diff options
Diffstat (limited to 'vorlesungen/punktgruppen')
| -rw-r--r-- | vorlesungen/punktgruppen/crystals.py | 72 |
1 files changed, 38 insertions, 34 deletions
diff --git a/vorlesungen/punktgruppen/crystals.py b/vorlesungen/punktgruppen/crystals.py index cbae3d0..7eef9b8 100644 --- a/vorlesungen/punktgruppen/crystals.py +++ b/vorlesungen/punktgruppen/crystals.py @@ -130,7 +130,7 @@ class Geometric2DSymmetries(Scene): self.wait() cyclic = MathTex( - r"Z_n = \langle r \rangle" + r"C_n = \langle r \rangle" r"= \left\{\mathbb{1}, r, r^2, \dots, r^{n-1} \right\}") cyclic_title = Tex(r"Zyklische Gruppe") @@ -237,7 +237,7 @@ class Geometric2DSymmetries(Scene): dot.move_to(2 * RIGHT) figure = VGroup(circle, dot) - group_name = MathTex(r"S^1") + group_name = MathTex(r"C_\infty") # create circle self.play(Create(circle)) @@ -252,7 +252,7 @@ class Geometric2DSymmetries(Scene): self.play(Rotate(figure, PI/4), Write(group_name)) self.play(Uncreate(figure)) - nsphere = MathTex(r"S^1 = \left\{z \in \mathbb{C} : |z| = 1\right\}") + 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) @@ -269,6 +269,7 @@ class Geometric2DSymmetries(Scene): class Geometric3DSymmetries(ThreeDScene): def construct(self): self.improper_rotation() + self.icosahedron() def improper_rotation(self): # changes the source of the light and camera @@ -279,11 +280,12 @@ class Geometric3DSymmetries(ThreeDScene): square = Square() square.set_fill(PINK, opacity=.5) - self.play(Create(square)) - self.wait() + self.play(SpinInFromNothing(square)) + self.wait(2) - self.play(Rotate(square, PI/2)) - self.wait() + for i in range(4): + self.play(Rotate(square, PI/2)) + self.wait(.5) self.move_camera(phi= 75 * DEGREES, theta = -80 * DEGREES) @@ -306,7 +308,7 @@ class Geometric3DSymmetries(ThreeDScene): axis = Line3D(start=[0,0,-2.5], end=[0,0,2.5]) - axis_name = MathTex(r"r \in Z_4") + 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) @@ -316,8 +318,9 @@ class Geometric3DSymmetries(ThreeDScene): self.wait() cyclic_sphere = VGroup(*cyclic_slices) - self.play(Rotate(cyclic_sphere, PI/2)) - self.wait() + for i in range(4): + self.play(Rotate(cyclic_sphere, PI/2)) + self.wait() # reflection plane self.play(FadeOut(cyclic_sphere), FadeIn(square)) @@ -367,18 +370,21 @@ class Geometric3DSymmetries(ThreeDScene): self.play(FadeOut(square), Create(dihedral_sphere)) - self.play(Rotate(dihedral_sphere, PI/2)) - self.play(Rotate(dihedral_sphere, PI, RIGHT)) - - self.play(Rotate(dihedral_sphere, PI/2)) - self.play(Rotate(dihedral_sphere, PI, RIGHT)) + for i in range(2): + self.play(Rotate(dihedral_sphere, PI/2)) + self.play(Rotate(dihedral_sphere, PI, RIGHT)) + self.wait() self.wait(5) + def icosahedron(self): + pass + + class AlgebraicSymmetries(Scene): def construct(self): self.cyclic() - self.matrices() + # self.matrices() def cyclic(self): # show the i product @@ -391,30 +397,31 @@ class AlgebraicSymmetries(Scene): for part in product: self.play(Write(part)) + self.wait() - 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() + self.wait(2) # 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\}") + r"C_4 &= \left\{ \mathbb{1}, r, r^2, r^3 \right\}") self.play(ReplacementTransform(group, grouppow[0])) - self.wait() + self.wait(2) 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 = MathTex(r"\phi: C_4 \to G \\") morphism.shift(UP) self.play(Write(morphism)) + self.wait() # show an example mappings = MathTex( @@ -425,34 +432,34 @@ class AlgebraicSymmetries(Scene): mappings.next_to(morphism, DOWN) self.play(Write(mappings)) - self.wait() + self.wait(3) 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})") + 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, 2.5 * LEFT), - ApplyMethod(hom_bracegrp.shift, 2.5 * LEFT)) + 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}(G, Z_4)").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() - # 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) @@ -462,7 +469,7 @@ class AlgebraicSymmetries(Scene): 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) + iso_text_short = MathTex("C_4 \cong G").next_to(iso_brace.get_tip(), RIGHT) self.play( ReplacementTransform(hom_brace, iso_brace), @@ -478,9 +485,6 @@ class AlgebraicSymmetries(Scene): 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) @@ -523,7 +527,7 @@ class AlgebraicSymmetries(Scene): FadeOutAndShift(morphgrp, RIGHT)) modulo = MathTex( - r"\phi: Z_4 &\to (\mathbb{Z}/4\mathbb{Z}, +) \\" + 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) |