xkcd.WTF!?

Explanation ➲

This comic imagines which mathematical symbols would be good in a fight if they were made corporeal in two (or three) dimensions.

Generally, objects with longer reach and pointier ends wound up on the right ("more useful") side of the scale, and symbols with less reach and more curves wound towards the left ("less useful") side. A straight line is farthest to the "more dangerous" side; however, the straight line does not appear to be any thicker or thinner, or pointier, than any of the other lines that would make it more "useful" (It should be noted that this chart seems to fall afoul of what Eliezer Yudkowsky (who also wrote HPMoR) calls the intent to kill: that humans tend to define "winning a fight" and "useful" as causing some form of bodily harm on their opponent despite survival and purely defensive strategies being an equally valid goals.).

Below the chart, with the symbols listed in order of usefulness, eight characters wield eight of the symbols. See the table below for the meaning of each symbol.

The comic invokes surreal humor by suggesting that mathematical symbols could be handled as physical objects in the real world. Another component of the humor is the implication that it is useful to prepare to use mathematical symbols in a fight, even though mathematicians, who use mathematical symbols, usually do not conduct their debates violently (though some stories suggest that Hippasus was killed by his fellow Pythagoreans for his proof that irrational numbers exist), and even if they did, they wouldn't use large reproductions of their symbols as weapons.

Note that Black Hat and Danish are using two of the most useful symbols, and appear to have teamed up against their opponents, resulting in an unfair advantage they would both absolutely exploit.

The title text refers to a treble clef, which is not a mathematical symbol but rather a musical symbol. The note of concern in the text suggests musical symbols may be viewed in such fights as exotic or especially dangerous. See also the last entry in the table below.

Here is a list of symbols and their explanation, from left to right (by rightmost edge) : ℝ θ ∅ ∞ > ∪ ∝ ∈ ∀ π ∂ + ≠ Ψ # ~ △ ζ ⇒ ⋅ ℵ Γ ρ √ → ∮ ∫ ⊥ ⇀ ––

Symbol	Meaning	Notes on using in a fight
ℝ	The set of real numbers	Megan seems to be struggling with a giant version of this symbol because of its weight. A very strong person might find it slightly more useful as a bludgeon.
θ	Theta	This symbol is not very sharp, and White Hat is unable to use it in combat. It would not work as a good shield due to the hole in the symbol, but it possibly could be thrown. (The later version of Xena's chakram had a bar in the middle.)
∅	Empty set	Similarly shaped to theta, but identified as being slightly more useful, presumably because the parts of the bar that extend beyond the circle can be used as thrusting weapons. Alternatively, the round nature of the symbol makes it easier to handle for defense, by holding on to the line and using the edge of the circular part to block.
∞	Infinity	Could be gripped between the rings and used as a blunt instrument.
>	Greater than	Could be used to thrust with the point, but lacks a grip by which to do so. If held by one leg, the angle and balance is wrong to use it as a pick. Could also be used as a boomerang.
∪	Union	Could be used as a boomerang, which is notoriously less practical in real life than in fiction.
∝	Direct proportionality (not to be confused with lowercase alpha)	Could be brandished by the ring and used to thrust with the prongs, but the two prongs extending at 90-degree angles make thrusting motions awkward.
∈	Member of (set theory)	Cueball is holding this in the manner of a Klingon "Bat'leth" from the Star Trek franchise. The Bat'leth is infamous among swordfighters for being rather impractical, and that the Klingon warrior race would have been better suited using swords like humans. It seems that Randall agrees, as the ∈ is quite far on the left of the chart.
∀	"For all"	Could be held by the crossbar clenched in a fist, and then used as a push dagger, with the legs acting as a guard. Or could be held by the legs. Compared to the similar Δ (delta) the point is sharper, the grip is closer to the point so it would be easier to control (keep the weapon from twisting). It is unclear why it is listed as much less effective than the delta. Perhaps the guard legs are too long?
π	Pi	Could be used as a hammer, but the two handles would make doing so impractical.
∂	Partial derivative or boundary operator	It's not clear why Randall ranks this symbol as so much less effective than the similarly-shaped 𝜌, but the curl in the "tail" of the ∂ would give it a shorter lever arm when swung and would disrupt the balance.
+	Plus sign	Maybe thrown like a shuriken? It could also be used like Kylo Ren's lightsaber.
≠	Not equal to	Could be gripped by the crossbar, using the lengths to puncture. Notably, the equal sign (=) is absent from the chart, likely because the lengths are separate and would not be used as a single entity.
Ψ	Psi	It could be used as a slightly-less-functional trident or pitchfork, with a shorter handle. More particularly, it resembles a sai (which, funnily enough, is close to how "Psi" is pronounced).
#	Cardinality, connected sum (knot theory), or primorial.	Blondie uses this symbol, gripping it by two of the prongs on one side. The other three prongs could thus be swung at opponents, but being sharp on all sides would pose a threat to the user as well as the opponent.
~	Tilde, meaning "approximately", "equivalent", or several transforms of a function. Also used as negation in mathematical logic and in many programming languages.	A potentially dangerous throwing weapon. Could also be similar to a wavy bladed dagger (Kris) or a sword (Flame-bladed sword).
Δ	Delta	Could be held with one bar clenched in a fist, and then used as a push dagger. It is unclear why listed as much more effective than ∀. Also, delta could be thrown like a star.
ζ	Zeta	Could be sharpened into a scythe or curved saber, depending on the curvature and length of the 'tail' at the bottom.
⇒	Material consequence or logical consequence, meaning "implies"	The point is nice, but having two poles (or one board-like thick handle, if the shape is drawn "closed" at the back) would be difficult to wield. A collection of them might make a nice defensive pike line.
⋅	Decimal point or multiplication sign	Functionally a ball, and could therefore be thrown as a projectile weapon, or scattered on the ground as a trip hazard.
ℵ	Aleph number	The irregular shape of the symbol leads to edges and points on all sides; thus it could be brandished or thrown in the manner of a shuriken.
Γ	Uppercase Greek letter gamma	If this letter is formed with serifs, it could be used as an axe or hook, and if it is made sans-serif, it would make a powerful pick or war hammer.
ρ	Italic rho	Ponytail is brandishing this symbol against Danish, apparently using it like a club or hammer and striking with the curve.
√	Radical symbol	Randall has drawn this sign with a long overbar, which makes it useful like a pole arm. Black Hat has chosen this symbol in his fight against Hairy.
→	Implies; X→Y means that if X is true, then Y is also true	Danish is brandishing this symbol against Ponytail, apparently using it like a spear.
∮	Contour integral	This symbol is drawn with a very tight center, giving it an almost identical figure to the regular integral symbol. Presumably, the added bulk for marginal additional use earns it a rating of 'less effective'.
∫	Integral	Could be used in the manner of a shepherd's crook, or as a billhook. Could be used as a spear thrower, combined with ⋅ or ⇀.
⊥	Up tack or falsum, indicating a false proposition in logic, perpendicular lines in geometry, or the bottom element in a partial order	Used as a war hammer (held by the long stem) so the opponent would be struck with either leg of the top of the T. Held by the short legs (like a wishbone), could be used as a two handed sword. (If tipped with a knapped stone head, it could become a falsum point.)
⇀	Vector notation	Could be used as a single-barbed spear or, if held by the other end, similarly to a sabre. The multi-functionality is probably why it is listed as more useful than the 'implies' arrow.
––	Fraction bar (division) or overline (complex conjugate or mean) or minus sign.	Hairy is brandishing this symbol against Black Hat. The single shaft would allow it to be used in the manner of a quarterstaff, bō, or other stick-fighting weapon. Of the symbols shown in the comic, Randall considers this one the most useful in a fight, presumably because of its greater reach than the vector arrow and its simplicity compared to the radix root symbol. Alternatively, could be handled like a pike, with a reach greater than that of an implication or vector arrow.
𝄞	Treble clef (title text)	Mentioned in the title text; this is not a mathematical symbol, but a musical symbol. The treble clef is a much more complicated symbol than those used in mathematics, hence the musician's "confidence" in his weapon. The curve at the bottom could be used as a hook, the upper curl could be used as a blunt weapon, and the tight curl of the center would serve as a better defensive shield than theta.