Here is the beginning of a catalogue of other science fiction that includes real mathematics.

10/14/12 The Butlerian Jihad, Brian Herbert and Kevin J. Anderson, 2002.

There are two stereotypical mathematicians in this book; Tio Holtzman, the established scholar, and Norma Cenva, the Rossak mathematical prodigy. Holtzman is the over-the-hill has-been, who uses slaves to do the calculations for his theories. Do the authors have bad memories of their graduate studies? Cenva, a deformed outsider, does the real work. "God is the mathematician of the universe. There is an ancient correlation known as the Golden Mean, a pleasing ratio of form and structure that is found in this leaf, in seashells, and in the living creatures of many planets. It is the tiniest part of the key, known since the time of the Greeks and Egyptians of Earth. They used it in their architecture and pyramids, in their Pythagorean pentagram and Fibonacci sequence. ... But there is so much more." [p91]

"Anything imagined can be made real ... given sufficient genius." -Tio Holtzman. But this is obviously untrue - Escher, Klein bottles, paradoxes, etc. [p119]

"They (the slave solvers/computers) are replaceable, so don’t give them personalities or temperaments. Only the solutions matter to us. An equation has no personality." - Tio Holtzman. "It seemed to her that the more esoteric orders of mathematics did indeed have personalities, that certain theorems and integrals required finesse and consideration that simple arithmetic never demanded." - Norma Cenva. [p269] If mathematics is a creation of man’s reason, then the mathematics definitely reflects the personalities of the creators. If mathematics is discovered truth, then the discoveries attract certain types of personalities to work in those areas.

7/26/12 Implied Spaces, Walter Jon Williams, 2009. Unlike anything I’ve read before, it throws every sci-fi concept into one big conglomeration and races forward. You want computers and AI? It’s got it. You want swords and sorcerers? It’s got it. You want physics, wormholes, and multiverses? It’s got it. You want avatars, space weapons, aliens, individual combat, galactic war? It’s got it! Unexpectedly, in the midst of all this, is quite a bit of religion and consideration of the "Existential Crisis."  (see the entry in Sci-Fi and religion)

Mathematically, the only trace is the use of "Q.E.D." without explanation for the reader.

6/23/12 The Algebraist, Iain M. Banks, 2004. Every nook and cranny of the galaxy is full of life, but this story deals mostly with a human scholar of Dwellers, the life forms that populate gas giants. His quest is to search for a possibly mythical text, The Algebraist, which is "all about mathematics, navigation as a metaphor, duty, love, longing, honour, long voyages home ... that sort of stuff." Banks’ writing style is as dense and detailed as befits such a complex universe. His sentences extend to paragraphs, his lists can’t stop at two or three examples but invariably explode to ten or twelve descriptors. There is sex, there is violence, there is politics, there is humor, but don’t let the title fool you, there is no Algebra!

However, there are a few scatterings of mathematics; "by any algebra of justice under any sun", "Lagrange point", "an algorithm for elegance", "fractally spiraled", "and some algebra ciphered into the base code".

"It looked like algebra." What does algebra look like? Especially an alien algebra?

I liked a couple of lines that could get the reader to refresh his basics. "the number of genuine galaxy-spanning wars didn’t make it to double figures. In base eight!" And if they had, how many would there be? :-) "7.35 x 10^8 seconds ago" I almost started to calculate that, but then he said, "about twenty years earlier." "I suspect good luck will be necessary, if not sufficient."

The holy grail of the plot is "some extra set of coordinates, or even a single mathematical operation, a transform, which, when applied to any given set of coordinates in the original list, somehow magically derived the exact position"

6/9/12 Although Tony Ballantyne’s trilogy, Recursion (2004), Capacity (2005), Divergence (2007) deals with the questions of reality and virtual reality, of intelligence and artificial intelligence, and other computer science based ideas, it is also full of mathematical references. The definition of recursion does not use the simple examples of n! = n*(n-1)! or x^n = x*x^(n-1), but is more algorithmic and philosophical; can something be created from nothing? However, his description of faster than light travel, the "hyperdrive", dives right in: "Start by imagining a four-dimensional section of an eleven-dimensional sphere. Now deform that section over any non-Euclidean space ... I mean, I won’t go on; you get the picture. The human mind can’t contain the concepts." I’m still working on that challenge. :-) I got too involved in the story to remember to make note of other mathematical references, but I remember that half a page of C++ code was included.

Capacity is a great sequel, blending in the necessary background information smoothly as the story develops. Dark plants, Schroedinger boxes, and black velvet bands are marvelous villains. As the black velvet bands shrink to nothing, he says, "They can’t vanish if they have a hole in the middle. Basic topology. What’s the smallest a ring can be?" Answering another of the author’s questions; if there were more of me than I thought then maybe I could enjoy this book more two or three times as much as I already did. Some basic terms are defined as he describes an "old-fashioned" way of representing images as a random array of bytes, a 2-D picture format, a bitmap. He asks if intelligence is just a mental application of Godel’s incompleteness theorem. Escher is also thrown in as a presumably familiar artist, as he should be to anyone interested in mathematics.

Divergence starts slowly with too many characters, but then zooms off into the depths of the story with all its philosophical issues and amazing details. "There are different levels of programming languages, so why not one specifically for the soul?" Well, that line should actually go into my blog on religion in sci-fi. As you would expect from an author who taught math and IT, Ballantyne mentions stellated icosahedrons and dodecahedrons, the Sierpinski Gasket and the Mandelbrot set, the golden ratio, Riemannian transforms, and Hilbert space as comfortably as another author would describe clouds as luminous white airships. He even includes the formula for Heisenberg’s uncertainty principle! One of the characters sails off in a spaceship called the Fourier Transform, on which the classical mathematical impossibilities are no longer impossible: creating a formal way for determining a proof, finding an even number that is not the difference of two primes, having a recursive set for everything, a solution for an NP complete problem, and all the other NP problems tumbling into P.

11/06/11 Second Foundation, Isaac Asimov, 1953. The end of the trilogy contains a number of interesting details. On page 84 he heads the chapter with this entry from the Encyclopedia Galactica: "Mathematics: The synthesis of the calculus of n-variables and of n-dimensional geometry is the basis of what Seldon once called ‘my little algebra of humanity’" But algebra is deterministic and Asimov’s real construct is statistical. On page 90 he states, "It is enough for a Pyschohistorian, as such, to know his Biostatistics and his Neurochemical Electromathematics. Some know nothing else and are fit only to be statistical technicians. But a Speaker must be able to discuss the Plan without mathematics."

There is an interesting, to me, section where he describes a bit of "doctoral hazing" of one of his student Psychohistorians. On page 89 the student manages an answer, "It is a Rigellian integral, using a planetary distribution of a bias indicating the presence of two chief economic classes on the planet, or maybe a Sector, plus an unstable emotional pattern." Having just finished reading a few books on integration, I enjoyed the morphing of Riemannian integral into Rigellian integral. Another amusing sci-fi device was the description on page 99. "The Analytical Rule must be considered a distant relation - as a skyscraper to a shack - of that kindergarten toy, the logarithmic Slide Rule." Of course, having written this in 1953 today’s reader can’t blame him for that guess. I never did really "get" the logarithmic slide rule. Luckily computers, or Radio Shack calculators, came along just in time.

On a topic related to the "multiminds" in my own series Asimov, page 86, says, "Through the development of the mathematics necessary to understand the facts of neural physiology and the electro-chemistry of the nervous system, which themselves had to be, had to be, traced down to nuclear forces." My writing traces consciousness down to the Turing machine structures within DNA, but Asimov lived and wrote in the atomic age. And further, on page 178, he says, "I solved the mathematics of the Second Foundation in the sense that I evolved a function that would predict the necessary combination of neuronic paths that would allow for the formation of an organ such as I have just described - but, unfortunately, the function is too complicated to be solved by any of the mathematical tools at present known."

10/25/11 Foundation and Empire, Isaac Asimov, 1952. As far as the math goes, the psycho-history is supposed to be all mathematical formulae but details are never given. He states in the prologue, "Psycho-history ... could forecast reactions to stimuli with something of the accuracy that a lesser science could bring to the forecast of a rebound of a billiard ball." Now that is getting a bit personal! Since my thesis, "Visibility With Reflection", and my work in Computational Geometry was all about forecasting the rebounding of billiard balls I think I am offended. :-)

10/12/11 Foundation, Isaac Asimov, 1951. An all human galactic empire? Also, as to be expected, a number of other little oddities: a lot of smoking, too much hard copy messages. However, psychohistory is great!

Asimov describes psychohistory as, "that branch of mathematics which deals with the reactions of human conglomerates to fixed social and economic stimuli." It enables a universe wide prophesy that is not revealed in its details to the people. It is based on mathematical and statistical analysis of large group psychology and sociology, but all without much detail about the computers involved.

The developer of psychohistory, Hari Seldon, makes a universal prophecy that is not prophesied to the people. Asimov describes the Foundation and Empire over long periods of time using the construct of Seldon Crises. Most conflict is resolved non-violently, so there is a lot of political intrigue.

The Foundation turns science into a religion in order to protect themselves from the ignorant masses. That approach is successful, but by the end of the book it is being challenged by other power sources such as trade and economics.

In my universe, the Foreverones can take prophecy down to the individual level, and TheBook is his Encylcopedia Galactica with electronic distribution.

7/10/11 Beowulf’s Children, Larry Niven, Jerry Pournelle, Steven Barnes, 1995. There is no mathematical discourse in this book. Human colony on Avalon fights for survival against the exobiology.

6/17/11 A Case of Conscience, James Blish, 1958. There is no mathematical discourse in this story, but some really great mathematical lines thrown in here and there. For example, "It is more than an astronomical coincidence – that tired old metaphor for numbers that don’t seem very large any more – it is a transfinite coincidence. It would take the shade of Cantor himself to do justice to the odds against it." Blish mentions transfinite numbers once or twice more, but the main emphasis is biology and especially religion and logic. "Your Lithian is a creature of logic. ... He is as rational as a machine. Indeed, the only way we can distinguish the Lithian from an organic computer is his possession and use of a moral code. And that, I beg you to observe, is completely irrational. It is based upon a set of axioms, a set of propositions which were ‘given’ from the beginning." Much of the rest of this section is a logical argument by the priest. In fact, this book is a perfect example of the genre I tried to place my work into. I think I will start collecting examples of "religion in SciFi."

Starship Troopers, Robert A. Heinlein, 1959. Contains the common complaints by the hero that mathematics is too difficult. But, his character admits, "Math is hard work and it occupies your mind – and it doesn’t hurt to learn all you can of it, no matter what rank you are; everything of any importance is founded on mathematics.

Imperial Earth, Arthur C. Clarke, 1976. Pentominoes, group theory, ... "I was introduced to polyominoes by Martin Gardner’s Scientific American Book of Mathematical Puzzles and Diversions, which, however, fiendishly refrains from giving the solution to the 20 x 3 rectangle. In his definitive book Polyominoes, Solomon W. Golomb takes mercy on his readers. In the hope of preventing a few nervous breakdowns, I reproduce his answer herewith: UXPILNFTWYZV.

Anyone who wishes to construct this rectangle from the twelve pentominoes should have no difficulty in matching them with the letters they (sometimes approximately) resemble. It is easy to see that the second of the (only) two solutions is obtained by rotating a seven element central portion.

Dr. Golomb, who is now professor of Electrical Engineering and Mathematics at the University of Southern California, has also invented an ingenious game called Pentominoes (distributed in North America by Hallmark Cards and in Europe by Zimpfer puzzles). It has more openings than chess. In an earlier version of 2001: A Space Oddysey, Stanley Kubrik shot Hal playing the game against the astronauts."

Duncan’s grandmother tests him with pentominoes ... how many different patterns can you make putting five squares together in a plane ... fit them in to a rectangular box with dimensions 10 by 6 ... more than one answer ... 2,339 actually ... 20 by 3 would take a computer six million million years to show all combinations (what about a newer computer?) ...12!x2^21 =1,004,539,160,000,000 ...