diff options
Diffstat (limited to 'vorlesungen')
| -rw-r--r-- | vorlesungen/punktgruppen/crystals.py | 81 |
1 files changed, 49 insertions, 32 deletions
diff --git a/vorlesungen/punktgruppen/crystals.py b/vorlesungen/punktgruppen/crystals.py index dd55bbe..76dee1f 100644 --- a/vorlesungen/punktgruppen/crystals.py +++ b/vorlesungen/punktgruppen/crystals.py @@ -66,9 +66,7 @@ class Geometric2DSymmetries(Scene): # show some rotations dot = Dot(UP + RIGHT) - figure = VGroup() - figure.add(square) - figure.add(dot) + figure = VGroup(square, dot) rot = MathTex(r"r") self.play(Write(rot), Create(dot)) @@ -184,9 +182,7 @@ class Geometric2DSymmetries(Scene): rot = MathTex(r"r^4 = \mathbb{1}") rot.next_to(square, DOWN * 3) - figure = VGroup() - figure.add(dot) - figure.add(square) + figure = VGroup(dot, square) self.play(Write(rot), Create(dot)) for i in range(4): @@ -201,9 +197,7 @@ class Geometric2DSymmetries(Scene): axis = DashedLine(2 * LEFT, 2 * RIGHT) self.play(Create(dot), Create(axis), Write(action)) - figure = VGroup() - figure.add(dot) - figure.add(square) + figure = VGroup(dot, square) for i in range(2): self.play(Rotate(figure, PI/2)) @@ -317,6 +311,7 @@ class AlgebraicSymmetries(Scene): self.wait() self.cyclic() + self.matrices() def cyclic(self): # show the i product @@ -374,9 +369,7 @@ class AlgebraicSymmetries(Scene): for part in homomorphism: self.play(Write(part)) - hom_bracegrp = VGroup() - hom_bracegrp.add(morphism) - hom_bracegrp.add(homomorphism) + hom_bracegrp = VGroup(morphism, homomorphism) self.play( ApplyMethod(grouppow.shift, 2.5 * LEFT), @@ -384,21 +377,21 @@ class AlgebraicSymmetries(Scene): 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)) + 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() - iso_bracegrp.add(hom_bracegrp) - iso_bracegrp.add(isomorphism) + iso_bracegrp = VGroup(hom_bracegrp, isomorphism) iso_brace = Brace(iso_bracegrp, RIGHT) iso_text = Tex("Isomorphismus").next_to(iso_brace.get_tip(), RIGHT) @@ -412,44 +405,68 @@ class AlgebraicSymmetries(Scene): self.wait() # create a group for the whole - morphgrp = VGroup() - morphgrp.add(iso_bracegrp) - morphgrp.add(iso_brace) - morphgrp.add(iso_text_short) + morphgrp = VGroup(iso_bracegrp, iso_brace, iso_text_short) self.play( - FadeOutAndShift(grouppow, UP), - FadeOutAndShift(morphgrp, DOWN)) + ApplyMethod(grouppow.to_edge, LEFT), + ApplyMethod(morphgrp.to_edge, LEFT)) + # self.play( + # FadeOutAndShift(grouppow, UP), + # FadeOutAndShift(morphgrp, DOWN)) # draw a complex plane - plane = ComplexPlane() - plane.axis_config["number_scale_val"] = 1 - self.play(Create(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] ))) - self.play( - *map(Create, roots), - *map(Write, plane.get_coordinate_labels(1, -1, 1j, -1j))) - self.wait() - 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(FadeOut(square), FadeOut(arrow), *map(FadeOut, roots)) + 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): + question = MathTex(r"D_n \cong \,? \\ S_n \cong \,? \\ A_n \cong \,?") + question.scale(1.5) + + self.play(Write(question)) + + self.wait(3) |