Unit Circle
They're continuing to search for a square with the same area as the circle, as efforts to construct one have run into difficulties.
They're continuing to search for a square with the same area as the circle, as efforts to construct one have run into difficulties.
A unit circle is a mathematical concept which is a circle whose radius is one (with no units). Or put another way, the unit circle's radius is itself a unit of measure, hence the name. Thus when doing math problems with a unit circle, all other distances are therefore in terms of the circle's radius: a line with length 3 is three times the radius, a line of length 1/2 is half the radius, and so on. This is very useful in many geometry problems.
This comic shows an expedition of some experts (White Hat, Ponytail, Miss Lenhart (the mathematician), Cueball and Megan) having located a "real unit circle": a physical object which somehow is this mathematical idea. Cueball is holding a set of vernier calipers, precise instruments used to provide an exact measurement of the unit circle. By measuring the "real unit circle", mathematicians could then provide its measurement in whatever ordinary unit they choose, such as centimeters or inches, to textbooks which describe the unit circle. The notion of defining a unit in terms of an actual physical object is actually quite reasonable, as the meter was officially defined as a length of a specific platinum–iridium bar from 1889 to 1960 and the kilogram was defined by the mass of a specific physical object until 2019. Doing so with the unit circle would be entirely pointless, however, as the entire purpose of the unit circle is to define mathematical relationships, which can be generalized to any unit, rather than being restricted to a given length.
The title text refers to the old geometry problem of squaring the circle, where one starts with a circle with a known area - for a unit circle, π - and tries to create a square with the same area, traditionally using nothing more than an idealized compass and straightedge. Such a square would have edges measuring √π units in length, and once it was proven that π is a transcendental number, it was definitively known that squaring a circle is impossible. This causes problems for the comic's team of mathematicians, who wished to create such a square to go along with its unit circle but must instead rely upon finding one, presumably using the same approach they used to find this circle.
Note that this is not the definition of a unit square in mathematics: a real unit square, should one also exist in the comic's context, would have edges the same length as the unit circle's radius, and not have the same area as the unit circle or the conceptual equal area square that this comic mentions. Having found a physical unit square artefact would have been as useful as this unit circle, for many purposes (it would have defined the length of the unit identically; or better, as it seems that the circle's diameter will be measured, which then needs to be halved to discover its radius, although sufficiently accurate measurement of its perimeter also reveals something about the nature of pi and/or tau), whereas the square counterpart of the unit circle would only be useful for 'unit' purposes already specifically involving the root of pi (as length) or pi (as related to area). Though conspicuously equipped to measure the archetypal unit circle's diameter, or a square's edge-length, the expedition is not so clearly prepared to check the circumference (e.g. with a surveyor's steel tape) or directly quantify its (or any square's) area.