Interactive Manim + LaTeX Visual Encyclopedia

Visual Math & CS Concept Wiki

Every foundational concept in AI, Linear Algebra, Calculus, and Signal Processing explained with interactive 3Blue1Brown-style Manim canvas animations, inspectable Manim Python source code, and step-by-step KaTeX derivations.

Runnable Code Answers Engineering Cheat Sheets

Showing 8 of 8 interactive Manim + LaTeX concept pagesLive 60fps HTML5 Canvas + KaTeX Symbolic Proofs

Signal Processing

FourierEpicyclesScene10m

The Fourier Transform: Complex Plane Winding, Epicycles & Spectral Decomposition

Decompose any time-domain signal into rotating complex exponentials via Euler's formula and inner products.

\hat{f}(\xi) = \int_{-\infty}^{\infty} f(t)\, e^{-2\pi i \xi t}\,dt, \qquad f(t) = \int_{-\infty}^{\infty} \hat{f}(\xi)\, e^{2\pi i \xi t}\,d\xi

#Fourier Transform#FFT#Complex Analysis

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Linear Algebra

EigenTransformationScene9m

Eigenvectors & Eigenvalues: Invariant Axes of Linear Transformations

Visualize how matrices shear space while keeping characteristic eigenvector directions on their own span.

A\mathbf{v} = \lambda\mathbf{v} \iff (A - \lambda I)\mathbf{v} = \mathbf{0} \iff \det(A - \lambda I) = 0

#Linear Algebra#Eigenvectors#Eigenvalues

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Deep Learning & Transformers

BackpropComputationalGraphScene10m

Backpropagation & Reverse-Mode Automatic Differentiation on Computational Graphs

Trace upstream adjoint gradients backward through DAG nodes in O(1) passes regardless of parameter count.

\frac{\partial \mathcal{L}}{\partial x_i} = \sum_{j \in \text{children}(i)} \frac{\partial \mathcal{L}}{\partial z_j} \cdot \frac{\partial z_j}{\partial x_i}

#Backpropagation#AutoDiff#Chain Rule

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Calculus & Optimization

GradientDescentTrajectoryScene9m

Gradient Descent, Polyak Momentum & Adam Optimization Geometry

Understand ravine oscillation damping, exponential moving averages of moments, and decoupled weight decay (AdamW).

\theta_{t+1} = \theta_t - \eta \left(\frac{\hat{m}_t}{\sqrt{\hat{v}_t} + \epsilon} + \lambda \theta_t\right)

#Optimization#Gradient Descent#AdamW

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Probability & Information Theory

BayesGeometricSquareScene8m

Bayes' Theorem: Geometric Area Proportions, Likelihood Ratios & Posterior Updating

Replace rote formula memorization with 3Blue1Brown's geometric probability-square restriction view.

P(H \mid E) = \frac{P(E \mid H)\,P(H)}{P(E)} = \frac{P(E \mid H)\,P(H)}{P(E \mid H)P(H) + P(E \mid \neg H)P(\neg H)}

#Probability#Bayes Theorem#Statistics

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Linear Algebra

SVDUnitCircleDecompositionScene10m

Singular Value Decomposition (SVD) & Low-Rank Matrix Approximation (LoRA)

Factorize any rectangular matrix into rotation-stretch-rotation (A = U Sigma V^T) and prove the Eckart-Young theorem.

A = U \Sigma V^\top = \sum_{i=1}^{r} \sigma_i\, \mathbf{u}_i \mathbf{v}_i^\top, \qquad A_k = \arg\min_{\text{rank}(B)=k} \|A - B\|_F = \sum_{i=1}^{k} \sigma_i\, \mathbf{u}_i \mathbf{v}_i^\top

#SVD#Linear Algebra#LoRA

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Deep Learning & Transformers

AttentionSubspaceCircuitScene11m

Attention Geometry: Softmax Kernel Smoothing & Query-Key Subspace Routing

Visualize Multi-Head Attention as differentiable dictionary lookup and low-rank bilinear form W_Q W_K^T.

\text{MHA}(X) = X + \sum_{h=1}^{H} \text{softmax}\!\left(\frac{X W_Q^{(h)} (W_K^{(h)})^\top X^\top}{\sqrt{d_k}}\right) X W_V^{(h)} W_O^{(h)}

#Attention#Transformers#LLMs

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Probability & Information Theory

MarkovStationaryFlowScene9m

Markov Chains, Ergodicity & Stationary Distributions (PageRank)

Analyze stochastic transition matrices, the Perron-Frobenius theorem, and random-surfer teleportation.

\boldsymbol{\pi} P = \boldsymbol{\pi}, \qquad G = \alpha P + (1 - \alpha)\frac{1}{N}\mathbf{1}\mathbf{1}^\top

#Markov Chains#Stochastic Processes#Linear Algebra

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Custom Manim Course Synthesis

Want a custom Manim animation and LaTeX proof for any theorem?

PAITS's ManimGenerator synthesizes custom mathematical scenes alongside Lesson 1.1 in under 45 seconds.

Generate Custom Visual Course