explain xkcd - User contributions [en]

2034: Equations

2018-11-14T19:08:06Z

Athyndmion: /* Explanation */ "the the typical angel bracket" -> "the typical angle bracket"

{{comic
| number = 2034
| date = August 17, 2018
| title = Equations
| image = equations.png
| titletext = 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.
}}
==Explanation==
{{incomplete|TODO: some simplified explanations. The title text is bad explained. Do NOT delete this tag too soon.}}

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. In many disciplines, the mathematical description of a large area is summed up in a small number of equations, such as Maxwell's equations for electromagnetism. In similar fashion, the equations here purport to encompass the whole of their given field.

===Simplified Explanations===

;All kinematics equations
:Kinematics is the study of the motion of objects. More specifically, it describes how the location, velocity, and acceleration of an object vary over time. The equation shown contains two of these standard kinematic variables, velocity ''v'' and time ''t'', in addition to several quantities (''E'', ''K0'', and ''ρ'') that are completely unrelated to kinematics.

;All number theory equations
:Number theory is a branch of mathematics concerned primarily with the study of integers. However, the equation shown contains the non-integer Euler's constant ''e'' (approximately 2.718). It also uses the Greek letter π as an integer-valued variable, even though the symbol π is used in mathematics almost exclusively to denote the well-known ''non''-integer circle constant (approximately 3.14159). Even with π treated as a variable here, one of its uses in the equation is still nonsensical. <math>\pi-\infty</math> uses ∞ as if it were a specific number, which it is not, thus giving an undefined result.

;All fluid dynamics equations
:Fluid dynamics try to cover the movement of fluids, and also gases, in a given environment. It has to include turbulences which this formula presents by some other meanings. In fact there is no simple formula to explain fluid dynamics and for example it's impossible to predict the exact shape of a forming cloud over time.

;All quantum mechanics equations
:This formula uses the typical angle bracket and vertical bar notation used in quantum mechanics. The uppercase letter ''A'' probably indicates ''all'', but only in respect to two dimensions (x,y). An equation covering ''all'' would need at least the missing third dimension and of course the time to cover some dynamics.

;All chemistry equations
:Randall implies that all chemistry is just combustion of chemicals, demonstrated with an incorrect form of a 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. While the H in "HEAT" is the chemical symbol of the element hydrogen, none of the letters E, A, or T are symbols of any actual elements. Also, to account for the second hydrogen in "H(2)EAT" on the products side, the oxygen gas on the reactants side has been altered to be hydroxide, a strong base that would not facilitate traditional combustion.

;All quantum gravity equations
:At first this isn't a equation at all because there is no equal sign or similar between the terms. The notation SU(2) again refers to a two dimensional system but also to symmetries in quantum mechanics like particles always have their anti-particles. Quantum gravity in general hasn't been discovered yet.

;All gauge theory equations
:Gauge theories in physics try to describe the dynamics of elementary particles in the field of the {{w|Quantum field theory}} and lead to the famous {{w|Standard Model}} of particle physics.

;All cosmology equations
:The {{w|Hubble's law}} is a simple explanation how the expansion of the universe appears to work, but there is also a Nobel Prize worthy {{w|Dark energy}} trying to explain the the moving speeds of distant galaxies. All theories are far away to explain the Cosmos at all.

;All truly deep physics equations
:TODO: missing simple explanation

===Technical Explanations===
;All kinematics equations
:<math>E = K_0t + \frac{1}{2}\rho vt^2</math>
:{{w|Kinematics}} describes the motion of objects without considering mass or forces. The latter is described by {{w|Kinetics (physics)|kinetics}}. The two fields get frequently confused due to the similarity of words.

:This equation here literally states: "Energy equals a constant <math>K_0</math> 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 <math>K_0</math> is a constant power applied to the system, but this symbol would more normally be used to denote an initial energy, in which case multiplying by <math>t</math> would be wrong. Alternatively, the term is similar to <math>k_B T</math> (sometimes written as ''kT''), a term that often appears in {{w|Statistical_mechanics|statistical mechanics}} equations, where ''kB'' (or ''k'') is {{w|Boltzmann_constant|the Boltzmann constant}}, and ''T'' is the {{w|Thermodynamic_temperature|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 <math> \frac{1}{2}\rho v^2 </math> in [http://hyperphysics.phy-astr.gsu.edu/hbase/pber.html the Bernoulli equation] for fluids. (More properly, this is the kinetic energy ''density'' in the fluid).

:The whole equation appears to be a play on the kinematics formula: <math>s = v_0t + \frac{1}{2}\ at^2</math>, where distance travelled (''s'') by a constantly accelerating object is determined by initial velocity (''v0''), 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
:<math>K_n = \sum_{i=0}^{\infty}\sum_{\pi=0}^{\infty}(n-\pi)(i+e^{\pi-\infty})</math>
:{{w|Number theory}} is a branch of mathematics primarily studying the properties of integers.

:Said in English, the equation can be read: "The ''n''th K-number is equal to the sum for all ''i'' from 0 to infinity of the sum for all π from 0 to infinity of ''n'' minus π, multiplied by ''i'' plus ''e'' raised to the power of π minus infinity." (''i'' here is an iteration variable, not the imaginary number constant; ''e'' is Euler's number, approximately 2.718). A twofold misconception can be seen here. The first is the use of π as a variable instead of the circle constant (3.14...). This might be a jab at how in number theory letters and numbers are used interchangeably, but where some letters are suddenly fixed constants.

:Further confusion comes from the use of unusual mathematical models. While the term <math>e^{\pi-\infty}</math> is meaningless when considered in standard ("high school") mathematics, it is valid when considered on the {{w|extended real number line}}, a concept unfamiliar to most non-mathematicians and uncommon in number theory. Naively, this would signify that (with the use of π as a variable) the exponent would range from negative infinity to zero. In fact, assuming ''e'' really does mean Euler's constant (or at least a real number strictly greater than 1) the term would be zero for every π < ∞. Ultimately, the sum diverges for every ''n''.

:The close proximity of the letters i, e and π also evokes {{w|Euler's identity}} <math>e^{i\pi}+1=0</math> (also written <math>e^{i\pi}=-1</math>), without actually using it, especially since both π and i are used as variables here.

;All fluid dynamics equations
:<math>\frac{\partial}{\partial t}\nabla\cdot \rho = \frac{8}{23}
\int\!\!\!\!\!\!\!\!\!\;\;\bigcirc\!\!\!\!\!\!\!\!\!\;\;\int
\rho\,ds\,dt\cdot \rho\frac{\partial}{\partial\nabla}
</math>
:{{w|Fluid dynamics}} describes the movement of non-solid material. In particular for gases, the density <math>\rho</math> is often the most interesting quantity (for liquids, this is often just constant). A unique feature of fluid-dynamic equations is the presence of {{w|Advection|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 <math>\nabla</math>, 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 {{w|Magic number (programming)|unexplained numbers}} <math>8</math> and <math>23</math>, probably alluding to the {{w|Heat capacity ratio|specific heat ratio}} which is often written out as the fraction <math>\tfrac{7}{5}</math>, whereas most other physics equations [[899: Number Line|avoid including any plain numbers higher than 4]].

;All quantum mechanics equations
:<math>|\psi_{x,y}\rangle = A(\psi) A(|x\rangle \otimes |y\rangle)</math>
:{{w|Quantum mechanics}} is a fundamental theory in physics which describes nature at scales of atoms and below. It typically uses the {{w|Bra–ket notation|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. Since 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 the identity or is its own inverse such as a bit-flip or Hermitian operator, this equation is therefore incorrect.

;All chemistry equations
:<math>\mathrm{CH}_4 + \mathrm{OH} + \mathrm{HEAT} \rightarrow \mathrm{H}_2\mathrm{O} + \mathrm{CH}_2 + \mathrm{H}_2 \mathrm{EAT}</math>
:A {{w|Chemical equation|chemical equation}} represents a chemical reaction as a formula, with the reactant entities 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 those on the right side. The energy produced or absorbed in this process is not included in that formula.

:This 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.<math>\mathrm{HEAT}</math> is normally shorthand for {{w|activation energy}}, but in Randall's version it's jokingly used as a chemical ingredient and becomes <math>\mathrm{H}_2\mathrm{EAT}</math>, taking the hydrogen atom freed by the combustion equation shown. The proper methane combustion equation would be: <math>\mathrm{CH}_4 + 2 \mathrm{O}_2 \rightarrow 2 \mathrm{H}_2\mathrm{O} + \mathrm{CO}_2</math>

:While <math>\mathrm{OH}</math> often appears in chemical equations in the form of a negatively charged hydroxide group (<math>\mathrm{OH}^-</math>), the left side of the equation involves a bare <math>\mathrm{OH}</math>, possibly the highly unstable hydroxyl radical (although this would typically be written with a leading dot, e.g. <math>\bullet\mathrm{OH}</math>). Similarly, the right side contains an unstable methylene radical which would generally only appear as an intermediate rather than a product.

;All quantum gravity equations
:<math>\mathrm{SU}(2)\mathrm{U}(1) \times \mathrm{SU}(\mathrm{U}(2))</math>
:This is more similar to expressions which appear in {{w|Grand_Unified_Theory|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 {{w|Lie_Group|Lie Groups}}. A real example would be the Standard Model of particle physics which has symmetry according to <math>\rm{SU(3)\times SU(2) \times U(1)}</math>. Here, <math>\rm{SU}</math> and <math>\rm{U}</math> 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, zebra, donkey or other equine here.

:Randall's version clearly involves some similar groups although without the <math>\times</math> symbol it is hard to work out what might be happening. A term like <math>\rm{SU(U(2))}</math> 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
:[[File:All gauge theory equations.png]]
:In physics, a {{w|Gauge theory|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 {{w|Yang–Mills_theory#Quantization|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 {{w|Einstein_notation|Einstein Sum}}.

:The left-hand side <math>S_g</math> is the symbol for some {{w|Action_(physics)|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 <math>\frac{1}{2\bar{\varepsilon}}</math> looks like a constant of proportionality which often appears in gauge theories. The factor of <math>i = \sqrt{-1}</math> is not unusual as many of these equations use complex numbers. The <math>\eth</math> symbol looks similar to a <math>\partial</math> partial derivative symbol especially as the {{w|Dirac_equation#Covariant_form_and_relativistic_invariance|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 <math>\mu</math> or <math>\nu</math>) 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 <math>p^{abc}_v</math> would likely come from some <math>\rm{SU(3)}</math> transformation (related to the strong nuclear force). Since <math>S_g</math> 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, this is similar to the mistake of messing up units, for instance setting a distance equal to a mass.

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

:The second section looks like the inequalities used to show how the equation varies with the shape of the Universe, based on the value of the curvature parameter <math>\Omega_k</math>. A value of 0 indicates a flat Universe (this is more or less what we observe) while 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
:[[File:All truly deep physics equations.png]]
:<math>\hat H</math> 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 <math>\hat L = \hat K - \hat U </math> with K being the kinetic energy and U the potential. Hamiltonian mechanics uses the equation <math>\hat H = \hat K + \hat U </math>. 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.

The title text states 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.

==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=0∞∑π=0∞(n-π)(i-eπ-∞)
:All number theory equations

:∂/∂t ∇ ⋅ ρ = 8/23 (∯ ρ ds dt ⋅ ρ ∂/∂∇)
:All fluid dynamics equations

:|ψx,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

{{comic discussion}}

[[Category:Science]]
[[Category:Physics]]
[[Category:Math]]
[[Category:Chemistry]]
[[Category:Astronomy]]

2072: Evaluating Tech Things

2018-11-14T19:05:40Z

Athyndmion: just typography

{{comic
| number = 2072
| date = November 14, 2018
| title = Evaluating Tech Things
| image = evaluating_tech_things.png
| titletext = Also known as the Black Mirror-Mythbusters scale.
}}

==Explanation==
{{incomplete|Created by a DRONE FLYING INTO A TORNADO. Please mention here why this explanation isn't complete. Do NOT delete this tag too soon.}}

Many things are possible today thanks to technology, and while often the "wow factor"—how cool it is that we can do that—kicks in right away, there can also be deep potential longer term consequences for humanity. Things like {{w|atomic fission}}, the {{w|Internet}}, {{w|CRISPR}} technology, are amazing things we have learned how to do, but they also have the potential, in some cases already realized, of massively affecting human life (i.e. nuclear annihilation, instant wide-scale communication, elective genetic engineering), both for the better and for the worse.

Here Cueball, upon hearing of a cool idea he hadn't thought of before, mentally measures it on a scale to decide if he can be excited about it, or should be worried about how it might affect humanity. After weighing it out, he decides it's just plain cool and it will not adversely affect humanity at all. In the comic, it appears this mental decision took awhile, judging by the multiple panels showing him thinking, even absentmindedly rubbing his chin, before he gives his response.

The title text refers to this mental weighing also being known as the Black Mirror-Mythbusters scale. "{{w|Black Mirror}}" and "{{w|Mythbusters}}" are both TV shows that explore science and technology. Black Mirror, on the one end of the scale, explores the unintended and often dire consequences of many of our more influential technologies—the horrible stuff that can happen—whereas Mythbusters, on the other end of the scale, explores the fun side of technology and see what kinds of cool things can or cannot be done.

==Transcript==
{{incomplete transcript|Do NOT delete this tag too soon.}}
:[Megan and Cueball face each other]
:Megan: Sooner or later, someone is going to fly a drone into a tornado and post the footage to YouTube.

:[Zoom in on Cueball imagining a semicircular dial with a moving pointer currently fixed vertically in the mid value. The left-most value indicating his opinion to be "This raises big questions about technology and society" and the right-most being "Haha, cool!"]

:[Panel with just Cueball and the imaginary dial above his head, the pointer having shifted a small amout to the right]

:[Same panel with Cueball, his right hand pensively on his chin, the pointer having shifted an equally small amount to the left of the mid value]

:[Panel with Cuball, Megan and the imaginary dial above, the pointer now all the way to the right]
:Cueball: Haha, cool!

{{comic discussion}}

1983: Clutter

2018-04-20T19:25:36Z

Athyndmion: /* Explanation */ little cosmetic cleaning

{{comic
| number = 1983
| date = April 20, 2018
| title = Clutter
| image = clutter.png
| titletext = I found a copy of The Life-Changing Magic of Tidying Up, but the idea of reading it didn't spark joy, so I gave it away.
}}

==Explanation==
{{incomplete|I'd say this is about done, but you could explain the motivations for each cleanup more in depth if you want.}}

As the graph shows, the amount of junk sitting around [[Randall]]'s house is on an ever-increasing trend. Thus, it will continue to pile up and cause problems.

Randall cleans up sometimes, thinking that he is returning to the same baseline amount of stuff each time, but it is not actually effective enough to keep up with the cluttering trend, and hence his worry.

The four places on the graph where the amount of stuff decreases reference common times when people clean up and get rid of junk or excess stuff. This includes:

* The satisfaction many people feel from getting rid of things and making the remaining items look neat.
* Moving, a time when most people will get rid of items they no longer need and use, rather than packing them up and moving them to a new home where they will once again cause clutter.
* Spring cleaning. In many areas with a harsh winter, it is common to clean in the spring when it is warm enough to open windows for dusting, after months of building up smoke or soot from fires to keep the house warm. In other cultures where the year starts in the spring (a time of rebirth) there are traditions of cleaning up before the start of the new year.

Although not mentioned in the quotes, it is also common in the United States to clean up and donate items (for instance to Goodwill) on December 31st, right before the New Year, to gain the charitable donation benefit on their taxes for that year.

The title text refers to the book ''The Life-Changing Magic of Tidying Up: The Japanese Art of Decluttering and Organizing'' by {{w|Marie Kondo}}. The main concept of the book is that one should gather all belongings and only retain items that "spark joy". Ironically, the thought of reading the book didn't spark joy for Randall so he decided to donate it. Thus, one of the few things that he did get rid of was something that if he had kept and put into practice could have helped him actually reduce his clutter.

==Transcript==
{{incomplete transcript|Do NOT delete this tag too soon.}}
:[There is a panel containing a line graph. The x- and y-axes are labeled "time" and "amount of stuff in my house" respectively.]
:[The y-value generally increases as x increases. There are a few labeled exceptions where the y-value decreases slightly. From left to right:]
:"I need to clean up."
:"I've really let junk build up. Feels good to clear it out."
:"I hate moving, but at least it's a chance to finally get rid of all this excess stuff."
:"Ah, spring cleaning!"

:[Caption below the panel:]
:I'm starting to worry about my strategy for dealing with clutter.

{{comic discussion}}

[[Category:Line graphs]]