a list compiled by Alex Kasman (College of Charleston)
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In his presentation entitled "An Examination of Some Supposed Mathematical Impossibilities" before the Royal Society, Professor Marmion demonstrates that he can do three geometric constructions that mathematicians have proved are impossible:
Of course, many of the famous mathematicians and scientists in the audience are skeptical, but they do not know that Marmion's miraculous abilities stem from his discoveries regarding the fourth dimension and the ancient Egyptian princess who has been reincarnated as his own daughter. The threat of ancient Egyptian horrors in Victorian England is the main concern of this thriller, but I will focus in this review on the mathematical aspects. Both in fiction and in reality, there are "crackpots" who aim to show that mathematicians are wrong to claim that squaring the circle, doubling the cube and angle trisection are impossible. What they often miss is that they are impossible given a rather specific set of rules. In the sense that mathematics is axiomatic, all of math follows certain rules. But, the claim that trisecting an angle is impossible refers specifically to Compass and Straightedge Construction. Under these rules, one has only these two tools with which to work, and it really has been proved mathematically that certain tasks like trisecting a given angle cannot be achieved under those rules. Of course, if one changes the "rules of the game", then these things are no longer impossible tasks. (Given an angle of size θ, we can simply declare it to be decomposed into three angles of size θ/3, for instance. But, that is not a compass and straight edge construction!) One might think, therefore, that what Griffith is describing is merely that Marmion has changed the underlying axioms when he moved to working in "N^4" (his notation for 4dimensional space.) But, I do not believe this is the intent. Towards the beginning of the book, his daughter suggests that fourdimensional geometry renders everything possible:
Furthermore, one of the audience members at his presentation objects that Marmion has found contradictions in mathematics itself, in that these things can be proved to be impossible and yet are also demonstrated to be possible. So, it seems to me that Griffith expects us to believe that having made discoveries about the fourth dimension, Marmion has not simply changed the rules but is no longer constrained by the concepts of "possible" and "impossible" in any way. Clearly, this science fiction story is a bit "over the top" and difficult to believe. However, it is rather well written as compared with other "pulp" science fiction. Moreover, considering its early publication date, it is quite prescient, containing passages which bring to mind World War I, Kurt Gödel's work on consistency of mathematics, and Minkowski space. I am very grateful to Elaine Freedman for bringing it to my attention. As a side note, it may be interesting to keep in mind that George Chetwyn GriffithJones (18571906) was known as an explorer as well as being famous in Britain for his science fiction. The entire novel can be read for free in various locations on the Internet, including Gutenberg.org. 
Buy this work of mathematical fiction and read reviews at amazon.com. 
(Note: This is just one work of mathematical fiction from the list. To see the entire list or to see more works of mathematical fiction, return to the Homepage.) 

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(Maintained by Alex Kasman, College of Charleston)