探索精選的 Manim 動畫範例、視覺化證明、圖形動畫與可重用數學影片靈感，適用於教學、學習與內容創作。
28 個動畫 · 7 個主題
Demonstrates backpropagation in a 3-layer neural network. A forward pass first computes activations left to right, then a loss is computed. Gradients flow backward through the network as a glowing wave, with the chain rule applied at each layer. The key partial derivative formula is displayed.
Demonstrates binary search tree operations step by step: inserting nodes to build a BST, searching for a value with left/right comparisons shown, detecting an unbalanced tree, and performing an AVL left rotation to restore balance. Height and balance factor labels update throughout.
Explains the CAP theorem through an animated distributed system with 3 nodes. A network partition is triggered, and the animation shows the trade-off: choosing CP sacrifices availability while choosing AP allows divergence. Real-world systems are labeled with their CAP classification.
Demonstrates consistent hashing on a circular hash ring (0–360 degrees). Server nodes are placed on the ring, keys are assigned clockwise, and the animation shows that adding or removing a server only redistributes a minimal fraction of keys. Virtual nodes are introduced to improve load distribution.
34 個動畫 · 11 個主題
This animation visualizes the infinite geometric series 1 + 1/2 + 1/4 + 1/8 + ... by repeatedly halving a unit square. Each term is shown as a filled rectangle that fits beside the previous ones like puzzle pieces, and the partial sums are displayed numerically, all converging toward the limit of 2.
This animation introduces linear functions of the form y = mx + b by visually demonstrating how the slope (m) and y-intercept (b) affect the graph. A line rotates as the slope changes from -2 to 2, then shifts vertically as the intercept changes from -3 to 3. A rise/run right-triangle indicator reinforces the geometric meaning of slope.
This animation establishes the relationship between exponential and logarithmic functions. It plots y = 2^x and y = log₂(x) side-by-side reflected across y = x, demonstrating inverse symmetry. A concrete numeric example (2³ = 8 → log₂(8) = 3) is shown with perpendicular guide lines, making the inverse relationship tangible.
This animation contrasts permutations (order matters) with combinations (order doesn't matter) using a concrete example of choosing 2 items from {A, B, C}. A tree diagram visually enumerates all 6 permutations, then groups are collapsed to show only 3 combinations, making the divide-by-r! factor intuitive.
7 個動畫
A neuron diagram with a voltage-time graph showing all phases of an action potential. Ion animations (Na+ rushing in, K+ rushing out) are synchronized with the graph. The signal propagates down the axon at the end.
A visual simulation of natural selection showing a population of bugs on a background. Camouflaged bugs survive while conspicuous bugs are removed by predators. Over three generations the population frequency shifts toward the camouflaged trait, tracked by a live bar chart.
A two-phase animation of the central dogma. Phase 1 shows transcription inside the nucleus: DNA unzips and mRNA is synthesized. Phase 2 shows translation at the ribosome: tRNA delivers amino acids matching mRNA codons and the protein chain elongates.
An animated construction of the DNA double helix, starting from two parallel backbone strands and progressively adding base pairs. The structure then twists into the recognizable double helix and base pairing rules are displayed.
8 個動畫
Teaches the law of conservation of mass by balancing two chemical equations step by step. Shows atom-count tracking on both sides of the equation, then demonstrates adding coefficients to balance H₂ + O₂ → H₂O and CH₄ + O₂ → CO₂ + H₂O.
Demonstrates Le Chatelier's Principle using the Haber process (N₂ + 3H₂ ⇌ 2NH₃). Shows three stresses applied to a system at equilibrium and how the equilibrium position shifts in response to each stress using arrow-size animations.
Introduces the pH scale from 0 to 14 with a color-gradient bar and real-world substance examples. Explains the logarithmic relationship between pH and hydrogen ion concentration, and animates how H⁺ concentration changes across the scale.
Demonstrates covalent bonding by showing two hydrogen atoms sharing electrons to form H₂, then extends to water (H₂O) where oxygen shares electrons with two hydrogens. Electron clouds overlap to illustrate shared pairs.
6 個動畫
Demonstrates the power of compound interest by comparing three growth scenarios over 30 years: no interest (flat), simple interest at 5% (linear), and compound interest at 5% (exponential). Illustrates the compounding formula and a concrete $1000 @ 7% example showing $7,612 after 30 years.
Illustrates how inflation erodes purchasing power over time through multiple visual metaphors: a shrinking coin, a bar chart showing how many goods $100 buys across decades, the rising CPI index, and the Rule of 72 applied to purchasing power halving at 3% annual inflation (~24 years).
Visualizes how supply and demand curves interact to form a market equilibrium. Demonstrates the effects of a demand shift (rightward) and a supply shift (leftward) on equilibrium price and quantity, building core microeconomics intuition.
Introduces the Prisoner's Dilemma, a foundational concept in game theory. Shows a 2x2 payoff matrix, animates two prisoners making independent decisions, explains why (Defect, Defect) is the Nash Equilibrium even though mutual cooperation is Pareto superior. Highlights dominant strategies and the Nash Equilibrium condition.
22 個動畫
A ball on a string orbits in a circle. The velocity vector (tangent) and centripetal force vector (inward) are shown continuously updating. When the string breaks, the ball flies off tangentially, illustrating why centripetal force is required.
Two collision scenarios on a frictionless surface demonstrate momentum conservation. An elastic collision shows ball 1 stopping and ball 2 moving on. An inelastic collision shows the balls sticking together. Momentum vectors p = mv are shown before and after each collision.
Electric field lines are shown radiating from a positive charge and converging on a negative charge. A dipole field shows lines flowing from + to -. A test charge placed in the field experiences a force along the field line direction.
A water pipe analogy introduces voltage, current, and resistance intuitively. A circuit diagram then shows these quantities. Increasing voltage drives more current. A V-I graph shows the linear relationship.
