From 3b99e1259fea76103adc3c25b3af2da6d98635ed Mon Sep 17 00:00:00 2001 From: Nao Pross Date: Wed, 14 Apr 2021 17:14:20 +0200 Subject: Add some animations --- vorlesungen/punktgruppen/crystals.py | 269 +++++++++++++++++++++++++++++++++++ 1 file changed, 269 insertions(+) create mode 100644 vorlesungen/punktgruppen/crystals.py (limited to 'vorlesungen/punktgruppen/crystals.py') diff --git a/vorlesungen/punktgruppen/crystals.py b/vorlesungen/punktgruppen/crystals.py new file mode 100644 index 0000000..0f97a0f --- /dev/null +++ b/vorlesungen/punktgruppen/crystals.py @@ -0,0 +1,269 @@ +from manim import * + +import math as m +import numpy as np + +# 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"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() + figure.add(square) + figure.add(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_corner, UP)) + + actions = map(MathTex, [r"1", r"r", r"r^2", r"r^3", r"r^4", r"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\{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"C_n = \langle r \rangle" + r"= \left\{1, r, r^2, \dots, r^n \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 = 1,", + r"r^4 = 1,", + r"(\sigma r)^2 = 1 \rangle") + + self.play(Write(group), run_time = 2) + self.wait() + self.play(ApplyMethod(group.to_corner, UP)) + self.play(FadeIn(square)) + + axis = DashedLine(2 * LEFT, 2 * RIGHT) + sigma = MathTex(r"\sigma^2 = 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 = 1") + rot.next_to(square, DOWN * 3) + + figure = VGroup() + figure.add(dot) + figure.add(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 = 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() + figure.add(dot) + figure.add(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 = 1 \rangle \\" + r"&= \left\{" + r"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 = 1 \rangle \\" + r"&= \left\{" + r"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() + + 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=-20 * DEGREES) + + cube = Cube() + axes = ThreeDAxes() + + axis = np.array([2,2,2]) + line = DashedLine(-axis, axis) + + self.play(Create(axes), Create(cube)) + self.play(Create(line)) + + self.play(ApplyMethod(cube.rotate, PI, axis / abs(axis))) + + self.begin_ambient_camera_rotation(rate=0.1) + self.wait(7) + + +class AlgebraicSymmetries(Scene): + def construct(self): + pass -- cgit v1.2.1