Contributed by
Vijay Fafat
JBS Haldane once wrote a wonderful article, “On Being the Right Size”, which can be found in James Newman’s “The World of Mathematics, Vol 2”. It encapsulates beautifully the idea that biologically, each function is associated with a “right scale”, a domain of size which must be respected; that one cannot arbitrarily change one biological parameter and expect all else to be equal. As he wrote:
“In a large textbook of zoology before me I find no indication that the eagle is larger than the sparrow, or the hippopotamus bigger than the hare, though some grudging admissions are made in the case of the mouse and the whale. But yet it is easy to show that a hare could not be as large as a hippopotamus, or a whale as small as a herring. For every type of animal there is a most convenient size, and a large change in size inevitably carries with it a change of form.” 
Fletcher Pratt used this idea very cleverly in reverse in the story, “The Square Cube Law”. As the Editor describes it, “The square cube law sometimes gives engineers gray hair, but it has given our Fletcher Pratt the springboard nudge for a very different kind of story.” The novella is a moneyheist set in the future where technology has evolved to have regular rocket transport and resized cloning (though oddly, the Federal Reserve still transports physical money geographically...). How the seemingly impossible heist is pulled off and what leads to the denoument is worth reading. I won’t describe the details, but reproduce a spoiler:
(quoted from The Square Cube Law)
“IT WAS like this,” said Jones, lifting his glass and squinting through it at the azure barlights of the Caveme Bleu. “The squarecube law' was responsible for the whole business.” “Why don’t they repeal it, then?” said O’Neill. “That would be difficult,” said Jones. “It’s international. [...]was just smart enough to figure out how to use it to do something that couldn’t be done, but not smart enough to escape the consequences of what he'did.” [...] “The squarecube law goes roughly something like this: as you increase the size, or mass, of an animal by the cube of its previous figure, its strength only goes up to the square. A man thirty feet high would be almost too weak to walk , around ; that is, if he had the same proportions. On the other hand very small animals, like a mouse or a marmoset, are prodigiously strong for their size, and when you get down to an ant, it can walk around with a load of ten or twenty times its own weight. Try lifting fifteen hundred pounds some time.”

The story is a sequel to Fetcher Pratt’s “Double Jeopardy” and was published in the June 1952 issue of Thrilling Wonder Stories.
Finally, we should point out that one famous Google interview question goes something like this:
“Suppose you are shrunk down to the size of a bug but your density remains constant. You are dropped into a blender of height 25 cm. The blender will be turned on in one minute. How do you react?” The innovative answer, following Haldane’s article, is supposed to be, “No problem. I can jump out of the blender because my muscle strength to body weight ratio is high enough for me to jump many times my reduced height”.

The socalled "square cube law" essentially is the fact that when an object's dimensions are scaled up, the surface area is proportional to the square of the length while volume is proportional to the cube of the length. This is a mathematical fact in that it is true of any geometric object separate from any of the laws of physics or biology. For example, note that if one changes units from meters to centimeters then any length would get multiplied by 100 but any associated surface area (now measured in cm^{2}) would get multiplied by 10000 and any volume (measured in cm^{3}) would get multiplied by 1000000!
This has consequences in physics and biology (such as the fact that an insect whose size was doubled would have only four times as much surface area to absorb oxygen but would have eight times as much muscle mass requiring oxygenation). And those consequences are also often referred to as the "square cube law", though I would at that point no longer consider those things to be in the realm of mathematics.
This simple law is often misstated and misunderstood. In fact, even the editor's remark accompanying Pratt's novella "The Square Cube Law" gets it wrong! They wrote "The trouble lies in the square cube law, which states if you square your size you cube your weight", which is not correct.
The observation that smaller creatures are proportionally stronger is an empirical fact, one mentioned frequently in AntMan films and comics. However, nobody seems to consider "AntMan" to be mathematical fiction. (I certainly don't.) Although Pratt's novella mentions cubes and squares (briefly in the text and also in the title), it never mentions surface area or volume which are the subjects of the mathematical version of the law. Since it only really mentions physical/biological aspects I'm not sure I would have considered it to be "mathematical fiction". But, Vijay Fafat (a frequent contributor to this site) clearly does and that's good enough for me.
Contributed by
Vijay Fafat (in response to the previous paragraph):
My intuitive thought is that Mathfiction falls into at least 6 categories:
1. Flimsy: In these stories, a passing reference to a mathematician or mathematics is taken as a license to indulge in very loosely connected fantasy. e.g. Unless the story makes a good attempt at explaining higher dimensions, topology, etc. or the thought process of a mathematician thinking about these concept, the 4D stories like those of Bob Olsen fall into this category.
2. About Mathematicians & Mathematical Process: Here, the characterization of a fictional or a nonfictional mathematician takes center stage. Mathematicians questioning the value of their work and their struggles can form rich structures of narrative in such stories.
3. Integral Mathematics: There are many stories where a mathematical problem or process is an essential aspect of the story. e.g. "Devil and Simon Flagg", "Subway called Mobius" or stories related to Fermat's Last Theorem, etc. Mathematics is an indivisible part of the story and distinct from (2) above, not an addon plot device.
4. Applied Mathematics: A story like "A Botts and the Moebius Strip" or "The Cold Equations" or a few stories by Greg Egan look at the application of mathematics  in engineering, space travel, chemistry, etc. These are clever uses of mathematics in applied situations where you know that math is sharing an equal seat at the story table with other aspects of the story (In the context of Chemistry, I am reminded of the wonderful story, "The Blue Water" by William Lemkin)
5. About Mathematics itself or Alternative Mathematics: These are the most precious and most difficult ones to write, IMO. These stories delve into the nature of mathematics itself  like Kasman's "Unreasonable Effectiveness", Egan's "Luminous" and "Dark Integers", Ted Chiang's "Division by Zero" and Colin Kapp's "Getaway from Getawehy". They carry a heavy dose of philosophy in general and explore mathematical philosophies in particular, making for a fascinating embedding of metamathematics in literary settings.
6. Fictional NonFiction with Mathematics: Nonfiction work on mathematics which is entirely imaginary. Very rare, and I don't have a good example of this, though Dewedeny's "Planiverse" comes very, very close but for its setting as a fiction story. It is very elaborate, contains diagrams, explanations, sidebars  the works. There may be examples of this genre integrated in some stories involving aliens and their mathematics but I have not seen much of it ("Flatland" and "Sphereland" are more category 2 and 3, not (6), which is a difficult subgenre).
Stories may straddle these boundaries or simultaneously fall into multiple categeories (and I'm sure there are many that may not fit the above taxonomy, requiring new grouping.)
So coming to Fletcher Pratt, I had a strong feeling that it fell in category 4. Without the specific application of a mathematical fact  the squarecube law  to Biology and human physiology as espoused by JBS Haldane, the story falls apart (thus straddling category 3). The mathematical aspect of the story is not that there are references to mathematical operations of squaring and cubing as much as the very scholarly observation rising from Haldane that one cannot realistically just make up giants in nature without contradicting a basic mathematical fact (and if you went in reverse, you could create midgets with superhuman strength.) Indeed, I would rate Fletcher Pratt's story much higher on mathematical scale than "Into the Fourth", for example, for this reason.

