Flatland: A Romance of Many Dimensions

by Edwin A. Abbott

Flatland: A Romance of Many Dimensions, though written in 1884, is still considered useful in thinking about multiple dimensions. It is also seen as a satirical depiction of Victorian society and its hierarchies.

A square, who is a resident of the two-dimensional Flatland, dreams of the one-dimensional Lineland. He attempts to convince the monarch of Lineland of the possibility of another dimension, but the monarch cannot see outside the line. The square is then visited himself by a Sphere from three-dimensional Spaceland, who must show the square Spaceland before he can conceive it. As more dimensions enter the scene, the story's discussion of fixed thought and the kind of inhuman action which accompanies it intensifies.

Genres: Fiction, Science Fiction, Classics, Mathematics, Fantasy, Philosophy, Literature

Kindle Edition

First published January 1, 1884

About the author

People best know British theologian and writer Edwin Abbott Abbott for his imaginative satirical novella Flatland: A Romance of Many Dimensions (1884).

This English schoolmaster authored of the mathematical satire.

He was educated at the city of London school and at college of Saint John, Cambridge, where he as fellow took the highest honors in classics, mathematics, and theology. In 1862, he took orders. After holding masterships at school of king Edward, Birmingham, and at Clifton college, he succeeded G.F. Mortimer as headmaster of the City of London School in 1865 at the early age of 26 years. He was Hulsean lecturer in 1876.

He retired in 1889, and devoted himself to literary and theological pursuits. Liberal inclinations of Abbott in theology were prominent both in his educational views and in his books. His Shakespearian Grammar (1870) is a permanent contribution to English philology. In 1885 he published a life of Francis Bacon. His theological writings include three anonymously published religious romances - Philochristus (1878), Onesimus (1882), and Sitanus (1906).

More weighty contributions are the anonymous theological discussion The Kernel and the Husk (1886), Philomythus (1891), his book The Anglican Career of Cardinal Newman (1892), and his article "The Gospels" in the ninth edition of the Encyclopædia Britannica, embodying a critical view which caused considerable stir in the English theological world. He also wrote St Thomas of Canterbury, his Death and Miracles (1898), Johannine Vocabulary (1905), Johannine Grammar (1906). Flatland was published in 1884.

Sources that say he is the brother of Evelyn Abbott (1843 - 1901), who was a well-known tutor of Balliol College, Oxford, and author of a scholarly history of Greece, are in error.

Reviews

[Robert · Rating 5 out of 5]

When you read this book, keep two things in mind. First, it was written back in 1880, when relativity had not yet been invented, when quantum theory was not yet discovered, when only a handful of mathematicians had the courage (yet) to challenge Euclid and imagine curved space geometries and geometries with infinite dimensionality. As such, it is an absolutely brilliant work of speculative mathematics deftly hidden in a peculiar but strangely amusing social satire.

Second, its point, even about itself, is still as apropos today as it was then. We still do not really know what the true dimensionality of the Universe is. It seems somehow unlikely that it is just "four", even in terms of spacetime dimensions. String theory talks seriously about thousands of dimensions. Quantum theory implements very seriously infinite numbers of dimensions. And yet we are still stuck in our 3 space dimensions mentally, hardly able to visualize the 4 in which we live "properly" unless we study theoretical physics for a decade or three, and utterly unable to mentally imagine those four embedded in a veritable Hilbert's Grand Hotel of dimensions.

Ultimately, this is a book about keeping an open mind. A really open mind -- avoiding the trap of scientific materialism and the trap of theistic idealism and the trap of any other favorite -ism you might come up with. Our entire visible space-time continuum could be nothing more than a single thin page in an infinitely thick book of similar pages, that book one of an infinite number of similar books on an infinite shelf, that shelf but one such shelf in an infinite bookcase of shelves, that bookcase but one in an infinite library of bookcases, that library but one... but by now you get the idea.

We have a hard time opening our minds up to the enormous range of possibilities, preferring to live our lives mentally trapped in a single tiny period on just one of those pages, in pointland. We may be quite unable to actually perceive the space in which our tiny point is embedded, but our minds are capable of conceiving it, and Abbot's lovely parable is a mind-expanding work to those who choose to read it that way.

rgb

[Stephen · Rating 3 out of 5]

Take a classically styled, 19th century satire about Victorian social mores…dress it up in dimensional geometry involving anthropomorphized shapes (e.g., lines, squares, cubes, etc.)…bathe it in the sweet, scented waters of social commentary…and wrap it all around humble, open-minded Square as protagonist.

The result is Flatland, a unique “classic” parked at the intersection of a number of different genres, thus pinging the radar of a wider than normal audience to appreciate (or detest) it. Since I’m recommending the book, I’m really hoping for the former, as I do not want to incur a cyber-flogging (or worse) from my fellow “goodreaders.”

So...um...math.

Let’s get this out of the way right now. As I alluded to in my intro, this book contains MATH. Now I hesitate to even mention that, because of the potential angst that subject causes many of my friends. I certainly don’t want people going all



…and dashing away in a panic.

Rest easy and increase your calm, the math is very minor. It’s really limited to discussions of geometric figures in the context of how many spatial dimensions they inhabit. Damn, that didn’t sound good either….just trust me, you won’t need a slide rule, an abacus or a lifeline to Stephen Hawking to read the book.

However, with that said, while the math is not tricky, some of the concepts can be a little brain twisty to try and visualize. Thus, I want to caution that when you get to the section where a three dimensional “Sphere” is explaining a universe containing only one dimension to our two-dimensional protagonist, you should….IMMEDIATELY…DISCONTINUE…READING…until you have:

1. burned some incense,
2. poured a big tumbler of whiskey, and
3. eaten a few “peyote” brownies, because the SHIT is about to get…



PLOT SUMMARY:

Written in 1884, the story is told by “A. Square,” who lives in Flatland, a world of two-dimensions, which means length and width, but no depth (just like the Kardashians). The men of Flatland are multi-sided polygons, and the more sides an individual has, the greater their social standing. On the other hand, women are all simple lines and have no voice in the governing of the society.

Yep...the Flatlanders are chauvinists.

The book begins with “A Square” describing his life as part of the “professional class” and providing details on daily life in Flatland. This section serves as a In reality, this is a pretty good satire on Victorian London society, the social caste system and gender inequality.

Later, “A Square” dreams of a one-dimensional world called Lineland, where the inhabitants exist as simple points along a straight line, as there is no other width or depth. I seriously hope you have that tumbler of whiskey and some brownies close by because you are going to need them. What follows is a fun, but somewhat confusing discussions during which “A Square” tries to explain the two-dimensional world to the king of Lineland.

Eventually, our protagonist wakes up back in Flatland, only to find that he is now being visited by a Sphere from a three-dimensional universe…whiskey…peyote…now. Sphere takes our flatlander on a mind-expanding, eye opening journey to witness the wonders and mysteries of the higher and higher dimensions (3rd, 4th, 5th, etc.). Afterwards, “A Square” returns to Flatland to teach the wonders of such “enlightened” dimensions to his fellow flatlanders, the result of which is…

…nope…no spoilers here!

THOUGHTS:

As I sit here, sober and “mostly” peyote free, I think I enjoyed the “ideas and concepts” of the story more than the actual plot. The writing was fine, but nothing that struck me as particularly eloquent. However, I’ve the concepts of the story have stayed with me and I have actually become more appreciative of the material as time has gone by.

Overall, I liked the book. I think it’s worth reading, but more for the interesting ideas and mental gymnastics that the narrative puts you through than for the simply enjoyment of the plot. Still, a worthwhile read, and since it’s actually a novella, you can get through it quickly without a large time commitment.

3.0 stars. Recommended!

[Taufiq Yves · Rating 4 out of 5]

This mathematical sci-fi novel, Flatland, written over 100 years ago, is more than just science fiction; its "dimension" transcends mathematical space, and it is also a satirical novel of the Victorian era.

Written from the perspective of a "Square" in a 2-dimensional Flatland, the book tells about the social customs and life of Flatland and how the "Square" painfully recognizes the 3-dimensional world when he visits the 3-dimensional space. This reflects the difficulty of our 3-dimensional beings in understanding the 4-dimensional space.

Sounds childish?

But it was written over a 100 years ago, and its ideas were very avant-garde at the time. Flatland is a classic in Western culture, but less discussed in Asian contexts, possibly because the level of sci-fi is not hard enough for modern people, or because the translation has lost a lot of its beauty.

When I first read it, I found it very satirical. It has a strict hierarchy, with "circles" at the top, followed by "polygons", "hexagons", "squares", "equilateral triangles", "isosceles triangles", and finally "line segments" representing women. And it is full of discrimination against women: In Flatland, women have no intellectual power, no forethought, and scarcely any memory. In all houses, there must be special low-level doors for women to enter.

So, this book feels very much like Orwell’s 1984, using an imaginary story to satirize the current society.

Edwin A. Abbott, was a British theologian, educator, and classicist who was good at mathematics and had many mathematician friends. He also published several books on Shakespeare studies and was dedicated to promoting British literature in secondary education. Therefore, in Flatland, there are scattered quotes from Shakespeare. He didn't use quotation marks or footnotes, but simply changed a few words. If you are familiar with Shakespeare, you will admire the appropriateness of these quotations every time you encounter them. However, those without this background will miss a lot of the fun.

For example, he used the pseudonym A. Square for the first edition and narrated it in the first person as Mr. Square. In fact, his original name, Edwin Abbott - Abbott, is exactly "A squared", with the middle name coming from his mother, and his father and mother were relatives with the same surname.

On the cover of the first edition, he even drew a map of Flatland himself, on which it says: "O day and night, but this is wondrous strange", "And therefore as a stranger give it welcome." These 2 lines are from the conversation between Horatio and Hamlet in Hamlet. Note the change from strange to stranger here. Both Shakespeare and Abbott subtly played with the substitution of words to encourage readers to accept strange new things: if friendly people welcome strangers, then they should also accept things that are strange. These two lines are used very cleverly in Flatland: If we friendly welcome strangers from other high - dimensional or low - dimensional worlds, then we should also accept that their worlds are different from ours. In Flatland, Mr. Square met strangers from other "countries" and was able to understand the culture of "other countries".

In the first part introducing Flatland, he quotes from Romeo and Juliet: "Be patient, for the world is broad and wide." - The brilliance of this quote lies in the fact that the setting of Flatland is only 2-dimensional: breadth and width. That is, this sentence is both a lazy sermon and a precise and concrete description. In the second part, Other Worlds (where Mr. Square visits the 3 - dimensional and 1 - dimensional worlds), Abott quotes from The Tempest: "O brave new world, that has such people in it." This is a sentence exclaimed by Miranda when she first saw outsiders after being exiled on the island since childhood. But he changed all the singulars to plurals in Flatland, corresponding to the new worlds of 3 dimensions and even higher dimensions. (This famous quote from Shakespeare also inspired the title of Brave New World, but with a satirical meaning.)

There are also many sentences in Flatland that use alliteration. The mention of shadows vaguely corresponds to Plato's cave: Mr. Square once left the "cave" for the new world, but was arrested and imprisoned as a heretic for propagating new knowledge (returning to the cave again).

In addition to all these cultural bags, Flatland has become a favorite of many, mainly because of its wonderful description of the 2 - dimensional space. Mathematics teachers list it as a reference for the subject and write notes and commentaries on it. The most famous one is Ian Stewart's sequel, Flatterland: Like Flatland, Only More So.

A thought provoking & fun novel.

4.2 / 5 stars.

[Fergus, Weaver of Autistic Webs · Rating 4 out of 5]

I will start my own brief multi-dimensional romance on this book - but dangerously, perhaps, exploring elements of my own experiences out into Spaceland that led me back, like Abbott, to my home-sweet-home of Flatland - with a caveat that I am a rather odd duck.

Being born with ASD didn't help at all.

We Aspies, like Supertramp, always Take the Long Way Home. The prolonged ingenuous Springtime of The Soul. And the soul rejoices in the bright laughter of - as Mallarme says, the "Jeu Suprême" - as all reality curls up comfortably at its Ironic Edges.

But the simplicity of higher mathematics in high school soon became in my eyes an overgrown jungle. It was, to me, unworkable, because we Aspies come from a psychological Flatland: to the world, Dullsville.

So my apologies at the outset to my GR friends who were math majors.

No, I took this book at a personal EXISTENTIAL level! So please: take this review very much with a grain of salt!
***

As you probably know, I like to often change the subject. My adventures in Spaceland taught me that when my interlocutors dug deep, their interjections were to me like speeding bullets. I dodged them: I knew they would spell my mediocre death-in-life: I didn't want to go there.

Didn't even want to go to Spaceland - no - but the philosophical skills my teenaged reading skills gave me PUSHED me into a sophisticated outer spaceland.

That sophistication fed my nascent knowledge.

So when I went to college, I explored Spaceland heuristically. And of course - that path leads us into the Den of the Minotaur.

Yikes.

Anyway, after King Minos' foul beast had thoroughly chewed me up and expectorated me... alone, I found myself back home in Flatland!
***

An awakening refused leads back to simplicity, as Abbott found -

And it's where I am now writing this:

In "a state of simplicity (which cost me) not less than EVERYTHING."

And won me Flatland forever...

Flatland - AKA God's Peace - a patient, silent witness to an Ugly World.

I'm my reviews, I'm a precious prima donna who always sings her heart out, and who picks up every last one of the mountains of roses cast atop her.

But in real life I'm a frightfully well-medicated old buzzard of a senior citizen who cherishes every moment of his obscurely FLAT life.

Classic Freudian Wish Fulfilment!

And that's my Secret Life of Walter Mitty.
***

Well, that was my ASD take on this playful novel.

Had I been born a much more scrupulous mathematical man, I would have seen the ironic logic of the book Totally Differently.

So now let's hear, for you more learned folks, a mathematician's explanation of why we are fated to always return to Flatland (a Much better simple take than mine, but then, I dropped Math in High School)!

https://youtu.be/JQq_w38j9v0

[فؤاد · Rating 5 out of 5]

حجمستان
یادداشتی بر داستان «پختستان» نوشتۀ ادوین ابوت

معرفی
ادوین ابوت رمان پختستان را در سال ۱۸۸۴ در انگلستان دوران ویکتوریا منتشر کرد: دوران نظام اجتماعی و اخلاقی خشک، و ازلی‌ابدی.
رمان از دو فصل تشکیل شده: در فصل اول راوی که مربعی از اهالی پختستان است سرزمین دو بعدی خود را توصیف می‌کند، شکل و شمایل هندسی و فیزیکی پختستان، این چگونه همدیگر را می‌بینند، همدیگر را می‌شناسند. همچنین از ساختار اجتماعی و سیاسی پختستان می‌گوید، اختلاف طبقاتی و تنش‌های بین اشکال مختلف، انقلاب‌ها و سرکوب‌ها، تبعیض‌های جنسیتی، و...
در فصل دوم، مربع برای نخستین بار با دنیاهای جدیدی رو به رو می‌شود، دنیاهایی که حتی تصورشان برای اهالی پختستان ناممکن است و صحبت کردن از آن، جرم. اگر فصل اول انتقاد از وضع کنونی است، فصل دوم را می‌توان نوید امکان تغییر به وضعی جدید دانست، وضعی جدیدی که همه ناممکنش می‌دانند، تنها از آن جهت که هنوز یاد نگرفته‌اند به جهت درست نگاه کنند، و به محض آن که یاد بگیرند، تعجب خواهند کرد که چطور در تمام این مدت از دیدن آن عاجز بودند.

شبی تابستانی ادوین ابوت در کتابخانه نشسته بود و به مسأله‌ی بغرنجی فکر می‌کرد. مسأله از این قرار بود که آیا می‌توان بعدی مکانی غیر از سه بعد تصور کرد که در جهتی غیر از جهاتی که می‌شناسیم امتداد داشته باشد؟ و اگر چنین جهان چهار بعدی ممکن باشد، در مقایسه با ما چه شکلی خواهد داشت؟

در همین فکرها بود، که ناگهان اتفاق شگفت‌انگیزی افتاد. در وسط اتاق مطالعه‌اش، نقطه‌ای پدید آمد، در مقابل چشمان حیرت‌زده‌اش به سرعت رشد کرد، به جنینی بدل شد، سپس کودکی شد و نوجوانی و جوانی، و در عرض چند ثانیه، پیرمردی در مقابل او ایستاده بود.

ادوین ابوت وحشت‌زده فریاد کشید و به قفسه‌ی کتاب‌ها چنگ زد تا خودش را از افتادن نگاه دارد و چند کتاب با سر و صدا زمین ریخت. زنش از این همه داد و فریاد هراسان به اتاق دوید و پرسید: «چه خبر شده؟»
ابوت بی آن که بتواند حرف بزند پیرمرد را نشان داد و زن که تازه متوجه حضور پیرمرد محترم و موقری در کتابخانه‌ی شوهرش شده بود، با رعایت آداب چند کلمه‌ای با او خوش و بش کرد و رو به ابوت گفت: «ادوین، نمی‌فهمم سبب این رفتار بدون نزاکت چیست؟ از آقای محترم دعوت کن بنشیند. من هم‌اکنون چای می‌آورم.» و از اتاق خارج شد.

پیرمرد رو کرد به ابوت، که هنوز به قفسه چنگ زده بود، و گفت: «معذرت می‌خواهم که این طور ظاهر شدم. همیشه وارد شدن به حجمستان دردسر ساز است.»

ابوت با لکنت گفت: «شما که هستید؟»

«ببخشید، فراموش کردم خودم را معرفی کنم. من متوازی الانسان هستم. در جهان چهار بعدی که از آن می‌آیم نام من این است.»

«متوازی الانسان؟»

«بله، خب این طور تصور کنید که تمام وجود شما در جهان سه بعدی، فقط یکی از اضلاع ما در جهان چهار بعدی است. همان طور که تمام وجود مربع در جهان دو بعدی، فقط یکی از پهلوهای مکعب در جهان سه بعدی است. من هم چون از انسان‌هایی دو به دو متوازی ساخته شده‌ام، متوازی الانسان نامیده می‌شوم.»

ابوت دهانش را چنان باز کرده بود که انگار در نیمه‌راه یک ناسزا متوقف شده. پیرمرد گفت: «بیایید، بگذارید نشان‌تان دهم.» و قبل از آن که ابوت بتواند کاری بکند، دست او را گرفت و او را در جهتی، نه بالا و پایین، نه چپ و راست، نه درون و بیرون، بلکه در جهتی به کلی متفاوت بلند کرد، از جهان سه بعدی برداشت و بیرون برد. ابوت از وحشت جهان چهار بعدی یک بند فریاد می‌زد، مخصوصاً وقتی چشمش به میهمانش افتاد که به شکلی غیر قابل توصیف، از انسان‌هایی دو به دو متوازی ساخته شده بود و هر انسان با تمام سه بعدش تنها یکی از اضلاع او را تشکیل می‌داد.

متوازی الانسان صبر کرد تا وحشت ابوت به انس بدل شود، و آن گاه برای او شرح داد که چطور در تمام این مدت جهانی چهار بعدی درست جلوی چشم تمام اهالی حجمستان بود، اما هیچ کس سر را به جهت درست نمی‌گرداند تا آن را ببیند، جهتی غیر از بالا و پایین و چپ و راست. و از ابوت خواست که دیگر اهالی حجمستان را از وجود این جهان عظیم مطلع کند، جهانی که حجمستان با تمام عظمتش در قیاس با آن، همچون نقش روی دیوار است.

ابوت به جهان سه بعدی نگاه کرد که همچون نقش روی دیوار، بدون حجم به نظر می‌رسید. از آن جا می‌توانست داخل کتابخانه‌اش را ببیند، انگار که سقفی به کار نباشد، با آن که سقف بود، و همسرش که سینی چای به دست سرگردان در میان اتاق ایستاده بود و سر در نمی‌آورد که مردها کجا رفته‌اند. گفت: «اما هیچ کس حرف من را درک نخواهد کرد.»

متوازی الانسان گفت: «خب چرا یک داستان نمی‌نویسی تا فهمش راحت‌تر شود؟»

«چه داستانی؟»

«نمی‌دانم. مثلاً داستان یک مربع ساکن جهان دو بعدی، که روزی کره‌ای از جهان سه بعدی در خانه‌اش ظاهر می‌شود و سعی می‌کند بعد سوم را برای او توضیح دهد.»

ابوت به تمام چند صد چشم متوازی الانسان نگاه کرد، و به نشان پذیرش سر تکان داد. از همین حالا عنوان داستان را انتخاب کرده بود: «پختستان».






این یادداشت برای سایت پلاتو نوشته شده، به این آدرس:
لینک

[Cecily · Rating 3 out of 5]

Imagine losing or gaining a spatial dimension

Living in a 3D world, my mind was pleasingly warped when I watched this 7-minute Action Lab video, explaining what a 4D ball would look like in 3D. I sent it to Apatt who likened it to And He Built a Crooked House, which in turn, reminded me of a book I’d heard of, Flatland.

Reading them one after the other was enjoyably challenging. This is a review, and star rating, of both.

And He Built a Crooked House, by Robert Heinlein

This is an early short story (1941) by a big name in sci-fi. It's about a 4D construction.

Quintus Teal is a young architect who thinks of a house as “a machine for living, a vital process, a live dynamic thing”. His big idea is to use the fourth spatial dimension to “put an eight-room house on the land now occupied by a one-room house. Like a tesseract”. He goes on to explain a lot of geometry, and I would have been bemused if I hadn’t already seen illustrations of what a tesseract, aka hypercube, would look like.

We’re all familiar with a cube and its net:


Then add a dimension for a hypercube and its net:


For more of an explanation, see video links at the end of this review.

Teal achieves his dream “by using strong girders and folding money” and takes his friends, Mr and Mrs Bailey to see it.
“That's the grand feature about a tesseract house, complete outside exposure for every room, yet every wall serves two rooms and an eight-room house requires only a one-room foundation.

But four dimensions don’t fit easily into a three dimensional world, and things become strange. Stranger than any staircases or buildings that Escher drew.

Flatland, by Edwin A Abbott

About a 2D world and published in 1884.

This short book uses geometry as an analogy for a socio-political satire of Victorian attitudes to class and social mobility; gender (in)equality; biological determinism, evolution, and eugenics; religion and the supernatural. It probably works better for those with more knowledge of and interest in the period than I have. Fortunately, my edition had lots of notes, and Abbot included several illustrations.

The key concept is that in a two-dimensional world, a three-dimensional figure is observed as a cross-section. If it's moving, it is seen as a series of cross-sections, and can appear to appear and disappear out of nowhere.

In the first half, the narrator, A Square, explains the very hierarchical social system of the two-dimensional Flatland. The more sides a shape has, and the more regular it is, the higher up the scale it is. “Delicate females” are are mere lines, “wholly devoid of brain-power, and have neither reflection, judgement, nor forethought, and hardly any memory”, but Abbot himself was a priest and headmaster who believed in women’s education, including at university.

The second half has more of a story: “my initiation into the mysteries of Space” and “the Gospel of Three Dimensions”. A Square develops a theoretical and then practical understanding of other spatial dimensions. The trouble is, that’s unimaginable and heretical in Flatland.

The progression of understanding of different spatial dimensions is well explained, with a few illustrations: A square is “a Line of Lines” and a sphere is “many Circles in one”. However, Abbott’s love of initial capital letters was a little distracting, and in part of the second section the language becomes overly Biblical and deferential (thee, thou, My Lord etc).


Image: “The Weeping Woman”, in which “Picasso used cubist forms of fragmentation to depict the face in a series of angular planes”. (Source)

Clearly explained information about 2D and 4D

To understand the fourth dimension, you need a firm grasp of the zeroth, first, second, and third dimensions. Unless you have a mathematical background, I strongly suggest you watch one or two of these videos before reading either story. I found it helpful to see the same ideas explained in slightly different ways.

• The fourth dimension explained, via second and third, in six minutes, HERE.

• Hypercubes explained and illustrated, without sound, in five minutes, HERE.

• Five-minute TED animation, summarising the story of Flatland and explaining 2D and 4D, HERE.

• Animation of the story of Flatland, narrated by Carl Sagan, in four minutes, HERE.

• Animation of the story of Flatland, made in 1965, eleven minutes, HERE.

• Carl Sagan explains Flatland and goes on to the fourth dimension, in nine minutes, HERE.

The two stories

• You can read Flatland, free, in a variety of digital formats, on Gutenberg, HERE.

• You also can find the text of And He Built a Crooked House online.


Image: “Mr Osborne, may I be excused? My brain is full.” (The Far Side, by Gary Larson)

See also

• If it’s all too much, there’s Norton Juster’s delightful picture book, The Dot and the Line, which I reviewed HERE. I included illustrations and a link to an Oscar-winning animation.

• Malcolm wrote two short stories about very high-tech homes in The Martian Chronicles, which I reviewed HERE. Usher II is a subversive dystopian comedy that's also a tribute to Poe's Fall of the House of Usher. There Will Come Soft Rains starts off almost slapstick, but takes a very tragic turn.

[Michael Finocchiaro · Rating 4 out of 5]

A curious little novella about a man a two-dimensional world thinking literally out of the box. First he explains his world in which the angles you have the higher social status you have in Flatland - Circles being the highest rank. He meets someone from Lineland (one-dimensional) who is incapable of understanding Flatland and he meets Sphere from Spaceland (three dimenions) and he is able himself to comprehend the difference between "up" and "North". However, Sphere cannot extrapolate to 4+ dimensions and when the protagonist returns to Flatland and tries to explain Spaceland, he is imprisoned as a heretic.

The text is a social criticism on the rigid thinking of hierarchal social ranks, the dogmatism and often anti-scientific bent of religion, and also has a feminist bent to it as well. A fascinating and mind-bending little book that has not aged a day after almost a century and a half.

[Dan · Rating 5 out of 5]

This book should not be read in hopes of finding an entertaining story. As a novel, it's terrible. It's plot (if you can call it that) is simple and contrived. But, it wasn't written as a novel.

Flatland is a mathematical essay, meant to explain a point: that higher dimensions (more than length, depth and width) may be present in our universe, but if they are, it will be nearly impossible for us to understand them.

The story itself consists of a two dimensional world (Flatland), in which there are people of assorted shapes. These shapes live regular lives, just as we do. The protagonist (a square), is visited by a sphere, which tries to explain to him the existence of a third dimension. This proves difficult, though, because to the square in flatland, the sphere appears to be nothing more than a circle that can expand, contract, disappear and reappear.

In the course of the explanation, the book also describes "Lineland," a one dimensional world where the inhabitants would also have trouble understanding dimensions above their own.

This book's excellence lies in the way it takes a complex topic and breaks it down into a metaphor that can be more easily understood. It argues quite well that if there is a fourth dimension, it probably isn't "time."

This book isn't one that will win wide-spread acclaim from the general reading community. For those of us who enjoy higher math, though, it's excellent.

[Apatt · Rating 5 out of 5]

“I used to be a renegade, I used to fool around
But I couldn't take the punishment and had to settle down
Now I'm playing it real straight, and yes, I cut my hair
You might think I'm crazy, but I don't even care
Because I can tell what's going on
It's hip to be square”
Huey Lewis And The News - Hip To Be Square

According to IMDB, several film adaptations have been made of Flatland, but no blockbusting Pixar / DreamWorks extravaganza just yet. If they do make one I can’t imagine a more appropriate theme song than the above Huey Lewis And The News number.

Flatland is set in a two-dimensional world and narrated in the first person by a square (or “A Square” as appears on the original edition’s book cover). In the first half of the book Square gives us a tour of his world where women are straight lines and, if you are symmetrical, the more sides you have the better. This means that circles are the elite of this society because they are really polygons with zillions of super tiny sides. Irregular polygons are abominations and isosceles are plebeians.

Special laws are applied to women because they are capable of accidentally stabbing people to death due to their pointiness. Use of colours is banned because they can be used as disguises. How these geometric persons move around without legs is deliberately left unexplained (with a bit of "lampshading"). The second half of the book tells the remarkable story of Square’s adventures in lands of different dimensions, one, three and even zero (no trip to the fourth dimension, though; no time, probably). Guided by an enigmatic Sphere who seems to have popped up out of nowhere (and who Square initially mistook to be a circle), these trips to other planes of existence enables Square to not only think outside the box but to introduce him to the existence of boxes. This was a steep learning curve for him but he adapts like a champ and becomes a more rounded individual because of it.

Flatland is a very odd novella it is part allegory, part satire, part geometry lessons, part spec fic. I generally avoid reading geometry books because they are full of problems I don’t want to consider (screw the hypotenuse, man!). However, for Flatland I don’t mind making an exception, for once I find the flat characters entirely acceptable and even find the more apparently rounded character to be arrogant and clearly obtuse in their outlook, if not in appearance. The satirical look at the class system makes this all too real issue painfully acute. One thing that blows my mind a bit is that prior to reading the book I visualized it as a story of different geometric shapes moving around going about their business. However, the denizens of the Flatland cannot actually see these different shapes. As the Square (or Edwin Abbott Abbott) mentions early in the book you have to imagine looking at these shapes with your line of sight on the same level as their surface. Mr. Abbott explains it very clearly as follows:

“Place a penny on the middle of one of your tables in Space; and leaning over it, look down upon it. It will appear a circle. But now, drawing back to the edge of the table, gradually lower your eye (thus bringing yourself more and more into the condition of the inhabitants of Flatland), and you will find the penny becoming more and more oval to your view, and at last when you have placed your eye exactly on the edge of the table (so that you are, as it were, actually a Flatlander) the penny will then have ceased to appear oval at all, and will have become, so far as you can see, a straight line.”


So all they really ever see is straight lines of different lengths, however, they can distinguish the different geometrical shapes by hearing, by touch (done by the working class only), and by sight with the aid of fog for estimating depths (different angles appear to fade differently into fog). In the one-dimensional Lineland everybody looks like a point and sideways movement is impossible; as for the zero-dimensional Pointland, there is only one denizen and he is weird!

I really enjoyed Flatland, it is bizarre and thought-provoking; it definitely gave me a new perspective on life. The treatment of women may seem a little sexist but E.A. Abbott is perhaps satirizing sexism rather than perpetuating it. I definitely recommend you read Flatland before you flatline.


Notes:
• Audiobook credit: Wonderfully read for Librovox (i.e. free) by Ruth Golding. (link)

• There are quite a few diagrams scattered over the book, drawn by AbbottX2 himself, they illustrate the geometrical concepts nicely. These should be in all editions as they are intrinsic to the story.

• There is one error in the book where Square mentions a cellar: “So I endeavoured to reassure her by some story, invented for the occasion, that I had accidentally fallen through the trap-door of the cellar, and had there lain stunned.”. You can't have a bloody cellar if you only have two dimensions and you can’t “fall through” anything.

• Another error (I think) is the existence of cupboards in Flatland. If there is no depth or verticality you can't have cupboards!

• The 3D world is called Spaceland, it is not our world. Their most popular singer is probably Britney Sphere. (͡° ͜ʖ ͡°)

• I initially thought this book was a collaboration between two abbots.

Quotes:
Yet even in our best regulated and most approximately Circular families I cannot say that the ideal of family life is so high as with you in Spaceland. There is peace, in so far as the absence of slaughter may be called by that name.

In a word, to comport oneself with perfect propriety in Polygonal society, one ought to be a Polygon oneself. Such at least is the painful teaching of my experience.

Doubtless, the life of an Irregular is hard; but the interests of the Greater Number require that it shall be hard. If a man with a triangular front and a polygonal back were allowed to exist and to propagate a still more Irregular posterity, what would become of the arts of life?

You, who are blessed with shade as well as light, you, who are gifted with two eyes, endowed with a knowledge of perspective, and charmed with the enjoyment of various colours, you, who can actually SEE an angle, and contemplate the complete circumference of a circle in the happy region of the Three Dimensions—how shall I make clear to you the extreme difficulty which we in Flatland experience in recognizing one another's configuration?

Hipster Square

[Duane Parker · Rating 4 out of 5]

I give it an extra star for it's originality, it's uniqueness. The concept was genius, Abbott was probably a math genius himself. However, as a work of literature it does not hold up well. It has a shadowy similarity to Gulliver's Travels, but falls well short of that Swift classic.

[Kinga · Rating 4 out of 5]

This was one crazy, opium fuelled, brilliant book about geometry and different dimensions and I am going to explain it the best way I can but Edwin A Abbott does it so much better.

Here is a story of Square who is a square and lives in a two dimensional world of geometrical figures. The first part of the book talks about the social breakdown of the Flatland and it is a thinly disguised satire on the Victorian society. People are divided into classes according to their geometry and the worst off are women who are not even figures; they are just straight lines. They have few rights and no one actually takes their intellect seriously. On the other hand they are dangerous because being straight lines they can easily pierce any figure. A woman from behind looks just like a dot, you might miss her until it’s to late and she has stabbed you. Different parts of Flatland developed different strategies for dealing with the danger, from not allowing women to leave their houses, to forcing them to constantly wiggle their bums, so they are visible from far. They should also sound a ‘peace-cry’ when out and about, in case anyone missed the wiggling bum. Seriously children, don’t do drugs. It makes you write things like that.

The second part of the book gets more interesting as it delves deeper into the concept of dimensions. As I said, our hero lives in a two-dimension reality. Try to imagine such a world. You probably see it as a piece of paper with various figures drawn on it. Of course, that’s how a creature from 3D world would see it. You’re looking at it from above, i.e from the third dimension. If a 2D world was your entire reality you would only be able to see lines and dots. Your eyes would be on the same level as the figures and you would see everything in one dimension and infer the second dimension because you can move in it and you have learnt it through experience.

The same way we can’t actually see the third dimension but we can tell it’s there. We know we can move in three dimensions and we know about perspective, light, shadow, etc. It is easier for us to understand a two-dimension reality than it is to imagine a four-dimension one. We can see it perfectly when our Square visits a one dimensional land and he laughs at it and tries to explain to the King that there is more to life than just looking at a dot in front of you. There is another dimension where there are not only dots but lines as well. The King of course laughs him off. Yet, when Square is confronted by Sphere who tells him about the third dimension and shows him ‘tricks’ that the third dimension allows him to do, Square is just as incredulous.

Even though the mathematics tells him there must be another dimension (and another, and another), he can’t quite believe it until Sphere shows him a little bit of a 3D world. Then he is a convert, and he quickly assumes there must be more dimensions. Fourth and fifth and ad infinitum. I think while reading this I got as close as I would ever get to understanding and imagining a 4D world. If in a 3D world we can see the insides of everything of a 2D world, then I suppose a in 4D world we would be able to actually SEE all three dimensions, all the insides of everything. My brains hurts. Am I making any sense? I thought I could see it but now it’s been a week after I finished reading the book and had those vivid dreams about the fourth dimension. The vision pales. I still believe in it but I can no longer grasp it. Just like the poor Square, back in his 2D-Land, thrown in prison for preaching revolution, still believes in the third dimension, but can no longer conjure the image of a Sphere in his head. Sometimes he feels he can almost see it again for half a second, and then it’s gone.

[Djali · Rating 4 out of 5]

5 per la seconda parte e 3 per la prima, che a tratti risulta un po’ noiosa essendo una sorta di elenco delle regole di Flatlandia.
In ogni caso, geniale!

[Julie Demar · Rating 4 out of 5]

Un racconto fantastico a più dimensioni con una velata (neanche poi troppo) critica alla chiusura mentale. Geniale.

[Nandakishore Mridula · Rating 5 out of 5]

At the outset... the 5 stars are entirely subjective. I love maths, I love playing mathematical games, I love philosophising about maths. So this book is perfect for me. But if maths is not your cup of tea, you may not enjoy it as much as I did.

I first read about this book in one of Martin Gardner's "Mathematical Games" anthologies, and was enthralled by the concept. (In fact, he discusses two books: Flatland by Edwin A. Abbot and An Episode of Flatland by Charles Hinton written with the same premise. He says Hinton's book is better, and I have managed to locate an online version recently, but have not had time to read it so far.)

We live in a world of three dimensions. It is easy for us to deal with one dimension (the line), two dimensions (the plane) and three dimensions (space). But can we conceptualise a fourth dimension? It is well-nigh impossible, for our whole being is tied up on this three-dimensional paradigm.

Abbot's fictional world is two-dimensional. The characters move about on a flat landscape. They cannot imagine a third dimension. The narrator of the story, A. Square, is living the relatively comfortable life of a country gent until he is snatched up into "Spaceland" by a sphere, a three-dimensional being. He has a view of his land from a three-dimensional perspective, and Square is never the same again. He comes back to preach the concept of Space to his fellow countrymen and is promptly incarcerated in an asylum as a lunatic.

There is no story in this short novella: it is more of a mathematical exploration and social commentary. The first part uses the Flatland society to poke fun at Victorian norms, and is quite entertaining. The inhabitants of Flatland are all geometrical figures: social pedigree is conferred by the number of sides one has, the lowliest being the isoceles triangles (the soldiers) and the highest being the cirles (the priests). (The circle is a special instance of a polygon with an infinite number of sides.) The male children of a member of one class are usually born with one more side than the parent, so social climbing is possible. However, the women are all single lines: they can't aspire to be anything other than "women"! There are also irregular polygons, who are social misfits.

Abbot explains at length the geography and history of his society. The "Chromatic Revolution" where an attempt to overthrow the established order by a scheming "irregular" is scuttled by a clever circle, through an inspiring speech in parliament worthy of Mark Antony, is especially hilarious.

In the second part, the story submerges itself in the philosophy of maths. The protagonist has a vision of "Lineland", a world of a single dimension: he tries to explain Flatland to the King of that realm, but with little success. Then, our hero has a visit from a Sphere, an inhabitant of "Spaceland", and he faces the same problem in comprehending the third dimension as the king of Lineland had in comprehending the second (later, the Sphere demonstrates the same shortsight when Square moots the possibility of a fourth dimension).

Square is transported into Spaceland by Sphere, and suddenly he can see Flatland from the outside: including the inside of the houses and the intestines of the inhabitants, all at the same time! He also comprehends that the magical ability of a Spaceland denizen to move in and out of Flatland wherever he/ she wishes is nothing but a question of simple three-dimensional geometry. Square also is witness to a parliarmentary meeting where the Sphere makes a surprise appearance, to try to convince the rulers of Flatland about the existence of space, but to no avail. The preaching of space is a state crime in Flatland, with the penalty of either death or life in confinement(according to the social status of the individual)- the ultimate fate of the narrator of the story.

Yet even though he is destined to spend his remaining life in an asylum, Square is not willing to let go of his vision of Space. Once seen, he is transformed for life.

Abbot, a teacher and theologician, uses his knowledge of philosophy and mathematics not only to create a satire, but also to raise big questions about the limitations of perception in general. It is an extremely enjoyable read, and the issues it raises will stay with you even after you finish it.

Since it is available online free from Gutenberg, I suggest everyone to give it a try.

[Charles van Buren · Rating 1 out of 5]

Very strange

Review of free Kindle edition
ASIN: B0083ZRQR4
124 pages

I have long considered FLATLAND to be an overrated mostly boring book. However it is supposed to be a classic, admired and enjoyed by many people some of whom are considered to be or consider themselves to be intellectuals. So I thought, maybe it's me. Maybe a deficiency in my ability to grasp the fine qualities of the book prevents me from understanding how great it is. Or maybe my imagination is sub par. Then I read a passage about FLATLAND from THE POT THIEF WHO STUDIED EDWARD ABBEY by J. Michael Orenduff. Orenduff, a former college professor with his PhD in Mathematical Logic, a former university president and chancellor is now a full-time author known mainly for writing the Pot Thief Mystery Series. He should understand FLATLAND as a mathematician and as an author. Here is what his characters say about FLATLAND:

“All work is 3-D,” said Martin.
We stared at him.
He looked at me. “Remember that book Flatland you made me read?”
“I didn’t make you read it.”
“When a white college student visits a fourteen-year-old dropout on the rez and suggests a book, that’s the same as making me read it.”
“But you liked it, right?”
“Yeah, because it made me feel smarter than the guy who wrote it.”
“How so?” asked Sharice.
“He says the men who live in Flatland are polygons. The fewer sides a man has, the lower he is on the social scale. So triangles are the lowest level, squares are higher, pentagons higher and so on. But he also says they can see each other and interact, which is impossible. Because if they were truly two-dimensional, they would have no sides, so there would be nothing to see.”
“You could see them from the top,” said Susannah, “and from that vantage point, you could also see how many sides they have.”
“No. To see them from on top, you’d have to be above them. But there is no up in Flatland. And there is no down. There is only north, south, east and west. So they wouldn’t even know other men existed.”
“They would when they bumped into them,” she said.
He shook his head and placed two pennies on the table, sliding them until they touched. “These pennies can bump into each other because they have sides. But imagine them without sides. I don’t mean just really flat. I mean no side dimension at all. The men in Flatland can’t bump into each other because they have no sides.”
“Why do you keep calling them men? Aren’t there women in Flatland?”
“Sure. The author says they are straight lines.
”She shook her head. “Sheesh. I might have guessed. The women are the lowest life-forms because they have only one dimension.”
“Right. And he makes the same sort of mistake in describing them, saying that seen from straight-on they look like a point. But you can’t see the end of a line because that would require that it have some height. Lines have only length. You could see them from below or above but not in a world that has only two dimensions.”
“He says something else about women,” I noted. “Because of their lack of intelligence, they accidentally pierce and kill people without even knowing it. But ten minutes later they can’t remember it happening.”
Sharice stared at me. “And you made him read that?”
“For the math part. I knew he was smart enough not to believe the stuff about women.”
“He was also smart enough not to believe the stuff about math. A guy who thinks you can see something that doesn’t have sides …wait, they can’t see anything anyway. If they had eyes, they would have to be on their surface, because they have no sides. So the only direction their eyes could look would be up. But there is no up.”
“See,”I said, trying to move beyond my having forced a misogynist book on Martin, “You’re also smarter than the author.”

I agree with Mr. Orenduff and his fictional characters. That's my position and I'm sticking to it.

There are no illustrations in the free Kindle edition. However, Amazon offers the Distinguished Chiron Edition with illustrations by the author. This is supposed to be some kind of improvement of the original 1884 edition. The illustrations which I examined seem to confirm Mr. Orenduff's opinions by depicting a very flat three dimensional world instead of a two dimensional one.

[Blaine · Rating 3 out of 5]

I call our world Flatland, not because we call it so, but to make its nature clearer to you, my happy readers, who are privileged to live in Space.

Imagine a vast sheet of paper on which straight Lines, Triangles, Squares, Pentagons, Hexagons, and other figures, instead of remaining fixed in their places, move freely about, on or in the surface, but without the power of rising above or sinking below it, very much like shadows-only hard with luminous edges-and you will then have a pretty correct notion of my country and countrymen.

In Flatland, everyone is a two-dimensional being. Each person’s place in the social order is determined by their number of sides and perfect angles (any physical deformity is usually met with death). At the top is the Priestly order of near-circular polygons. The Nobility starts with six-sided hexagons and above. Pentagons and squares—our narrator is literally named “A. Square”—form the Professional Class. After the middle class equilateral triangles, there are the soldiers and working-class isosceles triangles. Finally, at the very bottom of society, are all women, who are isosceles triangles of such a narrow angle that they are effectively straight lines, and forced to utter a “peace-cry” at all times so they’re not invisible and stabby to the menfolk.

Flatland is really two books in one, and one is much better than the other. Let’s dispense with the bad first. The book was apparently intended and understood in its time as a satire of Victorian era society. Sure, and its strict hierarchy certainly has some parallels. But the rampant misogyny is utterly absurd. And this isn’t just a modern critique. When the author released a revised second edition—in 1884!— he felt compelled to include a part apology/part defense for the harsh depiction of women in the story. So yeah, any value it has as a satire of Victorian era society is largely swamped by its sexist nonsense.

But as a geometrical thought experiment, Flatland is really cool. The story dives deeply into what a two-dimensional world would look like to its inhabitants. Then the second part of the story involves a visit to Flatland by a sphere from Spaceland, who tries to convince A. Square that there’s a three-dimensional world beyond Flatland. We see visions of one-dimensional Lineland, zero-dimensional Pointland, and consider the possibility of a four-dimensional land where instead of squares becoming cubes, cubes would become something grander, the never-named tesseract:



I’m glad I read this strange little story. The discussions about geometry in Flatland are highly imaginative. I’d recommend it to anyone capable of disregarding the grossly outdated societal notions.

[B. A. Balfour · Rating 3 out of 5]

3.5 stars.

This is a very personal rating. I think I would’ve enjoyed this far more if I’d read it 10-20 years ago. The parts that were a pleasant surprise existed only to facilitate all the things I largely already knew.

This is by no means a bad book. On the one hand, this is a remarkable work of creativity by a man whose passion for his subject shines on every page. On the other, it may be better suited to capture the imagination of people in the early stages of their love for maths, geometry and higher dimensions. It simply came a little too late for me.

[Jon Nakapalau · Rating 3 out of 5]

I have to be honest...did not get as much from this book as I could have because of my decayed math skills (not that there was ever much there to decay). But it was a "bucket-list" book that I thought was quite inventive. Think this would be a really good STEM book for 8th graders; if they can understand the concepts here then I think they are ready for high school math.

[Carol, She's so Novel ꧁꧂ · Rating 4 out of 5]

For the most part I hate maths, other than stats & arithmetic, but I loved this absolutely mad book!

My copy didn't come with the line drawings, but they are available on http://www.geom.uiuc.edu/~banchoff/Fl... I only found this site after I finished my read last night. I was happy with my imagination travelling with A Square trying to puzzle out his universe!

I wish I had discovered this book when I was at intermediate school. I was decent at maths until Year 10 and using my love of words may have made me try harder with maths - although I don't think anything could have made me & trig friends!

Knocked off a star for an often patronising attitude to women. Abbott may have thought he was being funny.

I didn't.

[Tanabrus · Rating 4 out of 5]

Un breve testo geniale e didattico, sorprendente pensando che è stato redatto nel 1884 (e infatti era talmente avanti che all'epoca passò sottotraccia, venendo poi ripreso qualche decennio dopo).

Il quadrato che vive in Flatlandia, mondo a due dimensioni, prima ci racconta il suo mondo e la sua vita, chi vi abita, come funzionano i rapporti sociali e quali leggi governano le loro vite. E qui si va di sottile satira e palate di fantasia.
Poi arriva l'elemento che cambia tutto, il visitatore dalla terza dimensione che poco a poco cambia la prospettiva del Quadrato aprendogli gli occhi all'universo in tre dimensioni.
E facendolo così bene da portarlo a ipotizzare ulteriori dimensioni, inizialmente rifiutate sdegnosamente dalla Sfera (riottosa a vedere la figura bidimensionale smettere di vederlo quale creatura suprema) e poi accettate e concesse, quasi con una vena di orgoglio per l'allievo.

Una lezione di matematica e geometria, l'ipotesi di dimensioni ulteriori per noi inconcepibili come l'altezza per le figure bidimensionali (o la lunghezza per quelle unidimensionali, o la pluralità per quelle a zero dimensioni) e l'invito a rimanere con una mente aperta, a non cedere a facili dogmi, a cercare la verità.

Divertente e profetico.

[Paul · Rating 4 out of 5]

What a fantastic little thought-experiment, only really half-disguised as a story. Through his witty little parable, Abbott manages to explore the physical, mathematical, societal, philosophical and theological without once spoon-feeding his readers (OK, maybe there's a little bit of spoon-feeding in the earlier chapters).

It's only a shame, then, that this is without a doubt the most misogynist book I've ever read in my forty-odd years... Oh, well; I suppose nothing's perfect...

[George Kaslov · Rating 4 out of 5]

Quite a charming allegory for the English society of the time, and boy does it show it's age. This is basically covered by everyone who reviewed this book, so I am not going to talk about that. What I noticed and I haven't seen anybody mention this yet, is the fact that at the time when this book was written Darwinian evolution has already grasped popular imagination. Just look how he talked about careful pairings between men and women to produce an equilateral triangle and then how each generation after that is achieved gets more sides until it reaches their version of perfection that is the circle. As I am aware people looked towards evolution with quite an optimism at the time and started envisioning utopias that will come to existence with careful work, selection and patience. Just look at the squares enlightenment at the prospect of 3 then 4 and as many dimensions it can possibly go.

Now this book, by it's writing style would get 3 stars, but no one can write something that after reading it makes me spend a night thinking about tesseracts (4 dimensional cubes) and glomes (4 dimensional spheres) and not be rewarded. Both mindfuckery and awesomeness.

[Wee Lassie · Rating 5 out of 5]

This is without a doubt, the weirdest book I've ever read. Took me a little while to get into but once I did, I couldn't put the thing down. I would heartely recomend this to anyone not put off by the idea of shapes being the main characters.

[Carolyn · Rating 1 out of 5]

Read this piece of misogyny in high school. Yes, I understand that it's a product of its time. Yes, I have heard the warnings not to "misconstrue" Flatland's "seeming" misogyny, because it's not only a satire of Victorian attitudes toward women, it's satire that's positively "Swiftian", apparently.

Well! In that case, I'm totally revising my opinion before I haul off and pierce a square with my pointy one-dimensionality.

Or rather, on second thought, I don't buy it. Swift was funny. This book isn't.

The thing is, there's a difference between allegory and satire. This book is at its heart an allegory: It uses the language of geometry as a way to figuratively describe Victorian society, sort of, but mostly it's interested in using Victorian society as a way to describe geometry. Its so-called "satire" of the class structure of the society, as expressed in shapes, extends to not much more than stating that the number of sides of a being represents his status in the society: Circles are nobility, pentagons are doctors, squares are lawyers, equilateral triangles are merchants, isosceles triangles are working-class. Women aren't regular polygons, like upstanding men; they aren't even irregular polygons, like the criminals of Flatland. Women are practically pointy lines - just barely existing in two dimensions, parallelograms so narrow as to almost exist in only one. They're considered dangerous, irrational, driven purely by emotion, and so forth. Then there's the Female Code, in which these dangerous pointy line-like women have to use separate doorways and have to keep up a Peace Cry whenever in the presence of men, so as to alert the menfolk to the imminent danger of their presence. Best of all, the book is full of catty asides about their brainlessness, such as "with the usual hastiness and unreasoning jealousy of her sex", "I knew that to any woman in Flatland the narrative of my adventures must needs be unintelligible", "the passion of the moment predominates, in the Frail Sex, over every other consideration", etc., etc.

Here's a fun one:

"Then [the Sphere] proceeded to initiate me into mysteries yet higher than those I had witnessed, shewing me how to construct Extra-Solids by the motion of Solids, and Double Extra-Solids by the motion of Extra-Solids, and all "strictly according to Analogy," all by methods so simple, so easy, as to be patent even to the Female Sex."

Nice! I see what you did there, dude.

None of this, to me, reads satirically. As I said, it's allegorical, which means that to the author, it represents a certain truth. I get that this book is an early example of science fiction, and that many people choose to hold it in reverence for that reason. I can see why mathematicians might like it; it's not like math-oriented fiction is thick upon the ground.

But while I can appreciate how schoolmaster Abbott wanted to use his allegory to explain dimensions to bored schoolchildren, frankly I don't see any non-misogynistic reason to differentiate women in this particular way. If it's a comment on classes in Victorian society, well, women are of as many different classes as men. If it's a satire of the ways in which women are constricted by that society, I'm not sure why it's necessary to only point to this through occasional catty asides about their lack of understanding and brainlessness, without delving into any other aspects of their condition in a restrictive society. He just took a few potshots every now and then, and went back to explaining dimensions. This is supposed to be Swiftian satire? Not so much. Not satire at all, from what I can tell. For that matter, there isn't a whole lot of clever veiled commentary on the lot of the different classes in Flatland, either - just that they are represented by different numbers of sides. Yeah. ALLEGORY.

And I've read the preface to the second and third editions, which is considered by some to be the clue that indicates the author didn't really believe what he wrote on that front. It was written a few years after the initial publication, from the "perspective" of the sphere from Spaceland, which, existing in an additional dimension, is considered to be more enlightened than the square that "authored" the book. So, in this preface, the sphere says:

"It has been objected that he is a woman-hater; and as this objection has been vehemently urged by those whom Nature's decree has constituted the somewhat larger half of the Spaceland race, I should like to remove it, so far as I can honestly do so. ... Acting, therefore, as his interpreter and summarizer, I gather that in the course of an imprisonment of seven years he has himself modified his own personal views, both as regards Women and as regards the Isosceles or Lower Classes. Personally, he now inclines to the opinion of the Sphere that the Straight Lines are in many important respects superior to the Circles. But, writing as a Historian, he has identified himself (perhaps too closely) with the views generally adopted by Flatland, and (as he has been informed) even by Spaceland, Historians; in whose pages (until very recent times) the destinies of Women and of the masses of mankind have seldom been deemed worthy of mention and never of careful consideration."

Bolded emphasis mine. The square has modified his own personal views, the sphere says. Uh huh. In other words, those original views were indeed misogynistic? (And what the heck does he mean by "in many important respects", I wonder...I went back to double-check this, and I see absolutely nothing about what Spheres supposedly think of women. Apparently the satire doesn't extend to three dimensions. Oh, whatever, it's just tiresome.)

Look, I don't honestly know what views this man held in his own life; I haven't seen any non-Flatland writings of his. But in their absence, and as Abbott was a schoolmaster, he would perhaps have appreciated the application of the philosophical principle known as Ockham's razor to this question. Contrary to popular belief, the principle of Ockham's razor doesn't really suggest that the simplest explanation is best: It suggests that, when considering competing hypotheses, the one that makes the fewest assumptions should be selected. It seems to me that people have to bend over backwards to defend this book against charges of misogyny; calling it out as frankly misogynist (and yes, a product of its time), on the other hand, requires far fewer assumptions about the author's intention. So until some additional data turn up that would allow empirical testing of either hypothesis, I don't see any reason to buy the more tortured explanation, which ends up sounding simply apologistic.

[Carlo Mascellani · Rating 4 out of 5]

Trama originale. Un po' lenta la prima parte, sin troppo veloce quella finale, ma non importa. Esclusione, pregiudizio, chiusura mentale e conseguente condanna di tutte e tre la fanno da padrone in questa storia godibile. Non sono in sé nuove le tematiche (basti pensare a Swift), ma il modo in cui sono trattate e la profondità con la quale vengono sviscerate le ragioni che sostengono la necessità di un fecondo relativismo culturale in luogo di una sterile dittatura assolutistica. Consigliato.

[Luana · Rating 3 out of 5]

Questo libricino è talmente matto e bello che dovete andare a comprarlo, e siccome so che le mie maniere dispotiche, e poco convincenti (purtroppo non ho le doti dei venditori di cose inutili che riescono a farti comprare di tutti convincendoti che siano utili)forse non vi indurranno all'acquisto, armatevi di convincitudine vostra, e andate a comprarvelo, o a rubarlo, o a prenderlo in prestito dalla biblioteca.
Perché? Leggete quanto segue

Se come me siete sempre stati delle perfette schiappe in matematica e geometria questo è il libro che vi illuminerà di sapienza, una volta finito di leggerlo vorrete andare dai vostri vecchi professori/maestri, sbattere loro Flatlandia in testa e urlare MA CHE ACCIDENTI TI COSTAVA SPIEGARMELO COSì SEMPLICE?? . Purtroppo però temo si tratti di un provvedimento anticostituzionale, dunque, anziché prendervela con vecchi pedagoghi incapaci, potrete gentilmente ringraziare Abbott che nel 1882 ha scritto questo libro non so con quali intenti (devo ancora leggere saggi/articoli/, ma era troppo forte la voglia di parlarvene), ma di sicuro aveva capito che nel 1991 sarebbe nata una asina matematica come me, quindi ha voluto far parlare figure geometriche.

Se al contrario siete divini, comprendete la matematica e la geometria, avete l'occulto dono del far di conto e del comprendere combinazioni di lati, angoli, e quant'altro, tranquilli, a Flatlandia ce n'è anche per voi.
Per quanto siate saccenti, dubito infatti che in uno dei vostri compiti perfetti delle superiori o esami da 30/30 con lode dell'Università, le figure geometriche abbiano mai preso a muoversi o a parlare.

Ma non è tutto qui. Se pensavate che Geometria e Politica e Filosofia potessero difficilmente incastrarsi tra loro, vi sbagliavate. Perché nella Geometria di Flatlandia la Politica sessista e razzista discrimina le Donne, gli Isosceli, gli Irregolari e assurge a razza perfetta i Circoli. Quando poi qualcuno vede o sente, anche per sua non colpa, qualcosa che non andava visto o sentito, viene fatto sparire. Che cosa assurda, si potrebbe pensare. E invece no. Succede anche da noi. Quindi tranquilli, a Flatlandia ci ritroverete molto.

Non incazzatevi troppo con Abbott se ce l'aveva con le donne.
Insomma, Dante ha reagito bene. Ma a volte gli approcci sessuali non vanno a finire in Commedie, sopratutto non in Divine.

La mia cartolina da Flatlandia smette qui di essere priva di raziocinio, scusate, ma sono ancora troppo esaltata.
Se decidete di farci un salto, tuttavia, chiedetemi l'indirizzo, perché colleziono cartoline, e da Flatlandia le ricevo ben volentieri.

[Matteo Fumagalli · Rating 5 out of 5]

Videorecensione: https://www.youtube.com/watch?v=i-UCR...

[TAP · Rating 2 out of 5]

This may be allegory and/or satire, but I was hella bored. Chapter 1 is genius. The rest did nothing for me.

Concept > execution.

[Liam O'Leary · Rating 5 out of 5]

My 4th best read of 2020.
An incredible, timeless book.

Flatland should be prescribed reading for mathematics in secondary school, because it teaches so much conceptually using basic mathematics that everyone can understand. My interpretation below might clarify what I mean.

It demonstrates the utility of mathematics is abstraction, not measurement. It's what the units 'represent' that enables and motivates quantitative calculations. By counting something we bring a perception into existence, and we count when we care about and can demonstrate that it can be counted by more than one observer. Counting is also a subject decision insofar as we attribute a value to the variable counted, and a meaning to what its counting could mean. 'What counts has been counted'... in a nutshell?

It also demonstrates that mathematical models can model social systems. The strength of mathematical models of social systems is that they avoid any obvious sociocultural bias and so enables things that would normally be socially censored to be claimed as theoretical propositions. In short, it enables discourse about stereotypes and demonstrates the possibility of universal social or moral values. I feel a lot of the social norms of Flatland still apply today, in a way.

The importance of Spaceland I think is to demonstrate how it is nearly impossible to mathematically, or universally, demonstrate evidence of realities outside of measurable reality, because they begin and end in the subjective experience of the individual. Which is used as an argument both for and against religion. I think Flatland as a theological text favours spiritual belief in its lowly depiction of atheism or 'man as God' in Pointland as a less diverse system, geometrically speaking. I guess in a way it's calling existentialists navel-gazers insofar as they take their perception to solipsistic levels that is perhaps theologically speaking commits the blasphemy of 'self-as-a-God'. This shows why Flatland is so brilliant — it can simply, concisely and thoroughly show a complex understanding of human nature — such as how theists fail to explain their conviction, and atheists fail not to reject any conviction that conflicts with their original sense of perception. Most people don't get get that far in a debate in an hour let alone in less than 100 pages!

Ok... I think I've made my point. I think most people read this as some 'goofy maths' essay, but I see it as a social commentary that somehow covers the archetypes of society and the perception of reality in less than 100 pages and is somehow still ahead of its time a century later.

[Roy Lotz · Rating 4 out of 5]

For why should you praise, for example, the integrity of a Square who faithfully defends the interests of his client, when you ought in reality rather to admire the exact precision of his right angles? Or again, why blame a lying Isosceles, when you ought rather to deplore the incurable inequality of his sides?

This is one of those delightful little books, so difficult to review because its charms require no toil to appreciate, and also because the book is so short you might as well read it and skip the reviews. So I’ll keep my remarks brief.

The charm of the book lies in its conceit, rather than its execution. Indeed, though certainly able, Abbott is not an expert writer; nor does he pretend to be. The genius of this book is in the simple beauty of its premise: What would life be like for a square living in a two-dimensional world?

Abbott wrings a remarkable amount out of this simple question. First, he gives us a satire of Victorian culture—perhaps the less enduring part of this work, though certainly keen and ruthless in its modest way. To me, the most interesting point Abbott makes in his satire has to do with education. The residents of Flatland spend all their time learning various methods to identify the shapes of others. For if you are living on a two-dimensional plane, telling a square from a circle is no easy matter, as they all appear to you as flat lines. And this question of shapes is important in Flatland, as one’s status depends on one’s shape, with irregulars being the lowest and circles the highest. So was Abbott implying that, when we educate our own youngsters, most of this “education” consists in merely teaching them to navigate our own social hierarchy?

But of course, the more fascinating part of the work has to do with dimensions. How would the possibility of two dimensions appear to a one-dimensional creature? Incomprehensible. And how would the prospect of three dimensions seem to a two-dimensional creature? Nonsense. To the residents of Flatland, tales of cubes and spheres appear like so much absurd metaphysics. Abbott uses this point to show how narrow is our mental framework, how completely blind we are to realities outside our everyday, commonsense world. Doing so, Abbott elevates this work from novelty to true art. For after satirizing the world we know, he gives us a glimpse of a world beyond.