def construct(self):
GraphScene.construct(self)
self.generate_spanning_tree()
self.generate_treeified_spanning_tree()
branches = self.spanning_tree.split()
branches_copy = deepcopy(branches)
treeified_branches = self.treeified_spanning_tree.split()
tree = TextMobject("``Tree''").to_edge(UP)
spanning_tree = TextMobject("``Spanning Tree''").to_edge(UP)
self.add(*branches)
self.play(
FadeOut(Mobject(*self.edges + self.vertices)),
Animation(Mobject(*branches)),
)
self.clear()
self.add(tree, *branches)
self.wait()
self.play(*[
Transform(b1, b2, run_time=2)
for b1, b2 in zip(branches, treeified_branches)
])
self.wait()
self.play(*[FadeIn(mob) for mob in self.edges + self.vertices] + [
Transform(b1, b2, run_time=2)
for b1, b2 in zip(branches, branches_copy)
])
self.accent_vertices(run_time=2)
self.remove(tree)
self.add(spanning_tree)
self.wait(2)
def construct(self):
time_per_branch = 0.5
text = TextMobject(
"""
In any tree:
$$E + 1 = V$$
"""
)
gs = GraphScene(SampleGraph())
gs.generate_treeified_spanning_tree()
branches = gs.treeified_spanning_tree.to_edge(LEFT).split()
all_dots = [Dot(branches[0].points[0])]
self.add(text, all_dots[0])
for branch in branches:
self.play(ShowCreation(branch), run_time=time_per_branch)
dot = Dot(branch.points[-1])
self.add(dot)
all_dots.append(dot)
self.dither()
self.remove(*all_dots)
self.play(
FadeOut(text),
FadeIn(Mobject(*gs.edges + gs.vertices)),
*[Transform(*pair) for pair in zip(branches, gs.spanning_tree.split())]
)
def construct(self):
GraphScene.construct(self)
self.generate_dual_graph()
self.add(TextMobject("Duality").to_edge(UP))
self.remove(*self.vertices)
def special_alpha(t):
if t > 0.5:
t = 1 - t
if t < 0.25:
return smooth(4 * t)
else:
return 1
kwargs = {"run_time": 5.0, "rate_func": special_alpha}
self.play(*[
Transform(*edge_pair, **kwargs)
for edge_pair in zip(self.edges, self.dual_edges)
] + [
Transform(Mobject(*[self.vertices[index]
for index in cycle]), dv, **kwargs)
for cycle, dv in zip(self.graph.region_cycles, self.dual_vertices)
])
self.wait()
def construct(self):
GraphScene.construct(self)
rot_kwargs = {"radians": np.pi / 3, "run_time": 5.0}
vertices = [point / 2 + OUT if abs(point[0]) == 2 else point + IN for point in self.points]
cube = Mobject(*[Line(vertices[edge[0]], vertices[edge[1]]) for edge in self.graph.edges])
cube.rotate(-np.pi / 3, [0, 0, 1])
cube.rotate(-np.pi / 3, [0, 1, 0])
dots_to_vertices = TextMobject("Dots $\\to$ Vertices").to_corner()
lines_to_edges = TextMobject("Lines $\\to$ Edges").to_corner()
regions_to_faces = TextMobject("Regions $\\to$ Faces").to_corner()
self.clear()
# self.play(TransformAnimations(
# Rotating(Dodecahedron(), **rot_kwargs),
# Rotating(cube, **rot_kwargs)
# ))
self.play(Rotating(cube, **rot_kwargs))
self.clear()
self.play(*[Transform(l1, l2) for l1, l2 in zip(cube.split(), self.edges)])
self.dither()
self.add(dots_to_vertices)
self.play(*[ShowCreation(dot, run_time=1.0) for dot in self.vertices])
self.dither(2)
self.remove(dots_to_vertices, *self.vertices)
self.add(lines_to_edges)
self.play(ApplyMethod(Mobject(*self.edges).highlight, "yellow"))
self.dither(2)
self.clear()
self.add(*self.edges)
self.add(regions_to_faces)
self.generate_regions()
for region in self.regions:
self.highlight_region(region)
self.dither(3.0)
def construct(self):
GraphScene.construct(self)
self.generate_dual_graph()
self.add(TextMobject("Duality").to_edge(UP))
self.remove(*self.vertices)
def special_alpha(t):
if t > 0.5:
t = 1 - t
if t < 0.25:
return smooth(4*t)
else:
return 1
kwargs = {
"run_time" : 5.0,
"rate_func" : special_alpha
}
self.play(*[
Transform(*edge_pair, **kwargs)
for edge_pair in zip(self.edges, self.dual_edges)
] + [
Transform(
Mobject(*[
self.vertices[index]
for index in cycle
]),
dv,
**kwargs
)
for cycle, dv in zip(
self.graph.region_cycles,
self.dual_vertices
)
])
self.dither()
def construct(self):
GraphScene.construct(self)
paths = [(1, 2, 4, 5, 6), (6, 7, 1, 3)]
non_paths = [[(0, 1), (7, 8), (5, 6)], [(5, 0), (0, 2), (0, 1)]]
valid_path = TextMobject("Valid \\\\ Path").highlight("green")
not_a_path = TextMobject("Not a \\\\ Path").highlight("red")
for mob in valid_path, not_a_path:
mob.to_edge(UP)
kwargs = {"run_time": 1.0}
for path, non_path in zip(paths, non_paths):
path_lines = Mobject(
*[
Line(self.points[path[i]], self.points[path[i + 1]]).highlight("yellow")
for i in range(len(path) - 1)
]
)
non_path_lines = Mobject(
*[Line(self.points[pp[0]], self.points[pp[1]]).highlight("yellow") for pp in non_path]
)
self.remove(not_a_path)
self.add(valid_path)
self.play(ShowCreation(path_lines, **kwargs))
self.dither(2)
self.remove(path_lines)
self.remove(valid_path)
self.add(not_a_path)
self.play(ShowCreation(non_path_lines, **kwargs))
self.dither(2)
self.remove(non_path_lines)
def construct(self):
GraphScene.construct(self)
account = ImageMobject("facebook_silhouette", invert = False)
account.scale(0.05)
logo = ImageMobject("facebook_logo", invert = False)
logo.scale(0.1)
logo.shift(0.2*LEFT + 0.1*UP)
account.add(logo).center()
account.shift(0.2*LEFT + 0.1*UP)
friends = TexMobject(
"\\leftarrow \\text{friends} \\rightarrow"
).scale(0.5*EDGE_ANNOTATION_SCALE_VAL)
self.clear()
accounts = [
deepcopy(account).shift(point)
for point in self.points
]
self.add(*accounts)
self.dither()
self.annotate_edges(friends)
self.dither()
self.play(*[
CounterclockwiseTransform(account, vertex)
for account, vertex in zip(accounts, self.vertices)
])
self.dither()
self.play(*[
Transform(ann, edge)
for ann, edge in zip(self.edge_annotations, self.edges)
])
self.dither()
def construct(self):
GraphScene.construct(self)
self.generate_spanning_tree()
self.generate_treeified_spanning_tree()
branches = self.spanning_tree.split()
branches_copy = deepcopy(branches)
treeified_branches = self.treeified_spanning_tree.split()
tree = TextMobject("``Tree''").to_edge(UP)
spanning_tree = TextMobject("``Spanning Tree''").to_edge(UP)
self.add(*branches)
self.play(FadeOut(Mobject(*self.edges + self.vertices)), Animation(Mobject(*branches)))
self.clear()
self.add(tree, *branches)
self.dither()
self.play(*[Transform(b1, b2, run_time=2) for b1, b2 in zip(branches, treeified_branches)])
self.dither()
self.play(
*[FadeIn(mob) for mob in self.edges + self.vertices]
+ [Transform(b1, b2, run_time=2) for b1, b2 in zip(branches, branches_copy)]
)
self.accent_vertices(run_time=2)
self.remove(tree)
self.add(spanning_tree)
self.dither(2)
def construct(self):
GraphScene.construct(self)
self.generate_regions()
self.generate_dual_graph()
region_mobs = [ImageMobject(disp.paint_region(reg, self.background), invert=False) for reg in self.regions]
for region, mob in zip(self.regions, region_mobs):
self.highlight_region(region, mob.get_color())
outer_region = self.regions.pop()
outer_region_mob = region_mobs.pop()
outer_dual_vertex = self.dual_vertices.pop()
internal_edges = filter(
lambda e: abs(e.start[0]) < SPACE_WIDTH
and abs(e.end[0]) < SPACE_WIDTH
and abs(e.start[1]) < SPACE_HEIGHT
and abs(e.end[1]) < SPACE_HEIGHT,
self.dual_edges,
)
external_edges = filter(lambda e: e not in internal_edges, self.dual_edges)
self.dither()
self.reset_background()
self.highlight_region(outer_region, outer_region_mob.get_color())
self.play(*[Transform(reg_mob, dot) for reg_mob, dot in zip(region_mobs, self.dual_vertices)])
self.dither()
self.reset_background()
self.play(ApplyFunction(lambda p: (SPACE_WIDTH + SPACE_HEIGHT) * p / np.linalg.norm(p), outer_region_mob))
self.dither()
for edges in internal_edges, external_edges:
self.play(*[ShowCreation(edge, run_time=2.0) for edge in edges])
self.dither()
def construct(self):
GraphScene.construct(self)
account = ImageMobject("facebook_silhouette", invert=False)
account.scale(0.05)
logo = ImageMobject("facebook_logo", invert=False)
logo.scale(0.1)
logo.shift(0.2 * LEFT + 0.1 * UP)
account.add(logo).center()
account.shift(0.2 * LEFT + 0.1 * UP)
friends = TexMobject("\\leftarrow \\text{friends} \\rightarrow").scale(
0.5 * EDGE_ANNOTATION_SCALE_FACTOR)
self.clear()
accounts = [deepcopy(account).shift(point) for point in self.points]
self.add(*accounts)
self.wait()
self.annotate_edges(friends)
self.wait()
self.play(*[
CounterclockwiseTransform(account, vertex)
for account, vertex in zip(accounts, self.vertices)
])
self.wait()
self.play(*[
Transform(ann, edge)
for ann, edge in zip(self.edge_annotations, self.edges)
])
self.wait()
class GraphWidget(QtGui.QGraphicsView):
def __init__(self, index, parent=None):
super(GraphWidget, self).__init__(parent)
self.index = index
self.createWidgets()
self.updateWidgets()
def mouseMoveEvent(self, event):
super(GraphWidget, self).mouseMoveEvent(event)
pt = event.pos()
self.statusBarShowMessage('Coordinate (%2.f, %2.f)' % (pt.x(), pt.y()))
def createWidgets(self):
self.scene = GraphScene(self)
self.scene.setItemIndexMethod(QtGui.QGraphicsScene.NoIndex)
self.scene.setSceneRect(-200, -200, 400, 400)
def updateWidgets(self):
self.setObjectName('GraphWidget-%d' % self.index)
self.setScene(self.scene)
self.setRenderHint(QtGui.QPainter.Antialiasing)
self.scale(0.8, 0.8)
self.setMinimumSize(400, 400)
self.setWindowTitle(
QtGui.QApplication.translate('GraphWidget-%d' % self.index,
'Polygon Area',
None,
QtGui.QApplication.UnicodeUTF8))
self.setMouseTracking(True)
def statusBarShowMessage(self, st):
qst = QtCore.QString(st)
mainWindow = QtGui.qApp.mainWindow
mainWindow.statusBar().showMessage(qst)
def construct(self):
GraphScene.construct(self)
self.generate_dual_graph()
self.generate_regions()
randy = Randolph().shift(LEFT)
morty = Mortimer().shift(RIGHT)
name = TextMobject("Mortimer")
name.shift(morty.get_center() + 1.2*UP)
randy_path = (0, 1, 3)
morty_path = (-2, -3, -4)
morty_crossed_lines = [
Line(self.points[i], self.points[j]).highlight("red")
for i, j in [(7, 1), (1, 5)]
]
kwargs = {"run_time" : 1.0}
self.clear()
self.add(randy)
self.dither()
self.add(morty, name)
self.dither()
self.remove(name)
small_randy = deepcopy(randy).scale(RANDOLPH_SCALE_VAL)
small_morty = deepcopy(morty).scale(RANDOLPH_SCALE_VAL)
small_randy.move_to(self.points[randy_path[0]])
small_morty.move_to(self.dual_points[morty_path[0]])
self.play(*[
FadeIn(mob)
for mob in self.vertices + self.edges
] + [
Transform(randy, small_randy),
Transform(morty, small_morty),
], **kwargs)
self.dither()
self.highlight_region(self.regions[morty_path[0]])
for last, next in zip(morty_path, morty_path[1:]):
self.play(WalkPiCreature(morty, self.dual_points[next]),**kwargs)
self.highlight_region(self.regions[next])
self.dither()
for last, next in zip(randy_path, randy_path[1:]):
line = Line(self.points[last], self.points[next])
line.highlight("yellow")
self.play(
WalkPiCreature(randy, self.points[next]),
ShowCreation(line),
**kwargs
)
self.dither()
self.play(*[
ApplyMethod(
line.rotate_in_place,
np.pi/10,
rate_func = wiggle)
for line in morty_crossed_lines
], **kwargs)
self.dither()
def construct(self):
GraphScene.construct(self)
self.generate_dual_graph()
self.add(TextMobject("Dual Graph").to_edge(UP).shift(2 * LEFT))
self.play(*[
ShowCreation(mob) for mob in self.dual_edges + self.dual_vertices
])
self.wait()
def construct(self):
GraphScene.construct(self)
point_path = [self.points[i] for i in self.path]
randy = Randolph()
randy.scale(RANDOLPH_SCALE_VAL)
randy.move_to(point_path[0])
for next, last in zip(point_path[1:], point_path):
self.play(WalkPiCreature(randy, next), ShowCreation(Line(last, next).highlight("yellow")), run_time=2.0)
self.randy = randy
def construct(self):
GraphScene.construct(self)
randy = Randolph().move_to((-3, 0, 0))
name = TextMobject("Randolph")
self.play(Transform(randy, deepcopy(randy).scale(RANDOLPH_SCALE_VAL).move_to(self.points[0])))
self.dither()
name.shift((0, 1, 0))
self.add(name)
self.dither()
def construct(self):
GraphScene.construct(self)
self.generate_dual_graph()
self.add(TextMobject("Dual Graph").to_edge(UP).shift(2*LEFT))
self.play(*[
ShowCreation(mob)
for mob in self.dual_edges + self.dual_vertices
])
self.dither()
def construct(self):
GraphScene.construct(self)
randy = Randolph()
randy.scale(RANDOLPH_SCALE_VAL).move_to(self.points[0])
dollar_signs = TextMobject("\\$\\$")
dollar_signs.scale(EDGE_ANNOTATION_SCALE_VAL)
dollar_signs = Mobject(*[
deepcopy(dollar_signs).shift(edge.get_center())
for edge in self.edges
])
unneeded = TextMobject("unneeded!")
unneeded.scale(EDGE_ANNOTATION_SCALE_VAL)
self.generate_spanning_tree()
def green_dot_at_index(index):
return Dot(
self.points[index],
radius = 2*Dot.DEFAULT_RADIUS,
color = "lightgreen",
)
def out_of_spanning_set(point_pair):
stip = self.spanning_tree_index_pairs
return point_pair not in stip and \
tuple(reversed(point_pair)) not in stip
self.add(randy)
self.accent_vertices(run_time = 2.0)
self.add(dollar_signs)
self.dither(2)
self.remove(dollar_signs)
run_time_per_branch = 0.5
self.play(
ShowCreation(green_dot_at_index(0)),
run_time = run_time_per_branch
)
for pair in self.spanning_tree_index_pairs:
self.play(ShowCreation(
Line(
self.points[pair[0]],
self.points[pair[1]]
).highlight("yellow"),
run_time = run_time_per_branch
))
self.play(ShowCreation(
green_dot_at_index(pair[1]),
run_time = run_time_per_branch
))
self.dither(2)
unneeded_edges = filter(out_of_spanning_set, self.graph.edges)
for edge, limit in zip(unneeded_edges, range(5)):
line = Line(self.points[edge[0]], self.points[edge[1]])
line.highlight("red")
self.play(ShowCreation(line, run_time = 1.0))
self.add(unneeded.center().shift(line.get_center() + 0.2*UP))
self.dither()
self.remove(line, unneeded)
def construct(self):
GraphScene.construct(self)
tweaked_graph = deepcopy(self.graph)
for index in 2, 4:
tweaked_graph.vertices[index] += 2.8*RIGHT + 1.8*DOWN
tweaked_self = GraphScene(tweaked_graph)
edges_to_remove = [
self.edges[self.graph.edges.index(pair)]
for pair in [(4, 5), (0, 5), (1, 5), (7, 1), (8, 3)]
]
connected, planar, graph = TextMobject([
"Connected ", "Planar ", "Graph"
]).to_edge(UP).split()
not_okay = TextMobject("Not Okay").highlight("red")
planar_explanation = TextMobject("""
(``Planar'' just means we can draw it without
intersecting lines)
""", size = "\\small")
planar_explanation.shift(planar.get_center() + 0.5*DOWN)
self.draw_vertices()
self.draw_edges()
self.clear()
self.add(*self.vertices + self.edges)
self.dither()
self.add(graph)
self.dither()
kwargs = {
"rate_func" : there_and_back,
"run_time" : 5.0
}
self.add(not_okay)
self.play(*[
Transform(*pair, **kwargs)
for pair in zip(
self.edges + self.vertices,
tweaked_self.edges + tweaked_self.vertices,
)
])
self.remove(not_okay)
self.add(planar, planar_explanation)
self.dither(2)
self.remove(planar_explanation)
self.add(not_okay)
self.remove(*edges_to_remove)
self.play(ShowCreation(
Mobject(*edges_to_remove),
rate_func = lambda t : 1 - t,
run_time = 1.0
))
self.dither(2)
self.remove(not_okay)
self.add(connected, *edges_to_remove)
self.dither()
def construct(self):
GraphScene.construct(self)
self.generate_spanning_tree()
self.generate_dual_graph()
self.generate_regions()
randy = Randolph().scale(RANDOLPH_SCALE_VAL)
morty = Mortimer().scale(RANDOLPH_SCALE_VAL)
randy.move_to(self.points[0])
morty.move_to(self.dual_points[0])
attempted_dual_point_index = 2
region_ordering = [0, 1, 7, 2, 3, 5, 4, 6]
dual_edges = [1, 3, 4, 7, 11, 9, 13]
time_per_dual_edge = 0.5
self.add(randy, morty)
self.play(ShowCreation(self.spanning_tree))
self.dither()
self.play(
WalkPiCreature(
morty,
self.dual_points[attempted_dual_point_index],
alpha_func=lambda t: 0.3 * there_and_back(t),
run_time=2.0,
)
)
self.dither()
for index in range(len(self.regions)):
# if index > 0:
# edge = self.edges[dual_edges[index-1]]
# midpoint = edge.get_center()
# self.play(*[
# ShowCreation(Line(
# midpoint,
# tip
# ).highlight("red"))
# for tip in edge.start, edge.end
# ], run_time = time_per_dual_edge)
self.highlight_region(self.regions[region_ordering[index]])
self.dither(time_per_dual_edge)
self.dither()
cycle_index = region_ordering[-1]
cycle = self.graph.region_cycles[cycle_index]
self.highlight_region(self.regions[cycle_index], "black")
self.play(
ShowCreation(
Mobject(
*[
Line(self.points[last], self.points[next]).highlight("green")
for last, next in zip(cycle, list(cycle)[1:] + [cycle[0]])
]
)
)
)
self.dither()
def construct(self):
GraphScene.construct(self)
point_path = [self.points[i] for i in self.path]
randy = Randolph()
randy.scale(RANDOLPH_SCALE_FACTOR)
randy.move_to(point_path[0])
for next, last in zip(point_path[1:], point_path):
self.play(WalkPiCreature(randy, next),
ShowCreation(Line(last, next).highlight("yellow")),
run_time=2.0)
self.randy = randy
def construct(self):
GraphScene.construct(self)
self.generate_spanning_tree()
self.generate_dual_graph()
self.generate_regions()
randy = Randolph().scale(RANDOLPH_SCALE_FACTOR)
morty = Mortimer().scale(RANDOLPH_SCALE_FACTOR)
randy.move_to(self.points[0])
morty.move_to(self.dual_points[0])
attempted_dual_point_index = 2
region_ordering = [0, 1, 7, 2, 3, 5, 4, 6]
dual_edges = [1, 3, 4, 7, 11, 9, 13]
time_per_dual_edge = 0.5
self.add(randy, morty)
self.play(ShowCreation(self.spanning_tree))
self.wait()
self.play(
WalkPiCreature(
morty,
self.dual_points[attempted_dual_point_index],
rate_func=lambda t: 0.3 * there_and_back(t),
run_time=2.0,
))
self.wait()
for index in range(len(self.regions)):
# if index > 0:
# edge = self.edges[dual_edges[index-1]]
# midpoint = edge.get_center()
# self.play(*[
# ShowCreation(Line(
# midpoint,
# tip
# ).highlight("red"))
# for tip in edge.start, edge.end
# ], run_time = time_per_dual_edge)
self.highlight_region(self.regions[region_ordering[index]])
self.wait(time_per_dual_edge)
self.wait()
cycle_index = region_ordering[-1]
cycle = self.graph.region_cycles[cycle_index]
self.highlight_region(self.regions[cycle_index], "black")
self.play(
ShowCreation(
Mobject(*[
Line(self.points[last], self.points[next]).highlight(
"green")
for last, next in zip(cycle,
list(cycle)[1:] + [cycle[0]])
])))
self.wait()
def construct(self):
GraphScene.construct(self)
randy = Randolph()
randy.scale(RANDOLPH_SCALE_FACTOR).move_to(self.points[0])
dollar_signs = TextMobject("\\$\\$")
dollar_signs.scale(EDGE_ANNOTATION_SCALE_FACTOR)
dollar_signs = Mobject(*[
deepcopy(dollar_signs).shift(edge.get_center())
for edge in self.edges
])
unneeded = TextMobject("unneeded!")
unneeded.scale(EDGE_ANNOTATION_SCALE_FACTOR)
self.generate_spanning_tree()
def green_dot_at_index(index):
return Dot(
self.points[index],
radius=2 * Dot.DEFAULT_RADIUS,
color="lightgreen",
)
def out_of_spanning_set(point_pair):
stip = self.spanning_tree_index_pairs
return point_pair not in stip and \
tuple(reversed(point_pair)) not in stip
self.add(randy)
self.accent_vertices(run_time=2.0)
self.add(dollar_signs)
self.wait(2)
self.remove(dollar_signs)
run_time_per_branch = 0.5
self.play(ShowCreation(green_dot_at_index(0)),
run_time=run_time_per_branch)
for pair in self.spanning_tree_index_pairs:
self.play(
ShowCreation(Line(self.points[pair[0]],
self.points[pair[1]]).highlight("yellow"),
run_time=run_time_per_branch))
self.play(
ShowCreation(green_dot_at_index(pair[1]),
run_time=run_time_per_branch))
self.wait(2)
unneeded_edges = filter(out_of_spanning_set, self.graph.edges)
for edge, limit in zip(unneeded_edges, range(5)):
line = Line(self.points[edge[0]], self.points[edge[1]])
line.highlight("red")
self.play(ShowCreation(line, run_time=1.0))
self.add(unneeded.center().shift(line.get_center() + 0.2 * UP))
self.wait()
self.remove(line, unneeded)
def construct(self):
GraphScene.construct(self)
self.generate_dual_graph()
self.generate_regions()
randy = Randolph().shift(LEFT)
morty = Mortimer().shift(RIGHT)
name = TextMobject("Mortimer")
name.shift(morty.get_center() + 1.2 * UP)
randy_path = (0, 1, 3)
morty_path = (-2, -3, -4)
morty_crossed_lines = [
Line(self.points[i], self.points[j]).highlight("red")
for i, j in [(7, 1), (1, 5)]
]
kwargs = {"run_time": 1.0}
self.clear()
self.add(randy)
self.wait()
self.add(morty, name)
self.wait()
self.remove(name)
small_randy = deepcopy(randy).scale(RANDOLPH_SCALE_FACTOR)
small_morty = deepcopy(morty).scale(RANDOLPH_SCALE_FACTOR)
small_randy.move_to(self.points[randy_path[0]])
small_morty.move_to(self.dual_points[morty_path[0]])
self.play(
*[FadeIn(mob) for mob in self.vertices + self.edges] + [
Transform(randy, small_randy),
Transform(morty, small_morty),
], **kwargs)
self.wait()
self.highlight_region(self.regions[morty_path[0]])
for last, next in zip(morty_path, morty_path[1:]):
self.play(WalkPiCreature(morty, self.dual_points[next]), **kwargs)
self.highlight_region(self.regions[next])
self.wait()
for last, next in zip(randy_path, randy_path[1:]):
line = Line(self.points[last], self.points[next])
line.highlight("yellow")
self.play(WalkPiCreature(randy, self.points[next]),
ShowCreation(line), **kwargs)
self.wait()
self.play(
*[
ApplyMethod(line.rotate_in_place, np.pi / 10, rate_func=wiggle)
for line in morty_crossed_lines
], **kwargs)
self.wait()
def construct(self):
GraphScene.construct(self)
tweaked_graph = deepcopy(self.graph)
for index in 2, 4:
tweaked_graph.vertices[index] += 2.8 * RIGHT + 1.8 * DOWN
tweaked_self = GraphScene(tweaked_graph)
edges_to_remove = [
self.edges[self.graph.edges.index(pair)]
for pair in [(4, 5), (0, 5), (1, 5), (7, 1), (8, 3)]
]
connected, planar, graph = TextMobject(
["Connected ", "Planar ", "Graph"]).to_edge(UP).split()
not_okay = TextMobject("Not Okay").highlight("red")
planar_explanation = TextMobject("""
(``Planar'' just means we can draw it without
intersecting lines)
""",
size="\\small")
planar_explanation.shift(planar.get_center() + 0.5 * DOWN)
self.draw_vertices()
self.draw_edges()
self.clear()
self.add(*self.vertices + self.edges)
self.wait()
self.add(graph)
self.wait()
kwargs = {"rate_func": there_and_back, "run_time": 5.0}
self.add(not_okay)
self.play(*[
Transform(*pair, **kwargs) for pair in zip(
self.edges + self.vertices,
tweaked_self.edges + tweaked_self.vertices,
)
])
self.remove(not_okay)
self.add(planar, planar_explanation)
self.wait(2)
self.remove(planar_explanation)
self.add(not_okay)
self.remove(*edges_to_remove)
self.play(
ShowCreation(Mobject(*edges_to_remove),
rate_func=lambda t: 1 - t,
run_time=1.0))
self.wait(2)
self.remove(not_okay)
self.add(connected, *edges_to_remove)
self.wait()
def construct(self):
buff = 0.5
self.graph.vertices = map(
lambda v : v + DOWN + RIGHT,
self.graph.vertices
)
GraphScene.construct(self)
self.generate_regions()
objects, notions = Mobject(*TextMobject(
["Objects \\quad\\quad ", "Thing that connects objects"]
)).to_corner().shift(0.5*RIGHT).split()
horizontal_line = Line(
(-SPACE_WIDTH, SPACE_HEIGHT-1, 0),
(max(notions.points[:,0]), SPACE_HEIGHT-1, 0)
)
vert_line_x_val = min(notions.points[:,0]) - buff
vertical_line = Line(
(vert_line_x_val, SPACE_HEIGHT, 0),
(vert_line_x_val,-SPACE_HEIGHT, 0)
)
objects_and_notions = [
("Facebook accounts", "Friendship"),
("English Words", "Differ by One Letter"),
("Mathematicians", "Coauthorship"),
("Neurons", "Synapses"),
(
"Regions our graph \\\\ cuts the plane into",
"Shared edges"
)
]
self.clear()
self.add(objects, notions, horizontal_line, vertical_line)
for (obj, notion), height in zip(objects_and_notions, it.count(2, -1)):
obj_mob = TextMobject(obj, size = "\\small").to_edge(LEFT)
not_mob = TextMobject(notion, size = "\\small").to_edge(LEFT)
not_mob.shift((vert_line_x_val + SPACE_WIDTH)*RIGHT)
obj_mob.shift(height*UP)
not_mob.shift(height*UP)
if obj.startswith("Regions"):
self.handle_dual_graph(obj_mob, not_mob)
elif obj.startswith("English"):
self.handle_english_words(obj_mob, not_mob)
else:
self.add(obj_mob)
self.dither()
self.add(not_mob)
self.dither()
def construct(self):
GraphScene.construct(self)
randy = Randolph().move_to((-3, 0, 0))
name = TextMobject("Randolph")
self.play(
Transform(
randy,
deepcopy(randy).scale(RANDOLPH_SCALE_FACTOR).move_to(
self.points[0]),
))
self.wait()
name.shift((0, 1, 0))
self.add(name)
self.wait()
def construct(self):
GraphScene.construct(self)
self.draw_vertices()
self.draw_edges()
self.dither()
self.clear()
self.add(*self.edges)
self.replace_vertices_with(Face().scale(0.4))
friends = TextMobject("Friends").scale(EDGE_ANNOTATION_SCALE_VAL)
self.annotate_edges(friends.shift((0, friends.get_height() / 2, 0)))
self.play(
*[CounterclockwiseTransform(vertex, Dot(point)) for vertex, point in zip(self.vertices, self.points)]
+ [Transform(ann, line) for ann, line in zip(self.edge_annotations, self.edges)]
)
self.dither()
def construct(self):
Not, quote, planar, end_quote = TextMobject([
"Not \\\\",
"``",
"Planar",
"''",
# "no matter how \\\\ hard you try"
]).split()
shift_val = Mobject(Not, planar).to_corner().get_center()
Not.highlight("red").shift(shift_val)
graphs = [
Mobject(*GraphScene(g).mobjects)
for g in [CubeGraph(),
CompleteGraph(5),
OctohedronGraph()]
]
self.add(quote, planar, end_quote)
self.wait()
self.play(FadeOut(quote),
FadeOut(end_quote),
ApplyMethod(planar.shift, shift_val),
FadeIn(graphs[0]),
run_time=1.5)
self.wait()
self.remove(graphs[0])
self.add(graphs[1])
planar.highlight("red")
self.add(Not)
self.wait(2)
planar.highlight("white")
self.remove(Not)
self.remove(graphs[1])
self.add(graphs[2])
self.wait(2)
def construct(self):
GraphScene.construct(self)
self.generate_dual_graph()
dual_cycle = DUAL_CYCLE
trapped_points = [0, 1]
morty = Mortimer().scale(RANDOLPH_SCALE_VAL)
morty.move_to(self.dual_points[dual_cycle[0]])
time_per_edge = 0.5
text = TextMobject("""
One of these lines must be included
in the spanning tree if those two inner
vertices are to be reached.
""").scale(0.7).to_edge(UP)
all_lines = []
matching_edges = []
kwargs = {"run_time" : time_per_edge, "rate_func" : None}
for last, next in zip(dual_cycle, dual_cycle[1:]):
line = Line(self.dual_points[last], self.dual_points[next])
line.highlight("red")
self.play(
WalkPiCreature(morty, self.dual_points[next], **kwargs),
ShowCreation(line, **kwargs),
)
all_lines.append(line)
center = line.get_center()
distances = map(
lambda e : np.linalg.norm(center - e.get_center()),
self.edges
)
matching_edges.append(
self.edges[distances.index(min(distances))]
)
self.play(*[
Transform(v, Dot(
v.get_center(),
radius = 3*Dot.DEFAULT_RADIUS,
color = "green"
))
for v in np.array(self.vertices)[trapped_points]
])
self.add(text)
self.play(*[
Transform(line, deepcopy(edge).highlight(line.get_color()))
for line, edge in zip(all_lines, matching_edges)
])
self.dither()
def construct(self):
GraphScene.construct(self)
self.generate_dual_graph()
self.remove(*self.vertices)
self.add(*self.dual_edges)
self.wait()
self.play(*[
Transform(*pair, run_time=2.0)
for pair in zip(self.dual_edges, self.edges)
])
self.wait()
self.add(
TextMobject("""
(Or at least I would argue they should \\\\
be thought of as the same thing.)
""",
size="\\small").to_edge(UP))
self.wait()
def construct(self):
GraphScene.construct(self)
self.draw_vertices()
self.draw_edges()
self.wait()
self.clear()
self.add(*self.edges)
self.replace_vertices_with(Face().scale(0.4))
friends = TextMobject("Friends").scale(EDGE_ANNOTATION_SCALE_FACTOR)
self.annotate_edges(friends.shift((0, friends.get_height() / 2, 0)))
self.play(*[
CounterclockwiseTransform(vertex, Dot(point))
for vertex, point in zip(self.vertices, self.points)
] + [
Transform(ann, line)
for ann, line in zip(self.edge_annotations, self.edges)
])
self.wait()
def construct(self):
GraphScene.construct(self)
self.generate_dual_graph()
self.remove(*self.vertices)
self.add(*self.dual_edges)
self.dither()
self.play(*[Transform(*pair, run_time=2.0) for pair in zip(self.dual_edges, self.edges)])
self.dither()
self.add(
TextMobject(
"""
(Or at least I would argue they should \\\\
be thought of as the same thing.)
""",
size="\\small",
).to_edge(UP)
)
self.dither()
def construct(self):
GraphScene.construct(self)
terms = cycles, spanning_trees, dual_graphs = [
TextMobject(phrase).shift(y*UP).to_edge()
for phrase, y in [
("Cycles", 3),
("Spanning Trees", 1),
("Dual Graphs", -1),
]
]
self.generate_spanning_tree()
scale_val = 1.2
def accent(mobject, color = "yellow"):
return mobject.scale_in_place(scale_val).highlight(color)
def tone_down(mobject):
return mobject.scale_in_place(1.0/scale_val).highlight("white")
self.add(accent(cycles))
self.trace_cycle(run_time = 1.0)
self.dither()
tone_down(cycles)
self.remove(self.traced_cycle)
self.add(accent(spanning_trees))
self.play(ShowCreation(self.spanning_tree), run_time = 1.0)
self.dither()
tone_down(spanning_trees)
self.remove(self.spanning_tree)
self.add(accent(dual_graphs, "red"))
self.generate_dual_graph()
for mob in self.mobjects:
mob.fade
self.play(*[
ShowCreation(mob, run_time = 1.0)
for mob in self.dual_vertices + self.dual_edges
])
self.dither()
self.clear()
self.play(ApplyMethod(
Mobject(*terms).center
))
self.dither()
def construct(self):
GraphScene.construct(self)
self.generate_dual_graph()
dual_cycle = DUAL_CYCLE
trapped_points = [0, 1]
morty = Mortimer().scale(RANDOLPH_SCALE_FACTOR)
morty.move_to(self.dual_points[dual_cycle[0]])
time_per_edge = 0.5
text = TextMobject("""
One of these lines must be included
in the spanning tree if those two inner
vertices are to be reached.
""").scale(0.7).to_edge(UP)
all_lines = []
matching_edges = []
kwargs = {"run_time": time_per_edge, "rate_func": None}
for last, next in zip(dual_cycle, dual_cycle[1:]):
line = Line(self.dual_points[last], self.dual_points[next])
line.highlight("red")
self.play(
WalkPiCreature(morty, self.dual_points[next], **kwargs),
ShowCreation(line, **kwargs),
)
all_lines.append(line)
center = line.get_center()
distances = map(lambda e: np.linalg.norm(center - e.get_center()),
self.edges)
matching_edges.append(self.edges[distances.index(min(distances))])
self.play(*[
Transform(
v,
Dot(v.get_center(),
radius=3 * Dot.DEFAULT_RADIUS,
color="green"))
for v in np.array(self.vertices)[trapped_points]
])
self.add(text)
self.play(*[
Transform(line,
deepcopy(edge).highlight(line.get_color()))
for line, edge in zip(all_lines, matching_edges)
])
self.wait()
def construct(self):
GraphScene.construct(self)
terms = cycles, spanning_trees, dual_graphs = [
TextMobject(phrase).shift(y * UP).to_edge() for phrase, y in [
("Cycles", 3),
("Spanning Trees", 1),
("Dual Graphs", -1),
]
]
self.generate_spanning_tree()
scale_factor = 1.2
def accent(mobject, color="yellow"):
return mobject.scale_in_place(scale_factor).highlight(color)
def tone_down(mobject):
return mobject.scale_in_place(1.0 /
scale_factor).highlight("white")
self.add(accent(cycles))
self.trace_cycle(run_time=1.0)
self.wait()
tone_down(cycles)
self.remove(self.traced_cycle)
self.add(accent(spanning_trees))
self.play(ShowCreation(self.spanning_tree), run_time=1.0)
self.wait()
tone_down(spanning_trees)
self.remove(self.spanning_tree)
self.add(accent(dual_graphs, "red"))
self.generate_dual_graph()
for mob in self.mobjects:
mob.fade
self.play(*[
ShowCreation(mob, run_time=1.0)
for mob in self.dual_vertices + self.dual_edges
])
self.wait()
self.clear()
self.play(ApplyMethod(Mobject(*terms).center))
self.wait()
def construct(self):
GraphScene.construct(self)
self.generate_regions()
self.generate_dual_graph()
cycle = [4, 2, 1, 5, 4]
enclosed_regions = [0, 2, 3, 4]
dual_cycle = DUAL_CYCLE
enclosed_vertices = [0, 1]
randy = Randolph()
randy.scale(RANDOLPH_SCALE_VAL)
randy.move_to(self.points[cycle[0]])
lines_to_remove = []
for last, next in zip(cycle, cycle[1:]):
line = Line(self.points[last], self.points[next])
line.highlight("yellow")
self.play(
ShowCreation(line),
WalkPiCreature(randy, self.points[next]),
run_time = 1.0
)
lines_to_remove.append(line)
self.dither()
self.remove(randy, *lines_to_remove)
for region in np.array(self.regions)[enclosed_regions]:
self.highlight_region(region)
self.dither(2)
self.reset_background()
lines = Mobject(*[
Line(self.dual_points[last], self.dual_points[next])
for last, next in zip(dual_cycle, dual_cycle[1:])
]).highlight("red")
self.play(ShowCreation(lines))
self.play(*[
Transform(v, Dot(
v.get_center(),
radius = 3*Dot.DEFAULT_RADIUS
).highlight("green"))
for v in np.array(self.vertices)[enclosed_vertices]
] + [
ApplyMethod(self.edges[0].highlight, "green")
])
self.dither()
def construct(self):
GraphScene.construct(self)
paths = [
(1, 2, 4, 5, 6),
(6, 7, 1, 3),
]
non_paths = [
[
(0, 1),
(7, 8),
(5, 6),
],
[(5, 0), (0, 2), (0, 1)],
]
valid_path = TextMobject("Valid \\\\ Path").highlight("green")
not_a_path = TextMobject("Not a \\\\ Path").highlight("red")
for mob in valid_path, not_a_path:
mob.to_edge(UP)
kwargs = {"run_time": 1.0}
for path, non_path in zip(paths, non_paths):
path_lines = Mobject(*[
Line(self.points[path[i]], self.points[path[i + 1]]).highlight(
"yellow") for i in range(len(path) - 1)
])
non_path_lines = Mobject(*[
Line(
self.points[pp[0]],
self.points[pp[1]],
).highlight("yellow") for pp in non_path
])
self.remove(not_a_path)
self.add(valid_path)
self.play(ShowCreation(path_lines, **kwargs))
self.wait(2)
self.remove(path_lines)
self.remove(valid_path)
self.add(not_a_path)
self.play(ShowCreation(non_path_lines, **kwargs))
self.wait(2)
self.remove(non_path_lines)
def construct(self):
GraphScene.construct(self)
rot_kwargs = {"radians": np.pi / 3, "run_time": 5.0}
vertices = [
point / 2 + OUT if abs(point[0]) == 2 else point + IN
for point in self.points
]
cube = Mobject(*[
Line(vertices[edge[0]], vertices[edge[1]])
for edge in self.graph.edges
])
cube.rotate(-np.pi / 3, [0, 0, 1])
cube.rotate(-np.pi / 3, [0, 1, 0])
dots_to_vertices = TextMobject("Dots $\\to$ Vertices").to_corner()
lines_to_edges = TextMobject("Lines $\\to$ Edges").to_corner()
regions_to_faces = TextMobject("Regions $\\to$ Faces").to_corner()
self.clear()
# self.play(TransformAnimations(
# Rotating(Dodecahedron(), **rot_kwargs),
# Rotating(cube, **rot_kwargs)
# ))
self.play(Rotating(cube, **rot_kwargs))
self.clear()
self.play(
*[Transform(l1, l2) for l1, l2 in zip(cube.split(), self.edges)])
self.wait()
self.add(dots_to_vertices)
self.play(*[ShowCreation(dot, run_time=1.0) for dot in self.vertices])
self.wait(2)
self.remove(dots_to_vertices, *self.vertices)
self.add(lines_to_edges)
self.play(ApplyMethod(Mobject(*self.edges).highlight, "yellow"))
self.wait(2)
self.clear()
self.add(*self.edges)
self.add(regions_to_faces)
self.generate_regions()
for region in self.regions:
self.highlight_region(region)
self.wait(3.0)
def construct(self):
GraphScene.construct(self)
self.generate_dual_graph()
self.generate_spanning_tree()
randy = Randolph()
randy.scale(RANDOLPH_SCALE_VAL)
randy.move_to(self.points[0])
morty = Mortimer()
morty.scale(RANDOLPH_SCALE_VAL)
morty.move_to(self.dual_points[0])
dual_edges = [1, 3, 4, 7, 11, 9, 13]
words = TextMobject(
"""
The red edges form a spanning tree of the dual graph!
"""
).to_edge(UP)
self.add(self.spanning_tree, randy, morty)
self.play(ShowCreation(Mobject(*np.array(self.edges)[dual_edges]).highlight("red")))
self.add(words)
self.dither()
def construct(self):
GraphScene.construct(self)
self.generate_dual_graph()
self.generate_spanning_tree()
randy = Randolph()
randy.scale(RANDOLPH_SCALE_FACTOR)
randy.move_to(self.points[0])
morty = Mortimer()
morty.scale(RANDOLPH_SCALE_FACTOR)
morty.move_to(self.dual_points[0])
dual_edges = [1, 3, 4, 7, 11, 9, 13]
words = TextMobject("""
The red edges form a spanning tree of the dual graph!
""").to_edge(UP)
self.add(self.spanning_tree, randy, morty)
self.play(
ShowCreation(
Mobject(*np.array(self.edges)[dual_edges]).highlight("red")))
self.add(words)
self.wait()
def construct(self):
buff = 0.5
self.graph.vertices = map(lambda v: v + DOWN + RIGHT,
self.graph.vertices)
GraphScene.construct(self)
self.generate_regions()
objects, notions = Mobject(*TextMobject([
"Objects \\quad\\quad ", "Thing that connects objects"
])).to_corner().shift(0.5 * RIGHT).split()
horizontal_line = Line(
(-SPACE_WIDTH, SPACE_HEIGHT - 1, 0),
(max(notions.points[:, 0]), SPACE_HEIGHT - 1, 0))
vert_line_x_val = min(notions.points[:, 0]) - buff
vertical_line = Line((vert_line_x_val, SPACE_HEIGHT, 0),
(vert_line_x_val, -SPACE_HEIGHT, 0))
objects_and_notions = [("Facebook accounts", "Friendship"),
("English Words", "Differ by One Letter"),
("Mathematicians", "Coauthorship"),
("Neurons", "Synapses"),
("Regions our graph \\\\ cuts the plane into",
"Shared edges")]
self.clear()
self.add(objects, notions, horizontal_line, vertical_line)
for (obj, notion), height in zip(objects_and_notions, it.count(2, -1)):
obj_mob = TextMobject(obj, size="\\small").to_edge(LEFT)
not_mob = TextMobject(notion, size="\\small").to_edge(LEFT)
not_mob.shift((vert_line_x_val + SPACE_WIDTH) * RIGHT)
obj_mob.shift(height * UP)
not_mob.shift(height * UP)
if obj.startswith("Regions"):
self.handle_dual_graph(obj_mob, not_mob)
elif obj.startswith("English"):
self.handle_english_words(obj_mob, not_mob)
else:
self.add(obj_mob)
self.wait()
self.add(not_mob)
self.wait()
def construct(self):
GraphScene.construct(self)
self.generate_regions()
self.generate_dual_graph()
region_mobs = [
ImageMobject(disp.paint_region(reg, self.background), invert=False)
for reg in self.regions
]
for region, mob in zip(self.regions, region_mobs):
self.highlight_region(region, mob.get_color())
outer_region = self.regions.pop()
outer_region_mob = region_mobs.pop()
outer_dual_vertex = self.dual_vertices.pop()
internal_edges = filter(
lambda e : abs(e.start[0]) < SPACE_WIDTH and \
abs(e.end[0]) < SPACE_WIDTH and \
abs(e.start[1]) < SPACE_HEIGHT and \
abs(e.end[1]) < SPACE_HEIGHT,
self.dual_edges
)
external_edges = filter(lambda e: e not in internal_edges,
self.dual_edges)
self.wait()
self.reset_background()
self.highlight_region(outer_region, outer_region_mob.get_color())
self.play(*[
Transform(reg_mob, dot)
for reg_mob, dot in zip(region_mobs, self.dual_vertices)
])
self.wait()
self.reset_background()
self.play(
ApplyFunction(
lambda p: (SPACE_WIDTH + SPACE_HEIGHT) * p / np.linalg.norm(p),
outer_region_mob))
self.wait()
for edges in internal_edges, external_edges:
self.play(*[ShowCreation(edge, run_time=2.0) for edge in edges])
self.wait()
def construct(self):
GraphScene.construct(self)
self.generate_regions()
self.generate_dual_graph()
cycle = [4, 2, 1, 5, 4]
enclosed_regions = [0, 2, 3, 4]
dual_cycle = DUAL_CYCLE
enclosed_vertices = [0, 1]
randy = Randolph()
randy.scale(RANDOLPH_SCALE_FACTOR)
randy.move_to(self.points[cycle[0]])
lines_to_remove = []
for last, next in zip(cycle, cycle[1:]):
line = Line(self.points[last], self.points[next])
line.highlight("yellow")
self.play(ShowCreation(line),
WalkPiCreature(randy, self.points[next]),
run_time=1.0)
lines_to_remove.append(line)
self.wait()
self.remove(randy, *lines_to_remove)
for region in np.array(self.regions)[enclosed_regions]:
self.highlight_region(region)
self.wait(2)
self.reset_background()
lines = Mobject(*[
Line(self.dual_points[last], self.dual_points[next])
for last, next in zip(dual_cycle, dual_cycle[1:])
]).highlight("red")
self.play(ShowCreation(lines))
self.play(*[
Transform(
v,
Dot(v.get_center(), radius=3 *
Dot.DEFAULT_RADIUS).highlight("green"))
for v in np.array(self.vertices)[enclosed_vertices]
] + [ApplyMethod(self.edges[0].highlight, "green")])
self.wait()
def construct(self):
time_per_branch = 0.5
text = TextMobject("""
In any tree:
$$E + 1 = V$$
""")
gs = GraphScene(SampleGraph())
gs.generate_treeified_spanning_tree()
branches = gs.treeified_spanning_tree.to_edge(LEFT).split()
all_dots = [Dot(branches[0].points[0])]
self.add(text, all_dots[0])
for branch in branches:
self.play(ShowCreation(branch), run_time=time_per_branch)
dot = Dot(branch.points[-1])
self.add(dot)
all_dots.append(dot)
self.wait()
self.remove(*all_dots)
self.play(
FadeOut(text), FadeIn(Mobject(*gs.edges + gs.vertices)), *[
Transform(*pair)
for pair in zip(branches, gs.spanning_tree.split())
])
def createWidgets(self):
self.scene = GraphScene(self)
self.scene.setItemIndexMethod(QtGui.QGraphicsScene.NoIndex)
self.scene.setSceneRect(-200, -200, 400, 400)
def __init__(self, graph, path, *args, **kwargs):
self.path = path
GraphScene.__init__(self, graph, *args, **kwargs)
def __init__(self, graph, path, *args, **kwargs):
self.path = path
GraphScene.__init__(self, graph, *args, **kwargs)
