Sphereland: A Fantasy About Curved Spaces and an Expanding Universe

by Dionijs Burger Jr., Cornelia J. Rheinboldt (Translator), Isaac Asimov (Introduction)

A short Science Fiction novel by a distinguished Dutch mathematician which entertains and instructs in the multi-dimensional geometries of curved space and the expanding universe.

Genres: Fiction, Mathematics, Science Fiction, Physics, Philosophy, Fantasy, Science Fiction Fantasy

208 pages, Paperback

First published January 1, 1965

Reviews

[Jafar · Rating 2 out of 5]

The sequel to Flatland. This one was written by a Dutch mathematician in 1960. Flatland was written in 1881, before relativity and quantum, and while ingenious in concept, its physics and geometry had obvious flaws. However, the first thing that Burger feels compelled to correct in this book is the status of women. This story is told by the grandson (a Hexagon) of the Square who narrated Flatland. He goes to great length ensuring us that things have changed in Flatland, and women, even though still humble straight lines, are no longer considered deprived of intelligence, and an isosceles can now freely associate with a high polygon, etc. I really didn’t care much for this book and its political and scientific correctness. Burger has taken somebody else’s book and just retouched it.

The main theme of this story is that space is curved. It turns out that Flatland is not a flat plane, but the surface of a sphere. The poor 2D Flatlanders can’t imagine how this could be because the third dimension is beyond their comprehension. They can’t see how their universe can be finite, but still have no boundaries. We have the same problem imagining our universe being finite but without boundaries. However, we can easily see how this could be the case for the surface of a sphere. We just can’t imagine the 4D hyper-sphere whose surface is our universe. And then we learn about the wicked things that a visitor from a fourth dimension can do in our 3D world. He can come out of nowhere and disappear into nowhere. He can see inside enclosed spaces, and can touch the inside of any 3D object. He can remove objects from completely enclosed areas without breaking anything. He can take a left shoe, turn it in the fourth dimension, and bring it back as a right shoe. He can take a chain and turn its links in the fourth dimension, thereby opening the chain without breaking the links. More mischievously, he can take you, turn you in the fourth dimension, and when you come back your entire world will be the mirror image of what it was before. Only if I had access to the fourth dimension!

[Jen Maybe · Rating 2 out of 5]

Certainly worse than Flatland. Flatland was structured in a way that was hopelessly Victorian, but struck me as satirical in its social hierarchies, because the theme of the book is rigidity of thinking and how lack of imagination keeps even the "enlightened" shackled to the world they know. Sphereland with a heavy hand tries to update that rigid hierarchy with something more modern. Still a lot of sexism and classism, but fair enough as far as goals go for a sequel. Then, with equally heavy hand, it goes into way too much detail about the society of Flatland, in ways that made me angry because they had nothing to do with geometry or satire. "Oh, reader of Flatland, I bet you were dying to know how the Cinderella story goes in this universe!" Nope. Not a detail I needed. Unless there was some cool math concepts behind how a shoe or a hidden identity works in two dimensions. But there aren't. They just...do Cinderella, but with a line and a polygon. Like, what are we doing here?? Is this still a speculative sci-fi book to help me grasp math and expand my imagination?? The author seems to have no clue what made Flatland a success, and generally to struggle with story. This book is perfect proof that fan fic predates the internet, and even back then most of it was shit. Skip to the parts with charts and graphs to get a little hit of that sweet geometric action (albeit belabored, dim and mostly repeated beats from Flatland), then put this thing in a dusty corner.

[Chase Zerkle · Rating 4 out of 5]

This fantasy about 2-d shapes is most definitely a math nerd book. Or could be fun to read to kids if that’s your thing. It seems like an analogy to what discovering scientific facts has been in reality. You have scientists (certain shapes) in a 2-d world trying to make sense out of strange phenomena. When their ideas are brought forth, even with facts they are turned down and ridiculed. (This happened to Galileo too btw) but being true scientists they continue their journey to discover more and more. They know that with time the other shapes will eventually come to terms that their ideas are more than fiction.

[Kenny · Rating 4 out of 5]

Using our view of a two-dimensional world as a model, Burger attempts to show us what we cannot see about the limitations of our own 3-D world from the perspective of a 4-D one. Moving constantly between the 1-D world (Lineland) to the 2-D world (Flatland), our current experience in the 3-D world (Sphereland) is the jumping-off point to begin comprehension of the 4-D world. Unfortunately, like the characters in the novel, we too are limited in our ability to visualize the dimension above us, but we can easily see the one below. Fascinating (if verbose) experiment in physics-storytelling. Next I'm reading Rudy Rucker's "Spaceland," which I gather is an attempt to more fully explain the 4th dimension, General Relativity, and beyond. What I hope to learn is whether space is curved around a universe-sized sphere or simply undulating like the waves on the ocean, the gravity wells caused by the mass of bodies.

[Nick Black · Rating 2 out of 5]

Edwin Abbott ought step up from his musty grave and administer a crushing curby on Dionys Burger for raping his quaint Flatland: A Parable of Many Dimensions. Let this be a lesson to all of us about trusting people with such ridiculous first names. Dionys? GOMBIZ.

[Jessica · Rating 5 out of 5]

Flatland and Sphereland - an iconic duo! This sequel excited me and got me thinking about our own human experience of the third dimension (and ponder about higher ones; it is positively frustrating not to be able to visualize them). The combination of math, science, and social commentary is a distinct and fascinating concoction.

[Shayan · Rating 4 out of 5]

بنظر من کتاب جالبی بود و ادامه در خوری برای کتاب قبلی به حساب میاد. نویسنده اصول و قوانین کتاب اصلی را حفظ کرده و یکسری داستان ها و مسائل منطقی و منطبق با دنیای خودمان به آن افزوده. این کتاب جنبه علمی بیشتری داره و برای دوستداران ریاضی و فیزیک مناسبه و حتی میتوان از آن مسائل فیزیکی و پاسخ های ممکن به آن ها را آموخت. طنز آن نسبت به کتاب اول خیلی کمتر بود ولی در کل از خواندنش خوشحالم.

[dani dani · Rating 4 out of 5]

I’ve not read Flatland but Sphereland was okay. Felt more like a narrative about scientific discovery, which I should have expected given it is called a fantasy. I hoped for a bit more insight on our 3D world and the 4D world but this did introduce in a light manner the concepts of a curved and expanding universe.

[Shawn M. · Rating 3 out of 5]

An interesting book that I thought wanted to do way too many things but rarely sticked to one thing. I liked the parts with the Sphere and the king of Lineland though. It was a likable book just pales in comparison to its predecessor Flatland.

[Marissa Alvarez · Rating 5 out of 5]

A continuation of Flatlands, but goes into even greater detail about angles and shapes in spherical geometry which was most interesting to me. I recommend creating a hyperbolic surface if you want to visualize their retelling of their worlds’ geometry

[Ruthie Planamenta · Rating 4 out of 5]

Certainly not as good as Flatland, but a very interesting sequel! This one takes on the implications of Einstein’s theories on space-time, again making such things as curved and expanding space accesible to people like me!

[Jessica · Rating 5 out of 5]

Qué fantasía

[Rick Hurkens]

No recollection. I guess this wasn't that interesnting as Flatland was.

[Carlos · Rating 3 out of 5]

This book is well written and it is entertaining, however, it failed to help me get a better understanding of the concept of the fourth dimension, which was the main reason I read it.

[Sky · Rating 5 out of 5]

Wiskunde👍2.0

[Sarbajit Ghosh · Rating 5 out of 5]

A nice afternoon read that quickly makes you a fan of the Flatland subgenre

[Christopher Heckman · Rating 2 out of 5]

This was on my to-read list for a long time. Unfortunately, it did not live up to expectations.

[John Markow · Rating 5 out of 5]

A fitting sequel to Flatland that succeeds in explaining some of Einstein’s work in the expanding universe. Certainly a geeky book, though, but a fun one.

[Greg · Rating 3 out of 5]

As a sequel, and expansion (no pun intended) on the ideas presented in Flatland I thought, eventually, this book had something to offer. It did provide me some insight into the ideas of curved space and an expanding universe. My critique is that it took too long to get there and there was a lot of superfluous elements like the reselling of fairy tales. I didn't think it used the Flatland setup in an inventive way. It essentially copied what happened in that book, multiple times over, changing the knowledge component each time so that was disappointing but their wise it is a good follow up.

[Jc · Rating 4 out of 5]

In the 1960s a Dutch mathematician, Dionys Burger, decided to attempt updating E.A.Abbott's 1890s classic fantasy on dimensional mathematics. Not quite as surprising and fun as the original Flatland, Sphereland nonetheless works well as a sequel. Burger attempts to include some of the changes in our view of spatial geometry since Einstein, while keeping to the simple dialogue approach of Abbott. If Flatland hadn't already seen the light of day nearly a century earlier, Sphereland would still stand well on its own. To quote another GoodReads reviewer, "this book is ridiculously cool."

[Yatharth Agarwal · Rating 5 out of 5]

“Light doesn’t travel in straight lines,” you may have heard, but this book will develop the motivation as well as your intuition for the concepts. There is no better way to understand the nature of our space than to follow Hexagon along on his journey.

The serious, almost forceful tone of the ‘updated’ social commentary seems to miss that the original was a satire (and quite a clever one at that), but even with the interspersed fairy tale adaptations, the book remains short and easy to read with a satisfying ending.

[Jean-Luc · Rating 3 out of 5]

The main character here is descended from the main character in "Flatland" by Edwin A. Abbott and the story picks up soon after the end of the first book. This time the inhabitants, with the help of the Sphere from before, learn their world isn't flat at all, it's curved. o_O

More educational and much more entertaining than the excreble "An Episode of Flatland" by Charles Howard Hinton. If you enjoyed Flatland, you will love this one.

[Bill · Rating 4 out of 5]

At the time of me writing this review, I have not yet read "Flatland" except for a few parts.

This sequel to "Flatland," written in 1960, is quite a worthy one. It expands on the Flatland universe, giving much better thought to gender roles, societal roles, and the notion of the real-world Big Bang Theory and the expanding universe, showing that our worlds, whether Flatland or the real universe, are more than what we see.

[Dru · Rating 5 out of 5]

This is about the only GOOD sequel to "Flatland". It takes the next logical step, which was to take the flat 2-D world of "Flatland" and have the denizens begin to infer, from large surveys, that their world is "curved" (on a sphere). They can't SEE the curvature, but can see that their largest triangles don't add up to 180 degrees. Brilliant! It helps understand how we 3-D creatures could "infer" that our world is curved in 4-D but not be able to SEE that curvature.

[Joe Labriola · Rating 4 out of 5]

Further building upon the canon of modern geometrical fantasy as established by the brilliantly insightful Edwin A. Abbott in his classic, "Flatland", "Sphereland" is a daring sequel by Dionijs Burger Jr. - a story that creatively if sometimes confusingly challenges what we think we know about the physical reality of the world around us, including our own dimension as well as those far beyond our limited perceptions as "3D" beings.

[H. Givens · Rating 3 out of 5]

Not as clever or as much fun as Flatland, or as well-written as The Planiverse. It doesn't address the concept with the witty social satire of Flatland, or the serious consideration of what two-dimensional science might look like in The Planiverse. The second half is rather interesting and has more of a storyline, dealing with the actual sphereland concept of the title, but the first half is random and boring.

[Stacia Service · Rating 3 out of 5]

Interesting concepts but I preferred Flatland. I felt like unnecessary story was added into this one and sometimes I just wanted the author to get to the point. They are complex theories and I like the way they are broken down, but I would prefer more time spent on explaining those than discussing the interactions of his family, community, etc. I didn’t really read this for the story, I read it for the science.

[Jaime K · Rating 3 out of 5]

This was a bit slower and not as entertaining as "Flatland" especially with the last 30 pages or so.
The square's grandson is the writer now, explaining that his grandfather's views of a 3rd dimension are now allowed, but his views of an expanding world on a curved plane are not. This delves into non-Euclidean geometry, so that was fun.

[Malli · Rating 3 out of 5]

A good, honest sequel to Edwin Abbott's Flatland. Dionys Burger uses the same characters A Square and Sphere plus adds a few new ones, including the hexagon, Puncto. The development of the plot sharpens one's understanding of the 4th dimension as well as builds an understanding of the Expanding Universe. At times it does get tedious in the middle and hence the 3 star

[Liz · Rating 5 out of 5]

flatland started it... sphereland really stretches your imagination's flexibility and the language it is written in is much more accessible. for dreamers, artists, mathematicians and anyone interested in being alive and musing about it!!!!!