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Last seen on: –The Sun – Two Speed Crossword – Oct 30 2020
Wall Street Journal Crossword – July 15 2020 – Passing the Bar
Wall Street Journal Crossword – September 20 2019 – For Good Measure

Random information on the term “Volume”:

The Banach–Tarski paradox is a theorem in set-theoretic geometry, which states the following: Given a solid ball in 3‑dimensional space, there exists a decomposition of the ball into a finite number of disjoint subsets, which can then be put back together in a different way to yield two identical copies of the original ball. Indeed, the reassembly process involves only moving the pieces around and rotating them without changing their shape. However, the pieces themselves are not “solids” in the usual sense, but infinite scatterings of points. The reconstruction can work with as few as five pieces.

A stronger form of the theorem implies that given any two “reasonable” solid objects (such as a small ball and a huge ball), the cut pieces of either one can be reassembled into the other. This is often stated informally as “a pea can be chopped up and reassembled into the Sun” and called the “pea and the Sun paradox”.

The reason the Banach–Tarski theorem is called a paradox is that it contradicts basic geometric intuition. “Doubling the ball” by dividing it into parts and moving them around by rotations and translations, without any stretching, bending, or adding new points, seems to be impossible, since all these operations ought, intuitively speaking, to preserve the volume. The intuition that such operations preserve volumes is not mathematically absurd and it is even included in the formal definition of volumes. However, this is not applicable here because in this case it is impossible to define the volumes of the considered subsets. Reassembling them reproduces a volume, which happens to be different from the volume at the start.

Volume on Wikipedia

Random information on the term “undefined”:

In mathematics, undefined has several different meanings, depending on the context. Such contexts include the following, among others:

In ancient times, geometers attempted to define every term. For example, Euclid defined a point as “that which has no part”. In modern times, mathematicians recognize that attempting to define every word inevitably leads to circular definitions, and therefore leave some terms, “point” for example, undefined (see primitive notion).

This more abstract approach allows for fruitful generalizations. In topology, a topological space may be defined as a set of points endowed with certain properties, but in the general setting the nature of these “points” is left entirely undefined. Likewise, in category theory a category consists of “objects” and “arrows”; again, these are primitive, undefined terms. This allows such abstract mathematical theories to be applied to very diverse concrete situations.

The expression 0/0 is undefined in arithmetic, as explained in division by zero (the expression is used in calculus to represent an indeterminate form).

undefined on Wikipedia