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Revision as of 19:09, 27 February 2013


Welcome to the explain xkcd wiki!

We have collaboratively explained 6 xkcd comics, and only 2028 (33800%) remain. Add yours while there's a chance!

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Equations
All electromagnetic equations: The same as all fluid dynamics equations, but with the 8 and 23 replaced with the permittivity and permeability of free space, respectively.
Title text: All electromagnetic equations: The same as all fluid dynamics equations, but with the 8 and 23 replaced with the permittivity and permeability of free space, respectively.

This comic gives a set of equations supposedly from different areas of science in mathematics, physics, and chemistry. To anyone not familiar with the field in question they look pretty similar to what you might find in research papers or on the relevant Wikipedia pages. To someone who knows even a little about the topic, they are clearly very wrong and only seem even worse the more you look at them.

Simplified Explanations

Ambox notice.png This explanation may be incomplete or incorrect: Created by a mere human. Do NOT delete this tag too soon.
All chemistry equations

This shows a parody of the common example chemistry equation of burning Methane and Oxygen (with added heat), to form water and carbon dioxide. However in this form "HEAT" is an actual molecule, rather than simply indicating the presence of heat to start the reaction. Thus the equation is modified to incorporate the fictional "HEAT" into the reaction.

TODO: other simplified explanations.


Technical Explanations

Ambox notice.png This explanation may be incomplete or incorrect: Created by an EQUATION. Do NOT delete this tag too soon.


All kinematics equations
E = K_0t + \frac{1}{2}\rho vt^2

Kinematics describes the motion of objects without considering mass or forces.

This equation here literally states: "Energy equals a constant K_0 multiplied by time, plus half of density multiplied by speed multiplied by time squared".

The first term here is hard to interpret: it could be correct if K_0 is a constant power applied to the system, but this symbol would more normally be used to denote an initial energy, in which case so multiplying by t would be wrong. Alternatively, the term is similar to k_B T (sometimes written as kT), a term that often appears in statistical mechanics equations, where kB (or k) is the Boltzmann constant, and T is the absolute temperature. In this latter case, the term would have units of energy, consistent with the left side of the equation.

The second term looks similar to the kinetic energy term  \frac{1}{2}\rho v^2 in the Bernoulli equation for fluids (or, more properly, the kinetic energy density in the fluid).

The whole equation appears to be a play on the kinematics formula: s = ut + \frac{1}{2}\ at^2, where distance travelled (s) by a constantly accelerating object is determined by initial velocity (u), time (t), and acceleration (a)

Kinematics is often one of the first topics covered in an introductory physics course, both at the high school and freshman college levels. As such, mixing in material from more advanced topics like statistical mechanics and the Bernoulli equation, even if done correctly, would be very confusing for a typical student learning kinematics.

All number theory equations
K_n = \sum_{i=0}^{\infty}\sum_{\pi=0}^{\infty}(n-\pi)(i-e^{\pi-\infty})

Number theory is a branch of mathematics primarily to the study the properties of integers.

Taken literally the equation says: "The nth K-number is equal to: the sum of all i from 0 to infinity, the sum of all pi from 0 to infinity; subtract pi from n, and multiply it with i minus e to the power of pi minus infinity". A twofold misconception can be seen here. The first is the reassignment of pi as a variable instead of the constant (3.14...). This might be a jab at how in number theory letters and numbers are used interchangeably, but where some letters are all of a sudden fixed constants. The second misconception is the use of infinity in the latter part of the formula. Naively this would signify that (with the reassigned pi values) the part in the power would range from minus infinity to zero. However, infinity is not a number and cannot be used as one without using a limit construct.

All fluid dynamic equations
\frac{\partial}{\partial t}\nabla\cdot \rho = \frac{8}{23}
\int\!\!\!\!\!\!\!\!\!\;\;\bigcirc\!\!\!\!\!\!\!\!\!\;\;\int
\rho\,ds\,dt\cdot \rho\frac{\partial}{\partial\nabla}

Fluid dynamics describes the movement of non-solid material. In particular for gases, the density \rho is often the most interesting quantity (for liquids, this is often just constant). A unique feature of fluid-dynamic equations is the presence of advection terms, which take the form of often strange-looking spatial derivatives. This equation turns this up to a new level by differentiating with respect to a differential operator \nabla, which does not make any sense at all. Also it has a contour integral which seems reminiscent to a closed-circle process like in a piston engine, but this does not really fit in the context (differential description of a gas), and it has a pair of unexplained numbers 8 and 23, probably alluding to the specific heat ratio which is often written out as the fraction \tfrac{7}{5}, whereas most other physics equations avoid including any plain numbers higher than 4.

The title text stating that the electromagnetism equation is the same as the fluid dynamics equation, but with the arbitrary 8 and 23 replaced with the permittivity and permeability of free space is likely because electromagnetism equations often have relations to fluid dynamics, and because those two constants appear in the vast majority of electromagnetism equations.

All quantum mechanic equations
|\psi_{x,y}\rangle = A(\psi) A(|x\rangle \otimes |y\rangle)

Quantum mechanics is a fundamental theory in physics which describes the nature at scales of atoms and below. It typically uses the bra–ket notation in its formulae.

This equation takes a state psi in the dimensions of x and y and equates it to an operator A performed on psi multiplied by the same operator performed on the tensor product of x and y. Seeing as the state psi is already the tensor product of the states x and y, this is equivalent to performing the same unknown operator twice on psi, and unless this operator is its own inverse such as a bit-flip or Hermitian operator, this equation is therefore incorrect.

All chemistry equations
\mathrm{CH}_4 + \mathrm{OH} + \mathrm{HEAT} \rightarrow \mathrm{H}_2\mathrm{O} + \mathrm{CH}_2 + \mathrm{H}_2 \mathrm{EAT}

A chemical equation is the symbolic representation of a chemical reaction in the form of symbols and formulae, wherein the reactant entities are given on the left-hand side and the product entities on the right-hand side. The number of each element on the left side must match them on the right side, the equation is balanced. The energy produced or absorbed in this process is not included in that formula.

This here is a modification of the combustion of methane. The correct form is often taught and a good example problem but obviously there are more chemistry problems.\mathrm{HEAT} is normally shorthand for activation energy, but in Randall's version it's jokingly used as a chemical ingredient and becomes \mathrm{H}_2\mathrm{EAT}, taking the hydrogen atom freed by the combustion equation shown. To deliver the punchline while maintaining proper stoichiometry, \mathrm{OH} (which should be \mathrm{OH}^-, since the oxygen keeps a free electron when it combines with a single hydrogen) is shown instead of \mathrm{O}_2. The proper methane combustion equation would be: \mathrm{CH}_4 + 2 \mathrm{O}_2 \rightarrow 2 \mathrm{H}_2\mathrm{O} + \mathrm{CO}_2

All quantum gravity equations
\mathrm{SU}(2)\mathrm{U}(1) \times \mathrm{SU}(\mathrm{U}(2))

This is more similar to expressions which appear in Grand Unified Theory (GUT) than general quantum gravity. Unlike some of the other equations, this one has no interpretation which could make it mathematically correct. This is similar to the notations used to describe the symmetry group of a particular phenomena in terms of mathematical Lie Groups. A real example would be the Standard Model of particle physics which has symmetry according to \rm{SU(3)\times SU(2) \times U(1)}. Here, \rm{SU} and \rm{U} denote the special unitary and unitary groups respectively with the numbers indicating the dimension of the group. Loosely, the three terms correspond to the symmetries of the strong force, weak force and electromagnetism although the exact correspondence is muddied by symmetry breaking and the Higgs mechanism.

Of course, an expression missing an "=" sign, is difficult to interpret as an "equation", because equations normally express an "equality" of some kind. Nobody knows whether Randal refers to a horse here (equidae)

Randall's version clearly involves some similar groups although without the \times symbol it is hard to work out what might be happening. A term like \rm{SU(U(2))} has no current interpretation in mathematics, if anyone thinks otherwise and possibly has a solution to the quantum gravity problem they should probably get in touch with someone about that.

All gauge theory equations
All gauge theory equations.png

In physics, a gauge theory is a type of field theory which is invariant to local transformations. The term gauge refers to any specific mathematical formalism to regulate redundant degrees of freedom.

This equation looks broadly similar to the sorts of things which appear in gauge theory such as the equations which define Yang-Mills Theory. By the time physics has got this far in, people have normally run out of regular symbols making a lot of the equations look very daunting. The actual equations in this field rarely go far beyond the Greek alphabet though and no-one has yet to try putting hats on brackets. The appearance of many sub- and superscripts is normal (this links to the group theory origins of these equations) and for the layperson it can be impossible to determine which additions are labels on the symbols and which are indices for an Einstein Sum.

The left-hand side S_g is the symbol for some action, in Yang-Mills theory this is actually used for a so-called "ghost action". On the right-hand side we have a large number of terms, most of which are hard to interpret without knowing Randall's thought processes (this is why real research papers should all label their equations thoroughly). The \frac{1}{2\bar{\varepsilon}} looks like a constant of proportionality which often appears in gauge theories. The factor of i = \sqrt{-1} is not unusual as many of these equations use complex numbers. The \eth symbol looks similar to a \partial partial derivative symbol especially as the Dirac Equation uses a slashed version as a convenient shorthand.

The rest of the equation cannot be mathematically correct as the choice of indices used does not match that on the left-hand side (which has none). In particle physics subscripts (or superscripts) of greek letters (usually \mu or \nu) indicate terms which transform nicely under Lorentz transformations (special relativity). Roman indices from the beginning of the alphabet relate to various gauge transformation propetries, the triple index seen on p^{abc}_v would likely come from some \rm{SU(3)} transformation (related to the strong nuclear force). Since S_g has none of these (and is thus a scalar which remains constant under these operations), we would need the right-hand side to behave in the same way. Most of the indices which appear are unpaired and so will not result in a scalar making the equation very wrong. For those not familiar with this type of equation, it is a similar mistake messing up units and setting a distance equal to a mass.

All cosmology equations
H(t) + \Omega + G \cdot \Lambda \, \dots \begin{cases} \dots > 0 & \text{(Hubble model)} \\ \dots = 0 & \text{(Flat sphere model)} \\ \dots < 0  & \text{(Bright dark matter model)} \end{cases}

This is a parody of equations defining the Hubble Parameter H(t) although it looks like Randall has become bored and not bothered to finish his equation. Such equations usually have several \Omega terms representing the contributions of different substances to the energy-density of the Universe (matter, radiation, dark energy etc.). In this context G could be Newton's constant and \Lambda is the cosmological constant (energy density of empty space) although seeing them appear multiplied and on the same footing as H is unusual (the dot is entirely unnecessary). Choosing to make H a function of time t and not of redshift z is also unusual.

The second section looks like the inequalities used to show how what shape the Universe, based on the value of the curvature parameter \Omega_k. A value of 0 indicates a flat Universe (this more or less what we observe) whilst a positive /negative value indicates an open /closed curved Universe. Randall's choice of labels further makes fun of the field as both a flat sphere and bright dark matter are oxymoronic terms which would involve some rather strange model universes.

All truly deep physics equations
All truly deep physics equations.png

\hat H is the Hamiltonian operator, which when applied to a system returns the total energy. In this context, U would usually be the potential energy. However, there is also a subscript 0 and a diacritic marking indicating some other variable. Much of physics is based on Lagrangian and Hamiltonian mechanics. The Lagrangian is defined as \hat L = \hat K - \hat U with K being the kinetic energy and U the potential. Hamiltonian mechanics uses the equation \hat H = \hat K + \hat U . The Hamiltonian must be conserved so taking the time derivative and setting it equal to zero is a powerful tool. The "principle of least action" allows most modern physics to be derived by setting the time derivative of the Lagrangian to zero.

Transcript

[Nine equations are listed, three in the top row and two in each of the next three rows. Below each equation there are labels:]
E = K0t + 1/2 ρvt2
All kinematics equations
Kn = ∑i=0π=0(n-π)(i-eπ-∞)
All number theory equations
∂/∂t ∇ ⋅ ρ = 8/23 (∯ ρ ds dt ⋅ ρ ∂/∂∇)
All fluid dynamics equations
x,y〉 = A(ψ) A(|x〉⊗ |y〉)
All quantum mechanics equations
CH4 + OH + HEAT → H2O + CH2 + H2EAT
All chemistry equations
SU(2)U(1) × SU(U(2))
All quantum gravity equations
Sg = (-1)/(2ε̄) i ð (̂ ξ0 +̊ pε ρvabc η0 )̂ f̵a0 λ(ʒ̆) ψ(0a)
All gauge theory equations
[There is a brace linking the three cases together.]
H(t) + Ω + G⋅Λ ...
... > 0 (Hubble model)
... = 0 (Flat sphere model)
... < 0 (Bright dark matter model)
All cosmology equations
Ĥ - u̧0 = 0
All truly deep physics equations


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