2586: Greek Letters
Greek Letters |
Title text: If you ever see someone using a capital xi in an equation, just observe them quietly to learn as much as you can before they return to their home planet. |
Explanation
Mathematics uses lots of Greek letters, typically using the same letter consistently to represent a particular constant or type of variable. This comic gives a (non-)explanation of what they typically mean, see below.
In the title text the joke about capital Xi from the main comic is continued. In the main comic those using Ξ (capital xi) greets us as Earth mathematicians, indicating they are not from Earth, but have come here to learn what we know of math. In the title text the idea that any one using Ξ must be aliens is made clear. So if you ever meet someone using this letter while doing math, then learn as much as you can by quietly observing them, before they return to their home planet. Either learn from their possible advanced math (that allowed them to construct a way to get from one star system to another), or learn about them as the aliens species they represent.
Previously Randall made a similar comic, 2520: Symbols, about math symbols.
Greek letters
- π (lowercase pi): This math is either very simple or impossible. — Typically used to refer to the constant ratio between a circle’s circumference and its diameter (approximately 3.14). In a common school curriculum, this constant first shows up in introductory geometry classes, which would be considered "simple" by advanced mathematicians. But often, pi can show up seemingly randomly in advanced equations that have nothing to do with a circle at first glance, such as in infinite series. And because pi is transcendental, it can sometimes be difficult to work with pi in those situations. The comic may also be a reference to the impossibility of squaring the circle.
- An alternate explanation is that the comic refers to how the symbol can sometimes be used as a variable where the 'p' sound might make sense, such as in the prime-counting function where it stands for "prime" or the Buckingham π theorem where it stands for "parameter." These uses can be confusing to students who have only ever seen a lowercase pi used for the circle constant. This has pushed college courses to use it less and less frequently for anything other than the circle constant so that now you are only likely to see π as something else in higher math. (More confusing still is the variant lowercase pi, so-called omega pi ϖ sometimes used for angular frequency instead of the more common (and very similar-looking) lowercase omega ω. In astronomy, ϖ is traditionally used to denote the longitude of perihelion.)
- In a field of math known as algebraic topology, it is standard to use π to denote the homotopy groups. Despite decades of study, the homotopy groups of spheres in high dimensions are unknown.
- Δ (capital delta): Something has changed. — Typically prefixes a variable to refer to a macroscopic change in or finite difference of that variable. For instance, Δv may be the finite change in velocity v over some finite time span, while Δ[f](x) represents the forward difference of f at x, defined as Δ[f](x) = f(x+1) - f(x).
- δ (lowercase delta): Something has changed and it's a mathematician's fault. — Used in calculus. In many areas of math, systems are studied by introducing small changes (perturbations) in input variables and observing how the system changes. The perturbations introduced are often written down as x → x + δx for some variable x being perturbed, where δx is the change being introduced. These are often applied in physics (perturbation theory, the principle of least action, Noether's theorem,…). Since this change was purposefully introduced by the mathematician instead of occurring naturally, it is, therefore, their fault.
- θ (lowercase theta): Circles! — Used in trigonometry. Typically used to refer to an angle, and is notably used in the polar coordinate system. The text refers to its close relationship with circles, on which the polar coordinate system is based. In European handwriting, the variant form ϑ is commonly used with the same meaning.
- ϕ (lowercase phi): O R B S — Typically used to refer to another angle other than one referred to by theta. It's used in spherical coordinates, and the text refers to how spheres, or orbs, are important in spherical coordinates. Lowercase phi has two forms in modern typography which are confused by this website's default font. In the comic, it has a complete circle with a vertical line passing through it, which is what Knuth called "phi." The alternate form, φ, is what Knuth called "variant phi" and can be written in a single stroke. Most fonts reverse the way these symbols are rendered. There is no difference in meaning between the symbols. Additionally, "O R B S" is written with spaces between each letter, possibly a reference to the linguistic phenomena of surreal memes and their tendency to add spaces between letters of "surreal-sounding" words like "orbs".
- ϵ (lowercase epsilon): Not important, don't worry about it. — Typically used to refer to a very small quantity. ϵ may be an error term in a statistical model (which is usually small if the model is useful), a remainder term in an approximation (same), or an arbitrarily small (positive) quantity in analysis. Although a total cumulative change of "ϵ" is negligible, in analysis, ϵ is most often applied in a context of an infinitesimal change occurring with infinite frequency. The study of ratios of quantities that approach zero gives rise to infinitesimal calculus. This Greek lowercase letter has two common modern variants, ϵ and ε. ϵ is called the "lunate epsilon" and may be more common in the U.S. A stylized version (∈) is used as the mathematical symbol for "is an element of." ε is what Knuth called the "variant epsilon" and is never used for the "element of" symbol but otherwise has identical meaning. Because epsilon represents an arbitrarily small (positive) quantity, there's no reason for anyone to worry about it from a practical standpoint.
- υ,ν (lowercase upsilon and nu): Is that a v or a u? Or...oh no, it's one of those. — Common in college-level physics and engineering equations. ν commonly represents wavenumber in physics as well as a wide variety of other variables, often with names starting in N (e.g. neutrino) or V (e.g. viscosity). Lowercase upsilon is rarely used, probably to avoid confusion. The symbols look remarkably similar to Latin u and v, to the point that they are nearly indistinguishable in some fonts; Randall has complained about this before in 2351: Standard Model Changes. The use of both terms together is most commonly seen when performing integration by parts, a famously counterintuitive method of integration (one of those).
- μ (lowercase mu): This math is cool but it's not about anything that you will ever see or touch, so whatever. — Used broadly in the abstract mathematical fields of category theory and measure theory. Also used in statistics for the mean (average). Physicists use Latin letters for the indices of the 3-vectors of classical physics and Greek indices, including μ, for the 4-vectors of special relativity. This leads to μ being ubiquitous in a field that is very far from everyday experience (where speeds approach the speed of light). It is also employed in statistics for the population mean, which is a quantity that the statistician never actually knows and frequently wants to estimate. Equations requiring a μ are thus impossible to apply directly. However, μ is used in physics for the coefficient of friction in the Coulomb model, typically used to approximate resistive forces between dry solids of different materials sliding past each other. A very common use of μ in science and engineering is as the symbol of the SI prefix micro- for a millionth. Unicode has officially added a point for μ as the "micro sign," distinct from its usual codepoint as the lowercase Greek letter mu.
- Σ (capital sigma): Thank you for purchasing Addition Pro®! — Typically used as a symbol for the sum of a series of numbers. The comic is making fun of summation, pointing out that it's essentially a complicated, "pro" version of simple addition. The capital sigma is often used as the icon for the all-important "sum" button in spreadsheet software. However, the sigma operator is often necessary for explicitly defining infinite sums, avoiding ambiguous notation like an ellipsis (...).
- Π (capital pi): ...and the Multiplication® expansion pack! — Typically used as a symbol for the product of a sequence of numbers. The joke is the same as for summation. Here, it is advertised as an "expansion pack," a term used for a piece of software that cannot stand alone but adds features to some existing software. Any paid spreadsheet or database program should already have the ability to perform multiplication. The ® symbol indicates that Multiplication is a registered trademark somewhere, which is unlikely, as the term is not unique. However, common words are registered as parts of longer trademarks rather often.
- ζ (lowercase zeta): This math will only lead to more math. — Frequently used for the Riemann zeta function in analytic number theory, a function of complex numbers which is challenging even to define and which is the focus of a famously unsolved problem in highly advanced mathematics. Zeta is used much less often in other contexts, such as the ζ-potential in colloidal chemistry, and even there it is likely to just lead to more math.
- β (lowercase beta): There are just too many coefficients. — This could be a reference to the typical usage of beta to represent coefficients of independent variables in the ordinary least squares regression model. Regression can potentially have a large number of independent variables, hence potentially many different betas (differentiated by a subscript, or compacted into matrix notation) would be used, while there is only ever a single zeroth-order coefficient α in these models. Alternatively, the comic might suggest whatever source this equation is from has run out of Latin letters to use as symbols, and is now going through the Greek letters.
- α (lowercase alpha): Oh boy, now this is math about something real. This is math that could kill someone. — As the first Greek letter, α is used for a tremendous variety of purposes in math. For example, it is used to represent the probability of a Type-I error (false positive) occurring in a hypothesis test. It could also possibly refer to the fine-structure constant which shows up in high energy physics, atomic physics, quantum electrodynamics, and at least one other xkcd comic. Alpha could also refer to angular acceleration, and a rapidly-rotating system is capable of killing people in a number of interesting ways. In aviation, α refers to angle of attack, which could cause a deadly aerodynamic stall if it gets too large. Another dangerous meaning for α comes from ionizing α-radiation: While it can be easily blocked by even a sheet of paper, it has been used for assassinations through ingestion.
- Ω (capital omega): Oooh, some mathematician thinks their function is cool and important. — The last letter of the Greek alphabet and thus often seen as momentous (the end, the final word, death). The capital letter has been used as the symbol for a variety of mathematical functions, the first uncountable ordinal, and Absolute Infinity. It is commonly used in physics and electrical engineering as the symbol for ohms, a unit for electrical resistance. Capital omega has produced a fascination in common culture, perhaps due to God reportedly describing himself as "the alpha and the omega" in the Book of Revelation or due to its highly distinctive shape. It is often used to represent something of grave or transcendent significance. So using it to name your function (instead of a conventional symbol like f or g) may mean you think the function is particularly important.
- ω (lowercase omega): A lot of work went into these equations and you are going to die here among them. — Used for the least transfinite number of ordinals, the order type of the natural numbers under '<'. The line about dying here among the transfinite equations may be in reference to the "eternity" of the infinite set it represents. It is also used in physics and electrical engineering for angular frequency, equal to 2π times the frequency, and thus it is ubiquitous in equations dealing with all sorts of wave phenomena. ω is also used for the angular velocity of a rotating system, defined by v = r×ω.
- σ (lowercase sigma): Some poor soul is trying to apply this math to real life and it's not working. — In statistics, σ commonly refers to the population standard deviation of a distribution. Many simplified statistical equations substitute the population standard deviation σ for the sample standard deviation s for simplicity, even when this is not justifiable. A common example is using the normal distribution to model the mean of several identically normally distributed variables instead of the T distribution. The variant ς is used at the end of Greek words (called the "final sigma") but is rarely used in math or science.
- ξ (lowercase xi): Either this is terrifying mathematics or there was a hair on the scanned page. — Randall comments that this looks like a strand of curly hair. Xi is used in the Riemann Xi function and sometimes as a variable or function symbol in higher math. It is famously difficult to write in a way that is consistent and clearly distinct from other symbols.
- γ (lowercase gamma): Zoom pew pew pew [space noises] zoooom! — Lowercase gamma is used for the Lorentz factor, an important variable in special relativity calculations. Its use implies that you are dealing with speeds approaching the speed of light and therefore with spaceships or other moving objects not confined to Earth. γ-rays are also the highest energy photons, so a space opera might have ships flying near the speed of light firing gamma-ray weapons that go PEW PEW. γ is also used as the symbol for the Euler-Mascheroni constant and occasionally as a variable or function name.
- ρ (lowercase rho): Unfortunately, the test vehicle suffered an unexpected wing separation event. — Used in statistics to measure the association between variables. Lowercase rho often represents volumetric mass density, such as the density of air that a wing might be traveling through. The density of a fluid is directly proportional to the Reynold's number, which dictates the sort of physics used to model motion through the fluid. Flying a plane in conditions with a Reynold's number well outside of what it was designed for could have catastrophic consequences. A variant symbol ϱ with the same meaning is common in European handwriting.
- Ξ (capital xi): Greetings! We hope to learn a great deal by exchanging knowledge with your Earth mathematicians. — Probably the least used Greek letter in math and physics despite being easy to write and recognize. According to the comic, anyone using this letter is likely a being from another planet. It does see very occasional use, such as in the Riemann xi function or as the symbol for a class of heavy baryons in particle physics. It resembles but is not to be confused with a "hamburger button" or a triple equals sign ≡. Coincidentally, it also resembles the Chinese or Japanese character for the number 3 (三). Randall thinks it most closely resembles alien writing.
- ψ (lowercase psi): You have entered the domain of King Triton, ruler of the waves. — Both capital and lowercase psi are shaped like tridents. In classical mythology, Triton is one of the gods of the sea, alongside his father Poseidon, and tridents are commonly associated with sea gods. In quantum mechanics, either psi is used to represent the wave function of a particle, leading to a pun. (Psi is also used in mathematics to represent the sum of the inverse of the Fibonacci numbers, the division polynomials, the supergolden ratio, and other purposes.)
Transcript
- [A list with 21 explanations of different Greek letters. To the left, the letter (in one case two letters) are shown, and then the explanation is written to the right in one or two lines (and in one case on three lines). Above these explanations, there is a header in a slightly larger font:]
- What Greek letters mean in equations
- π This math is either very simple or impossible.
- Δ Something has changed.
- δ Something has changed and it's a mathematician's fault.
- θ Circles!
- ϕ O R B S
- ϵ Not important, don't worry about it.
- υ,ν Is that a v or a u? Or...oh no, it's one of those.
- μ This math is cool but it's not about anything that you will ever see or touch, so whatever.
- Σ Thank you for purchasing Addition Pro®!
- Π ...and the Multiplication® expansion pack!
- ζ This math will only lead to more math.
- β There are just too many coefficients.
- α Oh boy, now this is math about something real. This is math that could kill someone.
- Ω Oooh, some mathematician thinks their function is cool and important.
- ω A lot of work went into these equations and you are going to die here among them.
- σ Some poor soul is trying to apply this math to real life and it's not working.
- ξ Either this is terrifying mathematics or there was a hair on the scanned page.
- γ Zoom pew pew pew [space noises] zoooom!
- ρ Unfortunately, the test vehicle suffered an unexpected wing separation event.
- Ξ Greetings! We hope to learn a great deal by exchanging knowledge with your Earth mathematicians.
- ψ You have entered the domain of King Triton, ruler of the waves.
Discussion
Pi also shows up in lots of extremely advanced equations as pi, not as something else, adding edit. 123.456.7890
zeta_0 is also used for the first transfinite ordinal that is unreachable through ((edit: by some random IP: "...finite application of...")) addition, multiplication, exponentiation, and epsilons subscripting. EDIT: phi is used for the Veblen hierachy. GcGYSF(asterisk)P(vertical line)e (talk) 05:11, 26 February 2022 (UTC)
- I wouldn't normally internally spellcheck/factcheck someone's signed Talk comment, as I think it's rude to do so (especially 'invisibly'), but an IP added some words to yours (without clear indication) probably with good reason but also with slightly bad typing. So I've highlighted their (corrected) addition, which at first sight seems a valid clarification but I haven't double-checked. And now this is me taking fully (IP-)signed ownership of what I changed. Would have been simpler for the prior editor just to have made a signed-reply, like this but far shorter, but they didn't! Ah well... 172.70.90.121 14:03, 1 March 2022 (UTC)
Don't you have an English saying: simple/easy as π? Nukio (talk) 05:51, 26 February 2022 (UTC)
- the saying is easy as pie as in the dessert. sometimes we write it easy as π as a nerdy joke. 162.158.107.230 08:08, 26 February 2022 (UTC)
- sqrt(-1) 2³ Σ π and it was delicious Elektrizikekswerk (talk) 08:30, 28 February 2022 (UTC)
Related: https://xkcd.com/2520/ 162.158.103.224 08:59, 26 February 2022 (UTC)
I've found a use for capital Xi: https://en.wikipedia.org/wiki/Harish-Chandra%27s_%CE%9E_function that seems to be from the field of Harmonic Analysis. Douira (talk) 14:50, 26 February 2022 (UTC)
Uppercase xi is used by Alan Turing in his essay *Systems of Logic Based on Ordinals* to represent ordinals. I consider this to be a possible example of an alien mathematician among us. 17:11 26 October 2022
The part that says the farad is "unusually large" is incredibly biased IMO. On the scale of planets its "unusually small", In fact, on the scale of EV's its even pretty normal. The writer is only considering small electronic circuits. Also the Henry is very well scaled to the Farad so how "unusual" is it really? 108.162.241.33 17:13, 26 February 2022 (UTC)
- Apologies for the incredible bias. You're right in saying that I'm only considering small electronic circuits; I haven't worked on power distribution systems or applications with large capacitor banks, so my only hands-on experience of components measured in whole farads would be supercapacitors. In consumer electronics, where capacitors are typically labelled in pico, nano or microfarads, the whole farad is rarely encountered. I do still think that capacitors are a good counter-example of items using Mu that you can see and touch, in so many modern electronic devices. But as my previous use of language was so divisive, I'll let someone else attempt to reintegrate the point, if they feel it's useful. Kazzie (talk) 16:11, 27 February 2022 (UTC)
Isn't the capital psi used for the wavefunction? GcGYSF(asterisk)P(vertical line)e (talk) 19:35, 26 February 2022 (UTC)
- Yes, but rarely. The lowercase ψ is much more common (AFAIK it dates back to Schrödinger himself.
How sad that there is no η! Missed chance to blame steam machine engineers for not trying harder to invent the perpetuum mobile. --172.70.242.177 20:01, 26 February 2022 (UTC)
The lowercase epsilon is used much more often for something else - usually to denote that the "variable" on the lefthanded side is a member of the "set" of the righthanded side of the lowercase epsilon. Of course, this is totally unimportant ;-).
- You are referring to the "element of" sign, which is distinct from lowercase epsilon (although based on it).
- Yes it is distinct, but then the used typeface in the comic looks more like the epsilon for "element of" then for the usual epsilon in analysis (ie. for definitions of continues functions).
I highly doubt that the use of Ξ has anything to do with it "looking like a UFO." Rather, I'd suggest it's because it's essentially never used, at least among the English speaking mathematicians in the US, and probably Europe. Douira went out of their way to find an example, and found something increadibly obscure, which supports the point. Why Ξ is rarely used is another question. Maybe because it could easily be confused for an E or Sigma, with lazy handwritting? Maybe because it's a Greek letter without a direct Latin counterpart, so doesn't correspond with the first letter of any common words? 162.158.63.49 22:50, 26 February 2022 (UTC)som
In my experience lower case eta, zeta, (and xi) most commonly show up as dummy variable in an integral. Any two may be used for a double integral and all three for a triple. Double and triple integrals are often quite terrifying, particularly when somebody cannot write all three symbols consistently and distinctly, so many integrals become "integral squiggle squiggle dee squiggle dee squiggle".172.70.174.219 10:10, 27 February 2022 (UTC)
π is also commonly used as the prime-counting function in number theory. Most problems regarding primes are usually considered hard, like the twin prime conjecture.
Lower-case sigma is also used in sigma-algebras, which is part of the theoretical background underlying statistics, among other things. I second that the lower-case epsilon drawn by Randall is the lunate variant that looks indistinguishable from the "is an element of" symbol and should probably get mentioned. On an unrelated note, there's a story of someone using capital xi at a math conference specifically to annoy some other mathematician who *really* didn't like them. 172.70.211.26 20:30, 27 February 2022 (UTC)
- Yeah I came to comment this, lower-case sigmas come up in sigma-algebras and are absolutely terrifying (bias) in that context. The joke about ‘someone trying very hard to apply this’ works with sigma-algebras in the context of measure theory -- someone trying to actually apply measure theory to a real problem. 162.158.129.117 10:48, 28 February 2022 (UTC)
Alpha is also used in aeronautics for the angle of attack of the airflow over a wing. Exceeding a critical angle of attack leads to an aerodynamic stall, which has been cause of many fatal accidents.
Uppercase phi looks like an obvious reference to this comic and author, as he normally uses the term orb to refer to spheres and balls (as part of the intrincate language of the characters), besides he normally uses that typographic resource of writing a word with its letters separated by spaces, i.e.: e x p e r i e n c e, in the example link. I'm missing the math context on why refering to orbs for uppercase phy, but it could be just because due to the form of the character. 172.70.86.54 10:28, 28 February 2022 (UTC)
The lowercase gamma symbol and description may also be a reference to the downward-looping flight path of enemy ships in the video-game Galaga, which zoom down the screen at the player's starfighter while shooting at them, then retreat and zoom back up. 172.70.131.122
Why was my remark on the impossibility of squaring the circle removed? (https://www.explainxkcd.com/wiki/index.php?title=2586:_Greek_Letters&diff=227689&oldid=227680) ThomasGauss (talk) 20:06, 28 February 2022 (UTC)
My sibling's in an advanced calculus course, when she saw the joke about lowercase omega she laughed for a different reason, remarking how accurate it was and how impossible it is to use in her class? (I can't remember exactly.) I don't understand what she meant, I'm in a lower class. Could somebody add an alternate explanation possibly? 123.456.7890
- Differential forms? 172.70.230.111 03:24, 2 March 2022 (UTC)
- Yes, that's it. I still have no idea how they work, though. 123.456.7890
O R B S are chanted with such gravitas in the Games Done Quick speedrunning fundraiser events. I would say this is a niche pull, but it seems up Randall's alley.
Why is Randall so good at making me crack up? Am I really that much of a nerd? (Okay granted I needed an explanation for some of this, hence my presence here, but still, Addition® and Multiplication® Pro® got me so...)--Twisted Code (talk) 17:10, 4 March 2022 (UTC)
It just occurred to me Poseidon could be written as Poψdon. Then his trident is in his name. Bwisey (talk) 13:11, 5 March 2022 (UTC)
One thing missed so far, the lowercase epsilon is also used in automata theory. There usually words (strings of symbols from an alphabet), which do contain lowercase epsilons, are equivalent to the same word, after removing any occurence of lowercase epsilon from the string. Ie. the lowercase epsilon there denotes the "empty symbol" (or "empty letter"). For further reading ie. look up non determitistic finite automata with (and without) epsilon transitions. Also in other branches of theoretical computer science the lowercase epsilon is usually used as "empty symbol" or "empty letter".
My favourite use of Greek letters in maths is that (in my experience) ψ is used as a backup for φ (so when you have already used phi, you use psi as the second one), and χ is sometimes used as a reserve for that. This means you can have an equation involving all three. This is perfectly clear on paper, but any discussion surrounding it is a nightmare, as they all sound exactly the same... Also this is generally φ rather than ϕ. The former is generally used for functions (specifically homomorphisms in groups) and the latter for spherical polars. Although they both have other uses, and that is a pretty vague rule... 172.70.86.44 18:21, 16 March 2022 (UTC)
the natural numbers under '<'
What does this mean?
Ordinal analysis
Why keep all the unnecessary ordinal-related stuff (beyond omega)? Ordinal analysis is an _extremely_ niche field - most professional mathematicians never even heard of it - so it is quite safe to assume Randall did not have it in mind. I don't think it helps explaining the comic any more than saying rho is the usual symbol for the Gaussian mass of a Euclidean lattice (i.e. not at all). -- Laurus (talk) (please sign your comments with ~~~~)
- Taking a close look at the comic, I think you're right. It pains me a bit, but I think you're right, the ordinals ought to go. Vandalbane (talk) 01:23, 27 May 2022 (UTC)
- I can't believe that the article is being edited again, over this, for the Nth time! 141.101.98.11 08:00, 27 May 2022 (UTC)
- Was there some kind of debate over it? Vandalbane (talk) 15:32, 27 May 2022 (UTC)
- Not that I'm aware of, that was just a pune on the term 'ordinal'. Here's the missing smiley that makes it more obvious. -> ;) 141.101.98.221 18:22, 27 May 2022 (UTC) ( :p <-- And here's the other one that this comment might need.)
- Was there some kind of debate over it? Vandalbane (talk) 15:32, 27 May 2022 (UTC)
- I can't believe that the article is being edited again, over this, for the Nth time! 141.101.98.11 08:00, 27 May 2022 (UTC)