Interactive Timeline of Mathematical and Physical Breakthroughs

Euclid's Five Postulates

1. A straight line segment can be drawn joining any two points
2. Any straight line segment can be extended indefinitely
3. Given any straight line segment, a circle can be drawn with the segment as radius
4. All right angles are congruent
5. Parallel Postulate: If a line intersects two other lines such that the sum of inner angles on one side is less than 180°, the two lines will intersect

Pythagorean Theorem

$$a^2 + b^2 = c^2$$

For a right triangle with legs $a$ and $b$, and hypotenuse $c$

Circle Properties

$$\text{Area} = \pi r^2$$ $$\text{Circumference} = 2\pi r$$

The Derivative

$$\frac{df}{dx} = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}$$

Measures instantaneous rate of change

The Integral

$$\int_a^b f(x) \, dx = \lim_{n \to \infty} \sum_{i=1}^{n} f(x_i) \Delta x$$

Measures area under curve

Fundamental Theorem of Calculus

$$\int_a^b f'(x) \, dx = f(b) - f(a)$$

Links differentiation and integration

Example Function

$$f(x) = ax^3 + bx^2 + cx + d$$ $$f'(x) = 3ax^2 + 2bx + c$$

First Law (Inertia)

An object at rest stays at rest, and an object in motion stays in motion, unless acted upon by an external force.

Second Law (F = ma)

$$\vec{F} = m\vec{a} = m\frac{d\vec{v}}{dt} = m\frac{d^2\vec{r}}{dt^2}$$

Force equals mass times acceleration

Third Law (Action-Reaction)

$$\vec{F}_{12} = -\vec{F}_{21}$$

For every action, there is an equal and opposite reaction

Universal Gravitation

$$F = G\frac{m_1 m_2}{r^2}$$

Where $G = 6.674 \times 10^{-11}$ N⋅m²/kg²

Maxwell's Equations

$$\nabla \cdot \vec{E} = \frac{\rho}{\epsilon_0}$$ (Gauss's law) $$\nabla \cdot \vec{B} = 0$$ (No magnetic monopoles) $$\nabla \times \vec{E} = -\frac{\partial \vec{B}}{\partial t}$$ (Faraday's law) $$\nabla \times \vec{B} = \mu_0 \vec{J} + \mu_0 \epsilon_0 \frac{\partial \vec{E}}{\partial t}$$ (Ampère-Maxwell law)

Wave Equation

$$\frac{\partial^2 \vec{E}}{\partial t^2} = c^2 \nabla^2 \vec{E}$$ $$c = \frac{1}{\sqrt{\mu_0 \epsilon_0}} = 3 \times 10^8 \text{ m/s}$$

Electromagnetic Wave

$$\vec{E}(z,t) = E_0 \cos(kz - \omega t + \phi) \hat{x}$$ $$\vec{B}(z,t) = \frac{E_0}{c} \cos(kz - \omega t + \phi) \hat{y}$$

Logistic Map

$$x_{n+1} = rx_n(1-x_n)$$

Simple equation that exhibits chaotic behavior for certain values of r

Lyapunov Exponent

$$\lambda = \lim_{n \to \infty} \frac{1}{n} \sum_{i=0}^{n-1} \ln|f'(x_i)|$$

Measures sensitivity to initial conditions. λ > 0 indicates chaos

Strange Attractor

A fractal set toward which a dynamical system evolves over time, characterized by sensitive dependence on initial conditions

Butterfly Effect

Small changes in initial conditions lead to large-scale and unpredictable variation in the future behavior of the system

Mandelbrot Set

$$z_{n+1} = z_n^2 + c$$

Where z₀ = 0 and c is a complex parameter. The set contains all points c for which the sequence remains bounded.

Julia Set

$$z_{n+1} = z_n^2 + c$$

For fixed c, the Julia set contains all starting points z₀ for which the sequence remains bounded.

Hausdorff Dimension

$$D = \lim_{\epsilon \to 0} \frac{\log N(\epsilon)}{\log(1/\epsilon)}$$

Generalizes the notion of dimension to non-integer values

Self-Similarity

A fractal appears similar to itself at all levels of magnification

Schrödinger Equation

$$i\hbar \frac{\partial \Psi}{\partial t} = \hat{H}\Psi$$

Fundamental equation governing quantum systems

Wave Function

$$|\Psi(x,t)|^2 = \text{probability density}$$

Born interpretation: |Ψ|² gives probability of finding particle at position x

Uncertainty Principle

$$\Delta x \Delta p \geq \frac{\hbar}{2}$$

Heisenberg's fundamental limit on simultaneous measurement precision

Particle in a Box

$$\Psi_n(x) = \sqrt{\frac{2}{L}}\sin\left(\frac{n\pi x}{L}\right)$$ $$E_n = \frac{n^2\pi^2\hbar^2}{2mL^2}$$

Special Relativity - Time Dilation

$$\Delta t = \frac{\Delta t_0}{\sqrt{1-\frac{v^2}{c^2}}} = \gamma \Delta t_0$$

Moving clocks run slower

Length Contraction

$$L = L_0\sqrt{1-\frac{v^2}{c^2}} = \frac{L_0}{\gamma}$$

Moving objects contract in direction of motion

Mass-Energy Equivalence

$$E = mc^2$$

Matter and energy are interchangeable

General Relativity - Einstein Field Equations

$$G_{\mu\nu} = \frac{8\pi G}{c^4}T_{\mu\nu}$$

Curvature of spacetime equals energy-momentum

Graph Theory Formulation

Given a planar graph G = (V, E), there exists a proper vertex coloring using at most 4 colors.

A proper coloring: χ(G) ≤ 4 for all planar graphs G

Chromatic Number

$$\chi(G) = \min\{k : G \text{ has a proper } k\text{-coloring}\}$$

The minimum number of colors needed

Euler's Formula for Planar Graphs

$$V - E + F = 2$$

Vertices minus Edges plus Faces equals 2

Reducible Configurations

Appel and Haken identified 1,936 unavoidable configurations, proving each was reducible by computer

Linearized Einstein Equations

$$\Box h_{\mu\nu} = -\frac{16\pi G}{c^4} T_{\mu\nu}$$

Wave equation for small perturbations in spacetime metric

Strain Amplitude

$$h \sim \frac{4G}{c^4} \frac{M v^2}{r}$$

Where M is mass, v is velocity, r is distance from source

Quadrupole Formula

$$P = \frac{G}{5c^5} \left\langle \dddot{Q}_{ij} \dddot{Q}^{ij} \right\rangle$$

Power radiated in gravitational waves

LIGO Detection

$$\Delta L = h \times L$$

Change in arm length proportional to strain h ≈ 10⁻²¹