symmetric_polynomials.py

#!/usr/bin/env python

from manimlib.imports import *

def get_title(title_str = "Història"):
    title = TextMobject(title_str)
    title.scale(1.5)
    title.to_edge(UP)
    return title

def create_image(source, scale=2.5):
    img = ImageMobject(source)
    img.scale(scale)
    return img

def create_label(img, desc):
    label = TextMobject(desc)
    label.scale(0.7)
    label.next_to(img, DOWN)
    return label

class IntroHistoria(Scene):
    def construct(self):
        title = get_title()

        self.play(Write(title))
        self.wait(1.5)

        pol = TexMobject("p(x) = ", "x^n + a_{n-1} x^{n-1} + \ldots + a_0")
        pol0, pol1 = pol.split();

        pol2 = TexMobject("\\prod_{i = 1}^n (x - ", "\\alpha_i", ")")
        pol2_1, alpha_pol, pol2_3 = pol2.split()

        vieta1 = TexMobject("a_{n-1} = - (", "\\alpha_1", " + \ldots + ", "\\alpha_n", ")")
        vieta1_i, vieta1_alpha1, vieta1_plus, vieta1_alphan, vieta1_post = vieta1.split()

        vietamore = TexMobject("\\vdots")

        vieta2 = TexMobject("a_0 = (-1)^n \\cdot (", "\\alpha_1", " \cdot \ldots \cdot ", "\\alpha_n", ")")
        vieta2_i, vieta2_alpha1, vieta2_prod, vieta2_alphan, vieta2_post = vieta2.split()

        general_vieta = TexMobject("(-1)^k a_{n-k} = \\sum_{1 \\le i_1 < i_2 < \\cdots < i_k \\le n} \\left( \\prod_{j=1}^k \\alpha_{i_j} \\right)")
        general_vieta_desc = TextMobject("Fórmules de Viète")

        vietas = VGroup(vieta1, vietamore, vieta2)
        vietamore.next_to(vieta1, DOWN, buff = 0.4)
        vieta2.next_to(vietamore, DOWN, buff = 0.4)
        vietas.next_to(pol, DOWN, buff = 1)
        flow = VGroup(pol, vietas)
        flow.center()

        # This should be done after the flow is centered:
        pol2.next_to(pol0, RIGHT)
        general_vieta.align_to(vietas, UP)
        general_vieta_desc.next_to(vietas, DOWN)

        alpha_pol2 = alpha_pol.copy()
        alpha_pol3 = alpha_pol.copy()
        alpha_pol4 = alpha_pol.copy()

        framebox = SurroundingRectangle(general_vieta, buff = .15)
        framebox.set_stroke(WHITE, 4)

        # Animations:
        self.play(Write(pol))
        self.wait(1.5)

        self.play(Transform(pol1, pol2))
        self.wait(1.5)

        self.play(ApplyMethod(
            alpha_pol.set_color, RED,
            rate_func = there_and_back,
            run_time = 2
        ))
        self.wait(1.5)

        self.play(
            Write(vieta1,
                run_time=1),
            Transform(alpha_pol, vieta1_alpha1),
            Transform(alpha_pol2, vieta1_alphan)
        )
        self.wait(1.5)

        self.play(FadeIn(vietamore))
        self.play(
            Write(vieta2,
                run_time=1),
            Transform(alpha_pol3, vieta2_alpha1),
            Transform(alpha_pol4, vieta2_alphan)
        )
        self.wait(1.5)

        self.play(Transform(vietas, general_vieta))
        self.wait(1.5)

        self.play(
            FadeIn(framebox),
            FadeInFrom(general_vieta_desc, UP)
        )
        self.wait(1.5)

class LucaPacioli(Scene):
    def construct(self):
        title = get_title()
        self.add(title)
        self.wait(1.5)

        luca = create_image("img/pacioli.jpg")
        luca.to_edge(LEFT, buff=1)
        luca_desc = create_label(luca, "Luca Pacioli (c. 1445-1509)")

        suma = create_image("img/suma.png")
        suma.to_edge(RIGHT, buff=1)
        suma_desc = create_label(suma, "\\textit{Summa de arithmetica} (1494)")

        self.play(
            FadeIn(luca),
            FadeInFrom(luca_desc, UP)
        )
        self.wait(1.5)

        self.play(
            FadeIn(suma),
            FadeInFrom(suma_desc, UP)
        )
        self.wait(1.5)

        cardano = create_image("img/cardano.jpg")
        cardano.to_edge(LEFT, buff=1)
        cardano_desc = create_label(cardano, "Girolamo Cardano (1501-1576)")

        arsmagna = create_image("img/arsmagna.png")
        arsmagna.to_edge(RIGHT, buff=1)
        arsmagna_desc = create_label(arsmagna, "\\textit{Ars Magna} (1545)")

        cardano_cita = TextMobject("``Differentia aequationum ", "verarum", " et ", "fictarui", " \\\\ semper est numerus quadrato''")
        cardano_pre, cardano_v, cardano_et, cardano_f, cardano_post = cardano_cita.split()
        cardano_cita.scale(0.825)
        cardano_cita.next_to(cardano, RIGHT)

        self.play(
            FadeOut(suma),
            FadeOut(suma_desc),
            FadeOut(luca),
            FadeIn(cardano),
            Transform(luca_desc, cardano_desc)
        )
        self.remove(luca_desc)
        self.add(cardano_desc)
        self.wait(1.5)

        self.play(
            FadeIn(arsmagna),
            FadeInFrom(arsmagna_desc, UP)
        )
        self.wait(1.5)

        self.play(
            FadeOut(arsmagna),
            FadeOut(arsmagna_desc),
            Write(cardano_cita)
        )
        self.wait(1.5)

        self.play(ApplyMethod(
            cardano_v.set_color, RED,
            rate_func = there_and_back,
            run_time = 1
        ))
        self.play(ApplyMethod(
            cardano_f.set_color, RED,
            rate_func = there_and_back,
            run_time = 1
        ))
        self.wait(1.5)

        viete = create_image("img/viete.jpeg")
        viete.to_edge(LEFT, buff=1)
        viete_desc = create_label(viete, "François Viète (1540-1603)")

        self.play(
            FadeOut(cardano_cita),
            FadeOut(cardano),
            FadeIn(viete),
            Transform(cardano_desc, viete_desc)
        )
        self.remove(cardano_desc)
        self.add(viete_desc)
        self.wait(1.5)

        girard = create_image("img/girard.jpg")
        girard.to_edge(LEFT, buff=1)
        girard_desc = create_label(girard, "Albert Girard (1595-1632)")

        self.play(
            FadeOut(viete),
            FadeIn(girard),
            Transform(viete_desc, girard_desc)
        )
        self.remove(viete_desc)
        self.add(girard_desc)
        self.wait(1.5)

        primer = TextMobject("Primer teorema de polinomis simètrics, \\\\ sense saber-ho")
        primer.scale(0.825)
        primer.next_to(girard, RIGHT)

        self.play(Write(primer))
        self.wait(1.5)

        invention = create_image("img/inventionnouvelle.png", 2)
        invention.to_edge(RIGHT, buff=1)
        invention_desc = create_label(invention, "\\textit{Invention nouvelle en l'algèbre} (1629)")

        self.play(
            FadeOut(primer),
            FadeIn(invention),
            FadeInFrom(invention_desc, UP)
        )
        self.wait(1.5)

        factions = create_image("img/factions.png", 1)
        factions.to_edge(RIGHT, buff=1)

        self.play(
            FadeOut(invention),
            FadeIn(factions),
            ApplyMethod(invention_desc.next_to, factions, DOWN)
        )
        self.wait(1.5)

        factions_tex = TexMobject("\\text{Faction } k = \\sum_{\\{ i_1, \ldots, i_k \\} \\in \\{ 1, 2, \ldots, n \\}} \\alpha_{i_1} \\cdot \\cdots \\cdot \\alpha_{i_k}")
        factions_tex.scale(0.95)
        factions_tex.next_to(girard, RIGHT)

        self.play(
            FadeOut(factions),
            Write(factions_tex)
        )
        self.wait(1.5)

        notthesame = TexMobject("\\sum_{i < j} \\alpha_i \\alpha_j ", "\\neq", " \\sum_{i} \\alpha_i^2")
        notthesame.next_to(girard, RIGHT).shift(1*RIGHT)

        notthesame2 = TexMobject("\\sum_{i < j < k} \\alpha_i \\alpha_j \\alpha_k ", "\\neq", " \\sum_{i} \\alpha_i^3")
        notthesame2.next_to(girard, RIGHT).shift(1*RIGHT)

        self.play(Transform(factions_tex, notthesame))
        self.remove(factions_tex)
        self.add(notthesame)
        self.wait(1.5)

        self.play(Transform(notthesame, notthesame2))
        self.remove(notthesame)
        self.add(notthesame2)
        self.wait(1.5)

        sommes = create_image("img/sommepuissances.png")
        sommes.to_edge(RIGHT, buff=1)

        self.play(
            FadeOut(notthesame2),
            FadeIn(sommes),
            ApplyMethod(invention_desc.next_to, sommes, DOWN)
        )
        self.wait(1.5)

        sommes2 = create_image("img/sommepuissances2.png", 1)
        sommes2.to_edge(RIGHT, buff=1)
        sommes2_desc = create_label(sommes2, "\\textit{A Short Account of the History of Symmetric} \\\\ \\textit{Functions of Roots of Equations}, \\\\ H. Gray Funkhouser (1930).")

        self.play(
            FadeOut(sommes),
            FadeIn(sommes2),
            Transform(invention_desc, sommes2_desc)
        )
        self.remove(invention_desc)
        self.add(sommes2_desc)
        self.wait(1.5)

        waring = create_image("img/waring.jpg")
        waring.to_edge(LEFT, buff=1)
        waring_desc = create_label(waring, "Edward Waring (1736-1798)")

        self.play(
            FadeOut(sommes2),
            FadeOut(sommes2_desc),
            FadeOut(girard),
            FadeIn(waring),
            Transform(girard_desc, waring_desc)
        )
        self.remove(girard_desc)
        self.add(waring_desc)
        self.wait(1.5)

class EdwardWaring(Scene):
    def construct(self):
        title = get_title()
        self.add(title)

        waring = create_image("img/waring.jpg")
        waring.to_edge(LEFT, buff=1)
        waring_desc = create_label(waring, "Edward Waring (1736-1798)")

        self.add(waring)
        self.add(waring_desc)

        th1 = TextMobject("Teorema 1. ", "???").set_color(GRAY)
        th2 = TextMobject("Teorema 2. ???").set_color(GRAY)
        th3 = TextMobject("Teorema 3. ", "???").set_color(GRAY)

        th1.shift(1*UP)
        th3.shift(1*DOWN)

        teoremes = VGroup(th1, th2, th3)
        teoremes.next_to(waring, RIGHT).shift(1*RIGHT)

        self.play(
            FadeInFrom(th1, UP),
            FadeIn(th2),
            FadeInFrom(th3, DOWN)
        )
        self.wait(1.5)

        th1_pre, th1_question = th1.split()
        th1_text = TexMobject("\\sum_{i} \\alpha_i^k = f_k(a_i) \\quad \\forall k \\in \\mathbb{N}")
        th1_text.scale(0.8)
        th1_text.next_to(th1_pre)

        self.play(
            Transform(th1_question, th1_text),
            ApplyMethod(th1_pre.set_color, WHITE)
        )
        self.wait(1.5)

        th3_pre, th3_question = th3.split()
        th3_text = TextMobject("Precursor al Teo. Fonamental \\\\ dels Polinomis Simètrics")
        th3_text.scale(0.7)
        th3_text.next_to(th3_pre)

        self.play(
            Transform(th3_question, th3_text),
            ApplyMethod(th3_pre.set_color, WHITE)
        )
        self.wait(1.5)

        meditationes = create_image("img/meditationes.png")
        meditationes.to_edge(RIGHT, buff=2)
        meditationes_desc = create_label(meditationes, "\\textit{Meditationes Algebraicae} (1770).")

        self.play(
            FadeOut(th1),
            FadeOut(th2),
            FadeOut(th3),
            FadeIn(meditationes),
            FadeInFrom(meditationes_desc, UP)
        )
        self.wait(1.5)

        conjectura = VGroup(
            TextMobject("$n$ es pot escriure com a màxim com a suma de:"),
            TextMobject("$\\cdot$ 4 quadrats"),
            TextMobject("$\\cdot$ 9 cubs"),
            TextMobject("$\\cdot$ 19 biquadrats"),
            TextMobject("$\\cdot$ en general, $m$ potències k-èssimes positives, \\\\ essent $m$ depenent de $k$ ($\\forall k \\in \\mathbb{N}$)"),
        )
        conjectura.scale(0.8)
        conjectura.arrange(DOWN, buff = MED_LARGE_BUFF, aligned_edge = LEFT)
        conjectura.to_edge(RIGHT, buff=0.75)

        self.play(
            FadeOut(meditationes),
            Write(conjectura)
        )
        self.wait(1.5)