FadeOut(hom_text_short)) + + # add the isomorphism part + isomorphism = Tex(r"\(\phi\) ist bijektiv") + isomorphism.next_to(homomorphism, DOWN).align_to(homomorphism, LEFT) + self.play(Write(isomorphism)) + + iso_bracegrp = VGroup(hom_bracegrp, isomorphism) + + iso_brace = Brace(iso_bracegrp, RIGHT) + iso_text = Tex("Isomorphismus").next_to(iso_brace.get_tip(), RIGHT) + iso_text_short = MathTex("Z_4 \cong G").next_to(iso_brace.get_tip(), RIGHT) + + self.play(Create(iso_brace)) + self.play(Write(iso_text)) + self.wait() + + self.play(ReplacementTransform(iso_text, iso_text_short)) + self.wait() + + # create a group for the whole + morphgrp = VGroup(iso_bracegrp, iso_brace, iso_text_short) + + self.play( + ApplyMethod(grouppow.to_edge, LEFT), + ApplyMethod(morphgrp.to_edge, LEFT)) + # self.play( + # FadeOutAndShift(grouppow, UP), + # FadeOutAndShift(morphgrp, DOWN)) + + # draw a complex plane + plane = ComplexPlane(x_min = -2, x_max = 3) + coordinates = plane.get_coordinate_labels(1, -1, 1j, -1j) + + roots = list(map(lambda p: Dot(p, fill_color=PINK), ( + [1, 0, 0], [0, 1, 0], [-1, 0, 0], [0, -1, 0] + ))) + + arrow = CurvedArrow( + 1.5 * np.array([m.cos(10 * DEGREES), m.sin(10 * DEGREES), 0]), + 1.5 * np.array([m.cos(80 * DEGREES), m.sin(80 * DEGREES), 0])) + arrowtext = MathTex("\cdot i") + arrowtext.move_to(2 / m.sqrt(2) * (UP + RIGHT)) + + square = Square().rotate(PI/4).scale(1/m.sqrt(2)) + square.set_fill(PINK).set_opacity(.4) + + figuregrp = Group(plane, square, arrow, arrowtext, *coordinates, *roots) + figuregrp.to_edge(RIGHT) + + self.play(Create(plane)) + self.play( + *map(Create, roots), + *map(Write, coordinates)) + self.wait() + self.play(FadeIn(square), Create(arrow), Write(arrowtext)) + + for _ in range(4): + self.play(Rotate(square, PI/2)) + self.wait(.5) + + self.play( + *map(FadeOut, (square, arrow, arrowtext)), + *map(FadeOut, coordinates), + *map(FadeOut, roots)) + self.play(Uncreate(plane)) + self.play( + FadeOutAndShift(grouppow, RIGHT), + FadeOutAndShift(morphgrp, RIGHT)) + + modulo = MathTex( + r"\phi: Z_4 &\to (\mathbb{Z}/4\mathbb{Z}, +) \\" + r"\phi(\mathbb{1} \circ r^2) &= 0 + 2 \pmod 4").scale(1.5) + self.play(Write(modulo)) + self.wait(2) + + self.play(FadeOut(modulo)) + self.wait(3) + + def matrices(self): 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a/vorlesungen/punktgruppen/script.pdf b/vorlesungen/punktgruppen/script.pdf Binary files differnew file mode 100644 index 0000000..293b248 --- /dev/null +++ b/vorlesungen/punktgruppen/script.pdf diff --git a/vorlesungen/punktgruppen/script.tex b/vorlesungen/punktgruppen/script.tex new file mode 100644 index 0000000..2a6d95c --- /dev/null +++ b/vorlesungen/punktgruppen/script.tex @@ -0,0 +1,63 @@ +\documentclass[a4paper]{article} + +\usepackage[cm]{manuscript} +\usepackage{xcolor} + +\newcommand{\scene}[1]{\noindent[ #1 ]\par} +\newenvironment{totranslate}{\color{blue!70!black}}{} + +\begin{document} + +\section{Intro} + +\section{Geometrie} +\begin{totranslate} +We'll start with geometric symmetries as they are the simplest to grasp. + +\scene{Intro} + To mathematically formulate the concept, we will think of symmetries as + actions to perform on an object, like this square. The simplest action, is to + take this square, do nothing and put it back down. Another action could be to + flip it along an axis, or to rotate it around its center by 90 degrees. + +\scene{Cyclic Groups} + Let's focus our attention on the simplest class of symmetries: those + generated by a single rotation. We will gather the symmetries in a group + \(G\), and denote that it is generated by a rotation \(r\) with these angle + brackets. + + Take this pentagon as an example. By applying the rotation \emph{action} 5 + times, it is the same as if we had not done anything, furthermore, if we + \emph{act} a sixth time with \(r\), it will be the same as if we had just + acted with \(r\) once. Thus the group only contain the identity and the + powers of \(r\) up to 4. + + In general, groups with this structure are known as the ``Cyclic Groups'' of + order \(n\), where the action \(r\) can be applied \(n-1\) times before + wrapping around. + + % You can think of them as the rotational symmetries of an \(n\)-gon. + +\scene{Dihedral Groups} + Okay that was not difficult, now let's spice this up a bit. Consider this + group for a square, generated by two actions: a rotation \(r\) and a + reflection \(\sigma\). Because we have two actions we have to write in the + generator how they relate to each other. + + Let's analyze this expression. Two reflections are the same as the identity. + Four rotations are the same as the identity, and a rotation followed by a + reflection, twice, is the same as the identity. + + This forms a group with 8 possible unique actions. This too can be generalized + to an \(n\)-gon, and is known as the ``Dihedral Group'' of order \(n\). +\end{totranslate} + +\scene{Symmetrische Gruppe} +\scene{Alternierende Gruppe} + +\section{Algebra} + +\section{Krystalle} + +\end{document} +% vim:et ts=2 sw=2: |