""" LinkedIn PDF carousel: "Why do we see harmonics? A mathematical answer." 4:5 slides (10.8 x 13.5 in), vector PDF. Every spectrum shown is computed with an FFT of the function on the slide, not drawn by hand. Run: python3 harmonics_carousel.py -> harmonics_carousel.pdf """ import textwrap import numpy as np import matplotlib matplotlib.use("Agg") import matplotlib.pyplot as plt from matplotlib.backends.backend_pdf import PdfPages SURFACE = "#111110" PANEL = "#1a1a19" INK = "#f0efec" INK2 = "#c3c2b7" MUTED = "#898781" GRID = "#2c2c2a" AXIS = "#383835" BLUE = "#3987e5" ORANGE = "#d95926" W, H = 10.8, 13.5 LEFT = 0.085 TOTAL = 17 plt.rcParams.update({ "font.family": "DejaVu Sans", "mathtext.fontset": "cm", "text.color": INK, "axes.edgecolor": AXIS, "xtick.color": MUTED, "ytick.color": MUTED, "axes.labelcolor": MUTED, }) # ---------------------------------------------------------------- functions N = 4096 t1 = np.arange(N) / N # exactly one period, T = 1 w0 = 2 * np.pi def sq(t): return np.sign(np.sin(w0 * t)) def saw(t): return 2 * (t - np.floor(t + 0.5)) def tri(t): return (2 / np.pi) * np.arcsin(np.sin(w0 * t)) def hw(t): return np.maximum(np.sin(w0 * t), 0) def sine(t): return np.sin(w0 * t) def harmonic_amps(f, n_max=10): X = np.fft.rfft(f(t1)) return 2 * np.abs(X[1:n_max + 1]) / N def real_sig(t): """Messy but periodic machine signal: imbalance (1X), misalignment (2X) and an impact once per revolution that rings and decays.""" tau_ = (t % 1.0 - 0.32) % 1.0 ring = np.exp(-tau_ / 0.15) * np.sin(2 * np.pi * 6.0 * tau_) y = 0.55 * np.sin(w0 * t) + 0.3 * np.sin(2 * w0 * t + 0.9) + 1.4 * ring return y / 1.16 def draw_mascot(fig, rect, happy=False, look=(0.8, 0.3)): """The series mascot: a sine-wave creature (original character).""" ax = fig.add_axes(rect) ax.set_axis_off() ax.set_xlim(-0.2, 1.2) ax.set_ylim(-0.75, 1.0) ax.set_aspect("equal") x = np.linspace(0, 1, 300) y = 0.42 * np.sin(2 * np.pi * x) ax.plot(x, y, color=BLUE, lw=34, alpha=0.12, solid_capstyle="round") ax.plot(x, y, color=BLUE, lw=13, solid_capstyle="round") th = np.linspace(0, 2 * np.pi, 40) gx, gy = look g = np.hypot(gx, gy) for xe in (0.15, 0.35): ye = 0.42 * np.sin(2 * np.pi * xe) + 0.17 r = 0.095 if happy: ax.plot(xe + r * np.array([-0.9, 0, 0.9]), ye + r * np.array([-0.3, 0.45, -0.3]), color=INK, lw=5, solid_capstyle="round") continue ax.fill(xe + r * np.cos(th), ye + r * np.sin(th), color=INK) ax.fill(xe + 0.45 * r * gx / g + 0.42 * r * np.cos(th), ye + 0.45 * r * gy / g + 0.42 * r * np.sin(th), color=SURFACE) s_ = np.linspace(-1, 1, 20) ax.plot(0.25 + 0.09 * s_, 0.42 - 0.2 + (0.05 + 0.07 * happy) * s_ ** 2, color=INK, lw=4, solid_capstyle="round") return ax def bubble(fig, x, y, text, size=20): fig.text(x, y, text, fontsize=size, color=INK, fontweight="bold", ha="center", va="center", linespacing=1.25, bbox=dict(boxstyle="round,pad=0.5", facecolor="#1d1d1b", edgecolor=BLUE, linewidth=2)) # ---------------------------------------------------------------- slide kit class Slide: def __init__(self, pdf, num, title, kicker=None): self.pdf = pdf self.fig = plt.figure(figsize=(W, H), facecolor=SURFACE) f = self.fig f.text(LEFT, 0.955, f"{num:02d} / {TOTAL:02d}", fontsize=15, color=MUTED, family="DejaVu Sans Mono") f.text(1 - LEFT, 0.955, "WHY WE SEE HARMONICS", fontsize=13, color=MUTED, ha="right", fontweight="bold") self.y = 0.905 if kicker: f.text(LEFT, self.y, kicker.upper(), fontsize=16, color=ORANGE, fontweight="bold", va="top") self.y -= 0.035 t = f.text(LEFT, self.y, title, fontsize=38, fontweight="bold", color=INK, va="top", linespacing=1.2) n_lines = title.count("\n") + 1 self.y -= n_lines * 38 * 1.2 / 72 / H + 0.025 f.add_artist(plt.Line2D([LEFT, LEFT + 0.08], [self.y, self.y], color=BLUE, lw=4, transform=f.transFigure)) self.y -= 0.035 # flow helpers --------------------------------------------------- def par(self, text, size=22, color=INK2, width=50, weight="normal", gap=0.022): lines = [] for p in text.split("\n"): lines += textwrap.wrap(p, width) or [""] body = "\n".join(lines) self.fig.text(LEFT, self.y, body, fontsize=size, color=color, va="top", linespacing=1.45, fontweight=weight) self.y -= len(lines) * size * 1.45 / 72 / H + gap def eq(self, tex, size=30, color=INK, height=None, x=None, ha="left", gap=0.02): h = height if height else size * 2.0 / 72 / H self.fig.text(x if x is not None else LEFT + 0.02, self.y - h / 2, tex, fontsize=size, color=color, va="center", ha=ha) self.y -= h + gap def box(self, text, size=22, color=INK, accent=BLUE, width=46): lines = [] for p in text.split("\n"): lines += textwrap.wrap(p, width) or [""] h = len(lines) * size * 1.45 / 72 / H + 0.035 f = self.fig f.add_artist(plt.Rectangle((LEFT, self.y - h), 1 - 2 * LEFT, h, transform=f.transFigure, facecolor=PANEL, edgecolor="none")) f.add_artist(plt.Rectangle((LEFT, self.y - h), 0.008, h, transform=f.transFigure, facecolor=accent, edgecolor="none")) f.text(LEFT + 0.035, self.y - 0.0175, "\n".join(lines), fontsize=size, color=color, va="top", linespacing=1.45, fontweight="bold") self.y -= h + 0.025 def axes(self, height, gap=0.03, label_room=0.03): ax = self.fig.add_axes([LEFT + 0.07, self.y - height, 1 - 2 * LEFT - 0.07, height - label_room], facecolor=SURFACE) style(ax) self.y -= height + gap return ax def wave_spec(self, f, title_w="Time (one period = T)", height=0.17, orders=10, color=INK): ax = self.axes(height) tt = np.linspace(0, 2, 2000) ax.plot(tt, f(tt), color=color, lw=3) ax.axhline(0, color=AXIS, lw=1) ax.set_xlim(0, 2) ax.set_xticks([0, 1, 2]) ax.set_xticklabels(["0", "T", "2T"]) ax.set_ylim(-1.25, 1.25) ax.set_yticks([-1, 0, 1]) ax.set_title(title_w, loc="left", fontsize=15, color=MUTED, pad=8) ax2 = self.axes(height) A = harmonic_amps(f, orders) k = np.arange(1, orders + 1) ax2.bar(k, A, width=0.38, color=[BLUE] + [ORANGE] * (orders - 1), zorder=3) for kk, a in zip(k, A): if a > 0.01: ax2.text(kk, a + 0.03, f"{a:.2f}", ha="center", fontsize=12, color=INK2) ax2.set_xlim(0.4, orders + 0.6) ax2.set_ylim(0, max(1.1, A.max() * 1.18)) ax2.set_xticks(k) ax2.set_xticklabels([f"{i}X" for i in k]) ax2.set_title("Spectrum: amplitude of each harmonic (FFT of the curve above)", loc="left", fontsize=15, color=MUTED, pad=8) return ax, ax2 def save(self): self.pdf.savefig(self.fig, facecolor=SURFACE) plt.close(self.fig) def style(ax): for s in ("top", "right"): ax.spines[s].set_visible(False) ax.spines["left"].set_color(AXIS) ax.spines["bottom"].set_color(AXIS) ax.tick_params(labelsize=14, length=0, pad=6) ax.grid(axis="y", color=GRID, lw=1) ax.set_axisbelow(True) # ---------------------------------------------------------------- slides def build(path="harmonics_carousel.pdf"): with PdfPages(path) as pdf: meta = pdf.infodict() meta["Title"] = "Why do we see harmonics? A mathematical answer" meta["Subject"] = "Fourier series and harmonics for vibration engineers" # 1 cover -------------------------------------------------------- fig = plt.figure(figsize=(W, H), facecolor=SURFACE) ax = fig.add_axes([0, 0.18, 1, 0.34], facecolor=SURFACE) ax.set_axis_off() tt = np.linspace(0, 2, 3000) ax.plot(tt, sine(tt), color=BLUE, lw=3, alpha=0.35) part = sum((4 / (np.pi * n)) * np.sin(w0 * n * tt) for n in range(1, 30, 2)) ax.plot(tt, part, color=INK, lw=3.5) ax.set_xlim(-0.05, 2.05) ax.set_ylim(-1.5, 1.5) fig.text(LEFT, 0.86, "Why do we see\nharmonics?", fontsize=62, fontweight="bold", va="top", linespacing=1.12) fig.text(LEFT, 0.635, "A mathematical answer,\nwith functions you can check yourself.", fontsize=26, color=INK2, va="top", linespacing=1.4) fig.text(LEFT, 0.09, "Fourier series · symmetry · smoothness · nonlinearity · the FFT", fontsize=16, color=MUTED) fig.text(1 - LEFT, 0.05, "Swipe →", fontsize=22, color=ORANGE, ha="right", fontweight="bold") draw_mascot(fig, [0.70, 0.47, 0.24, 0.17], look=(0.4, -0.9)) bubble(fig, 0.82, 0.69, "Let's take some\nsignals apart!", 18) pdf.savefig(fig, facecolor=SURFACE) plt.close(fig) # 2 the only assumption ------------------------------------------ s = Slide(pdf, 2, "Start with one\nassumption", "The setup") s.par("A rotating machine running at constant speed produces motion " "that repeats every revolution. Mathematically, the signal is periodic:") s.eq(r"$f(t+T) = f(t)$", size=44) s.par("T is the period (one revolution). It defines the fundamental " "frequency:") s.eq(r"$f_0 = \dfrac{1}{T}, \qquad \omega_0 = \dfrac{2\pi}{T}$", size=36, height=0.075) s.par("We assume nothing about the SHAPE of f(t). It can be smooth, " "flattened, spiky, lopsided. Only that it repeats.") s.box("Everything that follows, including where harmonics appear " "and how strong they are, follows from that one line.") s.save() # 3 which sines are allowed -------------------------------------- s = Slide(pdf, 3, "Which sines and\ncosines are allowed?", "Why integer multiples") s.par("If a sine or cosine at frequency ω is part of a signal with " "period T, it must also repeat after T:") s.eq(r"$\sin(\omega(t+T)) = \sin(\omega t) \;\;\Rightarrow\;\; \omega T = 2\pi n$", size=32) s.eq(r"$\Rightarrow\;\; \omega = n\,\omega_0, \qquad n = 1, 2, 3, \dots$", size=32) ax = s.axes(0.24) tt = np.linspace(0, 1, 1000) for n, c, lw, a in ((1, BLUE, 3, 1), (2, ORANGE, 2.5, 1), (3, ORANGE, 2, 0.6)): ax.plot(tt, np.sin(w0 * n * tt) + (3 - n) * 2.4, color=c, lw=lw, alpha=a) ax.text(1.02, (3 - n) * 2.4, f"{n}X ✓", color=INK2, fontsize=15, va="center") ax.plot(tt, np.sin(w0 * 1.5 * tt) - 2.4, color=MUTED, lw=2, ls=(0, (5, 4))) ax.text(1.02, -2.4, "1.5X ✗", color=MUTED, fontsize=15, va="center") ax.axvline(1, color=AXIS, lw=1.5) ax.set_xlim(0, 1.18) ax.set_xticks([0, 1]) ax.set_xticklabels(["0", "T"]) ax.set_yticks([]) ax.grid(False) ax.spines["left"].set_visible(False) ax.set_title("Only whole numbers of cycles fit into one period", loc="left", fontsize=15, color=MUTED, pad=8) s.box("Harmonics sit at EXACT integer multiples of running speed, " "not by convention, but because nothing else can fit.") s.save() # 4 Fourier series ----------------------------------------------- s = Slide(pdf, 4, "The Fourier series", "Fourier, 1822") s.par("Any reasonable periodic function can be written as a sum of " "exactly those allowed sines and cosines:") s.eq(r"$f(t) = a_0 + \sum_{n=1}^{\infty} \left[\, a_n \cos(n\omega_0 t) " r"+ b_n \sin(n\omega_0 t) \,\right]$", size=29, height=0.08) s.par("Each coefficient measures how much of f(t) looks like the n-th " "harmonic, a projection:") s.eq(r"$a_n = \dfrac{2}{T}\int_0^T f(t)\cos(n\omega_0 t)\,dt, \quad " r"b_n = \dfrac{2}{T}\int_0^T f(t)\sin(n\omega_0 t)\,dt$", size=25, height=0.08) s.par("This works because different harmonics are orthogonal: over one " "period they don't overlap at all.") s.eq(r"$\int_0^T \sin(m\omega_0 t)\,\sin(n\omega_0 t)\,dt = 0 \quad (m \neq n)$", size=28) s.box("The amplitude at nX on your spectrum is\n" "√(aₙ² + bₙ²). That's all a harmonic is.", accent=ORANGE) s.save() # 5 pure sine ------------------------------------------------------ s = Slide(pdf, 5, "Example 1:\na pure sine", "Baseline") s.eq(r"$f(t) = \sin(\omega_0 t) \;\;\Rightarrow\;\; b_1 = 1,\;\; " r"\mathrm{all\ others} = 0$", size=29) s.wave_spec(sine, height=0.19, color=BLUE) s.par("One line in the spectrum. Not approximately one: exactly one. " "A sine is already a harmonic, so the series has a single term.") s.save() # 6 square --------------------------------------------------------- s = Slide(pdf, 6, "Example 2:\na square wave", "Odd harmonics only") s.eq(r"$\mathrm{sgn}(\sin\omega_0 t) = \dfrac{4}{\pi}\left[\sin\omega_0 t " r"+ \dfrac{1}{3}\sin 3\omega_0 t + \dfrac{1}{5}\sin 5\omega_0 t + \dots\right]$", size=27, height=0.08) s.wave_spec(sq, height=0.18) s.par("Every odd harmonic, falling off as 1/n. Check: 4/π = 1.27, " "then 1.27/3 = 0.42, 1.27/5 = 0.25.\nAnd not a single even one. Why?") s.save() # 7 symmetry ------------------------------------------------------- s = Slide(pdf, 7, "Why only odd?\nSymmetry.", "Half-wave symmetry") s.par("The square wave has half-wave symmetry: the second half-cycle is " "the first one flipped upside down.") s.eq(r"$f(t + T/2) = -f(t)$", size=40) s.par("Now shift an even harmonic (n = 2, 4, ...) by half a period:") s.eq(r"$\sin\!\left(n\omega_0 (t + T/2)\right) = \sin(n\omega_0 t + n\pi) " r"= +\sin(n\omega_0 t)$", size=28) s.par("An even harmonic does NOT flip sign, so it can never contribute " "to a function that does. Its coefficient is forced to zero. " "Odd harmonics flip, so they survive.") s.box("Symmetric distortion → odd harmonics only.\n" "Break the symmetry → even harmonics appear.") s.save() # 8 sawtooth ------------------------------------------------------- s = Slide(pdf, 8, "Example 3:\na sawtooth", "All harmonics") s.eq(r"$f(t) = \dfrac{2}{\pi}\sum_{n=1}^{\infty} (-1)^{n+1}\," r"\dfrac{\sin(n\omega_0 t)}{n}$", size=30, height=0.085) s.wave_spec(saw, height=0.18) s.par("No half-wave symmetry, so every harmonic is present: " "0.64, 0.32, 0.21, 0.16 ... still 1/n, because the wave still " "has a jump.") s.save() # 9 triangle ------------------------------------------------------- s = Slide(pdf, 9, "Example 4:\na triangle wave", "Smoother shape") s.eq(r"$f(t) = \dfrac{8}{\pi^2}\sum_{n\ \mathrm{odd}} (-1)^{(n-1)/2}\," r"\dfrac{\sin(n\omega_0 t)}{n^2}$", size=30, height=0.085) s.wave_spec(tri, height=0.18) s.par("Same symmetry as the square wave, so again only odd harmonics. " "But no jump, only corners, and the harmonics fall as 1/n²: " "3X is 1/9 of 1X, 5X is 1/25.") s.save() # 10 smoothness rule ---------------------------------------------- s = Slide(pdf, 10, "Sharper shape,\nmore harmonics", "The decay rule") s.par("Integrate the coefficient formula by parts: each step pulls out " "a factor 1/n, until you hit a discontinuity. So:") rows = [("Jump in f(t)", r"$\sim 1/n$", "square, sawtooth"), ("Corner (jump in f′)", r"$\sim 1/n^2$", "triangle, clipped sine"), ("Jump in f″", r"$\sim 1/n^3$", "smoother still"), ("Infinitely smooth", r"$\ll 1/n^k$", "a pure sine")] for a, b, c in rows: s.fig.text(LEFT, s.y, a, fontsize=20, color=INK, va="top") s.fig.text(0.44, s.y + 0.004, b, fontsize=24 if "$" in b else 18, color=ORANGE, va="top") s.fig.text(0.70, s.y, c, fontsize=17, color=MUTED, va="top") s.y -= 0.042 s.y -= 0.01 ax = s.axes(0.20, gap=0.05) n = np.arange(1, 32, 2) Aq = harmonic_amps(sq, 31)[n - 1] At = harmonic_amps(tri, 31)[n - 1] ax.loglog(n, Aq, "o-", color=ORANGE, lw=2, ms=8, label="square (jump): 1/n") ax.loglog(n, At, "o-", color=BLUE, lw=2, ms=8, label="triangle (corner): 1/n²") ax.set_xlabel("harmonic number n (odd)", fontsize=15) ax.set_title("Harmonic amplitude, log–log", loc="left", fontsize=15, color=MUTED, pad=8) ax.grid(True, which="major", color=GRID) leg = ax.legend(frameon=False, fontsize=15, labelcolor=INK2) ax.tick_params(which="minor", length=0) s.box("Sharp features in time live at high frequency. Impacts and " "sudden stops push energy far up the harmonic series.") s.save() # 11 half-wave rectified ------------------------------------------- s = Slide(pdf, 11, "Example 5:\ncut a sine on one side", "One-sided truncation") s.eq(r"$\max(\sin\omega_0 t,\,0) = \dfrac{1}{\pi} + \dfrac{1}{2}\sin\omega_0 t " r"- \dfrac{2}{\pi}\sum_{n\ \mathrm{even}} \dfrac{\cos(n\omega_0 t)}{n^2-1}$", size=26, height=0.085) s.wave_spec(hw, height=0.18) s.par("Flatten one side and the symmetry breaks: 2X, 4X, 6X appear " "immediately (0.21, 0.04, 0.02). This is the extreme case of " "motion that is limited in one direction only.") s.save() # 12 real machine signal -------------------------------------------- s = Slide(pdf, 12, "Example 6:\na real machine signal", "In real life") s.par("Real signals are messier: imbalance, misalignment, and an " "impact every revolution that rings and decays.", gap=0.008) s.wave_spec(real_sig, height=0.135) s.par("No neat formula, but it repeats every revolution, so it is " "still a Fourier series: imbalance on 1X, misalignment on 2X, " "the ringing impact spread across 4X–7X.", gap=0.012) s.box("If it repeats and isn't a pure sine, you will see harmonics.", size=21) s.save() # 13 nonlinearity --------------------------------------------------- s = Slide(pdf, 13, "Where harmonics\nare born", "Nonlinearity") s.par("A linear system driven at ω₀ responds only at ω₀. Harmonics " "need a nonlinearity: a response not proportional to the input.", gap=0.012) s.eq(r"$y = x + \alpha x^2 + \beta x^3, \qquad x = \sin\omega_0 t$", size=30) s.par("Two identities you can check by hand:", gap=0.008) s.eq(r"$\sin^2\omega_0 t = \dfrac{1}{2} - \dfrac{1}{2}\cos 2\omega_0 t$", size=30, height=0.07) s.eq(r"$\sin^3\omega_0 t = \dfrac{3}{4}\sin\omega_0 t - \dfrac{1}{4}\sin 3\omega_0 t$", size=30, height=0.07) s.par("The x² term creates 2X (and a DC shift). The x³ term creates 3X.") s.box("Even-order (asymmetric) → even harmonics.\n" "Odd-order (symmetric) → odd harmonics.", size=21, width=50) s.par("Clearances, impacts, and stiffness that differs in each " "direction are all nonlinearities.", size=19, gap=0.01) s.save() # 14 FFT ------------------------------------------------------------ s = Slide(pdf, 14, "What the FFT\nactually computes", "Cooley & Tukey, 1965") s.par("Your analyser doesn't see a formula, it sees N samples x[m]. " "The FFT is a fast way to compute the discrete Fourier transform:") s.eq(r"$X[k] = \sum_{m=0}^{N-1} x[m]\, e^{-i 2\pi k m / N}$", size=34, height=0.085) s.par("If the record holds exactly P periods, harmonic n lands on bin " "k = nP, and (2/N)·|X[k]| equals the Fourier amplitude " "√(aₙ² + bₙ²). Same numbers as the series.") s.par("Two real-world caveats:", color=INK, weight="bold", gap=0.01) s.par("• Leakage: a non-integer number of periods smears peaks into " "neighbouring bins. A window (e.g. Hanning) controls it.\n" "• Aliasing: harmonics above fs/2 fold back to false " "frequencies. Anti-alias filtering prevents it.", size=19, width=54) s.save() # 15 takeaways ---------------------------------------------------- s = Slide(pdf, 15, "What this means\nfor the analyst", "Takeaways") items = [ ("Harmonics aren't extra signals.", "They are the Fourier description, in sines and cosines, of one " "periodic waveform that isn't a pure sine."), ("Position is forced by periodicity.", "Exactly n × running speed. Peaks off those lines come from " "something else."), ("Which harmonics tells you symmetry.", "Odd only: symmetric distortion. Evens present: something is " "one-sided."), ("How far they extend tells you sharpness.", "Long harmonic families mean jumps or impacts in the waveform."), ("Rule out the measurement first.", "Sensor saturation, a loose mount or a clipped input range " "distort the waveform too, and produce harmonics that aren't " "in the machine."), ] for head, body in items: s.par(head, size=22, color=INK, weight="bold", gap=0.006) s.par(body, size=18, color=INK2, width=60, gap=0.022) s.save() # 16 sources ------------------------------------------------------ s = Slide(pdf, 16, "Sources &\nfurther reading", "References") refs = [ ("J. Fourier (1822).", "Théorie analytique de la chaleur. Paris: Firmin Didot."), ("J. W. Cooley & J. W. Tukey (1965).", "An algorithm for the machine " "calculation of complex Fourier series. Mathematics of Computation, " "19(90), 297–301."), ("R. N. Bracewell.", "The Fourier Transform and Its Applications. " "McGraw-Hill."), ("E. O. Brigham (1988).", "The Fast Fourier Transform and Its " "Applications. Prentice Hall."), ("E. M. Stein & R. Shakarchi (2003).", "Fourier Analysis: An " "Introduction. Princeton University Press. (Decay of coefficients " "vs smoothness.)"), ("R. B. Randall.", "Vibration-based Condition Monitoring: " "Industrial, Automotive and Aerospace Applications. Wiley."), ] for a, b in refs: s.par(a, size=19, color=INK, weight="bold", gap=0.004) s.par(b, size=17, color=INK2, width=62, gap=0.014) s.save() # 17 teaser for Friday's video ----------------------------------------- s = Slide(pdf, 17, "Now watch them\ncome apart", "Coming Friday") s.par("Every function in this carousel, pulled apart into its harmonic " "lanes and rebuilt, in 3D, with sound.") import os if os.path.exists("teaser_frame.png"): # a still from the video, if you have one img = plt.imread("teaser_frame.png") h_img = (1 - 2 * LEFT) * W * img.shape[0] / img.shape[1] / H iax = s.fig.add_axes([LEFT, s.y - h_img, 1 - 2 * LEFT, h_img]) iax.imshow(img) iax.set_axis_off() s.y -= h_img + 0.04 draw_mascot(s.fig, [LEFT - 0.02, 0.05, 0.3, 0.22], happy=True) bubble(s.fig, 0.56, 0.17, "See you Friday!", 24) s.save() if __name__ == "__main__": build() print("done")