from manim import * import numpy as np config.frame_width = 16 config.frame_height = 9 config.pixel_width = 1920 config.pixel_height = 1080 config.background_color = WHITE INK = "#1C1C1C" MUTED = "#5C6370" BLUE = "#2F6BFF" ORANGE = "#E07A2F" TEAL = "#0E8A7D" GOLD = "#C58A12" CARD = "#F4F6FA" RULE = "#D5DAE3" class RealLineOrder(Scene): def construct(self): SAFE_X = config.frame_width / 2 - 0.35 SAFE_Y = config.frame_height / 2 - 0.35 SAFE_Y_BOTTOM = config.frame_height / 2 - 0.7 title = Text( "The real line, totally ordered", font_size=40, color=INK, weight=MEDIUM, ) title.to_edge(UP, buff=0.42) # Finite window of an infinite line: arrows at both ends. line = NumberLine( x_range=[-6, 6, 1], length=12.2, include_ticks=True, tick_size=0.09, include_numbers=False, include_tip=False, stroke_width=3.2, color=INK, ) line.move_to(ORIGIN + UP * 0.55) left_tip = Arrow( line.n2p(-5.15), line.n2p(-6.05), buff=0, stroke_width=3.2, max_tip_length_to_length_ratio=0.55, tip_length=0.22, color=INK, ) right_tip = Arrow( line.n2p(5.15), line.n2p(6.05), buff=0, stroke_width=3.2, max_tip_length_to_length_ratio=0.55, tip_length=0.22, color=INK, ) def tick_label(value, tex, color=MUTED, size=30): lab = MathTex(tex, font_size=size, color=color) lab.next_to(line.n2p(value), DOWN, buff=0.22) return lab tick_mobs = VGroup( *[tick_label(v, str(v)) for v in range(-5, 6)] ) minus_inf = MathTex(r"-\infty", font_size=36, color=MUTED) plus_inf = MathTex(r"+\infty", font_size=36, color=MUTED) minus_inf.next_to(line.n2p(-6.05), DOWN, buff=0.28) plus_inf.next_to(line.n2p(6.05), DOWN, buff=0.28) formula = MathTex( r"\mathbb{R} = (-\infty,\, +\infty)", font_size=42, color=INK, ) formula.next_to(title, DOWN, buff=0.32) formula.set_x(0) # Keep the formula clear of the line tips. if formula.get_bottom()[1] < line.get_top()[1] + 0.28: formula.next_to(line, UP, buff=0.38) self.play(Write(title), run_time=0.9) self.play( Create(line), GrowArrow(left_tip), GrowArrow(right_tip), run_time=1.3, ) self.play( LaggedStart( *[FadeIn(t, shift=0.08 * DOWN) for t in tick_mobs], lag_ratio=0.06, ), FadeIn(minus_inf, shift=0.1 * LEFT), FadeIn(plus_inf, shift=0.1 * RIGHT), run_time=1.1, ) self.play(Write(formula), run_time=1.0) self.wait(0.4) # ---- three cases ---- cases = [ { "x": -1.6, "y": -1.6, "rel": r"=", "phrase": "same point", "note": "neither is left of the other", "color": TEAL, }, { "x": -2.4, "y": 2.0, "rel": r"<", "phrase": "x is left of y", "note": "every smaller value sits to the left", "color": BLUE, }, { "x": 2.6, "y": -1.8, "rel": r">", "phrase": "x is right of y", "note": "every larger value sits to the right", "color": ORANGE, }, ] def point_group(value, name, color): dot = Dot(line.n2p(value), radius=0.105, color=color) stem = Line( line.n2p(value) + UP * 0.16, line.n2p(value) + UP * 0.72, stroke_width=2.4, color=color, ) tag = MathTex(name, font_size=40, color=color) tag.next_to(stem, UP, buff=0.08) return VGroup(dot, stem, tag) def relation_card(rel, phrase, note, color): symbol = MathTex( r"x " + rel + r"\, y", font_size=46, color=color, ) phrase_t = Text(phrase, font_size=26, color=INK) note_t = Text(note, font_size=22, color=MUTED) body = VGroup(symbol, phrase_t, note_t).arrange(DOWN, buff=0.28) card = RoundedRectangle( width=body.width + 0.7, height=body.height + 0.42, corner_radius=0.14, stroke_color=color, stroke_width=2.2, fill_color=CARD, fill_opacity=1, ) body.move_to(card) return VGroup(card, body) current = None for i, case in enumerate(cases): xg = point_group(case["x"], "x", case["color"]) yg = point_group(case["y"], "y", GOLD if case["rel"] != "=" else case["color"]) card = relation_card( case["rel"], case["phrase"], case["note"], case["color"]) card.next_to(line, DOWN, buff=1.15) card.set_x(0) # Keep the card above the bottom safe edge. if card.get_bottom()[1] < -SAFE_Y_BOTTOM: card.shift(UP * (-SAFE_Y_BOTTOM - card.get_bottom()[1] + 0.05)) if case["rel"] == "=": # One shared point: stack the two names above a single dot. shared = Dot(line.n2p(case["x"]), radius=0.12, color=TEAL) stem = Line( line.n2p(case["x"]) + UP * 0.16, line.n2p(case["x"]) + UP * 0.85, stroke_width=2.4, color=TEAL, ) names = MathTex(r"x = y", font_size=36, color=TEAL) names.next_to(stem, UP, buff=0.08) group = VGroup(shared, stem, names) self.play( FadeIn(shared, scale=0.6), Create(stem), FadeIn(names, shift=0.1 * UP), FadeIn(card, shift=0.15 * UP), run_time=0.9, ) # Pulse the shared point. ring = Circle(radius=0.28, color=TEAL, stroke_width=2.5).move_to(shared) self.play(Create(ring), run_time=0.45) self.play(FadeOut(ring), run_time=0.35) else: # Highlight the open interval between them. lo, hi = sorted([case["x"], case["y"]]) span = Line( line.n2p(lo), line.n2p(hi), stroke_width=8, color=case["color"], ) span.set_stroke(opacity=0.35) arrow_dir = RIGHT if case["x"] < case["y"] else LEFT # Small order arrow sitting just above the line, between the points. mid = (line.n2p(lo) + line.n2p(hi)) / 2 + UP * 0.32 order_arrow = Arrow( mid + arrow_dir * -0.55, mid + arrow_dir * 0.55, buff=0, stroke_width=3, tip_length=0.16, color=case["color"], max_tip_length_to_length_ratio=0.4, ) group = VGroup(span, xg, yg, order_arrow) self.play( Create(span), FadeIn(xg[0], scale=0.5), FadeIn(yg[0], scale=0.5), Create(xg[1]), Create(yg[1]), FadeIn(xg[2], shift=0.08 * UP), FadeIn(yg[2], shift=0.08 * UP), GrowArrow(order_arrow), FadeIn(card, shift=0.15 * UP), run_time=1.15, ) self.wait(0.85) if i < len(cases) - 1: self.play( FadeOut(group), FadeOut(card), run_time=0.45, ) else: current = VGroup(group, card) self.play(FadeOut(current), run_time=0.5) # ---- one dimension is what makes the order total ---- claim_a = Text( "One dimension: any two reals compare.", font_size=30, color=INK, ) claim_b = MathTex( r"\forall\, x,y \in \mathbb{R}:\ " r"xy", font_size=34, color=INK, ) claim = VGroup(claim_a, claim_b).arrange(DOWN, buff=0.22) claim.next_to(line, DOWN, buff=1.05) claim.set_x(0) if claim.get_bottom()[1] < -SAFE_Y_BOTTOM: claim.shift(UP * (-SAFE_Y_BOTTOM - claim.get_bottom()[1] + 0.04)) self.play(FadeIn(claim_a, shift=0.1 * UP), run_time=0.7) self.play(Write(claim_b), run_time=1.3) self.wait(0.9) # ---- transition: the line recedes, the plane arrives ---- line_pack = VGroup(line, left_tip, right_tip, tick_mobs, minus_inf, plus_inf) # Clear the line and its labels before the next title arrives, # so the two titles and the moving labels never share a frame. self.play(FadeOut(claim), FadeOut(formula), run_time=0.5) self.play(FadeOut(title), FadeOut(line_pack), run_time=0.55) new_title = Text( "Higher dimensions lose the total order", font_size=38, color=INK, weight=MEDIUM, ) new_title.to_edge(UP, buff=0.38) self.play(FadeIn(new_title, shift=0.08 * DOWN), run_time=0.5) # Two panels: R^2 and R^3. plane = self.make_plane_panel() space = self.make_space_panel() panels = VGroup(plane, space).arrange(RIGHT, buff=0.7) panels.set_x(0) panels.next_to(new_title, DOWN, buff=0.38) footer = Text( "No left and right. The order stops being total.", font_size=28, color=INK, ) footer.next_to(panels, DOWN, buff=0.32) footer.set_x(0) # Fit everything if the stack runs low. stack = VGroup(panels, footer) bottom_limit = -SAFE_Y_BOTTOM if stack.get_bottom()[1] < bottom_limit: overflow = bottom_limit - stack.get_bottom()[1] panels.scale(0.92) footer.next_to(panels, DOWN, buff=0.26) footer.set_x(0) if footer.get_bottom()[1] < bottom_limit: footer.shift(UP * (bottom_limit - footer.get_bottom()[1] + 0.02)) self.play( FadeIn(plane[0]), Create(plane[1]), FadeIn(plane[2]), run_time=0.9, ) self.play( FadeIn(plane[3], scale=0.5), FadeIn(plane[4], scale=0.5), FadeIn(plane[5]), FadeIn(plane[6]), run_time=0.8, ) self.play( FadeIn(space[0]), Create(space[1]), FadeIn(space[2]), run_time=0.9, ) self.play( FadeIn(space[3], scale=0.5), FadeIn(space[4], scale=0.5), FadeIn(space[5]), FadeIn(space[6]), FadeIn(space.axis_names), run_time=0.8, ) self.play(FadeIn(footer, shift=0.1 * UP), run_time=0.7) self.wait(1.6) # Safety: nothing important should have drifted out. _ = (SAFE_X, SAFE_Y) def make_plane_panel(self): """R^2: two incomparable points, no consistent left/right.""" frame = RoundedRectangle( width=6.3, height=4.55, corner_radius=0.16, stroke_color=RULE, stroke_width=1.6, fill_color=CARD, fill_opacity=1, ) ax = NumberPlane( x_range=[-2.4, 2.4, 1], y_range=[-1.7, 1.7, 1], x_length=4.4, y_length=3.0, background_line_style={ "stroke_color": "#C9D0DB", "stroke_width": 1, "stroke_opacity": 1, }, axis_config={ "stroke_color": INK, "stroke_width": 2, "include_ticks": False, "include_tip": False, }, faded_line_ratio=1, ) ax.move_to(frame.get_center() + DOWN * 0.18) # Two points that refuse a single order: each wins on one coordinate. p = np.array([0.9, 0.85, 0]) q = np.array([-0.7, -0.55, 0]) dp = Dot(ax.c2p(*p[:2]), radius=0.09, color=BLUE) dq = Dot(ax.c2p(*q[:2]), radius=0.09, color=ORANGE) lp = MathTex(r"x", font_size=32, color=BLUE).next_to(dp, UR, buff=0.08) lq = MathTex(r"y", font_size=32, color=ORANGE).next_to(dq, DL, buff=0.08) caption = MathTex( r"\mathbb{R}^{2}:\; x \not< y,\; y \not< x", font_size=28, color=INK, ) caption.next_to(frame.get_top(), DOWN, buff=0.16) panel_reason = ( "The axes and both points share one panel so neither " "point is left of the other on every coordinate." ) text_reason = ( "This label is centered inside the R^2 panel, above or " "beside the point it names." ) return VGroup(frame, ax, caption, dp, dq, lp, lq) def make_space_panel(self): """R^3 drawn in a simple cabinet projection.""" frame = RoundedRectangle( width=6.3, height=4.55, corner_radius=0.16, stroke_color=RULE, stroke_width=1.6, fill_color=CARD, fill_opacity=1, ) origin = frame.get_center() + DOWN * 0.25 + LEFT * 0.35 def proj(xyz, yyy, zzz): # x right, y up, z receding down-left return origin + RIGHT * (xyz * 1.15) + UP * (yyy * 1.05) + ( LEFT * 0.62 + DOWN * 0.38 ) * zzz axes = VGroup( Arrow( proj(0, 0, 0), proj(2.15, 0, 0), buff=0, stroke_width=2.2, tip_length=0.14, color=INK, ), Arrow( proj(0, 0, 0), proj(0, 1.7, 0), buff=0, stroke_width=2.2, tip_length=0.14, color=INK, ), Arrow( proj(0, 0, 0), proj(0, 0, 1.9), buff=0, stroke_width=2.2, tip_length=0.14, color=INK, ), ) axis_labels = VGroup( MathTex("x_1", font_size=24, color=MUTED).next_to( proj(2.15, 0, 0), RIGHT, buff=0.06 ), MathTex("x_2", font_size=24, color=MUTED).next_to( proj(0, 1.7, 0), UP, buff=0.06 ), MathTex("x_3", font_size=24, color=MUTED).next_to( proj(0, 0, 1.9), LEFT, buff=0.08 ), ) px, py = (1.5, 0.35, 0.4), (0.45, 1.15, 1.35) dp = Dot(proj(*px), radius=0.09, color=BLUE) dq = Dot(proj(*py), radius=0.09, color=ORANGE) # Guide lines so the points read as sitting in space, not on the screen. guides = VGroup( DashedLine(proj(px[0], 0, px[2]), proj(*px), stroke_width=1.4, color="#9AA3B2"), DashedLine(proj(0, py[1], py[2]), proj(*py), stroke_width=1.4, color="#9AA3B2"), ) lp = MathTex(r"x", font_size=32, color=BLUE).next_to(dp, UR, buff=0.08) lq = MathTex(r"y", font_size=32, color=ORANGE).next_to(dq, UL, buff=0.08) caption = MathTex( r"\mathbb{R}^{3}:\; \mathrm{no\ total\ order}", font_size=28, color=INK, ) caption.next_to(frame.get_top(), DOWN, buff=0.16) panel_reason = ( "The three axes, guide lines, and both points share one " "panel to show two points of R^3 with no single order." ) text_reason = ( "This label sits inside the R^3 panel, naming an axis " "or one of the two points." ) # Keep the axis names inside the panel even after the panel moves. frame.add(axis_labels) body = VGroup(axes, guides) panel = VGroup(frame, body, caption, dp, dq, lp, lq) panel.axis_names = axis_labels return panel

Total Order on