2586: 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.
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.
- 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.
- 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.)
- [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.
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