math-visualizer

# Math Visualizer Skill The Math Visualizer brings mathematical concepts to life through precise, beautiful animations that reveal the structure and relationships within mathematics. ## Mathematical Domains ### Supported Areas - **Algebra**: Equations, inequalities, polynomials - **Calculus**: Derivatives, integrals, limits, series - **Geometry**: Shapes, transformations, proofs - **Trigonometry**: Functions, identities, unit circle - **Linear Algebra**: Vectors, matrices, transformations - **Complex Analysis**: Complex numbers, transformations - **Number Theory**: Primes, sequences, patterns ## Rules ### [rules/equation-presentation.md](rules/equation-presentation.md) How to present equations with proper pacing and emphasis. ### [rules/color-coding-math.md](rules/color-coding-math.md) Consistent color schemes for mathematical elements. ### [rules/graphing-best-practices.md](rules/graphing-best-practices.md) Creating clear, informative function graphs. ### [rules/proof-visualization.md](rules/proof-visualization.md) Step-by-step proof animations that build understanding. ## Color Coding Standard | Element | Color | Hex | |---------|-------|-----| | Variables (x, y) | BLUE | #58C4DD | | Constants | YELLOW | #FFFF00 | | Operators | WHITE | #FFFFFF | | Key Terms | GREEN | #83C167 | | Equals/Results | GOLD | #FFD700 | | Negative/Subtract | RED | #FC6255 | ## Templates ### Equation Derivation ```python from manim import * class EquationDerivation(Scene): def construct(self): # Initial equation eq1 = MathTex(r"x^2 + 2x + 1 = 0") self.play(Write(eq1)) self.wait() # Transform step by step eq2 = MathTex(r"(x + 1)^2 = 0") eq3 = MathTex(r"x + 1 = 0") eq4 = MathTex(r"x = -1") # Show each transformation for new_eq in [eq2, eq3, eq4]: self.play(TransformMatchingTex(eq1, new_eq)) self.wait() eq1 = new_eq # Highlight final answer box = SurroundingRectangle(eq4, color=GREEN, buff=0.2) self.play(Create(box)) ``` ### Color-Coded Equation ```python from manim import * class ColorCodedEquation(Scene): def construct(self): # Equation with color-coded parts equation = MathTex( r"f(", r"x", r") = ", r"a", r"x^2", r" + ", r"b", r"x", r" + ", r"c" ) # Color code equation[1].set_color(BLUE) # x equation[3].set_color(YELLOW) # a equation[4].set_color(BLUE) # x^2 equation[6].set_color(YELLOW) # b equation[7].set_color(BLUE) # x equation[9].set_color(YELLOW) # c self.play(Write(equation)) # Explain each part labels = [ (equation[3], "coefficient"), (equation[1], "variable"), (equation[9], "constant") ] for part, label_text in labels: self.play(Indicate(part)) label = Text(label_text, font_size=24).next_to(part, DOWN) self.play(Write(label)) self.wait() self.play(FadeOut(label)) ``` ### Function Graph with Animation ```python from manim import * class FunctionGraph(Scene): def construct(self): # Create axes axes = Axes( x_range=[-4, 4, 1], y_range=[-2, 8, 1], x_length=8, y_length=5, axis_config={"include_tip": True} ) labels = axes.get_axis_labels(x_label="x", y_label="y") self.play(Create(axes), Write(labels)) # Function func = axes.plot(lambda x: x**2, color=BLUE) func_label = MathTex(r"f(x) = x^2", color=BLUE).to_corner(UR) self.play(Create(func), Write(func_label)) # Show derivative deriv = axes.plot(lambda x: 2*x, color=GREEN) deriv_label = MathTex(r"f'(x) = 2x", color=GREEN).next_to(func_label, DOWN) self.play(Create(deriv), Write(deriv_label)) # Tangent line demonstration x_tracker = ValueTracker(-2) tangent = always_redraw(lambda: axes.get_secant_slope_group( x=x_tracker.get_value(), graph=func, dx=0.01, secant_line_color=YELLOW, secant_line_length=4 )) dot = always_redraw(lambda: Dot( axes.c2p(x_tracker.get_value(), x_tracker.get_value()**2), color=RED )) self.play(Create(tangent), Create(dot)) self.play(x_tracker.animate.set_value(2), run_time=4) ``` ### 3D Mathematical Surface ```python from manim import * class Surface3D(ThreeDScene): def construct(self): # Set up camera self.set_camera_orientation(phi=75 * DEGREES, theta=-45 * DEGREES) # Create axes axes = ThreeDAxes( x_range=[-3, 3, 1], y_range=[-3, 3, 1], z_range=[-2, 2, 1] ) # Create surface surface = Surface( lambda u, v: axes.c2p(u, v, np.sin(u) * np.cos(v)), u_range=[-PI, PI], v_range=[-PI, PI], resolution=(30, 30), fill_opacity=0.7 ) surface.set_fill_by_value( axes=axes, colorscale=[(RED, -1), (YELLOW, 0), (GREEN, 1)] ) # Animate self.play(Create(axes)) self.play(Create(surface), run_time=3) self.begin_ambient_camera_rotation(rate=0.2) self.wait(5) ``` ### Geometric Proof ```python from manim import * class PythagoreanProof(Scene): def construct(self): # Create right triangle triangle = Polygon( ORIGIN, RIGHT * 3, RIGHT * 3 + UP * 4, color=WHITE, fill_opacity=0.3 ) # Labels a_label = MathTex("a").next_to(triangle, DOWN) b_label = MathTex("b").next_to(triangle, RIGHT) c_label = MathTex("c").move_to( (ORIGIN + RIGHT * 3 + UP * 4) / 2 + LEFT * 0.5 + UP * 0.3 ) self.play(Create(triangle)) self.play(Write(a_label), Write(b_label), Write(c_label)) # Show squares on each side sq_a = Square(side_length=3, color=BLUE, fill_opacity=0.5) sq_a.next_to(triangle, DOWN, buff=0) sq_b = Square(side_length=4, color=GREEN, fill_opacity=0.5) sq_b.next_to(triangle, RIGHT, buff=0) self.play(Create(sq_a), Create(sq_b)) # Area labels area_a = MathTex(r"a^2", color=BLUE).move_to(sq_a) area_b = MathTex(r"b^2", color=GREEN).move_to(sq_b) self.play(Write(area_a), Write(area_b)) # Conclusion theorem = MathTex(r"a^2 + b^2 = c^2").to_edge(UP) box = SurroundingRectangle(theorem, color=GOLD) self.play(Write(theorem), Create(box)) ``` ## LaTeX Quick Reference ### Common Expressions ```latex % Fractions \frac{a}{b} % Square root \sqrt{x} \sqrt[n]{x} % Summation \sum_{i=1}^{n} x_i % Integral \int_{a}^{b} f(x) \, dx % Limit \lim_{x \to \infty} f(x) % Matrix \begin{pmatrix} a & b \\ c & d \end{pmatrix} % Partial derivative \frac{\partial f}{\partial x} ``` ### Greek Letters ```latex \alpha \beta \gamma \delta \epsilon \theta \lambda \mu \pi \sigma \omega \Gamma \Delta \Theta \Lambda \Sigma \Omega ``` ## Best Practices 1. **Reveal equations gradually** - Build up complex equations piece by piece 2. **Use consistent notation** - Same symbol = same meaning throughout 3. **Annotate meaningfully** - Labels should clarify, not clutter 4. **Show, don't just state** - Animate the mathematical relationships 5. **Connect to intuition** - Bridge abstract math to visual understanding