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Science

The New Flatland 103

Posted by Hemos from the interesting-boook dept.
SilenceKit writes "The New York Times has a cool story today on a sequel to Flatland, the classic geometry/social satire which "it may be no exaggeration to say has been read by every self-respecting physicist, mathematician and science fiction writer." The new one, by Ian Stewart, is called "Flatterland" and is a tour of a century of strange geometry -- from fractals to "Minkowski space," whatever that is. The story (free registration required) is at the Times" I was loaned Flatland: A Romance Of Many Dimensions by one of my college profs - it's a great book, and this come from someone who really hated geometry (What bad high school teachers can do). It's still available on fatbrain - pretty good for a 19th century text about geometry, to still be in print.
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The New Flatland More

The New Flatland

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  • If all matter in the universe was in a point, or even a tiny volume, it would collapse into a black hole. Since nothing escapes from a black hole except an extremely tiny trickle of particles via Hawking radiation, the universe could never have undergone a "big bang". Either the "big bang" idea or the "black hole" idea is wrong.
    You forgot the third, correct answer: you don't understand general relativity.

    See the FAQ [corepower.com]:

    "Why did the universe not collapse and form a black hole at the beginning?"

    Sometimes people find it hard to understand why the big bang is not a black hole. After all, the density of matter in the first fraction of a second was much higher than that found in any star, and dense matter is supposed to curve space-time strongly. At sufficient density there must be matter contained within a region smaller than the Schwarzschild radius for its mass. Nevertheless, the big bang manages to avoid being trapped inside a black hole of its own making and paradoxically the space near the singularity is actually flat rather than curving tightly. How can this be?

    The short answer is that the big bang gets away with it because it is expanding rapidly near the beginning and the rate of expansion is slowing down. Space can be flat while space-time is not. The curvature can come from the temporal parts of the space-time metric which measures the deceleration of the expansion of the universe. So the total curvature of space-time is related to the density of matter but there is a contribution to curvature from the expansion as well as from any curvature of space. The Schwarzschild solution of the gravitational equations is static and demonstrates the limits placed on a static spherical body before it must collapse to a black hole. The Schwarzschild limit does not apply to rapidly expanding matter.

  • In addition to the afor mentioned "Geometry, Relativity, and the 4th Dimension", there is also _The Sex Sphere_. The story is about a hyperdimenstional being by the name of Babs, and her interaction with the charactors in the book. Lots of fun, kinda kinky at times...and all based upon math! Well worth the read. ttyl Farrell
  • Rudy Rucker [sjsu.edu], author of the Software, Wetware, Freeware, Realware series. He later wrote a somewhat more accessible introduction to thinking about dimension called The Fourth Dimension and How to Get There which I recommend highly. Rudy Rucker is lots of fun and most things he gets involved with are cool - check out his list of works [sjsu.edu].
    --
  • The planiverse was written by A. K. Dewdney, a prof a The University of Western Ontario. He was my thesis instructor and a very cool and intelligent guy. He wrote a column in Scientific American for a few years on computer and mathimatical puzzles and problems and is probably best known as the creator of Corewars. I highly suggest checking out some of his old columns and writings.
  • My mom works in a bookstore and knows that i liked Flatland (i have read it twice), so she gave me the advanced reading copy of Flatterland to try.

    While it starts off with some neat Terry Pratchet type humor and social satire, it quickly bogs down into nothing but math. While i am a big fan of quantum physics and the metaphysics of 'what is reality', i found this book to be too much math for my tastes. Too many examples and diagrams, all gone over and over and over until i couldn't take it any more and would jump ahead to the next mathematical theme. The humor at the beginning made me laugh out loud, but the rest of the humor was not to my liking and would have preferred he left that out.

    I gave my copy to a math major and she loved it right away, couldn't keep her nose out of it. Maybe this would be good for a young student interested in starting into the math world, as the examples are explained from several points of view and are definitely not skimmed over lightly.
  • Comment removed based on user account deletion
  • Yes, but Banks, unlike the previous poster, explicitly says he made all that stuff up to sound cool for his space opera.

    --

  • flatland on project gutenberg (Score:3)

    by DarkClown ( 7673 ) on Saturday April 21, 2001 @06:15AM (#275234) Homepage
    Here is a copy [ibiblio.org] of flatland online from project gutenberg. Great fun to read.
  • So, I usually don't like to dissent -- but I felt the need to say that while geometry is certainly a big part of Flatland, I don't think of it as a book about geometry.
    Flatland was published in the 1880s, and seems to me to contain some fairly powerful commentary on the class structure of Abbot's society. Sure, it's all about squares and circles, but as a good friend of mine said to her father once :
    "Sometimes, when people write about circles, they're actually writing about something else."
    Or, alternatively, this text from Flatland's dedication:
    This Work is Dedicated

    By a Humble Native of Flatland
    In the Hope that
    Even as he was Initiated into the Mysteries
    Of THREE Dimensions
    Having been previously conversant
    With ONLY TWO
    So the Citizens of that Celestial Region
    May aspire yet higher and higher
    To the Secrets of FOUR FIVE OR EVEN SIX Dimensions
    Thereby contributing
    To the Enlargment of THE IMAGINATION
    And the possible Development
    Of that most and excellent Gift of MODESTY
    Among the Superior Races
    Of SOLID HUMANITY

    that's my $0.02, anyway. --psykelus
  • I'm not sure it's fair to assume that Abbot was either. In fact, it doesn't seem unreasonable to believe that Abbot was intentionally commenting on / highlighting the (rampant) sexism and classism of his time.

    just my $0.02 -- psykelus
  • Flatland 3D: The Sequel

    ---
  • There's also The Planiverse by A. K. Dewdney (listed at Amazon - I just checked - although I can't paste the URL for the usual reasons). I read it many years ago; it's pretty good, as (as I recall) has some virtual/cyberspace tie-in.
  • i've compiled the book, including illustrations, into a newton book format. available here:

    http://www.newtontraveler.com/books.html [newtontraveler.com]

  • Minkowski is the gemoetrical model used in (special) relativity theorie to describe our world. So it's basically that 4d thing that "warps" and does other weird stuff. An intuitive description of this would really be a nice thing to have.
  • I have to agree with this poster. What matters in a scientific theory is not the formulation used, but what _predictions_ are made. Heaviside's equations are completely, absolutely identical in content to Maxwell's original formulation, and are MUCH more physically intuitive.

    A sad day for /. moderation.

    Bob
  • Of course it's a joke. Everyone knows it's turtles all the way down.

  • An Author I know (Score:3)

    by matth ( 22742 ) on Saturday April 21, 2001 @06:22AM (#275244) Homepage
    Jerry Thorwayn is an author of a book called "Proffessor Ovilstar's Infinate Paradox", and it deals with all kinds of math theorums, as well as these flat land principals. You can see it here: www.ovilstar.net [ovilstar.net]. He has written a whole book about this type of thing and inter-weaves romance and excitement into it. Kind of a sci-fi educational book :)


  • What does "OP" mean?
  • You're using "trivial" to mean "identity", or "Minkowski". I'm using it to mean "simple", that is, non-complicated, straightforward.

    Granted, bad choice of words, but that's why I specified what I meant by trivial.

  • Re:Geometry is the key. (Score:3)

    by barawn ( 25691 ) on Saturday April 21, 2001 @06:53AM (#275247) Homepage
    Dimensions are only orthogonal if the metric is trivial (i.e., diagonal).

    The "metric" as I stated above is a 4-dimensional tensor (matrix, essentially) which (basically) provides the coefficients in the distance measure.

    i.e., in 3-space, we have a distance measure of
    ds^2 = dx^2 + dy^2 +dz^2
    Here, the metric has a 1 in the dx-dx term, a 1 in the dy-dy term, and a 1 in the dz-dz term: in terms of a matrix, this would be just the three-dimensional unit matrix.

    General relativity says that the metric is affected by the presence of matter and energy, so therefore, depending on the spacetime configuration, the dimensions may not be orthogonal. Near a black hole, this is actually distinctly not true - there is significant radial/temporal mixing in certain coordinate systems. Granted, you can always define a coordinate system with orthogonal coordinates.

    In any case, you were mainly talking about special, rather than general, relativity here. The 4-dimensional metric being talked about is a Minkowski space - time is distinctly separate from any others, because in the Minkowski metric, we have
    ds^2 = dx^2 +dy^2 +dz^2 - c^2 dt^2
    note that time has a changed sign. We call this 3+1 spacetime.

    All special relativity does is define exactly how you transform from one coordinate system to the next - in this case, via Lorentz boosts. Lorentz boosts in one direction have both a spatial boost and a temporal dilation, which is what you are talking about. It doesn't mean the coordinates aren't orthogonal - it just means that when you boost from one frame to a faster-moving frame, both the boost direction and the temporal direction are affected.

    Short answer: you answered your question yourself - you said "rate of movement", but above you said "movement" - there's no reason to expect that "moving" is the same as "steady motion" - or, that a simple translation is the same as a velocity shift. In truth, it isn't.

    Think about it this way. It's true, time and space are orthogonal. If I do a time translation (i.e., in my coordinate system I redefine t = t + t0) it doesn't affect the x, y, or z coordinates. Likewise for spatial coordinates.

    But there's no a priori reason to expect that when you start talking about another frame of reference that is in constant motion with another that it shouldn't affect the other coordinates. After all, you're already mixing things: you have to talk about z = z0 + vt, for instance. Here, instead of the boost simply being z = z0 + vt, there's a Lorentz factor as well.
  • Could you please clarify? It sounds like from your balloon analogy that the current "universe" is like the surface of a balloon and that there are other balloons nested inside this balloon. From this you seem to conclude that there are 4 spatial dimentions (presumably three dimensions on the surface of the balloon and then a radial dimension which corresponds to traversing between the universes.) I must have something wrong though, because I have just stated a fallacy. If our universe if like the surface of a balloon, then it has only two spacial dimensions corresponding to radial and axial angles. Where have I gotten off course?
  • About 10 years ago, I ran across a book in a used book store that greatly reminded me of Flatland called "The Planiverse." It is not quite as mathematical as Flatland, but instead gets more into the biological-type details. The residents of the planiverse are not geometric shapes, but rather 2-dimensional organisms. The book explains complexities of their world (how do you have weather in a flat world, and how do you deal with water runoff?), their society (how would 2 dimensional paper, books, and libraries work?) and biology (if a flat organism has to eat, it cannot have a digestive track, otherwise it would be split into two pieces!).

    I remember really enjoying it, but I admit I was sixteen years old at the time and have not picked up the book since. You can find it at the usual places: http://www.amazon.com/exec/obidos/ASIN/0387989161/ o/qid=987870273/sr=8-1/ref=aps_sr_b_1_1/104-961239 4-9227167 [amazon.com]

    -E
  • I neglected to change the subject of that post before I submitted it. I just meant to say that there were at least two other "sequels" to Flatland that I knew of, that some people might be interested in reading.

    'Nuff said.

    ---

  • Actually, there have been two books already hailled as "sequels" to Edwin A. Abbot's Flatland.

    The first is Sphereland, written in 1960 by Dionys Burger. The English translation I have has a forward by Isaac Asimov [cool little useless tidbit].

    The second is Geometry, Relativity, and the Fourth Dimension, written by Rudolf v. B. Rucker in 1977. Aside from the <sarcasm>eloquent title</sarcasm> it follows in the same format as the previous two.

    The following is part of the blurb from Sphereland:

    This delightful fantasy by a distinguished Dutch mathematician both entertains and instructs in the multidimensional geometries of curved space and the expanding universe. Written in 1960 as a sequel to Edwin A. Aboot's Victorian satire, it is the story of a Hexagon, the brilliant grandson on Flatland's Square who in his lifetime is confronted with even greater dimensionality problems. In the process of solving them he corrects his grandfather on some points...
    And Geomerty, Relativity, and the Fourth Dimension:
    This is a highly readable, popular exposition of the fourth dimension and the structure of the universe. ... Find a perfect analogy in the situation of the geometrical characters in Flatland, Professor Rucker continues the adventures of the two-dimensional world visited by a three-dimensional world in terms of the fourth dimension. ...


    ---
  • You can get that book for 90 cents on Amazon or anywhere else that sells dover thrift editions.

    You can also get it for free online from Project Geutenberg [embnet.org]
    Any word on wether the new flatland will be sexist as hell? Reading the original was an interesting look into good old-school great chain of being type classism...

    The nobitity are polygonal, with rank being determined by number of sides. Commoners are isocoles triangles, with acute angles at the low and equilaterals at the high end of the social spectrum. Women, whatever their rank, are lines.

    I wonder how they are going to style the new one without being hopelessly sexist and classist while maintaining continuity...
  • I think I first read Flatland in High School, and I periodically reread it when the subject of 2d-3d-4d space analogies come up. I love it as much now as I did then. For the few remaining people who have never read the original or don't have their own copy, you can buy it at Amazon [amazon.com] for ninety cents! Or you can download it from Project Gutenberg here. [ibiblio.org]

    SuperID
    Free Database Hosting [freesql.org]

  • Yeah - I *love* The Planiverse - some of the gadgets in the many illustrations are fascinating.

    The major deviation from Flatland is that it's
    easier to think of Flatland as a 'top down'
    view of a 3D world - where Planiverse is a
    side view.

    Check out the 2D steam engine - it's quite amazing.

    The observations in Planiverse are much more thought-provoking than Flatland...just simple
    things like the fact that you can't build things
    using nails and screws in 2D, that wheels can't
    work, that wars in the Planiverse tend to focus
    on the ability of heroic individuals because the
    opposition can't just walk around the lone hero.

    It certainly makes you think much harder than
    Flatland does.
  • One thing that strikes me about flatland that I don't hear people talk about much is the stickingly effective way that Abott uses analogy to help deal with hard to grasp concepts.

    Don't understand what you can do in 4 dimensional space? No problem; just analyze what you can do in 3D that someone in 2D can't. Simply extrapolate from there.

    This book taught me about seeing reality from other's perspectives, about the power of analogy, and the value of approaching a problem starting from where the *other* person's viewpoints lie.

    -pos

    The truth is more important than the facts.
  • For inside the balloon is another ballon, and inside that one another. It is like an Onion, with all these different shells expanding outwards from the central fire of Creation.

    I knew it. The Onion [theonion.com] is the center of the universe!

    "Everything you know is wrong. (And stupid.)"

  • Actually, you were educated stupid and evil by evil educators. Academia is a religious cult empowerment of self word. Not even a god can deny that I have squared the circle of a static Earth and cubed the Earth sphere by rotating it once to a dynamic Time or Life Cube. Only a false god or academically brainwashed indoctrinated mindless moron would deny that the Earth lacks the top and bottom, the front and back, and 2-sides physical dimensions of a Cube that spirals a 4-season quad helix around the Sun - creating a swirling of 4 simultaneous years as in a separately created year for each of 4 seasons. You do not have the "guts" to seek Time Cube "Truth".

    beautiful!

    your fan WDK - WKiernan@concentric.net

  • I remember hurting my brain trying to "imagine" infinitesimals...
  • I remember acutally reading this book in high school and wondering what the fourth dimension would look like, then I got to college and found out that the fourth dimension is time and it was a big let-down. But it was fun to see how those in two dimensions couldn't imagine three dimensions and those in one dimension couldn't imagine two and so on down to the one point oblivious to all. I might have to read this again!
  • Everything2 seems slashdotted, but I think that link should have been to here [everything2.com].
  • dude you need to fix your web-page. the color scheme is nasty

  • His nickname says it all ... "Trevor Goodchild" ... the evil-mad/scientist/doctor from Aeon Flux? :)
  • Rate = distance covered / time taken

    If the distance covered is time, then it usually takes the same amount of time to cover that time. (Unless you happen to be moving close to the speed of light.) Therefore, we are moving into the future at a rate of one.

    That refers to one c, the physical constant of speed, the speed of light. We are all moving into the future at the speed of light.
  • "Lengths that can be imaginary."

    I know that part.

    "Nonzero vectors of zero lengths,"

    But how does this work?
  • Ah, of course - I forgot. Thanks.
  • Or even better, Project Gutenburg also has the illustrated [promo.net] version. I read Flatland for the first time in junior high, and I remember the wonder and mystery it evoked.
  • If spacetime travel, in any direction, is not even theoretically possible, why do all the First Posts on Slashdot appear in the middle of the lump?
  • mmm, yes. A good book, highly recommended! The metaphysics is a bit of mumbo-jumbo, but there's very little of it, and it actually makes sense for it to be in there. The "person who experienced the phenomenon" is trying to explain it as best he can, and concedes that he doubts he has the capability. The real meat of the book is the amazing things the author does with constructing a living, breathing, two-dimensional world. The "wheels and gears" are right there to see, while in Flatland, the mechanisms are hidden. (For instance, how exactly polygon lifeforms evolved isn't explained in Flatland; on the other hand, the author of Planiverse goes into great detail on this, and many other subjects.) _Definitely_ a recommended read.
  • Yeah, and monkeys might fly out of my butt. It's a joke guys, "Funny" not "Interesting".

    -Erik
  • someone who really hated geometry (What bad high school teachers can do)

    Or, what someone who did poorly in geometry and tries to pass off their own inadequecies on others can do. Seriously, Geometry is a subject; you can like it on its own, regardless of who presents it. I had what I thought was a bad professor for CS I and II but I don't hate CS, I hate the professor.
  • Ahem you sort of missed the mark on that "sexism" bit. Yes, by todays standards that is quite sexist. At the time of publishing however it was just another clever point being made by the author: it was a comment on Victorian society, which if you know much about it you'll realize that dubbing women as lines isn't necessarily too inaccurate (and this is not to say that the many sided "nobility" are any better). Read through the eyes of a 21st century American Flatland is quite sexist. Seen as a 19th century Englishman would see it and it's a comment on society as well as space-time.
  • I've got to add my "this is bullshit" comment to the chorus herein. The above poster is either joking or simply mad. If he's talking about Metaverses and the "destiny of our race, our species" to "transverse the universes", he's not really working on a PhD in Mathematics. (Which, parenthetically, he misspells as "D.Phill".)

    The comment in no way resembles currect cosmological or mathematical research.

  • Perhaps not. There is a lot of work on canonical quantum gravity which takes a pretty good stab at exactly that. (Whether it has succeeded is still an open question.)

    Well, that's true. Canonical quantization of Ashtekar's loop variables is a good stab, I agree. (In fact I hope they succeed just because I like to cheer for the underdogs vs anything stringy!). But it still is a difficult problem. Personally, I still find it a bit fuzzy in the intepretation of what is actually "momenta" in canonical quantization of gravity. But I am no expert in this field, so I guess I'd shut up now.

    (Btw, I just quickly skimmed the responses to OP, being one of the lazy ones who read at +1. So sorry if I repeated your rebuke to the OP.)
  • But that's just the 4-vector covariant index form. And your eequation is not the Maxwell equation, but just the definition of the Gauge Field Strength tensor F^{mu nu}. (For fun, your equation written in super-duper concise diff.forms is :

    F=dA where F is a 2-form and A is a one-form, and d is the exterior derivative.
  • Ok. This is now waaay over my head. I am but a puny funny gravity guy....:). (One day, when I have time, I'll think about these, but right now I contend myself with knowing the words not the details.)
  • I believe you score highly on the Crackpot Index [ucr.edu]

  • You could have at least said the the Maxwell theory of em fields was contradicted by the observation of particle phenomena (quanta). But you didn't, and even so that misses the point completely.

    When I said "classical field theory", that's what I meant. The quantum theory of EM is called "Quantum electrodynamics", and is part of "quantum field theory".

    Apparent paradoxes are used as an argument to justify an irrational, dualistic interpretation of the theory.

    If you mean the particle-wave duality, then it is not a paradox. Please do not confuse "not-intuitive" with "paradox". Quantum Field Theory is a hard subject to master, and it breaks down at some energy scales. But it is an excellent theory at low common day energy scales. QED predicts the fine structure constant to more than 10 decimal place, and no theory in this world is better than QED.

    However, at a closer look one can find that any paradox arises only from an inconsistent physical concept or other errors in logic. Goedel proved that from a logical system which contains a contradiction, absolutely any proposition may be proven. With a consistent theoretical interpretation (in any branch of science) no paradox should occur at all.

    Since your previous statement is wrong, this is non-sequitur. There is no paradox in QED. Every undergrad physics student know E&M is not self-consistent. (Read Feynman's lectures on physics Vol 2 for a beautiful exposition. The book is aimed at freshmen, so you should be able to figure it out.)

    "Quantum mechanics used only the Heaviside/ Gibbs externalized electromagnetics and completely missed Maxwell's internalized and ordered electromagnetics enfolded inside a structured scalar potential.Accordingly, QM [quantum mechanics] maintained its Gibbs statistics of quantum change, which is nonchaotic a priori. Quantum physicists by and large excluded Bohm's hidden variable theory, which conceivably could have offered the potential of engineering quantum change -- engineering physical reality itself.

    This, I gathered, is the whole problem. Bearden's generous used of big-sounding words and rhetoric masked the fact that this is a lot of BS. So, let's see :
    "externalized/internalized" : what does this mean?! There is no such thing in physics or math. "QM maintained its Gibbs statistic charge, which is no-chaotic apriori" : What's a Gibbs statistic charge? What is "no-chaotic"?! "Charge" is a conserved quantity in a field theory, which means that it is some physical thing (like eg. electric charges) that you can measure and is constant in time provided that there is no "current" (eg. electric current) flow into the system.
    I am too tired to say horrible things about the last sentence.

    I am sorry if I sounded sarcastic. But please read some better books if you want to understand this stuff. A good place to start is "Feynman Lectures on Physics Vol 1-3". And forget about crap like Bearden's book.

    As I said, physics is both financially and socially unrewarding. The price of admission into physics is many years of hard work and lots of frustration. But to be able to learn the stuff so that one can figure wonderful things out, and doing it correctly, is well-worth it. There is no short-cut to hard-core science.

  • Oh Cut that crap out. (Score:4)

    by efuseekay ( 138418 ) on Saturday April 21, 2001 @11:51AM (#275278)
    Never read Bearden's paper. From the OP, it's total crap.

    (a) I can write Maxwell's equation in whatever form I like : quaternions, vectors, component by component (go read Gell-Mann's book to see how cumbersome that is) covariant-indice form, and (most beautifully) in differential forms. It expresses a bunch of coupled differential equations that's all.

    (b) Maxwell's theory is a classical field theory. You can attach words like " unified ", "successful", and blah to it, and it's still a classical field theory. I don't understand what the heck the OP is rambling about "quartenions" blah. There is no point to be made. The use of "big words" and impressive looking references only serve to give the false impression that there is actually "something" there.
    (FYI, we now have quantum field theory, where we promote variables to operators and then impose canonical quantization on it. Which, btw, is totally incompatible with general relativity).

    I don't blame the OP for being ignorant about the finer points of physics (after all, it's a career choice which pays very little!). But the sad truth is that nowadays, there are so many "wannabes" like Bearden who writes crap and mislead the general public. The people should be careful about what they read.
  • Entertain me with geometrically flat women
  • Half of the book lampoons Victorian society and its strongly class-based structure. The imagination and the mathematical basis are fantastic, and when I read it as an undergrad math major it was great, but it was also readily apparent that Abbott's mathematical imagination and his social beliefs were of equal importance to him.

    In fact, the social (albeit historical nowadays) aspect is what might make this book highly relevant for slashdotters. Open source isn't cool because people say so, it's cool because it's the right thing to do. It's science and society. The same is true of Flatland.

  • I have only flipped through the new book (though there was another one published some time back called Sphereland or something of the sort -- it was sort of "Mr Tomkins in Wonderland" rewritten for the Square's hexagonal grandson). So I can't really comment.

    To those of you who haven't read the original, you should, if only to see how frightening the mores of late 19th century England really were. The culture of Flatland is one in which classism and sexism (and probably racism as well -- it's been a while since I read it) is hardcoded in the biology of its inhabitants. It doesn't put a great deal of effort into scientific logic (if you think about it, the Flatlanders basically live their lives floating in empty space), but its culture is vividly drawn.

    Sadly, Edwin Abbott Abbott's writing style doesn't measure up to his imagination -- he's very heavy-handed, with every corner of the story steeped in the telltale narrative overload of Delivering a Message. You get a peculiar picture of a man who is either strongly pro-status quo or violently against it but either way resigned to the belief that things are ordained to be as they were in his time and could never change.

    It's a peculiar little book -- I got my edition as a Dover Thrift Classic for a buck and I think it wound up in the custody of a high school theology teacher of mine (Catholic schooled and proud). I do recommend getting the cheapest printed version you can find; it's better with the pictures.

    If you're interested in a more scientifically thought out book about a flat universe, you might want to check out The Planiverse by A.K. Dewdney -- it's a story based on some old Scientific American columns he did about what life would really be like in a flat universe. It's a rather more interesting book that probably more resembles Dragon's Egg by Robert Forward than it does Flatland and follows the story of a laboratory full of grad students and researchers as their computer somehow makes contact with a two-dimensional universe through an inhabitant by the name of Yendred who leaves his relatively civilized home on a spiritual quest. It's quite heavy on details (but isn't that the point?) and a much easier read than Abbott.

    So now I guess I have to read two books...

    /Brian
  • You're a crackpot. You're mixing definitions of T. Guess what? When you don't set up the problem right, you can get all kinds of nonsensical answers. Your understanding of relativity is also pretty bad:

    In other words, relativists believe in time travel in one direction, toward the future.

    You might want to look into tachyons. No, it's not just a Star Trek notion, it's something that has had serious mathematical inqueries done to it, and rather than being prohibited by relativity, the notion came about because of relativity.

    "In conclusion, don't turn someone away who knocks at your door one day and claims to be your future great-great-great grandchild. They may be right."

    This quote that you used on your page as an example of crackpottery actually goes against what you claim physicists say! How would your great great great grandchild appear at your door if they could only travel forward in time?

    I could go on, but I think I've said enough for the moment.
  • "(a) I can write Maxwell's equation in whatever form I like : quaternions, vectors, component
    by component (go read Gell-Mann's book to see how cumbersome that is) covariant-indice
    form, and (most beautifully) in differential forms. It expresses a bunch of coupled differential equations that's all."

    You forgot my favorate form, TENSOR
    F^{mu nu} = dA^nu/dx_mu - dA^mu/dx_nu
  • It is turtles all the way down !
  • I first read Flatland AFTER I read Planiverse, and I was quite disappointed. I think Planiverse is a better, and very much less dated book.
  • here [nytimes.com] at channel.nytimes.com as usual.

    --
  • If nothing changes (==moves) in spacetime, doesn't it mean the future has already been determined? Then there would be no need for causal relations either.

    --
  • If you find this stuff about higher dimensions interesting, take a look at this article [bluesci.com]. Merely an introduction, but it goes on further into new theories, including the idea that gravity might be aware of higher dimensions (whereas other forces are not).

    --
  • Time is often considered a fourth dimension, but what was meant in the book was spatial. Einstein's General Relativity predicts that space is curved near massive objects, and it takes a fourth spatial dimension to curve the three-dimensional fabric of space. In fact the theory involves time as one of the dimensions so the four-dimensional spacetime is curved in a fifth dimension. See my article [bluesci.com] for more.

    --
  • In order to visualize the curvature of space, we need to embed it in a higher-dimensional space. But that's just an artifact of the visualization process.

    Interesting. I don't see any problem with this and I could say I agree. However, the new theories mentioned in the article are explicitly about real higher dimensions. They say gravity is much weaker than other fundamental forces because, in a small scale, gravitational flux is spread out in a higher number of dimensions. Here [lanl.gov] is the theory paper.

    In a way you could also argue that the three dimensions we know are merely a way of organizing our ideas about the surrounding world. If we never directly observe the dimension in which the spacetime is curved, it cannot be said to be real, but nevertheless it is a useful concept. You probably know there are many similar 'devices' in physics such as the quantum mechanical wavefunction. In the end reality is only what we perceive and even the three spatial dimensions are thus not real, for we only see 2-D projections of things. Now I better stop before going on deeper and referring to the Matrix ;-)

    --

  • For inside the balloon is another ballon

    Why? You state this without proof. Well, then my counter claim is that the universe isn't even a sphere, its a tube, and there is only one of them. Now we have two unprovable claims that lead to different conclusions.

    This shows why the universe must be Open

    The open-ness or closed-ness of the universe has nothing to do with geometric theory.

    This is a worthy long term destiny for us to aim for, and I urge that we start preparing now.

    Okay, this is a troll, you got me.

  • the blue on black you mean? or is it the links? i am changing the colors of links soon, because they are unreadable...but i like the blue and black...i think i might have to change the blue to a little lighter blue though....fill out a comments form from on the page.

  • this all reminds me of my tenth grade geometry teacher when he would take us on visits to flatland and planeland on the chalkboard. He would also often ask us to play touchy-feely with the triangles! Argh! The terrible memories are coming back...lets leave geometry IN THE TEXTBOOKS! Argh..just when i thought i had blocked those memories from my brain!

  • This also reminds me of a book (and website) a friend of mine made about adventures, romance and higher math. The website is at ovilstar.net [ovilstar.net] and it has some really interesting music and pictures on the site...the book is pretty interesting, and so are the essays...it is a book about an adventurer (professer ovilstar) and of course math and romance...it all mixes together really well....right?

    But really...check out ovilstar.net [ovilstar.net] and buy the book (only 4.95 and it promotes the selling of books online)

  • Thanks alot. I have it paperwise but it s great to also have it online.

  • Re:flatland on project gutenberg (Score:3)

    by EricEldred ( 175470 ) on Saturday April 21, 2001 @10:35AM (#275296) Homepage

    The Project Gutenberg text of Flatland was originally issued without the essential illustrations. Now it has them in ASCII form. If you prefer an HTML edition, Eldritch Press has it at http://www.eldritchpress.org/eaa/FL.HTM [eldritchpress.org].

    Abbott apparently based much of his satire on the work earlier published by Charles Howard Hinton. You can also read Hinton's works online at Eldritch Press, at http://www.eldritchpress.org/chh/hinton.html [eldritchpress.org].

    Hinton lead a colorful life for a mathematics professor. Rudy Rucker deserves our thanks for finding and collecting Hinton's contributions. Unfortunately, there were no reports that the year 2000 brought another visitor from that dimension to our world, as Hinton predicted.

    Does reading these books pique your interest about the Fourth Dimension? See Thomas F. Banchoff's excellent Scientific American Library paperback, "Beyond the Third Dimension," ISBN 0-7167-6015-0, 1990, 1996.

  • For all those wondering "WTF?" The internet seems to be rife with these people spouting nonsense and claiming they are talking about the forefront of physics. I'm a grad student in physics and I hear this sort of stuff in spam all the time (they just spam all the grads in the department.) This post really is nonsense, the anonymous coward post above "Re:Geometry is the key.[but not like this]" is a good summary of what is wrong with this post so I won't repeat.
  • Math geeks may want to look at this page:

    http://hypercomplex.com/research/emgrav/hypcx-p200 01015.html [hypercomplex.com]

    for a detailed introduction to Quaternions

    Check out the Vinny the Vampire [eplugz.com] comic strip

  • To repeat the referance:

    -- T.E. Bearden, "Possible Whittaker Unification of Electromagnetics, General Relativity, and Quantum Mechanics," (Association of Distinguished American Scientists 2311 Big Cove Road, Huntsville, Alabama, 35801)

    And here is the quote that I was able to dig up. But of course, since it rests on something that is over 100 years old, we can forget about it. You could have at least said the the Maxwell theory of em fields was contradicted by the observation of particle phenomena (quanta). But you didn't, and even so that misses the point completely. Apparent paradoxes are used as an argument to justify an irrational, dualistic interpretation of the theory. However, at a closer look one can find that any paradox arises only from an inconsistent physical concept or other errors in logic. Goedel proved that from a logical system which contains a contradiction, absolutely any proposition may be proven. With a consistent theoretical interpretation (in any branch of science) no paradox should occur at all.

    That being said here is the material I dug up

    " ... In discarding the scalar component of the quaternion, Heaviside and Gibbs unwittingly discarded the unified EM/G [electromagnetic/ gravitational] portion of Maxwell's theory that arises when the translation/directional components of two interacting quaternions reduce to zero, but the scalar resultant remains and infolds a deterministic, dynamic structure that is a function of oppositive directional/translational components. In the infolding of EM energy inside a scalar potential, a structured scalar potential results, almost precisely as later shown by Whittaker but unnoticed by the scientific community. The simple vector equations produced by Heaviside and Gibbs captured only that subset of Maxwell's theory where EM and gravitation are mutually exclusive. In that subset, electromagnetic circuits and equipment will not ever, and cannot ever, produce gravitational or inertial effects in materials and equipment.

    "Brutally, not a single one of those Heaviside/ Gibbs equations ever appeared in a paper or book by James Clerk Maxwell, even though the severely restricted Heaviside/Gibbs interpretation is universally and erroneously taught in all Western universities as Maxwell's theory.

    "As a result of this artificial restriction of Maxwell's theory, Einstein also inadvertently restricted his theory of general relativity, forever preventing the unification of electromagnetics and relativity. He also essentially prevented the present restricted general relativity from ever becoming an experimental, engineerable science on the laboratory bench, since a hidden internalized electromagnetics causing a deterministically structured local spacetime curvature was excluded.

    "Quantum mechanics used only the Heaviside/ Gibbs externalized electromagnetics and completely missed Maxwell's internalized and ordered electromagnetics enfolded inside a structured scalar potential. Accordingly, QM [quantum mechanics] maintained its Gibbs statistics of quantum change, which is nonchaotic a priori. Quantum physicists by and large excluded Bohm's hidden variable theory, which conceivably could have offered the potential of engineering quantum change -- engineering physical reality itself.

    "Each of these major scientific disciplines missed and excluded a subset of their disciplinary area, because they did not have the scalar component of the quaternion to incorporate. Further, they completely missed the significance of the Whittaker approach, which already shows how to apply and engineer the very subsets they had excluded.

    "What now exists in these areas are three separate, inconsistent disciplines. Each of them unwittingly excluded a vital part of its discipline, which was the unified field part. Ironically, then, present physicists continue to exert great effort to find the missing key to unification of the three disciplines, but find it hopeless, because these special subsets are already contradictory to one another, as is quite well-known to foundations physicists.

    "Obviously, if one wishes to unify physics, one must add back the unintentionally excluded, unifying subsets to each discipline. Interestingly, all three needed subsets turn out to be one and the same ..."

    In other words, you may be tossing the baby out with the bath water. But then, it's your baby. And theorectical physics is not my primary area of expertise.

    Check out the Vinny the Vampire [eplugz.com] comic strip

  • Multiple Dimensions (Score:4)

    by Alien54 ( 180860 ) on Saturday April 21, 2001 @07:19AM (#275300) Journal
    Before the advent of Relativity, Maxwell and others were very interested in the investigation of multiple dimensions. Math Geeks should search for quaternions, etc.

    In 1873, Maxwell succeeded in uniting a couple hundred years of electrical and magnetic scientific observations into a comprehensive, overarching electromagnetic theory of light vibrations ... carried across space by this "incompressible and highly stressed universal aetheric fluid ..." Maxwell's mathematical basis for his triumphant unification of these two great mystery forces of 19th Century physics were "quaternions" -- a term invented (adopted would be a more precise description) in the 1840s by mathematician Sir William Rowan Hamilton, for "an ordered pair of complex numbers" (quaternion = four).

    Complex numbers themselves, according to Hamilton's clarifications of long-mysterious terms such as "imaginary" and "real" numbers utilized in earlier definitions, were nothing more than "pairs of real numbers which are added or multiplied according to certain formal rules." In 1897, A.S. Hathaway formally extended Hamilton's ideas regarding quaternions as "sets of four real numbers" to the idea of four spatial dimensions, in a paper entitled "Quaternions as numbers of four-dimensional space," published in the Bulletin of the American Mathematical Society [4 (1887), 54-7]. It is obvious from Maxwell's own writings that, even before Hathaway's formalization, his choice of quaternions as mathematical operators for his electromagnetic theory was based on his belief that three-dimensional physical phenomena are dependent upon higher dimensional realities.

    In a tragedy for science, after Maxwell's death, two other 19th Century "mathematical physicists" -- Oliver Heaviside and William Gibbs -- "streamlined" Maxwell's original equations down to four simple (if woefully incomplete!) expressions. Because Heaviside openly felt the quaternions were "an abomination" -- never fully understanding the linkage between the critical scalar and vector components in Maxwell's use of them to describe the potentials of empty space ("apples and oranges," he termed them) -- he eliminated over 200 quaternions from Maxwell's original theory in his attempted "simplification."

    This means, of course, that the four surviving "classic" Maxwell's Equations -- which appear in every electrical and physics text the world over, as the underpinnings of all 20th Century electrical and electromagnetic engineering, from radio to radar, from television to computer science, if not inclusive of every "hard" science from physics to chemistry to astrophysics that deals with electromagnetic radiative processes -- never appeared in any original Maxwell' paper or treatise! They are, in fact--

    "Heaviside's equations!"

    You can check this out by read a highly revealing paper on the subject by another renowned British mathematical physicist of this century, Sir Edmund Whittaker, titled simply "Oliver Heaviside" (Bulletin of the Calcutta Mathematical Society, Vol. 20, 1928-29, p.202); or, another overview of Heaviside by Paul J. Nahin, "Oliver Heaviside: Sage in Solitude" (IEEE Press, New York, 1988, p.9, note 3.).

    The end result was that physics lost its promising theoretical beginnings to becoming truly "hyperdimensional" physics ... over a century ago ... and all that that implies.

    Lt. Col Thomas E. Bearden, retired army officer and physicist, has been perhaps the most vocal recent proponent for restoring integrity to the scientific and historical record regarding James Clerk Maxwell -- by widely promulgating his original equations; in a series of meticulously documented papers on the subject, going back at least 20 years, Bearden has carried on a relentless one-man research effort regarding what Maxwell really claimed. His painstaking, literally thousands of man-hours of original source documentation has led directly to the following, startling conclusion:

    Maxwell's original theory is, in fact, the true, so-called "Holy Grail" of physics ... the first successful unified field theory in the history of Science ... a fact apparently completely unknown to the current proponents of "Kaluza-Klein," "Supergravity," and "Superstring" ideas ....

    To investigate this further you should take a look at --

    "Possible Whittaker Unification of Electromagnetics, General Relativity, and Quantum Mechanics,"
    by T.E. Bearden
    (Association of Distinguished American Scientists
    2311 Big Cove Road, Huntsville, Alabama, 35801)

    Note, NOT available at Amazon

    Check out the Vinny the Vampire [eplugz.com] comic strip

  • There was an interesting book called "Planiverse" (If I remember correctly) that tracked the life of a two dimensional being. It tried to show how two dimensional beings might interact, explaining how digestion might work (to have a system like we have would split the 2D being in half), how people would pass each other (Their world looked more like Mario land, with gravity pulling you down to eart at the bottom, whereas in flatland you could roam over the entire flat space). It also covers some science, like how to change rotational speeds of 2D gears, and music, etc. All in all, fairly interesting.
  • Geometry is the descriptor of everyting all around us. I have been working on a mathematics D.Phill in which I explore new geometries. People often make the mistake that we live in a 4 dimensional universe (3 of space, 1 of time). Well, this is utterly untrue.

    The Universe can be considered 5 dimensional. Imagine that the universe is a sphere, for a moment. When it started off at the big bang, it was a point. Then it quite literally blew up, like an expanding balloon. This was called inflation.

    This is not the whole story, though. For inside the balloon is another ballon, and inside that one another. It is like an Onion, with all these different shells expanding outwards from the central fire of Creation. Extrapolate this to our 4 dimensional universe, and you can see that the Metaverse is 5 dimensional. There is some reason to believe that the metaverse will not be of infinite age, in which case there may be another 6th dimension of other metaverses constantly being created.

    This shows why the universe must be Open - it cannot contract again because it would collide with the universe underneath us. Eventually, our universe will slough off the edge of the onion, and that will be the end of us.

    Unless, that is, we start researching technologies to transverse the universes, and travel to the younger universes below us. Then eventually we can approach the everlasting point of creation at the nucleus of the onion.

    This is a worthy long term destiny for us to aim for, and I urge that we start preparing now. Why bother with the petty details of day to day life - we should be thinking of the destiny of our race, our species, our Universe, and striving to make tommorrow ours.

  • Yes, The Planiverse is a good book! I disagree with your statement that it is "not quite as mathematical as Flatland." It delves more into the physics of a real 2D world, whereas Flatland just makes up stuff when it needs to explain something, like how rain falls. (Since the missing dimension in Flatland is vertical, there is no gravity, so the author just posits a slight southward pull to get around the problem.)

    In The Planiverse, the missing dimension is horizontal, i.e., there is up and down. The author actually looks at equations in 2-space, and comes up with interesting results. For instance, inverse-square laws become simple inverse laws, which means escape velocity is infinite, and also that there are no ions. Reworking Schrodinger's equation for 2D makes the periodic table come out differently. The book also discusses the problems of designing computer logic for a 2D world.

    The weakest part is the plot and characterization. Some of it seems to be just to pad the story. For instance, there is a pointless discussion of political movements; they are similar to our own, and have nothing to do with 2D vs. 3D, and hence don't contribute to the story. But the author (Dewdney) is a science writer, not a fiction writer. So think of it as speculative sci-fi, with emphasis on the "sci" rather than the "fi," and then you can enjoy the book.

  • Rubbish. It's appealing, and often useful, to subscribe to the relativistic notion that the world line of a particle is a static curve in spacetime. However, it's just as valid to think of time evolution as motion through time. It's absolutely a matter of semantics, because the results obtained using each picture is the same. Notwithstanding your arrogant assumption that the greatest physicists of the last century have failed to share your insight, the static spacetime picture is known to every physics undergrad.

    I'm afraid your post comes off as a silly mix of conspiracy theory and pseudoscience. You may have had some physical insight, but it's buried under so much vagueness that it's impossible to tell.

  • As an amusing aside, we can replace the time t with an imaginary time called the "Euclidean time", T=it. This transformation, which is known as Wick rotation, produces a Euclidean spacetime, which is helpful because we are more used to working with Euclidean geometry.

    ds^2 = dT^2 + dx^2 + dy^2 + dz^2.

    This is not a fashionable approach to special relativity nowadays, as it does not follow the spirit of general relativity. Instead, we define the interval using the Minskowski metric, which has a minus sign in the tt component.

    But strangely enough, Wick rotation turns out to be a useful trick in performing quantum mechanical calculations using Feynman's path integral method. Roughly speaking, the action in the path integral gets transformed into a quantity called the "Euclidean Lagrangian," which is frequently much easier to sum.

  • What the ac said. Unfortuantely, the original post has now been modded up to (Score:4, Interesting), apparently on the basis that the poster claimed he is "working on a mathematics D.Phill" so his words obviously must be correct.

    Even if you neglect M-Theory, spacetime is actually three dimensional. You subtract, not add, a dimension. Look up the Holographic Principle, for example here [yetanotheruseless.com].

  • Flatland as a satire on Victorian England, where status was based on titles and money and women were considered things.

    The math part - just made it so obtuse censors allow it to be printed.
  • The story [ibiblio.org] is public domain and is freely available at project gutenberg [promo.net].
    Sindri Traustason
    "It takes two to lie, one to lie and one to listen"
  • I see three posts about Sphereland were posted within a minute of each other (10:23, 10:23, 10:24) .. obviously couldn't have been avoided .. :) Too bad, just plain bad luck.

  • I found myself anticipating, for the whole time I was reading your post, the part where you would claim that this "proves" that all science must be rubbish, that "science is just another religion", and that it also somehow therefore "proves" creationism without a doubt. Did you forget to include that part or something? Your post honestly sounds like something ripped straight out of talk.origins. Sheesh, what a load of tripe.

  • I think you're wrong. I think a good teacher/lecturer can really make a subject, and a really bad one can break it.

    I've had some pretty lousy CS lecturers (like you), but I don't hate CS. But the reason for this is that I already like programming and CompSci in general, and I'm guessing you did too. I liked it before studying it, and still like it afterwards.

    But this is obviously not going to be the case for every single subject. Take a subject like Combinatorics (2nd and 3rd year level) - I didn't already like the subject. In fact I knew essentially nothing about it when I first took it. But I ended up actually somewhat liking it, simply because we had a good lecturer, who enjoyed his subject, and knew how to make it interesting.

    The majority of students know extremely little about the subjects they're going to do at University - the experiences they have will very often be the introductions that these people will have to a subject. To try claim that the quality of the lecturing is not going to have an outcome on a students opinion of a subject is ridiculous - how can it not? The process of teaching is a dynamic two-way thing between student and teacher. A teacher who enjoys a subject can often show you the side of a subject that really is enjoyable. A teacher who doesn't, can't. You can't expect students to "magically" just pick up what is interesting and potentially enjoyable about subjects they know little or nothing about. In fact its basically by definition that students know little or nothing about a subject when they start out, otherwise it would kind of defeat the whole purpose.

    I already knew I liked programming before I went to study at University. So a lousy lecturer wasn't going to change that. But many of my subjects I had no existing opinions about. Obviosly you can't possibly *really like* every single subject - but if I think back to Univ, virtually every subject has aspects to it that would be at least interesting.

    You *might* be right about Hemos, it could well be that he just hated Geometry and finds it easier to blame the teacher. Many students do this. But to generalise this so far as to say that this is always the case, and that the teacher cannot make a difference, is flat out wrong.

  • For those of you who are updating your libraries, you might also want to check out Sphereland [fatbrain.com] by Dionys Burger. This is another sequel to Flatland which appearently talks about Einstein's theories of space-time in a Flatlander context. I say appearently because I read the book so long ago I have forgotten most of the contents, and had to be reminded by the review at Amazon.com.
  • Actually, anything before 1923 is in the public domain. Recent changes to the copyright laws just means that it's going to be many mire years before anything else enters the public domain!
  • I picked up this story today for a buck after reading the thread and was really suprised to find it more of a Victorian satire than an education on multi-dimensionality. Of course, I'm just finishing the first section now (which is presumably where most of the satire lies), so this will probably change. I read the reviews at amazon.com and was suprised at the ignorance of the reviewers. This is really blatent satire and it is rediculous to think otherwise, especially given the introduction (in the revised second edition). It is incredibly ironic to see women's rights advocates railing this book. Hell, I know next to nothing about Victorian-era England yet it was easy to pick up the social commmentary. I will say that mathematics have definitely improved somewhat in the last 120 years. The explanations are pretty simple and unneccesary for a modern reader (at least to the point I've reached).
  • The suggested price of the paperback is just a buck, so go to any of the horribly massive biblioplexes (borders, whatever) and it should be in the sci-fi section.
  • Wrote the poster:
    "This is a worthy long term destiny for us to aim for (travel[ing] to the younger universes below us), and I urge that we start preparing now. Why bother with the petty details of day to day life - we should be thinking of the destiny of our race, our species, our Universe, and striving to make tommorrow ours."

    This is either humor or an utterly bombastic pile of shit. But otherwise well written. =)

    My .02,

  • When do the works of George Gershwin attain the same status as "classical music" so schools can use the music to teach without being sued for royalties left and right?

    It will become interesting when composers and artists start to realize that nobody will bother studying their stuff in school until a LONG LONG time after its popularity has completely faded away.

    This raises the possibility that much of the culture of the 20th century will simply disappear into a legal black hole.

    This means that, some day, the last recordings of the big popular groups will simply grind themselves into dust and no one will remember who they were? Elton who? Britney what? Did Disney make movies way back then? Since the works, no longer legally copyable, will have no current fans, no one will care about preserving them. Want to be remembered as long as Beethoven or Shakespeare? Good luck. The copyright system is working against you.

  • The ultra cool completely flat version published by Arion Press, unfortanately is sold out. The full details can be found at Arion Press [arionpress.com] For those who wont click and go its http://www.arionpress.com/catalog/007.htm.
    "Format: 14 by 7 inches, accordion-fold of 56 folded panels", that makes it roughly 32 feet long when completely unfolded in all its flat glory.
  • The full text of flatland is available at http://www.geom.umn.edu/~banchoff/Flatland/ or http://www.information-resources.com/Library/libra ry73.html or http://www.jollyroger.com/xlibrary/Flatland:ARoman ceAE/Flatland:ARomanceAE1.html or if you use a TI-89, check this out http://www.ticalc.org/archives/files/fileinfo/151/ 15174.html
  • There is already a sequel to Fladland called Sphereland [amazon.com]. It was published 60 years after Flatland by a different author. Often the books are bundled together.
  • ...at least according to Iain Banks.

    Your "onion" shells are the first part of the cosmology of his Culture universe. But in the 6th dimension the universe is a torus; and as the onion shells expand and then contract they circle this torus, arriving back at their point of creation without ever colliding with the universes behind them. So each universe is closed, but an infinite number of them are created and destroyed at the center of the torus.

    Let's see, there is also an "energy grid" separating the universe-onion-shells which the Culture has not yet learned how to penetrate, though it comes in handy for powering faster-than-light spaceships.

    For more info on this I suggest reading one of these books by Iain Banks:

    The Player of Games
    Consider Phlebas
    Use of Weapons
    Excession

    There is also an online Banks fanzine called the Culture, and an online interview somewhere (do a search) where he explains this cosmology in great detail, and other items about the Culture which aren't explained fully in the books -- like how the characters get their names.

  • http://people.ne.mediaone.net/pshaughn/tandr.html its a comic strip, somewhat relevant to this book. FEAR THE PUDDING GHOSTS!
  • Excellent. You just figured out another reason why a time dimension is a myth. And if you can get this far, you can also ask yourself this question. If there is no time dimension, what's all this nonsense about time travel coming from Dr. Thorne and Sir Stephen? Did I hear anybody say "crackpottery"?
  • Q: Where do Flatlanders bury their dead? A: In a pine square.
  • Actually, that book by Rudy Rucker isn't much of a sequel to Flatland. Like almost all geometry books, it discusses the issue, but isn't framed in the Flatland world.
    However, Rucker DID write a direct sequel. It is a short story called "Message found in a copy of Flatland" and you can find it in "Transreal!" which is an anthology of his writings (and one of the best damn SF books I know). Check it out, ISBN 1-878914-00-6

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