There is certainly a lot of deep mathematics discussed in this `first
contact' novel, as well as a good deal of controversial physics and
psychology. Still, in the end, I did not find it especially
satisfying. It was as if all of this clever stuff was put together
into a farlessthanclever mess.
First, let me tell you about the math, page by page.
 On page 47 there is a discussion of factorization of large
numbers and the role that quantum computers can play. (However, since
Kyle believes in the many universe interpretation of quantum physics,
it is described in a ridiculous way: the computation is done by
infinitely many computers in infinitely many different universes.)
 On page 79 we are given the symbolic expression of the righthand
side of the Drake
equation for estimating the number of radiobroadcasting civilizations
in the galaxy as well as a description of the terms.
 Page 8184, back to this crazy sounding description of
factorization on quantum computer. In particular, it is argued that
since the number of computations involved in factoring the huge number
he asked his computer to factor is larger than the number of atoms in
the universe, it is proof that the computation is done in multiple
universes. (Not really, it could just argue that a single atom or
particle is capable of doing lots of traditional computations by
virtue of its quantum nature!) But, it doesn't matter, because it
only finds two factors (though he expects more) which (according to
Kyle) could have been done using only the number of atoms in this
universe.
 Page 92 has a diagram of a hypercube. This is a real
mathematical object which is to a cube what a cube is to a square. In
particular, it is what you get by taking a row of four cubes, sticking
four more on the four visible sides of the second one and then
identifying all of the remaining sides by `folding it' up. (This
object can only be embedded into spaces of dimension greater than four
so it is not something we can really see in our daily experience.
 On page 113 there is a table of the ratio of 2832 by its integer
factors. It is apparently a clue that she is supposed to recognize in
order to decode an alien message that 59 is the only prime factor of
2832 greater than 3.
 With the knowledge of the factorization of 2832 determined on
page 113 she is able to recognize that the message is a finite set of
tiles. On page 117 she mentions this to her physicist husband who
makes a remark (a nonsequitur as far as I'm concerned) about how it
is funny that the aliens would send a solvable, finite tiling problem when there
are problems about tiling the plane with infinitely many tiles that
are not clearly solvable. (This is a brief mention of some real math,
like the question of whether local rules can allow one to tile the
plane with Penrose tiles, but it is strange to mention it here and
pretty far fetched to say that this `echoes one of the big debates in
human consciousness.'
 One of the coolest mathematical discoveries of the late 20th
Century (IMHO) is the existence of `fake R^4's', or more specifically
the fact that there are no `fake R^n's' for any n other than
4. Let me put it like this: you can ask whether there is a space that is
topologically equivalent to ndimensional space (R^n) but not smoothly
equivalent to it. The surprising result is that there are such spaces
(topologically but not smoothly equivalent) only for n=4. This makes
R^4 a very special sort of space indeed. I always thought that this
should be worked into a science fiction story somehow.
Sawyer touches on this on page 121. He even has a two paragraph quote
from Ivars Peterson's Science News article on the subject. The
way he works it in to the story is that our heroine is decoding a
message from aliens and she figures there must be some assumption they
are making, something they will assume we know as well as they do.
She decides that they must be utilizing 4dimensional space because it
is so special, and she seems to be right because it all works out.
However, I find this rather implausable. It is true that
4dimensional space is special because it is the only one with the
property described above, but other dimensions are also singled out as
answers to certain questions! The unitball (the set of all points
one unit away from a fixed point) is discrete only in R^1.
Certain wave propogation phenomena (Huygens' Principle) apply only in
R^2. Etc, etc.
 There is a reference to Alan Turing on page 132 when it is
mentioned that one character in the book killed himself by eating a
poisoned apple (the same method that Turing used).
 On page 172 there is a discussion of Roger Penrose's theory of
the quantum mechanical nature of consciousness. The theory itself is
not presented in a mathematical formulation, but it is mentioned that
Penrose is a mathematician.
 Turing is mentioned again on page 173 in the context of the
Turing test for artificial intelligence.
 When Heather enters the machine that the aliens have
instructed her to build on page 192, we read that she was ``looking
through a twisted lens, a Mobius microscope, a topological telescope.
A hyperscope. And the hyperscope was allowing her to see the
fourdiemsional reality that surrounded her quotidian world, a reality
she'd been no more aware of than A Square  the hero of Abbott's
Flatland  had been aware of the three dimensional world surrounding
him."
 We learn on page 218 that we learn of seven colors in the rainbow
(ROYGBIV) because Isaac Newton liked the idea that there should be a
prime number of colors. (Is that true?)
 On page 260 we learn that ``Time isn't the fourth
dimension....The fourth dimension is a spatial direction, precisely
perpendicular to the other three.''
 On page 264 the standard description of how a lower dimensional
cross section of a connected object can have more than one connected
component: a donut passing through a tabletop can appear as two
separate disks. This is then applied to the overmind: all of our
minds appear separate, even though they are all actually connected if
you can travel into the fourth dimension.
This is used, on page 296, to eliminate the paradox of the
Schr"odinger's Cat thought experiment. (Well, if you're willing to
accept a special role for human knowledge in quantum physics, which I
still cannot bring myself to do.)
 Binary numbers appear on page 302 as part of an alien message.
We are supposed to believe that merely recognizing that the message is
made up of the binary numbers 6 and 14 would lead us to guess that it
is refering to carbon and silicon by atomic number.
The contrived plot of this novel concerns a husband and wife, Heather and Kyle, who
are learning to deal with the suicide of their daughter during a trial
separation when their surviving daughter hits them with a bombshell:
she accuses her father of having sexually abused her and her sister.
Doesn't sound much like a science fiction novel so far? Well, Kyle is
a physicist doing research in quantum computation and Heather is a
psychologist whose area of expertise is the Jungian overmind. Also,
there is a message from space which Heather is working to decode so
that she can get tenure (in a psych department?). Also, Kyle is being
approached by government agents who want to use his computer to decode
(coincidentally) a previous message from space that happened
(coincidentally) to have been intercepted by Heather's boyfriend who
(coincidentally) also commited suicide. Coincidentally, the picture
of a hypercube (an unfolded tesseract) that Kyle has hanging in his
office is exactly the key Heather needs to decipher the alien
message. The message turns out to be blue prints for a machine which
(coincidentally) allows her to travel through the Jungian human
overmind and (coincidentally) allows her both to resolve the sexual
abuse charges and figure out what the original message from
space said. I guess if you are a Jungian then all of these
coincidences (synchronicities?) and the whole idea that all humans
share a single mind might not be a problem for you. I'm afraid it was
a bit much for me.
As with Sawyer's other novels (See Calculating
God), there are many `inside joke' references to other science
fiction, especially Star Trek. This and the reasonably facile
presentation of real scientific ideas (even if they are extremely
controversial ones) make this book an enjoyable read even though it
appears to me in the end to collapse under its own weight. Contributed by
SarahMarie Belcastro
What distinguishes this book from many other SF works I've read is the breadth and multitude of references to popular culture from 19801998 (when the book was published), and how these popculture items are used in discussions of how our culture might change over the next twenty years. There are also many references to academic work (mainly Canadian) which influence the plot. (I was only able to come across one reference which appeared to be fictional, in fact.) That said, on to the math! The main item here is the hypercube  the characters fold a hypercube and examine it from the inside as well as outside. However, there's also a discussion of primes (and large primes, and factoring, for RSA)
There are also discussions of quantum computing, artificial intelligence, and Jungian psychology, all of which relate deeply to the plot of the book.

