*"[The universe]
is written in the language of mathematics, and its characters are triangles,
circles, and other geometric figures..."* — Galileo Galilei

By Joshua Bell.

*"How does gravity curve space? Well, imagine that the world is two
dimensional, like a rubber sheet..."*

Sure, you've heard that before. But have you really thought about life on that two dimensional sheet, or wondered where that analogy came from? When I was 8 years old an older cousin of mine introduced me to Flatland and Sphereland, and I was hooked. Of course, at the time, I missed the satire and was merely enthralled by the animate polygons, but it was a start.

- Books about Dimensions
- Fourth Dimension: Tetraspace - essays on multidimensional wheels, and more
- Bibliography for a 2005 Planiverse project by Stanford students.

**Fiction | Non-Fiction |
Video | Flatterland vs.
Sphereland | My Fiction**

While there are many books on the mathematics of higher dimensions, many of which use lower-dimensional analogies to explain various points, several works - both classical and modern - stand out as taking place in a fictional two-dimensional universe. More recently, some works take a look not just at a lower number of dimensions, but at a higher number of dimensions... or the same number, but with different qualities.

This first foray into a two-dimensional world doubles as both a lesson on the concept of higher dimensions and as a social satire on Victorian England. The narrator is A. Square, a four-sided polygon in a world of two-dimensional creatures (and thinking). A rigid social structure is in place, from women (mere lines) through the lower classes (isosceles triangles), regular polygons (such as our narrator) to the many-sided or nearly circular priest caste. All is orderly, rigid and unchanging - until a higher dimensional caller comes to visit.

Read online or download ebook:

Other readings:

In some ways an homage to the previous work, but set in a distinct— and more realistic—two-dimensional world known as Astria. The triangular inhabitants walk about on legs across the surface of their circular planet. The story introduces the world and some of the challenges facing two-dimensional creatures. Along with some of Hinton's other writings, this appears to portray the concept of higher dimensions in a spiritual or metaphysical light.

Much like Abbott's Flatland, the story is a satire, skewering colonialism, capitalism and the exploitation of the working class, the oligarchy and the military as a tool of social control, and more.

Read online or download ebook:

Other readings:

A sequel to Flatland, although written nearly a century later. This follows the adventures of A. Hexagon (grandson of A. Square) who lives in a world that has changed both socially and scientifically. Explorers proved that the world was indeed round, and the theory of higher dimensions is well known. But Flatland is still unready to cope with concepts such as the curvature or expansion of space.

Short fiction. The full text is available on the author's web site

Unlike earlier works, which use the technique of simplifying the world to teach an abstract concept (e.g. visualizing higher dimensions, the curvature of space), this work attempts to portray a plausible two-dimensional world and is best thought of as a thought-experiment towards that end. Physics, chemistry, biology, computation and culture are explored in detail. Whereas in Flatland the inhabitants are polygonal shapes sliding about on an idealized tabletop world, in The Planiverse the denizens walk on the one-dimensional surface of a circular planet (much like Hinton's Astrians), must climb over one another to pass, and build complex machines of springs and hinges. The story lasts just long enough to introduce the reader to the world and take them on a grand tour of a possible two-dimensional civilization.

Yendred - the protagonist of The Planiverse

Short fiction. I haven't read this. See the Wikipedia entry.

A delightful, illustrated student project taking the form of a parody of National Geographic, including interviews, a detailed map of Ajem Kollosh (from the Planiverse), book reviews, a message to readers from A. Square, and phony advertisements.

Another sequel to Flatland (but *not* to Sphereland). Victoria Line, the
great-great-granddaughter of A. Square, discovers her ancestor's suppressed
writings. In her time, higher dimensions are well accepted in theory but not as
a physical reality. With her guide she goes on to explore such notions as fractional dimensions, hidden spatial
dimensions, hyperbolic geometry, quantum weirdness, space-time, singularities
and time-travel, and perhaps the ultimate question - what is a geometry?

What if the story of Flatland took place with a four-dimensional being visiting our real three dimensional world, at the turn of the millennium (Y2K) during
the dot-com craze? And what if there was something sinister going on in hyperspace? That's the story of Spaceland, and unfortunately that's pretty much
all of the story. Just as in Flatland, the character visits lower dimensions to learn about hyperspace by analogy, and stumbles over issues such as
orientation and perspective. The author's attempt to spice things up with supposedly three dimensional characters falls flat. *(Sorry!)* There is
nothing new in this book; even the cameo at the end by a strange visitor from another two-dimensional universe seems out of place and unrewarding.

Greg Egan is a terrific science fiction author. A programmer and amateur mathematician and physicist, many of his works can often be summarized as: *an episode in a universe with differing laws of physics*.

The novels of the Orthogonal trilogy, 2011's The Clockwork Rocket, 2012's The Eternal Flame, and 2013's The Arrows of Time, take place in a familiar-seeming universe with four dimensions. But rather than the 3+1 (space+time) dimensons of ours, all four dimensions are interchangable. On a planet where one of the dimensions is experienced as time, life seems not too different than life on ours, although emitting light *creates* energy. But when this region of spacetime intersects with another where the arrow of time is on a different axis, the characters start to understand they are in terrible danger.

Egan's writing usually mixes world-building with an exploration of social issues, such as personal autonomy. Like many other books referenced here, developing the dialog and characters can often take a back seat to exploring the implications of novel physics, but his works are still an instant-buy for me. The ending is a bit rushed, but the conclusion to the trilogy is well earned.

The author also has an extensive web presence, and writes both detailed papers and interactive web pages that let readers explore the physics of the books. A good place to start is Plus, Minus: A Gentle Introduction to the Physics of Orthogonal.

What if... there were two dimensions of time?

Next, Egan turns his attention to exploring a world that still has four dimensions, but two are time-like. The characters are still three-dimensional, but one of those dimensions is not like the other two. The author teases with *"...the roles played by circles and spheres in our universe are taken by hyperbolas and hyperboloids, light can only travel in certain directions, and some rivers will flow uphill."* With this particular selection of dimensions, *x* and *y* are similar to those experienced by a Planiverse inhabitant (up and down aligned with gravity, forward and backward), as is the *t* arrow of time. But *u* is another spatial dimension, so the inhabitants can move sideways... but light cannot! And rotations through this dimension distort objects, making motion challenging.

Like many of Egan's other works, Dichronauts pits the protagonists against existential threats unique to their universe, and requires them to explore and utilize the unexpected characteristics of their world to find a path to survival. The characters also struggle with intriguing social issues, stemming from their symbiotic nature. Critiquing the story, the ending here is also a bit abrupt. Also, the choice was made to describe the disorienting experiences faced by the characters entirely subjectively, which becomes equally disorienting to the reader. There's no conceit in the book itself to explain what's happening as if to an omnisceient observer; I found it difficult to visualize what was happening until a second read and referencing Egan's materials on the web.

To learn more about this fascinating world, you can start at Double Plus, Double Minus: A Gentle Introduction to the Physics of Dichronauts.

A monograph published in limited numbers, this inspired articles in Scientific American and eventually led to the author to write The Planiverse.

This was followed by A Symposium on Two-Dimensional Science and Technology in 1981 and The Second Symposium on Two-dimensional Science and Technology in 1986.

Unfortunately, none of these appear to be available online or obtainable in other ways. The Preface to the Symposdium on Two-Dimensional Science and Technology (including the table of contents) and the paper Maxwell's Equations in Two Dimension are available at Jonathan (Y.) Stein's page - the author of the latter paper.

Gardner's review of Two-dimensional Science and Technology popularized two-dimensional physics to a wider audience with the wonderful illustrations typical of Scientific American at the time.

I still have the original issue! The essay has been reprinted in Gardner's The Last Recreations (1997) and in Martin Gardner: Adventures in Flatland (2022).

A PDF including the full text and illustrations of The Wonders of a Planiverse is available from the Mathematical Association of America.

I haven't read this one, but Ben Sandler wrote in with a short review:

It's sort of halfway between a textbook and a popular-level book, and it teaches all about different sorts of curved space, typically with examples in which Flatlanders explore a toroidal universe, a Klein bottle universe, and so on. (The book also deals unusual three-dimensional spaces. I found the "two-sided Moebius band" to be particularly interesting. The description involved ants like in Escher's famous Moebius band lithograph, but there were red ants on one side and black ants on the other...) In any event, although the latter chapters get harder, it's definitely worth reading as far as one can get.

Poor Ian Stewart - once he caught the Flatland bug while writing Flatterland, he couldn't stop himself. In The Annotated Flatland, Stewart dissects the people, history, meaning, and mathematics behind the original. A biography of Abbott starts things off, and practically every sentence in the original work spawns an annotation exploring some aspect of the environment which brought Flatland to be.

Two-dimensional worlds seem to call out for animated treatments, ~~but
surprisingly there aren't many~~ and they seem to be multiplying! Here
are the ones I know about:

Apparently it featured the voice talents of Dudley Moore. Disambiguation between this and the next adaptation c/o http://www.flatlandthefilm.com/faqs.html.

I only know about this one from a reference in Wikipedia and an entry in IMDB.

Based on Jeffrey Week's book (see above). Check out the web site: http://www.geom.uiuc.edu/video/sos/

A 15 minute short film - sadly, not available for public distribution, although the script is available at the home page.

Visit the web site at http://www.flatlandthemovie.com to purchase and view the trailer.

See the Wikipedia entry to learn more.

R. Perrin Ehlinger, voice artist for B-Square in the film, contacted me to let me know about this one.

See the Wikipedia entry to learn more.

Involved in a high speed collision on a Möbius strip, Professor Farnsworth, Fry, Leela and Bender are crushed into a two-dimensional universe. The episode makes multiple references to Flatland and especially The Planiverse, from talk of higher dimensions being treated as heresy to the problem with double-ended digestive systems. Multiple pieces of two-dimensional technology are shown, from doors to chains to horns. When returning to the third dimension, the crew passes through fractional dimensions, depicted as fractal shapes.

See the Wikipedia entry or the Futurama Fandom page to learn more.

Since someone asked, here are my thoughts, circa mid 2001:

My view of Sphereland is inevitably biased since I read both Flatland and Sphereland together at a young age. It always seemed the "canonical sequel" to me. Flatterland is also a sequel to Flatland both in terms of plot and intent, and definitely a product of the early 21st Century.

Flatland uses a (contemporary) Victorian setting to introduce the hot topic of the day - the notion of abstract higher and lower dimensions. The inhabitants of the 2D world and their culture are depicted through a social satire on the world of that day. It's a very concise book in that it doesn't actually try to teach you much, but - very much in the writing style of the day - merely slips a few new ideas in while you're enthralled by the tale.

Sphereland takes a "gentle" sequel approach; the story updates the setting with social reforms, bringing a (mostly) 20th Century equality, knowledge-base and technology to the culture, but plays the lessons it intends to teach about higher-spatial geometry against the strict environment of scientific dogma. Whereas Flatland, boiled down to the core, teaches the single lesson that other dimensions are conceivable as mathematical constructs, Sphereland introduces the notions of curved and expanding space and how they apply to the real world. Directly after reading Flatland, Sphereland seems the "obvious" sequel; same sort of tone, same sort of pacing, same use of analogy to introduce concepts.

Flatterland takes a "radical" sequel approach; it's a "hip and with it" 21st Century sequel; the protagonist isn't a lone scribe but a teenage girl who emails friends on the InterLine and gripes about her parents. The story moves quickly and is a wide-ranging overview of different geometries and even questions like "what the heck is a geometry anyway?", as well as diving into notions like space-time, quantum mechanics, etc. Rather than a gentle analogy or investigation by the characters, a helpful guide gives lessons about each new facet of mathematics.

Sphereland feels more like a timeless sequel to Flatland, but having had time to ponder, I think Flatterland is the true contemporary follow-on; it addresses pertinent questions now and using concepts familiar to the current audience.

Put another way: Would someone write Flatland the same way now? No - the protagonist fighting a repressive and close-minded system was a valuable tool at the time, but it's not the culture of the "educated elite" today and would get in the way of the lesson. So Flatland-the-book is an artifact of the 19th Century; Flatterland is a artifact of the 21st and does it rather well.

On the down side, it's so hip and with it that it won't age as gracefully. Already some of the cute bits are strained - the Space Girls, for example. In another year or so no-one will even get the joke. (That's a problem affecting much of our cultural output at the moment, however - c.f. the signaling protein named sonic hedgehog.) Also, it attempts to do too much and in not enough detail. I'd love to have spent more time visiting Platterland, visiting the Moobius Cow or pondering the plight of the two-and-a-half-gon and less time talking with the quantum cat or bartering with the Hawk King for a wormhole. But that's the 21st Century for you - the solution is left as a problem for the reader.

At one point I'd idly started toying with writing a sequel to Sphereland, called Fractalland. I'd barely gotten started and some notes made up about the rest of the book when I stumbled across Stewart's Flatterland. I was a tiny bit disappointed to see that someone had beaten me to it, but thrilled that another story had been added to the genre.

The story was to take the form of an epic quest to another planet, in the grand old tradition of science fiction. Voyage of the Space Beagle by A. E. Van Vogt would be an example of the micro-genre. Along the way the intrepid explorers would learn about new types of regular shapes, fractional dimensions, hidden dimensions and time dilation. Here's what I'd gotten jotted down before I hung up my keyboard:

I think that the first issue that I must clear up, before proceeding with the main narrative of this work, is that of my name. For readers familiar with the famous works Flatland: A Romance of Many Dimensions and Sphereland: A Fantasy about Curved Spaces and an Expanding Universe and thus acquainted with life on our planet, the notion of a feminine name attached to the label applied to an eight-sided polygon must appear at first as an oxymoron. Let me put your fears to rest: I am indeed a woman, and I am not an octagon in shape, but as slender and dainty a line as you might wish to imagine, thank you very much. Anne is my first name, and Octagon is the honored last name that I assumed upon marriage to my husband. Yes, it’s a bit of a silly and dated tradition, but if you knew my maiden name you would agree that I traded up.

While discussing my husband it is also best to get another important matter out of the way. Yes, he is the great-great-grandson of A. Square and the grandson of A. Hexagon, authors of those aforementioned works. He is rather embarrassed that his predilection for mathematics is not as great as that of his famous ancestors, his interests leaning much further towards works of art and culture than calculations and higher-dimensional math. That he sees this as a shortcoming is perhaps one of the reasons he was attracted to me when we met in college – at the time I was a budding student of mathematics.

- Blurb about social and technological changes since Sphereland
- Has a twin sister -
*wink wink*

- Radio contact
- Language difficulties
- “What do you look like?” – we say “our sides and angles are similar”, they say they’re also similar.

Perhaps the greatest moment of my life came when I learned that I was selected to be the Astronaut-Mathematician aboard our first mission to the planet. The need for a trained mathematician on the crew was driven by the fear that our language difficulties might prove insurmountable without a common framework, and mathematics might prove to be the universal basis from which to start.

- Inhabitants are Koch Snowflake-like lines.
- Reproduce by fragmenting
- (L-Systems?)
- Maybe inhabitants of the moon are encountered first, and they’re self-intersecting polygons?

- Kochians physics predicts "hidden" (rolled-up) dimensions
- Kochians help build a faster ship with fractal energizing plates
- Ship travels in a tenth the time, but they experience time dilation