Some of my favorite beautiful equations.

Apparently math/abstract beauty excites the same parts of the brain as sensory beauty:
"Mathematics: Why the brain sees maths as beauty" [http://www.bbc.co.uk/news/science-environment-26151062] [2014-02-12].

Start with counting: Equations of adding one. In programming, a common shortened notation is x++.

$$ 0 + 1 = 1 $$

$$ 1 + 1 = 2 $$

$$ 2 + 1 = 3 $$

$$ ... $$

Ratio [W].

$$ A:B $$

$$ \tfrac{A}{B} $$

Proportionality [W]. Assume that B and D are not zero.

$$ A:B::C:D $$

$$ A\,is\,to\,B\,as\,C\,is\,to\,D $$

$$ \tfrac{A}{B} = \tfrac{C}{D} = \tfrac{A + C}{B + D} $$

$$ AD = BC $$

The Polynomials [W] are expressions of constants and variables using only the operations of addition, subtraction, multiplication, and positive integer exponents (i.e. just multiplication). These are also known as polynomial equations or algebraic equations. A polynomial can be univariate (EG: \(x\)) or multivariate (EG: \(x + y\)). A ploynomial is also said to have a degree based on the largest exponent.

The Linear Equation [W] is a multivariate polynomial equation of the first degree. Linear equations deal with (gasp!) straight lines.

$$ Ax + By = C $$

The general/standard form above can be arranged to the slope-intercept form. M is the slope of the line, and K the intercept on the y-axis if x is zero.

$$ y = Mx + K $$

The Quadratic Equation [W] is a univariate polynomial equation of the second degree. These are graphed as parabolas.

$$ ax^2 + bx + c = 0 $$

where x represents a variable or an unknown, and a, b, and c are constants with a ≠ 0. (If a = 0, the equation is a linear equation, graphed as a line.)

The Pythagorean Theorem [W] is a multivariate polynomial equation of the second degree. The Pythagorean theorem deals with the three sides of a right triangle in Euclidean geometry.

$$ x^2 + y^2 = r^2 $$

$$ 3^2 + 4^2 = 5^2 $$

The Mean [W] has many different meanings, but it has to do with finding a representative number given a set of numbers. for a set of values. he 3 classical Pythagorean means [W] (Harmonic, Geomtric, and Arithmetic) are usually ordered thusly: \(\min \leq HM \leq GM \leq AM \leq QM \leq \max\). There are also weighted version of means.

The Generalized mean [W] (aka power mean) is a general mean function.

$$ M_p(x_1,\dots,x_n) = \left( \frac{1}{n} \sum_{i=1}^n x_i^p \right)^{\frac{1}{p}} $$

The Minimum [W].

$$ M_{-\infty}(x_1,\dots,x_n) = \lim_{p\to-\infty} M_p(x_1,\dots,x_n) = \min \{x_1,\dots,x_n\} $$

The Harmonic Mean (HM) [W].

$$ M_{-1}(x_1,\dots,x_n) = \frac{n}{\frac{1}{x_1}+\dots+\frac{1}{x_n}} $$

The Geometric Mean (GM) [W].

$$ M_0(x_1,\dots,x_n) = \lim_{p\to0} M_p(x_1,\dots,x_n) = \sqrt[n]{x_1\cdot\dots\cdot x_n} $$

The Arithmetic Mean (AM) [W]. When most people calculate the "average", they do it via the arithmetic mean.

$$ M_1(x_1,\dots,x_n) = \frac{x_1 + \dots + x_n}{n} $$

The Root Mean Sqaure [W]. Aka the quadratic mean (QM).

$$ M_2(x_1,\dots,x_n) = \sqrt{\frac{x_1^2 + \dots + x_n^2}{n}} $$

The Maximum [W].

$$ M_{+\infty}(x_1,\dots,x_n) = \lim_{p\to\infty} M_p(x_1,\dots,x_n) = \max \{x_1,\dots,x_n\} $$

The Golden Ratio [W] as represented by the Greek letter phi (φ) is also related to the Fibonacci Number [W]:

$$ \varphi = \frac{a + b}{a} = \frac{a}{b} = \frac{1 + \sqrt{5}}{2} = 1.6180339887... $$

$$ \varphi = \lim_{n\to\infty}\frac{F(n+1)}{F(n)} $$

$$ F_n = F_{n-1} + F_{n-2},\!\, $$

$$ 0,\;1,\;1,\;2,\;3,\;5,\;8,\;13,\;21,\;34,\;55,\;89,\;144,\; \ldots\; $$

Euler's Identity [W] from analytical mathematics. Three operations (addition, multiplication, subtraction) and five fundamental constants.

$$ e^{i \pi} + 1 = 0 $$

Scrap area:


Ccircle = 2πr
Acircle = πr2 
Asphere = 4πr2 
Vsphere = (4/3)πr3

τ and π

(148/296) + (35/70) = 1   This is fun because it has all 10 digits.

Bayes Theorem

Power Law and the Pareto principle (80-20 rule). 20% of the effort/experience yields 80% of the accuracy/skill.

Quarternions

Euler's Characteristic: V - E + F = 2

Beautiful math equations like the ones above are distinct from beautiful science equations. Pure math equation tend to describe an undeniable underlying reality, while science equations tend to approximate an underlying reality.

E = mc2
F = ma  Newton's 2nd Law
V = IR
PV = nRT  Ideal Gas Law
Maxwell's equations
Einstein's field equations

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