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authorAndreas Müller <andreas.mueller@ost.ch>2021-05-10 15:43:43 +0200
committerGitHub <noreply@github.com>2021-05-10 15:43:43 +0200
commit0d1ab0b01a3b0db1260d1bc287518fce52989f93 (patch)
tree728c6c602fbeecaf567ca4dc90c821cdbbb937b9 /vorlesungen/punktgruppen
parentnew slide (diff)
parentUpdate gitignore (diff)
downloadSeminarMatrizen-0d1ab0b01a3b0db1260d1bc287518fce52989f93.tar.gz
SeminarMatrizen-0d1ab0b01a3b0db1260d1bc287518fce52989f93.zip
Merge pull request #13 from NaoPross/master
Paper title & Presentation
Diffstat (limited to '')
-rw-r--r--vorlesungen/punktgruppen/.gitignore20
-rw-r--r--vorlesungen/punktgruppen/Makefile18
-rw-r--r--vorlesungen/punktgruppen/crystals.py611
-rw-r--r--vorlesungen/punktgruppen/media/images/nosignal.jpgbin0 -> 711846 bytes
-rw-r--r--vorlesungen/punktgruppen/poetry.lock743
-rw-r--r--vorlesungen/punktgruppen/pyproject.toml15
-rw-r--r--vorlesungen/punktgruppen/script.pdfbin0 -> 44991 bytes
-rw-r--r--vorlesungen/punktgruppen/script.tex214
-rw-r--r--vorlesungen/punktgruppen/slides.pdfbin0 -> 790926 bytes
-rw-r--r--vorlesungen/punktgruppen/slides.tex895
10 files changed, 2516 insertions, 0 deletions
diff --git a/vorlesungen/punktgruppen/.gitignore b/vorlesungen/punktgruppen/.gitignore
new file mode 100644
index 0000000..3633a3d
--- /dev/null
+++ b/vorlesungen/punktgruppen/.gitignore
@@ -0,0 +1,20 @@
+# directories
+__pycache__
+media/Tex
+
+media/images/crystal
+media/images/freezeframes
+
+media/videos
+media/audio
+
+media/Punktgruppen
+media/Punktgruppen.mp4
+
+build
+
+# files
+script.log
+slides.log
+slides.vrb
+missfont.log
diff --git a/vorlesungen/punktgruppen/Makefile b/vorlesungen/punktgruppen/Makefile
new file mode 100644
index 0000000..302e976
--- /dev/null
+++ b/vorlesungen/punktgruppen/Makefile
@@ -0,0 +1,18 @@
+TEX=xelatex
+TEXARGS=--output-directory=build --halt-on-error --shell-escape
+
+all: slides.pdf script.pdf media
+
+.PHONY: clean
+clean:
+ @rm -rfv build
+
+%.pdf: %.tex
+ mkdir -p build
+ $(TEX) $(TEXARGS) $<
+ $(TEX) $(TEXARGS) $<
+ cp build/$@ .
+
+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..4a9836a
--- /dev/null
+++ b/vorlesungen/punktgruppen/crystals.py
@@ -0,0 +1,611 @@
+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):
+ self.wait(5)
+
+ self.intro()
+ self.cyclic()
+ self.dihedral()
+ self.circle()
+
+ 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))
+ self.wait()
+
+ # 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))
+
+ 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(.5)
+ 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(3)
+
+ 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}", r"r"])
+
+ 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.wait()
+ newaction = next(actions, MathTex(r"r"))
+ self.play(
+ ReplacementTransform(action, newaction),
+ Rotate(figure, 2*PI/5, about_point=ORIGIN))
+ action = newaction
+ self.wait(2)
+
+ 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(5)
+
+ cyclic = MathTex(
+ r"C_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(5)
+
+ self.play(ApplyMethod(group.to_edge, UP))
+ self.play(FadeIn(square))
+ self.wait()
+
+ # flips
+ 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))
+
+ def circle(self):
+ circle = Circle(radius=2)
+ dot = Dot()
+ dot.move_to(2 * RIGHT)
+
+ figure = VGroup(circle, dot)
+ group_name = MathTex(r"C_\infty")
+
+ # create circle
+ self.play(Create(circle))
+ self.play(Create(dot))
+
+ # move it around
+ self.play(Rotate(figure, PI/3))
+ self.play(Rotate(figure, PI/6))
+ self.play(Rotate(figure, -PI/3))
+
+ # show name
+ self.play(Rotate(figure, PI/4), Write(group_name))
+ self.wait()
+ self.play(Uncreate(figure))
+
+ nsphere = MathTex(r"C_\infty \cong S^1 = \left\{z \in \mathbb{C} : |z| = 1\right\}")
+ nsphere_title = Tex(r"Kreisgruppe")
+ nsphere_title.next_to(nsphere, 2 * UP)
+
+ nsphere.scale(1.2)
+ nsphere_title.scale(1.2)
+
+ self.play(ReplacementTransform(group_name, nsphere))
+ self.play(FadeInFrom(nsphere_title, UP))
+
+ self.wait(5)
+ self.play(FadeOut(nsphere_title), FadeOut(nsphere))
+ self.wait(2)
+
+
+class Geometric3DSymmetries(ThreeDScene):
+ def construct(self):
+ self.improper_rotation()
+ self.tetrahedron()
+
+ def improper_rotation(self):
+ # changes the source of the light and camera
+ self.renderer.camera.light_source.move_to(3*IN)
+ self.set_camera_orientation(phi=0, theta=0)
+
+ # initial square
+ square = Square()
+ square.set_fill(PINK, opacity=.5)
+
+ self.play(SpinInFromNothing(square))
+ self.wait(2)
+
+ for i in range(4):
+ self.play(Rotate(square, PI/2))
+ self.wait(.5)
+
+ self.move_camera(phi= 75 * DEGREES, theta = -80 * DEGREES)
+
+ # create rotation axis
+ axis = Line3D(start=[0,0,-2.5], end=[0,0,2.5])
+
+ axis_name = MathTex(r"r \in C_4")
+ # move to yz plane
+ axis_name.rotate(PI/2, axis = RIGHT)
+ axis_name.next_to(axis, OUT)
+
+ self.play(Create(axis))
+ self.play(Write(axis_name))
+ self.wait()
+
+ # create sphere from slices
+ cyclic_slices = []
+ for i in range(4):
+ colors = [PINK, RED] if i % 2 == 0 else [BLUE_D, BLUE_E]
+ cyclic_slices.append(ParametricSurface(
+ lambda u, v: np.array([
+ np.sqrt(2) * np.cos(u) * np.cos(v),
+ np.sqrt(2) * np.cos(u) * np.sin(v),
+ np.sqrt(2) * np.sin(u)
+ ]),
+ v_min=PI/4 + PI/2 * i,
+ v_max=PI/4 + PI/2 * (i + 1),
+ u_min=-PI/2, u_max=PI/2,
+ checkerboard_colors=colors, resolution=(10,5)))
+
+ self.play(FadeOut(square), *map(Create, cyclic_slices))
+
+ cyclic_sphere = VGroup(*cyclic_slices)
+ for i in range(4):
+ self.play(Rotate(cyclic_sphere, PI/2))
+ self.wait()
+
+ new_axis_name = MathTex(r"r \in D_4")
+ # move to yz plane
+ new_axis_name.rotate(PI/2, axis = RIGHT)
+ new_axis_name.next_to(axis, OUT)
+ self.play(ReplacementTransform(axis_name, new_axis_name))
+
+ # reflection plane
+ self.play(FadeOut(cyclic_sphere), FadeIn(square))
+ plane = ParametricSurface(
+ lambda u, v: np.array([u, 0, v]),
+ u_min = -2, u_max = 2,
+ v_min = -2, v_max = 2,
+ fill_opacity=.3, resolution=(1,1))
+
+ plane_name = MathTex(r"\sigma \in D_4")
+ # move to yz plane
+ plane_name.rotate(PI/2, axis = RIGHT)
+ plane_name.next_to(plane, OUT + RIGHT)
+
+ self.play(Create(plane))
+ self.play(Write(plane_name))
+ self.wait()
+
+ self.move_camera(phi = 25 * DEGREES, theta = -75 * DEGREES)
+ self.wait()
+
+ condition = MathTex(r"(\sigma r)^2 = \mathbb{1}")
+ condition.next_to(square, DOWN);
+
+ self.play(Write(condition))
+ self.play(Rotate(square, PI/2))
+ self.play(Rotate(square, PI, RIGHT))
+
+ self.play(Rotate(square, PI/2))
+ self.play(Rotate(square, PI, RIGHT))
+ self.play(FadeOut(condition))
+
+ self.move_camera(phi = 75 * DEGREES, theta = -80 * DEGREES)
+
+ # create sphere from slices
+ dihedral_slices = []
+ for i in range(4):
+ for j in range(2):
+ colors = [PINK, RED] if i % 2 == 0 else [BLUE_D, BLUE_E]
+ dihedral_slices.append(ParametricSurface(
+ lambda u, v: np.array([
+ np.sqrt(2) * np.cos(u) * np.cos(v),
+ np.sqrt(2) * np.cos(u) * np.sin(v),
+ np.sqrt(2) * np.sin(u)
+ ]),
+ v_min=PI/2 * j + PI/4 + PI/2 * i,
+ v_max=PI/2 * j + PI/4 + PI/2 * (i + 1),
+ u_min=-PI/2 if j == 0 else 0,
+ u_max=0 if j == 0 else PI/2,
+ checkerboard_colors=colors, resolution=(10,5)))
+
+ dihedral_sphere = VGroup(*dihedral_slices)
+
+ self.play(FadeOut(square), Create(dihedral_sphere))
+
+ for i in range(2):
+ self.play(Rotate(dihedral_sphere, PI/2))
+ self.play(Rotate(dihedral_sphere, PI, RIGHT))
+ self.wait()
+
+ self.wait(2)
+ self.play(*map(FadeOut, [dihedral_sphere, plane, plane_name, new_axis_name]), FadeIn(square))
+ self.wait(3)
+ self.play(*map(FadeOut, [square, axis]))
+ self.wait(3)
+
+ def tetrahedron(self):
+ tet = Tetrahedron(edge_length=2)
+ self.play(FadeIn(tet))
+
+ self.move_camera(phi = 75 * DEGREES, theta = -100 * DEGREES)
+ self.begin_ambient_camera_rotation(rate=.1)
+
+ axes = []
+ for coord in tet.vertex_coords:
+ axes.append((-2 * coord, 2 * coord))
+
+ lines = [
+ Line3D(start=s, end=e) for s, e in axes
+ ]
+
+ self.play(*map(Create, lines))
+ self.wait()
+
+ for axis in axes:
+ self.play(Rotate(tet, 2*PI/3, axis=axis[1]))
+ self.play(Rotate(tet, 2*PI/3, axis=axis[1]))
+
+ self.wait(5)
+ self.stop_ambient_camera_rotation()
+ self.wait()
+ self.play(*map(Uncreate, lines))
+ self.play(FadeOut(tet))
+ self.wait(5)
+
+
+class AlgebraicSymmetries(Scene):
+ def construct(self):
+ self.wait(5)
+ 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(2)
+
+ # show C4
+ grouppow = MathTex(
+ r"G &= \left\{ 1, i, i^2, i^3 \right\} \\",
+ r"C_4 &= \left\{ \mathbb{1}, r, r^2, r^3 \right\}")
+ self.play(ReplacementTransform(group, grouppow[0]))
+ self.wait(2)
+
+ self.play(Write(grouppow[1]))
+ self.wait(4)
+
+ self.play(ApplyMethod(grouppow.to_edge, UP))
+
+ # define morphisms
+ morphism = MathTex(r"\phi: C_4 \to G \\")
+ morphism.shift(UP)
+ self.play(Write(morphism))
+ self.wait()
+
+ # 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, 2 * DOWN)
+
+ self.play(Write(mappings))
+ self.wait(3)
+ self.play(FadeOutAndShift(mappings, DOWN))
+
+ # more general definition
+ homomorphism = MathTex(
+ r"\phi(r\circ \mathbb{1}) &= \phi(r)\cdot\phi(\mathbb{1}) \\",
+ r"&= i\cdot 1")
+ homomorphism.next_to(morphism, DOWN).align_to(morphism, LEFT)
+ for part in homomorphism:
+ self.play(Write(part))
+ self.wait()
+
+ hom_bracegrp = VGroup(morphism, homomorphism)
+
+ self.play(
+ ApplyMethod(grouppow.shift, 3 * LEFT),
+ ApplyMethod(hom_bracegrp.shift, 3 * LEFT))
+
+ hom_brace = Brace(hom_bracegrp, direction=RIGHT)
+ hom_text = Tex("Homomorphismus").next_to(hom_brace.get_tip(), RIGHT)
+ hom_text_short = MathTex(r"\mathrm{Hom}(C_4, G)").next_to(hom_brace.get_tip(), RIGHT)
+
+ self.play(Create(hom_brace))
+ self.play(Write(hom_text))
+ self.wait()
+ self.play(ReplacementTransform(hom_text, hom_text_short))
+ self.wait()
+
+ # 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("C_4 \cong G").next_to(iso_brace.get_tip(), RIGHT)
+
+ self.play(
+ ReplacementTransform(hom_brace, iso_brace),
+ ReplacementTransform(hom_text_short, 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))
+
+ # draw a complex plane
+ plane = ComplexPlane(x_range = [-2.5, 2.5])
+ 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 = VGroup(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: C_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):
+ question = MathTex(
+ r"D_n &\cong \,? \\"
+ r"S_n &\cong \,? \\"
+ r"A_n &\cong \,?").scale(1.5)
+
+ answer = MathTex(
+ r"D_n &\cong \,?\\"
+ r"S_4 &\cong \mathrm{Aut}(Q_8) \\"
+ r"A_5 &\cong \mathrm{PSL}_2 (5)").scale(1.5)
+
+ self.play(Write(question))
+ self.wait()
+ self.play(ReplacementTransform(question, answer))
+
+ self.wait(3)
diff --git a/vorlesungen/punktgruppen/media/images/nosignal.jpg b/vorlesungen/punktgruppen/media/images/nosignal.jpg
new file mode 100644
index 0000000..2beeb8b
--- /dev/null
+++ b/vorlesungen/punktgruppen/media/images/nosignal.jpg
Binary files differ
diff --git a/vorlesungen/punktgruppen/poetry.lock b/vorlesungen/punktgruppen/poetry.lock
new file mode 100644
index 0000000..069d270
--- /dev/null
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diff --git a/vorlesungen/punktgruppen/pyproject.toml b/vorlesungen/punktgruppen/pyproject.toml
new file mode 100644
index 0000000..527eb57
--- /dev/null
+++ b/vorlesungen/punktgruppen/pyproject.toml
@@ -0,0 +1,15 @@
+[tool.poetry]
+name = "presentation"
+version = "0.1.0"
+description = ""
+authors = ["Nao Pross <np@0hm.ch>"]
+
+[tool.poetry.dependencies]
+python = "^3.7"
+manim = "^0.6.0"
+
+[tool.poetry.dev-dependencies]
+
+[build-system]
+requires = ["poetry-core>=1.0.0"]
+build-backend = "poetry.core.masonry.api"
diff --git a/vorlesungen/punktgruppen/script.pdf b/vorlesungen/punktgruppen/script.pdf
new file mode 100644
index 0000000..91993fb
--- /dev/null
+++ b/vorlesungen/punktgruppen/script.pdf
Binary files differ
diff --git a/vorlesungen/punktgruppen/script.tex b/vorlesungen/punktgruppen/script.tex
new file mode 100644
index 0000000..bc50e21
--- /dev/null
+++ b/vorlesungen/punktgruppen/script.tex
@@ -0,0 +1,214 @@
+\documentclass[a4paper]{article}
+
+\usepackage{amsmath}
+\usepackage{amssymb}
+
+\usepackage[cm]{manuscript}
+\usepackage{xcolor}
+
+\newcommand{\scene}[1]{\par\noindent[ #1 ]\par}
+\newenvironment{totranslate}{\color{blue!70!black}}{}
+
+\begin{document}
+\section{Das sind wir}
+\scene{Camera}
+
+\section{Ablauf}
+Zuerst werden wir Symmetrien in 2 Dimensionen anschauen, dann \"uberlegen wir
+kurz was es heisst f\"ur eine Symmetrie ``algebraisch'' zu sein. Von da aus
+kommt die dritte Dimension hinzu, die man besser mit Matrizen verstehen kann.
+Mit der aufgebauten Theorie werden wir versuchen Kristalle zu klassifizieren.
+Und zum Schluss kommen wir zu Anwendungen, welche f\"ur Ingenieure von
+Interesse sind.
+
+\section{intro}
+\scene{Spontan}
+
+\section{2D Geometrie}
+\scene{Intro}
+Wir fangen mit den 2 dimensionalen Symmetrien an, da man sie sich am
+einfachsten vorstellen kann. Eine Symmetrie eines Objektes beschreibt eine
+Aktion, welche nachdem sie auf das Objekt wirkt, das Objekt wieder gleich
+aussehen l\"asst.
+
+\scene{Viereck}
+Die einfachste Aktion, ist das Viereck zu nehmen, und wieder hinzulegen.
+Eine andere Aktion k\"onnte sein, das Objekt um eine Achse zu spiegeln,
+oder eine Rotation um 90 Grad.
+
+\scene{Zyklische Gruppe}
+Fokussieren wir uns auf die einfachste Klassen von Symmetrien: diejenigen die
+von einer reinen Drehung generiert werden. Wir sammeln diese in einer Gruppe
+\(G\), und notieren das sie von eine Rotation \(r\) generiert worden sind, mit
+diesen spitzen Klammern.
+
+Nehmen wir als Beispiel dieses Pentagon. Wenn wir \(r\) 5-mal anwenden, ist es
+dasselbe als wenn wir nichts gemacht h\"atten. Wenn wir es noch ein 6. mal
+drehen, entspricht dies dasselbe wie \(r\) nur 1 mal zu nutzen.
+
+\scene{Notation}
+So, die Gruppe setzt sich zusammen aus dem neutralen Element, und den Potenzen
+1 bis 4 von \(r\). Oder im allgemein Gruppen mit dieser Struktur, in welcher die
+Aktion \(n-1\) mal angewendet werden kann, heissen ``Zyklische Gruppe''.
+
+\scene{Diedergruppe}
+Nehmen wir nun auch noch die Spiegeloperation \(\sigma\) dazu. Weil wir jetzt 2
+Operationen haben, m\"ussen wir auch im Generator schreiben wie sie
+zusammenh\"angen. Schauen wir dann uns genauer diesen Ausdr\"uck an. Zweimal
+Spielegeln ist \"aquivalent zum neutralen Element, sowie 4 mal um 90 Grad
+drehen und 2 Drehspiegelungen, welche man auch Inversion nennt.
+
+\scene{Notation}
+Daraus k\"onnen wir wieder die ganze Gruppe erzeugen, die im allgemeinen den
+Symmetrien eines \(n\)-gons entsprechen.
+
+\scene{Kreisgruppe}
+Bis jetzt hatten wir nur diskrete Symmetrien, was nicht zwingend der Fall sein
+muss. Ein Ring kann man kontinuierlich drehen, und sieht dabei immer gleich
+aus.
+
+Diese Symmetrie ist auch als Kreisgruppe bekannt, die man sch\"on mit dem
+komplexen Einheitskreis definieren kann.
+
+\section{Algebra}
+\scene{Produkt mit \(i\)}
+\"Uberlegen wir uns eine spezielle algebraische Operation: Multiplikation mit
+der imagin\"aren Einheit. \(1\) mal \(i\) ist gleich \(i\). Wieder mal \(i\)
+ist \(-1\), dann \(-i\) und schliesslich kommen wir z\"uruck auf \(1\). Diese
+fassen wir in eine Gruppe \(G\) zusammen. Oder sch\"oner geschrieben:. Sieht das
+bekannt aus?
+
+\scene{Morphismen}
+Das Gefühl, dass es sich um dasselbe handelt, kann wie folgt formalisiert
+werden. Sei \(\phi\) eine Funktion von \(C_4\) zu \(G\) und ordnen wir zu
+jeder Symmetrieoperation ein Element aus \(G\). Wenn man die Zuordnung richtig
+definiert, dann sieht man die folgende Eigenschaft: Eine Operation nach eine
+andere zu nutzen, und dann die Funktion des Resultats zu nehmen, ist gleich wie
+die Funktion der einzelnen Operazionen zu nehmen und die Resultate zu
+multiplizieren. Dieses Ergebnis ist so bemerkenswert, dass es in der Mathematik
+einen Namen bekommen hat: Homorphismus, von griechisch "homos" dasselbe und
+"morphe" Form. Manchmal auch so geschrieben. Ausserdem, wenn \(\phi\) eins zu
+eins ist, heisst es \emph{Iso}morphismus: "iso" gleiche Form. Was man
+typischerweise mit diesem Symbol schreibt.
+
+\scene{Animation}
+Sie haben wahrscheinlich schon gesehen, worauf das hinausläuft. Dass die
+zyklische Gruppe \(C_4\) und \(G\) isomorph sind ist nicht nur Fachjargon der
+mathematik, sondern sie haben wirklich die selbe Struktur.
+
+\scene{Modulo}
+Das Beispiel mit der komplexen Einheit, war wahrscheinlich nicht so
+\"uberraschend. Aber was merkw\"urdig ist, ist das Beziehungen zwischen
+Symmetrien und Algebra auch in Bereichen gefunden werden, welche auf den ersten
+Blick, nicht geomerisch erscheinen. Ein R\"atsel für die Neugierigen: die Summe
+in der Modulo-Arithmetik. Als Hinweis: Um die Geometrie zu finden denken Sie
+an einer Uhr.
+
+\section{3D Geometrie}
+2 Dimensionen sind einfacher zu zeichnen, aber leider leben wir im 3
+dimensionalen Raum.
+
+\scene{Zyklische Gruppe}
+Wenn wir unser bekanntes Viereck mit seiner zyklischer Symmetrie in 3
+Dimensionen betrachten, k\"onnen wir seine Drehachse sehen.
+
+\scene{Diedergruppe}
+Um auch noch die andere Symmetrie des Rechteckes zu sehen, ben\"otigen wir eine
+Spiegelachse \(\sigma\), die hier eine Spiegelebene ist.
+
+\scene{Transition}
+Um die Punktsymmetrien zu klassifizieren orientiert man sich an einer Achse, um
+welche sich die meisten Symmetrien drehen. Das geht aber nicht immer, wie beim
+Tetraeder.
+
+\scene{Tetraedergruppe}
+Diese Geometrie hat 4 gleichwertige Symmetrieachsen, die eben eine
+Symmetriegruppe aufbauen, welche kreativer weise Tetraedergruppe genannt wird.
+Vielleicht fallen Ihnnen weitere Polygone ein mit dieser Eigenschaft, bevor wir
+zum n\"achsten Thema weitergehen.
+
+\section{Matrizen}
+\scene{Titelseite}
+Nun gehen wir kurz auf den Thema unseres Seminars ein: Matrizen. Das man mit
+Matrizen Dinge darstellen kann, ist keine Neuigkeit mehr, nach einem
+Semester MatheSeminar. Also überrascht es wohl auch keinen, das man alle
+punktsymmetrischen Operationen auch mit Matrizen Formulieren kann.
+
+\scene{Matrizen}
+
+Sei dann \(G\) unsere Symmetrie Gruppe, die unsere abstrakte Drehungen und
+Spiegelungen enth\"ahlt. Die Matrix Darstellung dieser Gruppe, ist eine
+Funktion gross \(\Phi\), von \(G\) zur orthogonalen Gruppe \(O(3)\), die zu
+jeder Symmetrie Operation klein \(g\) eine Matrix gross \(\Phi_g\) zuordnet.
+
+Zur Erinnerung, die Orthogonale Gruppe ist definiert als die Matrizen, deren
+transponierte auch die inverse ist. Da diese Volumen und Distanzen erhalten,
+natuerlich nur bis zu einer Vorzeichenumkehrung, macht es Sinn, dass diese
+Punksymmetrien genau beschreiben.
+
+Nehmen wir die folgende Operationen als Beispiele. Die Matrix der trivialen
+Operation, dass heisst nichts zu machen, ist die Einheitsmatrix. Eine
+Spiegelung ist dasselbe aber mit einem Minus, und Drehungen sind uns schon
+dank Herrn M\"uller bekannt.
+
+\section{Kristalle}
+\scene{Spontan}
+
+\section{Piezo}
+\scene{Spontan}
+
+\section{Licht}
+Als Finale, haben wir ein schwieriges Problem aus der Physik. Das Ziel dieser
+Folie ist nicht jedes Zeichen zu versehen, sondern zu zeigen wie man von hier
+weiter gehen kann. Wir mochten sehen wie sich Licht in einem Kristall verhaltet.
+Genauer, wir m\"ochten die Amplitude einer
+elektromagnetischer Welle in einem Kristall beschreiben.
+
+Das Beispiel richtet sich mehr an Elektrotechnik Studenten, aber die Theorie
+ist die gleiche bei mechanischen Wellen in Materialien mit einer
+Spannungstensor wie dem, den wir letzte Woche gesehen haben.
+% Ganz grob gesagt, ersetzt man E durch Xi und epsilon durch das Sigma.
+
+Um eine Welle zu beschreiben, verwenden wir die Helmholtz-Gleichung, die einige
+von uns bereits in anderen Kursen gel\"ost haben. Schwierig wird aber dieses
+Problem, wenn der Term vor der Zeitableitung ein Tensor ist (f\"ur uns eine Matrix).
+
+Zur Vereinfachung werden wir eine ebene Welle verwenden. Setzt man dieses E in
+die Helmholtz-Gleichung ein, erhält man folgendes zurück: ein Eigenwertproblem.
+
+Physikalisch bedeutet dies, dass die Welle in diesem Material ihre Amplitude in
+Abhängigkeit von der Ausbreitungsrichtung ändert. Und die Eigenwerte sagen
+aus, wie stark die Amplitude der Welle in jeder Richtung skaliert wird.
+
+Ich sagte, in jede Richtung skaliert, aber welche Richtungen genau?
+Physikalisch hängt das von der kristallinen Struktur des Materials ab, aber
+mathematisch können wir sagen: in Richtung der Eigenvektoren! Aber diesen
+Eigenraum zu finden, in dem die Eigenvektoren wohnen, ist beliebig schwierig.
+
+Hier kommt unsere Gruppentheorie zu Hilfe. Wir können die Symmetrien unseres
+Kristalls zur Hilfe nehmen. Zu jeder dieser Symmetrien lässt sich bekanntlich eine
+einfache Matrix finden, deren Eigenraum ebenfalls relativ leicht zu finden ist.
+Zum Beispiel ist der Eigenraum der Rotation \(r\), die Rotationsachse, für die
+Reflexion \(\sigma\) eine Ebene, und so weiter.
+
+Nun ist die Frage, ob man diese Eingenraume der Symmetrienoperationen
+kombinieren kann um den Eigenraum des physikalisches Problems zu finden.
+
+Aber leider ist meine Zeit abgelaufen in der Recherche, also müssen Sie mir 2
+Dingen einfach glauben, erstens dass es einen Weg gibt, und zweitens dass eher
+nicht so schlimm ist, wenn man die Notation einmal gelernt hat.
+
+Nachdem wir an, wir haben den Eigenraum U gefunden, dann können wir einen
+(Eigen)Vektor E daraus nehmen und in ihm direkt lambda ablesen. Das sagt uns,
+wie die Amplitude der Welle, in diese Richtung gedämpft wurde.
+
+Diese Methode ist nicht spezifisch für dieses Problem, im Gegenteil, ich habe
+gesehen, dass sie in vielen Bereichen eingesetzt wird, wie z.B.:
+Kristallographie, Festkörperphysik, Molekülschwingungen in der Quantenchemie
+und numerische Simulationen von Membranen.
+
+\section{Outro}
+\scene{Camera}
+
+\end{document}
+% vim:et ts=2 sw=2:
diff --git a/vorlesungen/punktgruppen/slides.pdf b/vorlesungen/punktgruppen/slides.pdf
new file mode 100644
index 0000000..e1769f8
--- /dev/null
+++ b/vorlesungen/punktgruppen/slides.pdf
Binary files differ
diff --git a/vorlesungen/punktgruppen/slides.tex b/vorlesungen/punktgruppen/slides.tex
new file mode 100644
index 0000000..cd3d7d7
--- /dev/null
+++ b/vorlesungen/punktgruppen/slides.tex
@@ -0,0 +1,895 @@
+\documentclass[12pt, xcolor, aspectratio=169, handout]{beamer}
+
+% language
+\usepackage{polyglossia}
+\setmainlanguage{german}
+
+% pretty drawings
+\usepackage{tikz}
+\usepackage{tikz-3dplot}
+
+\usetikzlibrary{positioning}
+\usetikzlibrary{arrows.meta}
+\usetikzlibrary{shapes.misc}
+\usetikzlibrary{calc}
+
+\usetikzlibrary{external}
+\tikzexternalize[
+ mode = graphics if exists,
+ figure list = true,
+ prefix=build/
+]
+
+% Theme
+\beamertemplatenavigationsymbolsempty
+
+% set look
+\usetheme{default}
+\usecolortheme{fly}
+\usefonttheme{serif}
+
+%% Set font
+\usepackage[p,osf]{scholax}
+\usepackage{amsmath}
+\usepackage[scaled=1.075,ncf,vvarbb]{newtxmath}
+
+% set colors
+\definecolor{background}{HTML}{202020}
+
+\setbeamercolor{normal text}{fg=white, bg=background}
+\setbeamercolor{structure}{fg=white}
+
+\setbeamercolor{item projected}{use=item,fg=background,bg=item.fg!35}
+
+\setbeamercolor*{palette primary}{use=structure,fg=white,bg=structure.fg}
+\setbeamercolor*{palette secondary}{use=structure,fg=white,bg=structure.fg!75}
+\setbeamercolor*{palette tertiary}{use=structure,fg=white,bg=structure.fg!50}
+\setbeamercolor*{palette quaternary}{fg=white,bg=background}
+
+\setbeamercolor*{block title}{parent=structure}
+\setbeamercolor*{block body}{fg=background, bg=}
+
+\setbeamercolor*{framesubtitle}{fg=white}
+
+\setbeamertemplate{section page}
+{
+ \begin{center}
+ \Huge
+ \insertsection
+ \end{center}
+}
+\AtBeginSection{\frame{\sectionpage}}
+
+% Macros
+\newcommand{\ten}[1]{#1}
+
+% Metadata
+\title{\LARGE \scshape Punktgruppen und Kristalle}
+\author[N. Pross, T. T\"onz]{Naoki Pross, Tim T\"onz}
+\institute{Hochschule f\"ur Technik OST, Rapperswil}
+\date{10. Mai 2021}
+
+% Slides
+\begin{document}
+\frame{
+ \titlepage
+ \vfill
+ \begin{center}
+ \small \color{gray}
+ Slides: \texttt{s.0hm.ch/ctBsD}
+ \end{center}
+}
+\frame{\tableofcontents}
+
+\frame{
+ \begin{itemize}
+ \item Was heisst \emph{Symmetrie} in der Mathematik? \pause
+ \item Wie kann ein Kristall modelliert werden? \pause
+ \item Aus der Physik: Licht, Piezoelektrizit\"at \pause
+ \end{itemize}
+ \begin{center}
+ \begin{tikzpicture}
+ \begin{scope}[
+ node distance = 0cm
+ ]
+ \node[
+ rectangle, fill = gray!40!background,
+ minimum width = 3cm, minimum height = 2cm,
+ ] (body) {\(\vec{E}_p = \vec{0}\)};
+
+ \node[
+ draw, rectangle, thick, white, fill = red!50,
+ minimum width = 3cm, minimum height = 1mm,
+ above = of body
+ ] (pos) {};
+
+ \node[
+ draw, rectangle, thick, white, fill = blue!50,
+ minimum width = 3cm, minimum height = 1mm,
+ below = of body
+ ] (neg) {};
+
+ \draw[white, very thick, -Circle] (pos.east) to ++ (1,0) node (p) {};
+ \draw[white, very thick, -Circle] (neg.east) to ++ (1,0) node (n) {};
+
+ \draw[white, thick, ->] (p) to[out = -70, in = 70] node[midway, right] {\(U = 0\)} (n);
+ \end{scope}
+ \begin{scope}[
+ node distance = 0cm,
+ xshift = 7cm
+ ]
+ \node[
+ rectangle, fill = gray!40!background,
+ minimum width = 3cm, minimum height = 1.5cm,
+ ] (body) {\(\vec{E}_p = \vec{0}\)};
+
+ \node[
+ draw, rectangle, thick, white, fill = red!50,
+ minimum width = 3cm, minimum height = 1mm,
+ above = of body
+ ] (pos) {};
+
+ \node[
+ draw, rectangle, thick, white, fill = blue!50,
+ minimum width = 3cm, minimum height = 1mm,
+ below = of body
+ ] (neg) {};
+
+ \draw[orange, very thick, <-] (pos.north) to node[near end, right] {\(\vec{F}\)} ++(0,1);
+ \draw[orange, very thick, <-] (neg.south) to node[near end, right] {\(\vec{F}\)} ++(0,-1);
+
+ \draw[white, very thick, -Circle] (pos.east) to ++ (1,0) node (p) {};
+ \draw[white, very thick, -Circle] (neg.east) to ++ (1,0) node (n) {};
+
+ \draw[white, thick, ->] (p) to[out = -70, in = 70] node[midway, right] {\(U > 0\)} (n);
+ \end{scope}
+ \end{tikzpicture}
+ \end{center}
+}
+
+\section{2D Symmetrien}
+%% Made in video
+{
+ \usebackgroundtemplate{
+ \includegraphics[height=\paperheight]{media/images/nosignal}}
+ \frame{}
+}
+
+\section{Algebraische Symmetrien}
+%% Made in video
+\frame{
+ \begin{columns}[T]
+ \begin{column}{.5\textwidth}
+ Produkt mit \(i\)
+ \begin{align*}
+ 1 \cdot i &= i \\
+ i \cdot i &= -1 \\
+ -1 \cdot i &= -i \\
+ -i \cdot i &= 1
+ \end{align*}
+ \pause
+ %
+ Gruppe
+ \begin{align*}
+ G &= \left\{
+ 1, i, -1, -i
+ \right\} \\
+ &= \left\{
+ 1, i, i^2, i^3
+ \right\} \\
+ C_4 &= \left\{
+ \mathbb{1}, r, r^2, r^3
+ \right\}
+ \end{align*}
+ \pause
+ \end{column}
+ \begin{column}{.5\textwidth}
+ Darstellung \(\phi : C_4 \to G\)
+ \begin{align*}
+ \phi(\mathbb{1}) &= 1 & \phi(r^2) &= i^2 \\
+ \phi(r) &= i & \phi(r^3) &= i^3
+ \end{align*}
+ \pause
+ %
+ Homomorphismus
+ \begin{align*}
+ \phi(r \circ \mathbb{1}) &= \phi(r) \cdot \phi(\mathbb{1}) \\
+ &= i \cdot 1
+ \end{align*}
+ \pause
+ %
+ \(\phi\) ist bijektiv \(\implies C_4 \cong G\)
+ \pause
+ %
+ \begin{align*}
+ \psi : C_4 &\to (\mathbb{Z}/4\mathbb{Z}, +) \\
+ \psi(\mathbb{1}\circ r^2) &= 0 + 2 \pmod{4}
+ \end{align*}
+ \end{column}
+ \end{columns}
+}
+
+\section{3D Symmetrien}
+%% Made in video
+{
+ \usebackgroundtemplate{
+ \includegraphics[height=\paperheight]{media/images/nosignal}}
+ \frame{}
+}
+
+\section{Matrizen}
+\frame{
+ \begin{columns}[T]
+ \begin{column}{.5\textwidth}
+ Symmetriegruppe
+ \[
+ G = \left\{\mathbb{1}, r, \sigma, \dots \right\}
+ \]
+ \pause
+ Matrixdarstellung
+ \begin{align*}
+ \Phi : G &\to O(3) \\
+ g &\mapsto \Phi_g
+ \end{align*}
+ \pause
+ Orthogonale Gruppe
+ \[
+ O(n) = \left\{ Q : QQ^t = Q^tQ = I \right\}
+ \]
+ \end{column}
+ \pause
+ \begin{column}{.5\textwidth}
+ \begin{align*}
+ \Phi_\mathbb{1} &= \begin{pmatrix}
+ 1 & 0 & 0 \\
+ 0 & 1 & 0 \\
+ 0 & 0 & 1
+ \end{pmatrix} = I \\[1em]
+ \Phi_\sigma &= \begin{pmatrix}
+ 1 & 0 & 0 \\
+ 0 & -1 & 0 \\
+ 0 & 0 & 1
+ \end{pmatrix} \\[1em]
+ \Phi_r &= \begin{pmatrix}
+ \cos \alpha & -\sin \alpha & 0 \\
+ \sin \alpha & \cos \alpha & 0 \\
+ 0 & 0 & 1
+ \end{pmatrix}
+ \end{align*}
+ \end{column}
+ \end{columns}
+}
+
+\section{Kristalle}
+\begin{frame}[fragile]{}
+ \begin{columns}
+ \onslide<1->{
+ \begin{column}{.5\textwidth}
+ \begin{center}
+ \begin{tikzpicture}[
+ dot/.style = {
+ draw, circle, thick, white, fill = gray!40!background,
+ minimum size = 2mm,
+ inner sep = 0pt,
+ outer sep = 1mm,
+ },
+ ]
+
+ \begin{scope}
+ \clip (-2,-2) rectangle (3,4);
+ \foreach \y in {-7,-6,...,7} {
+ \foreach \x in {-7,-6,...,7} {
+ \node[dot, xshift=3mm*\y] (N\x\y) at (\x, \y) {};
+ }
+ }
+ \end{scope}
+ \draw[white, thick] (-2, -2) rectangle (3,4);
+
+ \draw[red!80!background, thick, ->]
+ (N00) to node[midway, below] {\(\vec{a}_1\)} (N10);
+ \draw[cyan!80!background, thick, ->]
+ (N00) to node[midway, left] {\(\vec{a}_2\)} (N01);
+ \end{tikzpicture}
+ \end{center}
+ \end{column}
+ }
+ \begin{column}{.5\textwidth}
+ \onslide<2->{
+ Kristallgitter:
+ \(n_i \in \mathbb{Z}\),
+ }
+ \onslide<3->{
+ \(\vec{a}_i \in \mathbb{R}^3\)
+ }
+ \onslide<2->{
+ \[
+ \vec{r} = n_1 \vec{a}_1 + n_2 \vec{a}_2 \onslide<3->{+ n_3 \vec{a}_3}
+ \]
+ }
+ \vspace{1cm}
+
+ \onslide<4->{
+ Invariant unter Translation
+ \[
+ Q_i(\vec{r}) = \vec{r} + \vec{a}_i
+ \]
+ }
+ \end{column}
+ \end{columns}
+\end{frame}
+
+\begin{frame}[fragile]{}
+ \begin{columns}[T]
+ \begin{column}{.5\textwidth}
+ \onslide<1->{
+ Wie kombiniert sich \(Q_i\) mit der anderen Symmetrien?
+ }
+ \begin{center}
+ \begin{tikzpicture}[
+ dot/.style = {
+ draw, circle, thick, white, fill = gray!40!background,
+ minimum size = 2mm,
+ inner sep = 0pt,
+ outer sep = 1mm,
+ },
+ ]
+
+ \onslide<2->{
+ \node[dot] (A1) at (0,0) {};
+ \node[below left] at (A1) {\(A\)};
+ }
+
+ \onslide<3->{
+ \node[dot] (A2) at (2.5,0) {};
+ \node[below right] at (A2) {\(A'\)};
+
+ \draw[red!80!background, thick, ->]
+ (A1) to node[midway, below] {\(\vec{Q}\)} (A2);
+ }
+
+ \onslide<4->{
+ \node[dot] (B1) at (120:2.5) {};
+ \node[above left] at (B1) {\(B\)};
+
+ \draw[green!70!background, thick, ->]
+ (A1) ++(.5,0) arc (0:120:.5)
+ node[midway, above, xshift=1mm] {\(C_n\)};
+ \draw[red!80!background, dashed, thick, ->] (A1) to (B1);
+ }
+
+ \onslide<5->{
+ \node[dot] (B2) at ($(A2)+(60:2.5)$) {};
+ \node[above right] at (B2) {\(B'\)};
+
+ \draw[green!70!background, thick, dashed, ->]
+ (A2) ++(-.5,0) arc (180:60:.5);
+ \draw[red!80!background, dashed, thick, ->] (A2) to (B2);
+ }
+
+ \onslide<6->{
+ \draw[yellow!80!background, thick, ->]
+ (B1) to node[above, midway] {\(\vec{Q}'\)} (B2);
+ }
+
+ \onslide<10->{
+ \draw[gray, dashed, thick] (A1) to (A1 |- B1) node (Xl) {};
+ \draw[gray, dashed, thick] (A2) to (A2 |- B2) node (Xr) {};
+ \node[above left, xshift=-2mm] at (Xl) {\(x\)};
+ \node[above right, xshift= 2mm] at (Xr) {\(x\)};
+ }
+ \end{tikzpicture}
+ \end{center}
+ \end{column}
+ \begin{column}{.5\textwidth}
+ \onslide<7->{
+ Sei \(q = |\vec{Q}|\), \(\alpha = 2\pi/n\) und \(n \in \mathbb{N}\)
+ }
+ \begin{align*}
+ \onslide<9->{q' = n q \onslide<10->{&= q + 2x \\}}
+ \onslide<11->{nq &= q + 2q\sin(\alpha - \pi/2) \\}
+ \onslide<12->{n &= 1 - 2\cos\alpha}
+ \end{align*}
+ \onslide<13->{
+ Somit muss
+ \begin{align*}
+ \alpha &= \cos^{-1}\left(\frac{1-n}{2}\right) \\[1em]
+ \alpha &\in \left\{ 0, 60^\circ, 90^\circ, 120^\circ, 180^\circ \right\} \\
+ n &\in \left\{ 1, 2, 3, 4, 6 \right\}
+ \end{align*}
+ }
+ \end{column}
+ \end{columns}
+\end{frame}
+
+\begin{frame}[fragile]{M\"ogliche Kristallstrukturen}
+ \begin{center}
+ \begin{tikzpicture}[]
+ \node[circle, dashed, draw = gray,
+ thick, fill = background,
+ minimum size = 4cm] {};
+ \node[gray] at (.9,-1.2) {674};
+
+ \node[circle, draw = white, thick,
+ fill = orange!40!background,
+ xshift = -3mm, yshift = 2mm,
+ minimum size = 2.75cm,
+ outer sep = 1mm] (A) {};
+ \node[white, yshift = 2mm] at (A) {230};
+ \node[white, font=\large, above right = of A] (Al) {Raumgruppe};
+ \draw[white, thick, ->] (Al.west) to[out=180, in=60] (A);
+
+ \node[circle, draw = white, thick,
+ fill = red!20!background,
+ xshift = -5mm, yshift = -5mm,
+ minimum size = 1cm,
+ outer sep = 1mm] (B) {32};
+ \node[white, font=\large, below left = of B, xshift=-4mm] (Bl) {Kristallklassen};
+ \draw[white, thick, ->] (Bl.east) to[out = 0, in = 180] (B);
+ \end{tikzpicture}
+ \end{center}
+\end{frame}
+
+{
+ \usebackgroundtemplate[fragile]{
+ \begin{tikzpicture}[
+ overlay,
+ xshift = .45\paperwidth,
+ yshift = .47\paperheight,
+ classcirc/.style = {
+ draw = gray, thick, circle,
+ minimum size = 12mm,
+ inner sep = 0pt, outer sep = 0pt,
+ },
+ classlabel/.style = {
+ below right = 5mm
+ },
+ round/.style = {
+ draw = yellow, thick, circle,
+ minimum size = 1mm,
+ inner sep = 0pt, outer sep = 0pt,
+ },
+ cross/.style = {
+ cross out, draw = magenta, thick,
+ minimum size = 1mm,
+ inner sep = 0pt, outer sep = 0pt
+ },
+ ]
+ \matrix [row sep = 3mm, column sep = 0mm] {
+ \node[classcirc] (C1) {} node[classlabel] {\(C_{1}\)}; &
+ \node[classcirc] (C2) {} node[classlabel] {\(C_{2}\)}; &
+ \node[classcirc] (C3) {} node[classlabel] {\(C_{3}\)}; &
+ \node[classcirc] (Ci) {} node[classlabel] {\(C_{i}\)}; &
+
+ \node[classcirc] (Cs) {} node[classlabel] {\(C_{s}\)}; &
+ \node[classcirc] (C3i) {} node[classlabel] {\(C_{3i}\)}; &
+ \node[classcirc] (C2h) {} node[classlabel] {\(C_{2h}\)}; &
+ \node[classcirc] (D2) {} node[classlabel] {\(D_{2}\)}; \\
+
+ \node[classcirc] (D3d) {} node[classlabel] {\(D_{3d}\)}; &
+ \node[classcirc] (C2v) {} node[classlabel] {\(C_{2v}\)}; &
+ \node[classcirc] (D2h) {} node[classlabel] {\(D_{2h}\)}; &
+ \node[classcirc] (D3) {} node[classlabel] {\(D_{3}\)}; &
+
+ \node[classcirc] (C4) {} node[classlabel] {\(C_{4}\)}; &
+ \node[classcirc] (C6) {} node[classlabel] {\(C_{6}\)}; &
+ \node[classcirc] (D3dP) {} node[classlabel] {\(D_{3d}\)}; &
+ \node[classcirc] (S4) {} node[classlabel] {\(S_{4}\)}; \\
+
+ \node[classcirc] (S3) {} node[classlabel] {\(S_{3}\)}; &
+ \node[classcirc, dashed] (T) {} node[classlabel] {\(T_{}\)}; &
+ \node[classcirc] (C4h) {} node[classlabel] {\(C_{4h}\)}; &
+ \node[classcirc] (C6h) {} node[classlabel] {\(C_{6h}\)}; &
+
+ \node[classcirc, dashed] (Th) {} node[classlabel] {\(T_{h}\)}; &
+ \node[classcirc] (C4v) {} node[classlabel] {\(C_{4v}\)}; &
+ \node[classcirc] (C6v) {} node[classlabel] {\(C_{6v}\)}; &
+ \node[classcirc, dashed] (Td) {} node[classlabel] {\(T_{d}\)}; \\
+
+ \node[classcirc] (D2d) {} node[classlabel] {\(D_{2d}\)}; &
+ \node[classcirc] (D3h) {} node[classlabel] {\(D_{3h}\)}; &
+ \node[classcirc, dashed] (O) {} node[classlabel] {\(O_{}\)}; &
+ \node[classcirc] (D4) {} node[classlabel] {\(D_{4}\)}; &
+
+ \node[classcirc] (D6) {} node[classlabel] {\(D_{6}\)}; &
+ \node[classcirc, dashed] (Oh) {} node[classlabel] {\(O_{h}\)}; &
+ \node[classcirc] (D4h) {} node[classlabel] {\(D_{4h}\)}; &
+ \node[classcirc] (D6h) {} node[classlabel] {\(D_{6h}\)}; \\
+ };
+
+
+ \node[cross] at ($(C1)+(4mm,0)$) {};
+
+
+ \node[cross] at ($(C2)+(4mm,0)$) {};
+ \node[cross] at ($(C2)-(4mm,0)$) {};
+
+
+ \node[cross] at ($(C3)+( 0:4mm)$) {};
+ \node[cross] at ($(C3)+(120:4mm)$) {};
+ \node[cross] at ($(C3)+(240:4mm)$) {};
+
+
+ \node[cross] at ($(Ci)+(4mm,0)$) {};
+ \node[round] at ($(Ci)-(4mm,0)$) {};
+
+
+ \node[cross] at ($(Cs)+(4mm,0)$) {};
+ \node[round] at ($(Cs)+(4mm,0)$) {};
+
+
+ \node[cross] at ($(C3i)+( 0:4mm)$) {};
+ \node[cross] at ($(C3i)+(120:4mm)$) {};
+ \node[cross] at ($(C3i)+(240:4mm)$) {};
+ \node[round] at ($(C3i)+( 60:4mm)$) {};
+ \node[round] at ($(C3i)+(180:4mm)$) {};
+ \node[round] at ($(C3i)+(300:4mm)$) {};
+
+
+ \node[cross] at ($(C2h)+(4mm,0)$) {};
+ \node[cross] at ($(C2h)-(4mm,0)$) {};
+ \node[round] at ($(C2h)+(4mm,0)$) {};
+ \node[round] at ($(C2h)-(4mm,0)$) {};
+
+
+ \node[cross] at ($(D2)+( 20:4mm)$) {};
+ \node[cross] at ($(D2)+(200:4mm)$) {};
+ \node[round] at ($(D2)+(160:4mm)$) {};
+ \node[round] at ($(D2)+(340:4mm)$) {};
+
+
+ \foreach \x in {0, 120, 240} {
+ \node[cross] at ($(D3d)+({\x+15}:4mm)$) {};
+ \node[cross] at ($(D3d)+({\x-15}:4mm)$) {};
+ }
+
+
+ \foreach \x in {0, 180} {
+ \node[cross] at ($(C2v)+({\x+15}:4mm)$) {};
+ \node[cross] at ($(C2v)+({\x-15}:4mm)$) {};
+ }
+
+
+ \foreach \x in {0, 180} {
+ \node[cross] at ($(D2h)+({\x+15}:4mm)$) {};
+ \node[cross] at ($(D2h)+({\x-15}:4mm)$) {};
+ \node[round] at ($(D2h)+({\x+15}:4mm)$) {};
+ \node[round] at ($(D2h)+({\x-15}:4mm)$) {};
+ }
+
+
+ \foreach \x in {0, 120, 240} {
+ \node[cross] at ($(D3)+({\x+15}:4mm)$) {};
+ \node[round] at ($(D3)+({\x-15}:4mm)$) {};
+ }
+
+
+ \foreach \x in {0, 90, 180, 270} {
+ \node[cross] at ($(C4)+(\x:4mm)$) {};
+ }
+
+
+ \foreach \x in {0, 60, 120, 180, 240, 300} {
+ \node[cross] at ($(C6)+(\x:4mm)$) {};
+ }
+
+
+ \foreach \x in {0, 120, 240} {
+ \node[cross] at ($(D3dP)+({\x+15}:4mm)$) {};
+ \node[cross] at ($(D3dP)+({\x-15}:4mm)$) {};
+ \node[round] at ($(D3dP)+({\x+15+60}:4mm)$) {};
+ \node[round] at ($(D3dP)+({\x-15+60}:4mm)$) {};
+ }
+
+
+ \node[cross] at ($(S4)+(4mm,0)$) {};
+ \node[cross] at ($(S4)-(4mm,0)$) {};
+ \node[round] at ($(S4)+(0,4mm)$) {};
+ \node[round] at ($(S4)-(0,4mm)$) {};
+
+
+ \foreach \x in {0, 120, 240} {
+ \node[cross] at ($(S3)+(\x:4mm)$) {};
+ \node[round] at ($(S3)+(\x:4mm)$) {};
+ }
+
+
+ %% TODO: T
+
+
+ \foreach \x in {0, 90, 180, 270} {
+ \node[cross] at ($(C4h)+(\x:4mm)$) {};
+ \node[round] at ($(C4h)+(\x:4mm)$) {};
+ }
+
+
+ \foreach \x in {0, 60, 120, 180, 240, 300} {
+ \node[cross] at ($(C6h)+(\x:4mm)$) {};
+ \node[round] at ($(C6h)+(\x:4mm)$) {};
+ }
+
+
+ %% TODO: Th
+
+
+ \foreach \x in {0, 90, 180, 270} {
+ \node[cross] at ($(C4v)+(\x+15:4mm)$) {};
+ \node[cross] at ($(C4v)+(\x-15:4mm)$) {};
+ }
+
+
+
+ \foreach \x in {0, 60, 120, 180, 240, 300} {
+ \node[cross] at ($(C6v)+(\x+10:4mm)$) {};
+ \node[cross] at ($(C6v)+(\x-10:4mm)$) {};
+ }
+
+
+ %% TODO: Td
+
+
+ \foreach \x in {0, 180} {
+ \node[cross] at ($(D2d)+({\x+15}:4mm)$) {};
+ \node[round] at ($(D2d)+({\x-15}:4mm)$) {};
+
+ \node[round] at ($(D2d)+({\x+15+90}:4mm)$) {};
+ \node[cross] at ($(D2d)+({\x-15+90}:4mm)$) {};
+ }
+
+
+ \foreach \x in {0, 120, 240} {
+ \node[cross] at ($(D3h)+({\x+15}:4mm)$) {};
+ \node[cross] at ($(D3h)+({\x-15}:4mm)$) {};
+ \node[round] at ($(D3h)+({\x+15}:4mm)$) {};
+ \node[round] at ($(D3h)+({\x-15}:4mm)$) {};
+ }
+
+
+ %% TODO: O
+
+
+ \foreach \x in {0, 90, 180, 270} {
+ \node[cross] at ($(D4)+({\x+15}:4mm)$) {};
+ \node[round] at ($(D4)+({\x-15}:4mm)$) {};
+ }
+
+ \foreach \x in {0, 60, 120, 180, 240, 300} {
+ \node[cross] at ($(D6)+({\x+10}:4mm)$) {};
+ \node[round] at ($(D6)+({\x-10}:4mm)$) {};
+ }
+
+
+ % TODO Oh
+
+
+ \foreach \x in {0, 90, 180, 270} {
+ \node[cross] at ($(D4h)+(\x+15:4mm)$) {};
+ \node[cross] at ($(D4h)+(\x-15:4mm)$) {};
+ \node[round] at ($(D4h)+(\x+15:4mm)$) {};
+ \node[round] at ($(D4h)+(\x-15:4mm)$) {};
+ }
+
+
+ \foreach \x in {0, 60, 120, 180, 240, 300} {
+ \node[cross] at ($(D6h)+({\x+10}:4mm)$) {};
+ \node[cross] at ($(D6h)+({\x-10}:4mm)$) {};
+ \node[round] at ($(D6h)+({\x+10}:4mm)$) {};
+ \node[round] at ($(D6h)+({\x-10}:4mm)$) {};
+ }
+ \end{tikzpicture}
+ }
+ \begin{frame}[fragile]{}
+ \end{frame}
+}
+
+\section{Anwendungen}
+\begin{frame}[fragile]{}
+ \centering
+ \begin{tikzpicture}[
+ box/.style = {
+ rectangle, thick, draw = white, fill = darkgray!50!background,
+ minimum height = 1cm, outer sep = 2mm,
+ },
+ ]
+
+ \matrix [nodes = {box, align = center}, column sep = 1cm, row sep = 1.5cm] {
+ & \node (A) {32 Kristallklassen}; \\
+ \node (B) {11 Mit\\ Inversionszentrum}; & \node (C) {21 Ohne\\ Inversionszentrum}; \\
+ & \node[fill=red!20!background] (D) {20 Piezoelektrisch}; & \node (E) {1 Nicht\\ piezoelektrisch}; \\
+ };
+
+ \draw[thick, ->] (A.west) to[out=180, in=90] (B.north);
+ \draw[thick, ->] (A.south) to (C);
+ \draw[thick, ->] (C.south) to (D.north);
+ \draw[thick, ->] (C.east) to[out=0, in=90] (E.north);
+ \end{tikzpicture}
+\end{frame}
+
+\begin{frame}[fragile]{}
+ \begin{tikzpicture}[
+ overlay, xshift = 1.5cm, yshift = 1.5cm,
+ node distance = 2mm,
+ charge/.style = {
+ circle, draw = white, thick,
+ minimum size = 5mm
+ },
+ positive/.style = { fill = red!50 },
+ negative/.style = { fill = blue!50 },
+ ]
+
+ \node[font = {\large\bfseries}, align = center] (title) at (5.5,0) {Mit und Ohne\\ Symmetriezentrum};
+ \pause
+
+ \begin{scope}
+ \matrix[nodes = { charge }, row sep = 8mm, column sep = 8mm] {
+ \node[positive] {}; & \node[negative] (N) {}; & \node [positive] {}; \\
+ \node[negative] (W) {}; & \node[positive] {}; & \node [negative] (E) {}; \\
+ \node[positive] {}; & \node[negative] (S) {}; & \node [positive] {}; \\
+ };
+ \draw[gray, dashed] (W) to (N) to (E) to (S) to (W);
+ \end{scope}
+ \pause
+
+ \begin{scope}[xshift=11cm]
+ \foreach \x/\t [count=\i] in {60/positive, 120/negative, 180/positive, 240/negative, 300/positive, 360/negative} {
+ \node[charge, \t] (C\i) at (\x:1.5cm) {};
+ }
+
+ \draw[white] (C1) to (C2) to (C3) to (C4) to (C5) to (C6) to (C1);
+ \node[circle, draw=gray, fill=gray, outer sep = 0, inner sep = 0, minimum size = 3mm] {};
+ % \draw[gray, dashed] (C2) to (C4) to (C6) to (C2);
+ \end{scope}
+ \pause
+
+ %%
+ \node[below = of title] {Polarisation Feld \(\vec{E}_p\)};
+
+ %% hex with vertical pressure
+ \begin{scope}[xshift=11cm, yshift=-4.5cm]
+ \node[charge, positive, yshift=-2.5mm] (C1) at ( 60:1.5cm) {};
+ \node[charge, negative, yshift=-2.5mm] (C2) at (120:1.5cm) {};
+ \node[charge, positive, xshift=-2.5mm] (C3) at (180:1.5cm) {};
+ \node[charge, negative, yshift= 2.5mm] (C4) at (240:1.5cm) {};
+ \node[charge, positive, yshift= 2.5mm] (C5) at (300:1.5cm) {};
+ \node[charge, negative, xshift= 2.5mm] (C6) at (360:1.5cm) {};
+
+ \draw[white] (C1) to (C2) to (C3) to (C4) to (C5) to (C6) to (C1);
+ % \draw[gray, dashed] (C2) to (C4) to (C6) to (C2);
+
+ \foreach \d in {C1, C2} {
+ \draw[orange, very thick, <-] (\d) to ++(0,.7);
+ }
+
+ \foreach \d in {C4, C5} {
+ \draw[orange, very thick, <-] (\d) to ++(0,-.7);
+ }
+
+ \node[white] (E) {\(\vec{E}_p\)};
+ \begin{scope}[node distance = .5mm]
+ \node[red!50, right = of E] {\(+\)};
+ \node[blue!50, left = of E] {\(-\)};
+ \end{scope}
+ % \draw[gray, thick, dotted] (E) to ++(0,2);
+ % \draw[gray, thick, dotted] (E) to ++(0,-2);
+ \end{scope}
+ \pause
+
+ %% square with vertical pressure
+ \begin{scope}[yshift=-4.5cm]
+ \matrix[nodes = { charge }, row sep = 5mm, column sep = 1cm] {
+ \node[positive] (NW) {}; & \node[negative] (N) {}; & \node [positive] (NE) {}; \\
+ \node[negative] (W) {}; & \node[positive] {}; & \node [negative] (E) {}; \\
+ \node[positive] (SW) {}; & \node[negative] (S) {}; & \node [positive] (SE) {}; \\
+ };
+
+ \foreach \d in {NW, N, NE} {
+ \draw[orange, very thick, <-] (\d) to ++(0,.7);
+ }
+
+ \foreach \d in {SW, S, SE} {
+ \draw[orange, very thick, <-] (\d) to ++(0,-.7);
+ }
+
+ \draw[gray, dashed] (W) to (N) to (E) to (S) to (W);
+ \end{scope}
+ \pause
+
+ %% hex with horizontal pressure
+ \begin{scope}[xshift=5.5cm, yshift=-4.5cm]
+ \node[charge, positive, yshift= 2.5mm] (C1) at ( 60:1.5cm) {};
+ \node[charge, negative, yshift= 2.5mm] (C2) at (120:1.5cm) {};
+ \node[charge, positive, xshift= 2.5mm] (C3) at (180:1.5cm) {};
+ \node[charge, negative, yshift=-2.5mm] (C4) at (240:1.5cm) {};
+ \node[charge, positive, yshift=-2.5mm] (C5) at (300:1.5cm) {};
+ \node[charge, negative, xshift=-2.5mm] (C6) at (360:1.5cm) {};
+
+ \draw[white] (C1) to (C2) to (C3) to (C4) to (C5) to (C6) to (C1);
+ % \draw[gray, dashed] (C2) to (C4) to (C6) to (C2);
+
+ \draw[orange, very thick, <-] (C6) to ++(.7,0);
+ \draw[orange, very thick, <-] (C3) to ++(-.7,0);
+
+ \node[white] (E) {\(\vec{E}_p\)};
+ \begin{scope}[node distance = .5mm]
+ \node[blue!50, right = of E] {\(-\)};
+ \node[red!50, left = of E] {\(+\)};
+ \end{scope}
+ % \draw[gray, thick, dotted] (E) to ++(0,2);
+ % \draw[gray, thick, dotted] (E) to ++(0,-2);
+ \end{scope}
+ \pause
+
+
+ \end{tikzpicture}
+\end{frame}
+
+\frame{
+ \frametitle{Licht in Kristallen}
+ \begin{columns}[T]
+ \begin{column}{.45\textwidth}
+ \onslide<2->{
+ Helmholtz Wellengleichung
+ \[
+ \nabla^2 \vec{E} = \ten{\varepsilon}\mu
+ \frac{\partial^2}{\partial t^2} \vec{E}
+ \]
+ }
+ \onslide<3->{
+ Ebene Welle
+ \[
+ \vec{E} = \vec{E}_0 \exp\left[i
+ \left(\vec{k}\cdot\vec{r} - \omega t \right)\right]
+ \]
+ }
+ \onslide<4->{
+ Anisotropisch Dielektrikum
+ \[
+ (\ten{K}\ten{\varepsilon})\vec{E}
+ = \frac{k^2}{\mu \omega^2} \vec{E}
+ \implies
+ \Phi \vec{E} = \lambda \vec{E}
+ \]
+ }
+ \end{column}
+ \begin{column}{.55\textwidth}
+ \onslide<5->{
+ Eingenraum
+ \begin{align*}
+ U_\lambda &= \left\{ v : \Phi v = \lambda v \right\}
+ = \mathrm{null}\left(\Phi - \lambda I\right)
+ \end{align*}
+ }\onslide<6->{
+ Symmetriegruppe und Darstellung
+ \begin{align*}
+ G &= \left\{\mathbb{1}, r, \sigma, \dots \right\} \\
+ &\Phi : G \to O(n)
+ \end{align*}
+ }\onslide<7->{
+ Kann man \(U_\lambda\) von \(G\) herauslesen?
+ \only<7>{
+ \[
+ U_\lambda \stackrel{?}{=} f\left(\bigoplus_{g \in G} \Phi_g\right)
+ \]
+ }\only<8>{
+ \begin{align*}
+ \mathrm{Tr}\left[\Phi_r(g)\right]
+ &= \sum_i n_i \mathrm{Tr}\left[\Psi_i(g)\right] \\
+ |G| &= \sum_i\mathrm{Tr}\left[\Psi_i(\mathbb{1})\right]
+ \end{align*}
+ }
+ }
+ \end{column}
+ \end{columns}
+}
+
+% \begin{frame}[fragile]
+% \centering
+% \tdplotsetmaincoords{70}{110}
+% \begin{tikzpicture}[scale=2, tdplot_main_coords]
+% \node[draw=white, thick, minimum size = 3cm, circle] {};
+% % \foreach \x in {0, 120, 240} {
+% % }
+% \end{tikzpicture}
+% \end{frame}
+
+
+\end{document}