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 = "#0E1017" 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 TEXT_MAX_W = config.frame_width - 1.4 BLUE_S = "#58C4DD" GOLD_S = "#F0C674" GREEN_S = "#83C167" RED_S = "#E07A5F" def fit(m, w=None, h=None): w = w or TEXT_MAX_W h = h or (config.frame_height - 2.6) if m.width > w: m.scale_to_fit_width(w) if m.height > h: m.scale_to_fit_height(h) return m class ArchimedeDensite(Scene): def construct(self): self.titre1() self.axiome_A() self.axiome_Abis() self.preuve_A_implique_Abis() self.titre2() self.theoreme2() self.preuve_densite_1() self.preuve_densite_2() self.fin() # ---------- helpers ---------- def title_bar(self, txt): t = Text(txt, font_size=36, color=GOLD_S, weight=BOLD) t.to_edge(UP, buff=0.5) return t def body_top(self, title): return title.get_bottom()[1] - 0.4 def place_below(self, group, title, buff=0.45): group.next_to(title, DOWN, buff=buff).set_x(0) if group.get_bottom()[1] < -SAFE_Y_BOTTOM: group.scale_to_fit_height( max(0.5, group.height - (-SAFE_Y_BOTTOM - group.get_bottom()[1]))) group.next_to(title, DOWN, buff=buff).set_x(0) return group def make_line(self, xmin, xmax, step=1, width=12.5): return NumberLine( x_range=[xmin, xmax, step], length=width, include_numbers=True, font_size=24, color=GREY_B, decimal_number_config={"num_decimal_places": 0}, ) def clear_all(self, rt=0.7): if self.mobjects: self.play(FadeOut(Group(*self.mobjects)), run_time=rt) # ---------- 1 ---------- def titre1(self): t1 = Text("L'axiome d'Archim\u00e8de", font_size=58, color=BLUE_S, weight=BOLD) line = Line(LEFT * 3.2, RIGHT * 3.2, color=GOLD_S).next_to(t1, DOWN, buff=0.4) g = VGroup(t1, line).move_to(ORIGIN) self.play(Write(t1), run_time=1.4) self.play(Create(line), run_time=0.8) self.wait(1.0) self.play(FadeOut(g), run_time=0.7) # ---------- 2 ---------- def axiome_A(self): title = self.title_bar("Axiome (A)") fr = Text("Si x > 0 et si y est un r\u00e9el donn\u00e9,\nalors il existe n \u2208 N tel que n\u00b7x > y.", font_size=30, color=WHITE, line_spacing=0.8) sym = MathTex(r"(\forall x\in\mathbb{R}^{*}_{+})(\forall y\in\mathbb{R})(\exists n\in\mathbb{N}),\ nx>y", font_size=40, color=BLUE_S) band = VGroup(fr, sym).arrange(DOWN, buff=0.45) fit(band, TEXT_MAX_W, 3.0) band.move_to(ORIGIN) self.play(FadeIn(title, shift=DOWN * 0.2), run_time=0.6) self.play(Write(fr), run_time=2.0) self.play(Write(sym), run_time=1.8) self.wait(1.6) self.play(band.animate.scale(0.8).next_to(title, DOWN, buff=0.35).set_x(0), run_time=0.9) nl = self.make_line(0, 7, 1, 12.5) nl.move_to(DOWN * 1.1) self.play(Create(nl), run_time=1.0) x_val, y_val = 0.55, 4.3 ypt = Dot(nl.n2p(y_val), color=RED_S, radius=0.09) yline = DashedLine(nl.n2p(y_val), nl.n2p(y_val) + UP * 0.9, color=RED_S) ylab = MathTex("y", font_size=34, color=RED_S).next_to(yline, UP, buff=0.12) self.play(GrowFromCenter(ypt), Create(yline), FadeIn(ylab)) first = Line(nl.n2p(0), nl.n2p(x_val), color=GREEN_S, stroke_width=7) counter = MathTex("n=1", font_size=36, color=GREEN_S) counter.next_to(nl, DOWN, buff=0.75).set_x(0) xlab = MathTex("x", font_size=32, color=GREEN_S).next_to(first, DOWN, buff=0.15) self.play(Create(first), FadeIn(xlab), FadeIn(counter)) self.wait(0.5) self.play(FadeOut(xlab), run_time=0.3) segs = VGroup(first) n = 1 while n * x_val <= y_val: n += 1 seg = Line(nl.n2p((n - 1) * x_val), nl.n2p(n * x_val), color=GREEN_S, stroke_width=7) new_c = MathTex(f"n={n}", font_size=36, color=GREEN_S).move_to(counter) self.play(Create(seg), Transform(counter, new_c), run_time=0.3) segs.add(seg) concl = MathTex(rf"{n}\,x > y", font_size=40, color=GOLD_S) concl.next_to(counter, DOWN, buff=0.35).set_x(0) if concl.get_bottom()[1] < -SAFE_Y_BOTTOM: concl.shift(UP * (-SAFE_Y_BOTTOM - concl.get_bottom()[1])) self.play(Write(concl), Flash(nl.n2p(n * x_val), color=GOLD_S, line_length=0.3)) self.wait(1.8) self.clear_all() # ---------- 3 ---------- def axiome_Abis(self): title = self.title_bar("Axiome (A bis)") fr = Text("Pour tout z > 0, il existe un entier n \u2265 1\ntel que 1/n < z.", font_size=30, color=WHITE, line_spacing=0.8) sym = MathTex(r"(\forall z\in\mathbb{R}^{*}_{+})(\exists n\in\mathbb{N}^{*}):\ \frac{1}{n}0)(\exists n\in\mathbb{N}^{*}),\ \tfrac{1}{n} 1", font_size=34, color=BLUE_S), MathTex(r"n\neq 0\ \ \text{(si }n=0,\ 0\cdot z=0\not>1,\ \text{absurde)}", font_size=34, color=RED_S), MathTex(r"\text{On multiplie }nz>1\ \text{ par } n^{-1}>0", font_size=34, color=WHITE), MathTex(r"n^{-1}(nz) > n^{-1}\cdot 1 \quad\text{c.-\`a-d.}\quad z > \tfrac{1}{n}", font_size=34, color=GREEN_S), MathTex(r"\text{Donc (A)}\ \Longrightarrow\ \text{(A bis)}.\qquad \blacksquare", font_size=36, color=GOLD_S), ).arrange(DOWN, buff=0.34, aligned_edge=LEFT) fit(lines, TEXT_MAX_W, config.frame_height - 2.6) self.place_below(lines, title, 0.45) for ln in lines: self.play(FadeIn(ln, shift=RIGHT * 0.25), run_time=0.85) self.wait(0.55) self.wait(1.8) self.clear_all() # ---------- 5 ---------- def titre2(self): t = Text("Densit\u00e9 de Q dans R", font_size=56, color=BLUE_S, weight=BOLD) sub = MathTex(r"\mathbb{Q}\ \text{dense dans}\ \mathbb{R}", font_size=42, color=GOLD_S) g = VGroup(t, sub).arrange(DOWN, buff=0.5).move_to(ORIGIN) self.play(Write(t), run_time=1.3) self.play(FadeIn(sub, shift=UP * 0.2), run_time=0.8) self.wait(1.2) self.play(FadeOut(g), run_time=0.7) # ---------- 6 ---------- def theoreme2(self): title = self.title_bar("Th\u00e9or\u00e8me 2 (densit\u00e9 des rationnels)") fr = Text("Si a, b sont des r\u00e9els tels que a < b,\nalors il existe q \u2208 Q tel que a < q < b.", font_size=32, color=WHITE, line_spacing=0.8) sym = MathTex(r"a0", font_size=32, color=WHITE), MathTex(r"\text{Par (A bis) avec } z:=b-a:\ \exists n_0\in\mathbb{N}^{*},\ \tfrac{1}{n_0}0", font_size=30, color=RED_S).next_to(brace, UP, buff=0.1) self.play(GrowFromCenter(brace), FadeIn(blab)) self.wait(0.8) n0 = 8 step = 1.0 / n0 pas = Line(nl.n2p(0), nl.n2p(step), color=BLUE_S, stroke_width=10) pas.next_to(nl, DOWN, buff=0.95).align_to(nl.n2p(0), LEFT) plab = MathTex(r"\frac{1}{n_0}", font_size=32, color=BLUE_S).next_to(pas, RIGHT, buff=0.25) pg = VGroup(pas, plab) if pg.get_bottom()[1] < -SAFE_Y_BOTTOM: pg.shift(UP * (-SAFE_Y_BOTTOM - pg.get_bottom()[1])) self.play(Create(pas), FadeIn(plab)) self.wait(0.6) self.play(pg.animate.shift(RIGHT * (pa[0] - pas.get_left()[0])), run_time=1.2) self.wait(1.6) self.clear_all() # ---------- 7b : ensemble M, m0, chaîne d'inégalités ---------- def preuve_densite_2(self): title = self.title_bar("D\u00e9monstration (2) : l'ensemble M et m\u2080") self.play(FadeIn(title, shift=DOWN * 0.2), run_time=0.6) setM = MathTex(r"M:=\Big\{m\in\mathbb{N}\ :\ m\cdot\tfrac{1}{n_0}>a\Big\},\quad M\subseteq\mathbb{N}^{*},\ M\neq\emptyset\ \text{(par (A))}", font_size=32, color=BLUE_S) m0def = MathTex(r"m_0:=\min M\quad\Rightarrow\quad a<\frac{m_0}{n_0}", font_size=32, color=GREEN_S) head = VGroup(setM, m0def).arrange(DOWN, buff=0.3) fit(head, TEXT_MAX_W, 1.8) head.next_to(title, DOWN, buff=0.35).set_x(0) self.play(Write(setM), run_time=1.8) self.wait(0.6) nl = self.make_line(0, 2, 1, 12.0) nl.next_to(head, DOWN, buff=0.9).set_x(0) a, b = 0.62, 0.95 pa, pb = nl.n2p(a), nl.n2p(b) iv = Line(pa, pb, color=GOLD_S, stroke_width=9) la = MathTex("a", font_size=30, color=GOLD_S).next_to(Dot(pa), DOWN, buff=0.28) lb = MathTex("b", font_size=30, color=GOLD_S).next_to(Dot(pb), DOWN, buff=0.28) self.play(Create(nl), Create(iv), FadeIn(la), FadeIn(lb), run_time=1.2) n0 = 8 step = 1.0 / n0 ticks = VGroup() for k in range(0, 2 * n0 + 1): p = nl.n2p(k * step) ticks.add(Line(p + DOWN * 0.14, p + UP * 0.14, color=BLUE_S, stroke_width=3)) self.play(LaggedStartMap(Create, ticks, lag_ratio=0.04), run_time=1.6) m0 = int(np.floor(a / step)) + 1 walker = Dot(nl.n2p(0), color=BLUE_S, radius=0.09) cnt = MathTex("m=0", font_size=32, color=BLUE_S) cnt.next_to(nl, DOWN, buff=0.8).set_x(0) if cnt.get_bottom()[1] < -SAFE_Y_BOTTOM: cnt.shift(UP * (-SAFE_Y_BOTTOM - cnt.get_bottom()[1])) self.play(FadeIn(walker), FadeIn(cnt)) for m in range(1, m0 + 1): newc = MathTex(f"m={m}", font_size=32, color=BLUE_S).move_to(cnt) self.play(walker.animate.move_to(nl.n2p(m * step)), Transform(cnt, newc), run_time=0.3) self.play(walker.animate.set_color(GREEN_S).scale(1.3), Flash(nl.n2p(m0 * step), color=GREEN_S, line_length=0.3)) self.play(FadeIn(m0def, shift=UP * 0.2), run_time=0.9) self.wait(1.4) self.play(FadeOut(VGroup(nl, iv, la, lb, ticks, walker, cnt, setM, m0def)), run_time=0.8) chain = VGroup( MathTex(r"\frac{m_0}{n_0}=\frac{m_0-1}{n_0}+\frac{1}{n_0}", font_size=34, color=WHITE), MathTex(r"\frac{m_0-1}{n_0}+\frac{1}{n_0}\ \le\ a+\frac{1}{n_0}\quad(m_0-1\notin M)", font_size=34, color=BLUE_S), MathTex(r"a+\frac{1}{n_0}\ <\ a+(b-a)=b", font_size=34, color=RED_S), MathTex(r"a<\frac{m_0-1}{n_0}+\frac{1}{n_0}\le a+\frac{1}{n_0}

Densité des Rationnels