Flatland A romance of many dimensions We should investigate the significance of "romanza" as distinguished form a story, a novella, or other literary forms. The comment about "many" dimensions stops short of infinitely many, but certainly goes beyond four. This is also apparent in the frontispiece. With Illustrations by the Author, A SQUARE A SQUARE = A2 = Edwin (Abbott)2. Abbott inscribed his copy to Howard Candler "From the Square". Mary Abbott signed a 1926 copy "From A Square's daughter". Note that there is no period after the "A". (Edwin A. Abbott 1838-1926) The first edition that seems to include Abbott's name is the one published in June 1926, the year of his death, in December. The (pirated) American editions have no identification of the author, although his name was revealed within the first year. * * * "Oh day and night, but this is wondrous strange" is from Hamlet, Act I Scene IV. There is an allusion to the miraculous, and to the way we confront phenomena we recognize as being beyond ordinary experience. In early editions, the next line in the same scene appeared on the cover: "And therefore as a stranger give it welcome." (as a stranger = since it is a stranger) The exhortation is one of hospitality. We are not to shrink from the wondrous, but rather welcome it and learn from it. Some readers note a resemblance between the cloud of increasing dimension and the shape of Brazil, perhaps an exotic allusion. The letters in the title are drawn in perspective, with a vanishing point off the right side of the page. The sketch of the three-dimensional cube is either orthographic, with the back face the same size as the front, or in mild one point perspective. Note that Flatland deviates from the naming convention for the other dimensions. To fit with Pointland, Lineland, and Spaceland, the proper term would be "Planeland". "Fie, fie, how franticly I square my talk" is a punning reference from Titus Andronicus * * * To The Inhabitants of SPACE IN GENERAL And H. C. IN PARTICULAR Howard Candler it is. See the many files. When EAA signed over a prepublication copy of Flatland to his best friend, he wrote "To H. C., in particular>" That volume is at the Trinity College Library, Cambridge, donated by Christopher Candler, brother of Clare Phillipson, grandson of Howard Candler. This Work is Dedicated By a Humble Native of Flatland In the Hope that Even as he was Initiated into the Mysteries There is reference here to the cult of mystery religions, where anyone who attempts to reveal the secrets is punished. The Pythagoreans represent an example of this. Of THREE Dimensions Having been previously conversant The term "conversant" suggests enough familiarity to be on speaking terms with the concept, having the vocabulary necessary for the task. With ONLY TWO So the Citizens of that Celestial Region Treating higher dimensions as celestial is a common association of the transcendental with the unreachably high. A Square uses religious language to address the Sphere when he first realizes that he has come from an unknown direction. May aspire yet higher and higher To the Secrets of FOUR FIVE OR EVEN SIX Dimensions The frontispiece suggests even more, going up at least to ten dimensions (one of the favorite dimensions of modern physicists Thereby contributing To the Enlargement of THE IMAGINATION Imagination is a central concept in Abbott's teaching,as in "The Kernel and the Husk" And the possible Development Of that most rare and excellent Gift of MODESTY Among the Superior Races Modesty is particularly difficult for those who think of themselves as superior. They will be well served if they can imagine that there are others who look down on them even as they look down on their inferiors. Of SOLID HUMANITY It isn't clear what significance a Flatlander would attach to "solid". Is it a dimensional word, so that a plane figure would think of his countrymen as solid, or is the word reserved for three-dimensions, equivalent to the use of "figure" in Flatland? And does A Square think of himself as "planar humanity", or his "humanity" three-dimensional too? * * * PREFACE TO THE SECOND AND REVISED EDITION, 1884. The second edition came out a month after the first. BY THE EDITOR Actually Edwin Abbott Abbott himself, as can be inferred from the article in the Athenaeum, written in the name of A Square, and using many of the same wordings. There was no editor involved in the project, and this gives Abbott a chance to insert one more voice into the narrative. He already speaks from A Square and from the Sphere, and here is one more level. If my poor Flatland friend retained the vigour of mind which he enjoyed when he began to compose these Memoirs, I should not now need to represent him in this preface, in which he desires, firstly, to return his thanks to his readers and critics in Spaceland, whose appreciation has, with unexpected celerity, required a second edition of his work;Essentially this means that the first printing, probably of a thousand copies, sold out almost immediately. secondly, to apologize forcertain errors and misprints (for which, however, he is not entirely responsible); The errors include the substitution of "dodecahedron" for "dodecagon" in two places, possibly the action of a copy editor who knew only of the spaceland term. Abbott has to take responsibility for writing "Lineland" next to the plane viewed edge on in one of the illustrations, since that drawing is clearly inn his own hand. That error is continued in the pirated American edition on page 113, where the illustration is turned on its side, rather counter to its geometrical purpose. The first edition refers to an isosceles painting himself with the twelve colours of a dodecahedron, even though the word 'dodecagons' is used on the privious page (page 65 of the American edition). and, thirdly, to explain one or two misconceptions. But he is not the Square he once was. Years of imprisonment, and the still heavier burden of general incredulity and mockery, have combined with the natural decay of old age Abbott himself was 46 when Flatland was written. The Square is writing after seven years of imprisonment to erase from his mind many of the thoughts and notions, and much also of the terminology, which he acquired during his short stay in Spaceland. He has, therefore, requested me There is implied a continuing conversation between the imprisoned Square and his editor, even though he does not seem to have further contact with theSphere or other Spacelanders. He states explicitly that he is deprived of all contact except for occasional visits from his brother. to reply in his behalf to two special objections, one of an intellectual, the other of a moral nature. The first objection is, that a Flatlander, seeing a Line, sees something that must be thick to the eye as well as long to the eye (otherwise it would not be visible, if it had not some thickness); and consequently he ought (it is argued) to acknowledge that his countrymen are not only long and broad, but also (though doubtless in a very slight degree) thick or high. His objection is plausible, and, to Spacelanders, almost irresistible, so that, I confess, when I first heard it, I knew not what to reply. But my poor old friendąs answer appears to me completely to meet it. łI admit,˛ said he‹when I mentioned to him this objection‹łI admit the truth of your criticąs facts, but I deny his conclusions. It is true that we have really in Flatland a Third unrecognized Dimension called Śheight,ą just as it is also true that you have really in Spaceland a Fourth unrecognized Dimension, called by no name at present, but which I will call Śextra-heightą. A Square is stating as a fact something only implied by the argument. He does not necessarily believe that Flatlanders actually have this unobservable height in the third dimension, unless he also believes that there is a similar extension in the fourth and every subsequent dimension simultaneously. But we can no more take cognizance of our Śheightą than you can of your Śextra-heightą. Even I‹who have been in Spaceland, and have had the privilege of understanding for twenty-four hours The precise timing of the trip is not clear from the text itself, and it seems almost discordant to introduce standard measurements, like furlongs and inches and hours, in referring to this very different universe. The nature of the privileged "understanding" is also unclear. the meaning of Śheightą‹even I cannot now comprehend it, nor realize it by the sense of sight or by any process of reason; I can but apprehend it by faith. This is a key statement, much more explicit than professions of faith in Flatland. Faith is a way of apprending not only the unseeable but also that which transcends our reason. The words are well chosen--he cannot comprehend it, but he can apprehend it. This use of language is similar to the Victorian debate as to whether we could conceive of higher dimensions even if we could reason about them. łThe reason is obvious. Dimension implies direction, implies measurement, implies the more and the less. Now, all our lines are equally and infinitesimally thick (or high, whichever you like); consequently, there is nothing in them to lead our minds to the conception of that Dimension. No Śdelicate micrometerą‹as has been suggested by one too hasty Spaceland critic‹would in the least avail us; for we should not know what to measure, nor in what direction. When we see a Line, we see something that is long and bright; brightness, as well as length, is necessary to the existence of a Line; if the brightness vanishes, the Line is extinguished. Hence, all my Flatland friends‹when I talk to them about the unrecognized Dimension which is somehow visible in a Line‹say, ŚAh, you mean brightnessą: and when I reply, ŚNo, I mean a real Dimension,ą The implication here is that a dimension must imply a "physical" new direction, not just a new quantity that needs an additional number to measure it, although that is a reasonalby use of the term in modern times. they at once retort ŚThen measure it, or tell us in what direction it extendsą; and this silences me, for I can do neither. Only yesterday, when the Chief Circle (in other words our High Priest) came to inspect the State Prison and paid me his seventh annual visit, and when for the seventh time he put me the question, ŚWas I any better?ą I tried to prove to him that he was Śhigh,ą as well as long and broad, although he did not know it. But what was his reply? ŚYou say I am łhigh˛; measure my łhighness˛ and I will believe you.ą The pun, or metaphor, is part of our geometrization of hierarchical relations. "Your Highness" certainly is a term worth contemplating. What could I do? How could I meet his challenge? I was crushed; and he left the room triumphant. Compare the actual dialogue at the end of the book. "Crushed" has the sense of being reduced to a lower dimension. łDoes this still seem strange to you? Then put yourself in a similar position. Suppose a person of the Fourth Dimension, condescending to visit you, were to say, ŚWhenever you open your eyes, you see a Plane (which is of Two Dimensions) and you infer a Solid (which is of Three); but in reality you also see (though you do not recognize) a Fourth Dimension, which is not colour nor brightness nor anything of the kind, but a true Dimension, although I cannot point out to you its direction, nor can you possibly measure it.ą What would you say to such a visitor? Would not you have him locked up? This is a slight change from the actual situation. It is not the visitor who is locked up but rather the disciple. Also, the challenge formulation is reminiscent of the debates over the existence of the soul, which extends, if it extends at all, in a direction we do now recognize even when we see it. The concept of auras is significant here too--do they merely surround an object in one space or do they go on forever? Well, that is my fate: and it is as natural for us Flatlanders to lock up a Square for preaching the Third Dimension, as it is for you Spacelanders to lock up a Cube for preaching the Fourth. Alas, how strong a family likeness runs through blind and persecuting humanity in all Dimensions! Points, Lines, Squares, Cubes, Extra-Cubes‹we are all liable to the same errors, all alike the Slaves of our respective Dimensional prejudices, as one of your Spaceland poets has said‹ ŚOne touch of Nature makes all worlds akiną.˛1 The Shakespearean reference here is to "Troilus and Cressida" Act III, Scene III, Line 175. The analogy across dimensions of slavery to dimensional prejudices makes the point very clearly. On this, point the defence of the Square seems to me to be impregnable. I wish I could say that his answer to the second (or moral) objection was equally clear and cogent. lt has been objected that he is a woman-hater; and as this objection has been vehemently urged by those whom Natureąs decree has constituted the somewhat larger half of the Spaceland race, So women were considered a majority of the population in Victorian England. It isn't clear what particular women he had in mind, since the reviewers are not identified by gender, except indirectly where one writer says that a woman of his acquaintance can't figure out what the book is, as if it can be only one thing. I should like to remove it, so far as I can honestly do so. But the Square is so unaccustomed to the use of the moral terminology of Spaceland that I should be doing him an injustice if I were literally to transcribe his defence against this charge. 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. The feeling of the author, as expressed by the Sphere, is clear, and this introduction makes it even clearer. The qualities assoicated with women in Flatland are to be prized more than the extrapolated rational stance of the circles. No such justification is given for his revision of his opinion of the isosceles or lower classes, except a general enlightenment and perhaps a realization that he has more in common with them than with any being from a higher dimension. 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 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. Now there's a dig. Women's history was not so much in evidence then and not that much more now, according to some writers. Thinking of history in terms of heros and villains tends to ignore the contribution of the masses, even in the history of mathematics, according to Dirk Struik on the occasion of his hundredth birthday. In a still more obscure passage he now desires to disavow the Circular or aristocratic tendencies with which some critics have naturally credited him. While doing justice to the intellectual power with which a few Circles for many generations maintained their supremacy over immense multitudes of their countrymen, he believes that the facts of Flatland, speaking for themselves without comment on his part, declare that Revolutions cannot always be suppressed by slaughter; and that Nature, in sentencing the Circles to infecundity, has condemned them to ultimate failure‹łand herein,˛ he says, łI see a fulfillment of the great Law of all worlds, that while the wisdom of Man thinks it is working one thing, the wisdom of Nature constrains it to work another, and quite a different and far better thing.˛ Abbott himself was certainly not an elitist. As Head Master of the City of London School,he representied an ideal of access to education as a way of rising above ones social station. The fact that higher classes are less fecund than lower is taken to an extreme in the story. For the rest, he begs his readers not to suppose that every minute detail in the daily life of Flatland must needs correspond to some other detail in Spaceland; Compare "The Planiverse" for example, where the correspondence is much more complete. Even Hinton was more interested in such a correspondence than was Abbott, who did not busy himself with the physics for example. and yet he hopes that, taken as a whole, his work may prove suggestive as well as amusing, to those Spacelanders of moderate and modest minds who‹speaking of that which is of the highest importance, but lies beyond experience‹decline to say on the one hand, łThis can never be,˛ and on the other hand, łIt must needs be precisely thus, and we know all about it.˛ This definitely seems to be an appeal about the religious significance of the book. The attitudes described here are those that show up in religious debate. Compare to the introduction and dedication of "The Kernel and the Husk". * * * CONTENTS PART 1: THIS WORLD 1 Of the Nature of Flatland 2 Of the Climate and Houses in Flatland 3 Concerning the Inhabitants of Flatland 4 Concerning the Women 5 Of our Methods in Recognizing one another 6 Of Recognition by Sight 7 Concerning Irregular Figures 8 Of the Ancient Practice of Painting 9 Of the Universal Colour Bill 10 Of the Suppression of the Chromatic Sedition 11 Concerning our Priests 12 Of the Doctrine of our Priests PART II: OTHER WORLDS 13 How I had a Vision of Lineland 14 How I vainly tried to explain the nature of Flatland 15 Concerning a Stranger from Spaceland 16 How the Stranger vainly endeavoured to reveal to me in words the mysteries of Spaceland 17 How the Sphere, having in vain tried words, resorted to deeds 18 How I came to Spaceland and what I saw there 19 How, though the Sphere shewed me other mysteries of Spaceland, I still desired more; and what came of it 20 How the Sphere encouraged me in a Vision 21 How I tried to teach the Theory of Three Dimensions to to my Grandson, and with what success 22 How I then tried to diffuse the Theory of Three Dimensions by other means, and of the result Part I: This World łBe patient, for the world is broad and wide.˛ The sentence is spoken by Friar Laurence in "Romeo and Juliet". The use of two nearly synonymous terms is significant. Even if we think that the world has two dimensions, which one is breadth and which is width? Height, after all, is the only unambiguous one of the three. § 1.‹Of the Nature of Flatland I CALL our world Flatland, not because we call it so, In fact, the narrator never does tell the reader what his countrymen call their land. It is frequent enough that the name of a new land is not the one used by the original inhabitants. Thus the New World is not what native Americans would use to refer to the Western Hemispherebut to make its nature clearer to you, my happy fortunate, or blessedreaders, who are privileged to live in Space.The inhabitants of a two- dimensional world would not recognize that their universe is "flat" any more than we would characterize out three-dimensional world as flat. The term is chosen to appeal to readers in a higher space. Imagine a vast large enough so there is not consciousness of a bordersheet of paper on which straight Linesmore properly, line segments, Triangles, Squares, Pentagons, Hexagons, and other figures, instead of remaining fixed in their places, move freely aboutThis may be an allusion to the growing popularity of transformational geometry, concentrating upon properties of objects as they undergo deformations, in this case translations, formulated later in Felix Klein's "Erlanger Programm". The "free" motion is hampered by the fact that the figures cannot overlap or interpenetrate , on or in the surfaceThis is a crucial distinction. If the figures are "on" the surface, then we can distinguish a top and bottom side of the plane. If they are "in" the plane, there is no such distinction., but without the power of rising above or sinking below itThe words "above" and "below" would have no meaning to beings not conscious of a direction our of their universe. very much like shadowsThe concept of two-dimensional shadow beings had already been investigated by Plato, and in the nineteenth century by Gustav Fechner in Leipzig‹only hard not allowing penetration or overlapping, which is possible for shadowsand with luminous edgesIt is crucial that objects reflect some kind of light, and even inanimate objects should to that. But animate objects seem to exude light from within? Is that the import of the term "luminous"? Note that there is a continued connection with the unexplored world of the deep sea, where the sun does not shine, but where there are beings with their own luminescence. Was this phenomenon recognized in Victorian England? ‹and you will then have a pretty correct notion of my country and countrymen.A local way of putting it, as opposed to "my world" Alas, a few years ago, I should have said łmy universe˛or perhaps more likely "the" universe, knowing nothing further but now my mind has been opened in a totally new directionto higher not just more lofty in a metaphorical senseviewsHere view is used with multiple levels, as in Plato's Allegory of the Cave, where seeing and understanding are contrasted, leading up to a higher sort of sight of things. In such a country, you will perceive at once that it is impossible that there should be anything of what you call a łsolid˛ kindAlthough this was precisely a problem for some readers who objected that the only things that really exist have at least some thickness, i.e. extension into a third dimension; but I dare say you will suppose that we could at least distinguish by sight the Triangles, Squares, and other figures, moving about as I have described them.That is, in the same way that we look down and see these recognizable shapes moving about. On the contrary, we could see nothing of the kind, not at least so as to distinguish one figure from another. Nothing was would be? What is the significance of the change in mood? visible, nor could be visible, to us, except Straight Lines; and the necessity of this I will speedily demonstrate. 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), There are two ways to do this and they give somewhat different views. Abbott apparently has in mind that the viewer is positioned so that both eyes are aligned with thre horizontal plane. It is also possible to bend sideways so that one eye becomes level with the horizontal plane while the other remains above or below. Since Flatlanders are apparently monocular, the question does not occur to the narrator. and you will find the penny becoming more and more oval that is to say, elliptical. Any projection of a circle, orthographically or in perspective, will appear as an ellipse, unless viewed from directly above in which case it appears as a circle, or frome the side, as suggested here.to your view; and at last when you have placed your eye exactly on It is difficult to find the right preposition when different dimensions are involvedthe 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.Of course the coin has thickness, adding to the confusion of those who believe that this is the only way that something is viewable. The same thing would happen if you were to treat in the same way a Triangle, or Square, or any other figure cut out of pasteboardThis model of slightly thick objects moving along on a plane raises questions about the manner of locomotion.. As soon as you look at it with your eye on the edge on the table, you will find that it ceases to appear to you a figureA general term for a two-dimensional object, and that it becomes in appearance a straight line. Take for example an equilateral Triangle‹who represents with us a Tradesman of the respectable class.As opposed to a less than respectable non-equilateral triangle. Fig. 1 represents the Tradesman as you would see him while you were bending over him from above; figs. 2 and 3 represent the Tradesman, as you would see him if your eye were close to the level, or all but on the level of the table; and if your eye were quite on the level of the table (and that is how we see him in Flatland) you would see nothing but a straight line. It is curious that the illustration of the author has the base of the triangle increasing as the eye position moves closer to the edge of the plane.When I was in Spaceland I heard that your sailors have very similar experiences while they traverse your seas and discern some distant island or coast lying on the horizon. The far-off land may have bays, forelands, angles in and out to any number and extent; yet at a distance you see none of these (unless indeed your sun shines bright upon them revealing the projections and retirements by means of light and shade), nothing but a grey unbroken line upon the water. There is no reference here to the curvature of the earth, just a rather flat island in dim light, not reveailng the coastline feature. Well, that is just what we see when one of our triangular or other acquaintances comes toward us in Flatland. As there is neither sun with us, Compare other two- dimensional worlds, like Hinton's plane world and Burger's Sphereland, where there is a sun which gives light and around which planets revolve nor any light of such a kind as to make shadows, we have none of the helps to the sight that you have in Spaceland. If our friend comes closer to us we see his line becomes larger; if he leaves us it becomes smaller: but still he looks like a straight line; be he a Triangle, Square, Pentagon, Hexagon, Circle, what you will‹a straight Line he looks and nothing else. Change of field of view is the main clue, again from a monocular perspective. You may perhaps ask how under these disadvantageous circumstances we are able to distinguish our friends from one anotheror friends from enemies: but the answer to this very natural question will be more fitly and easily given when I come to describe the inhabitants of Flatland. For the present let me defer this subject, and say a word or two about the climate and houses in our country. § 2.‹Of the Climate and Houses in Flatland AS WITH you, so also with us, there are four points of the compass North, South, East, and West.Here we think of Flatland as horizontal, although we routinely place out maps and street diagrams on walls, in which case North is Up. There being no sun nor other heavenly bodies, it is impossible for us to determine the North in the usual way; but we have a method of our own. By a Law of Nature with us, there is a constant attraction to the South;This is easily understood as a gravitational effect if we think of Flatland as horizontal, or even as a slightly inclined plane.and, although in temperate climates this is very slight‹so that even a Woman in reasonable health can journey several furlongs This is the first indication of how vast Flatland is. northward without much difficulty‹yet the hampering effect of the southward attraction is quite sufficient to serve as a compass in most parts of our earth.As it might aid a diver to determine the direction back to the surface. Moreover, the rain (which falls at stated intervals) coming always from the NorthWeather is not all that well developed in Flatland, as it is for example in The Planiverse. , is an additional assistance; and in the towns we have the guidance of the houses, which of course have their side-walls running for The most part Pentagonal houses have their bases horizontal, so the side-walls are not vertical, though "or the most part" they run in a northerly direction.North and South, so that the roofs may keep off the rain from the North. In the country, where there are no houses, the trunks of the trees serve as some sort of guide.It isn't clear what the picture of a tree would be. We might think of some marine plant floating freely but maintaining its depth by some semi-automatic process. Should a tree be thought of as a vertical cross section of a tree in space? How do we represent roots, for example? In Space, moss grows on the south side of trees. In Flatland all we need to know is that the trunk is lower than the leaves. Altogether, we have not so much difficulty as might be expected in determining our bearings.Once again the comparison with fish in an aquarium might be helpful, with gravity being one force and pressure another. Yet in our more temperate regions, in which the southward attraction is hardly felt, walking sometimes in a perfectly desolate plain where there have been no houses nor trees to guide me, I have been occasionally compelled to remain stationary for hours together, waiting till the rain came before continuing my journey. This might be similar to sailors becalmed on an overcast night, waiting for a clearing to see the North Star once againOn the weak and aged, and especially on delicate Females, the force of attraction tells much more heavily than on the robust of the Male Sex, so that it is a point of breeding, if you meet a Lady in the street, always to give her the North side of the way‹by no means an easy thing to do always at short notice when you are in rude health and in a climate where it is difficult to tell your North from your South.Sir Walter Raleigh would be able to manage it. Windows there are none in our houses: for the light comes to us alike in our homes and out of them, by day and by night, equally at all times and in all places, whence we know not. Of course windows have more functions than merely letting in light. There is little attention to ventilation in Flatland. If we go back to the marine analogy, we might ask about the origin of saltiness, or of the oxygen dissolved in the water. It was in old days, with our learned men, an interesting and oft-investigated question, łWhat is the origin of light?˛ and the solution of it has been repeatedly attempted, with no other result than to crowd our lunatic asylums with the would-be solvers. Hence, after fruitless attempts to suppress such investigations indirectly by making them liable to a heavy tax, the Legislature, in comparatively recent times, absolutely prohibited them.Thought control seems consistent with what we would consider a repressive atmosphere in Flatland I‹ alas; I alone in Flatland‹know now only too well the true solution of this mysterious problem; The implication here is that the "light" actually comes from the third dimension. But what is the source here that would provide light "equally at all times and in all places"?but my knowledge cannot be made intelligible to a single one of my countrymen; and I am mocked at‹I, the sole possessor of the truths of Space and of the theory of the introduction of Light from the world of three Dimensions‹as if I were the maddest of the mad! A counterpart in the modern world might be the debate over the nature of gravity, and the theory that gravity is just a manifestation of the curvature of space. But a truce to these painful digressions: The plight of the narrator intrudes and maintains the interest in a section that otherwise woudl be dull?let me return to our houses. The most common form for the construction of a house is five-sided or pentagonal, as in the annexed figure.Or "model" we might say, since it is no mere diagram. The two Northern sides RO, OF, constitute the roof, and for the most part have no doors; on the East is a small door for the Women; on the West a much larger one for the Men; the South side or floor is usually doorless.Note that the labelling with non-symmetrical letters andwords presumes that Flatland is being viewed "from above". Square and triangular houses are not allowed, and for this reason. The angles of a Square (and still more those of an equilateral Triangle,) being much more pointed than those of a Pentagon, and the lines of inanimate objects (such as houses) being dimmer than the lines of Men and Women, The presence of any luminosity in inanimate objects is unusual, corresponding to the concept of a "soul" in animals and even plants as well as in human beings, not an immortal soul, but a soul nonetheless.it follows that there is no little danger lest the points of a square or triangular house residence might do serious injury to an inconsiderate or perhaps absent-minded traveller suddenly therefore, running against them: and as early as the eleventh century of our era This being roughly in correspondence with our own? The implication is that the time senses are similar in the plane and in space, so A Square and the Sphere can converse at the same speed. But what determines the passage of time in a place with no diurnal events? Here is a place where the speculation in "The Planiverse" is much more elaborate., triangular houses were universally forbidden by Law, the only exceptions being fortifications, powder-magazines, barracks, and other state buildings, which it is not desirable that the general public should approach without circumspection.Compare the sharp configurations in the plans of arsenals and forts One things that is unclear is what keeps the houses "together" is there is some downward gravitational pull. It is possible to posit interior supports, and a series of "locks" so the configuration does not collapse. The signicance of A and B is not clear. At this period, square houses were still everywhere permitted, though discouraged by a special tax. But, about three centuries afterwards, the Law decided that in all towns containing a population above ten thousand,Again, a strange idea, to think of the total population even though we see only a small number of the inhabitants.the angle of a Pentagon was the smallest house-angle that could be allowed consistently with the public safety. At some juncture it is important to mention that fact that the exterior angle at each vertex of an equilateral n-gon is 2pi/n and the interior angle is pi -2p/n, with a diagram to illustrate the proof of this result.The good sense of the community has seconded the efforts of the Legislature; and now, even in the country, the pentagonal construction has superseded every other. It is only now and then in some very remote and backward agricultural district that an antiquarian may still discover a square house.What about more elaborate houses, in the shape of other regular polygons for example, or even more general shapes? The argument against such would probably be that since houses aren't very bright, it is not useful for them to have much variation. If almost all houses are pentagonal, we know how to deal with them without having to figure out what shape they have. 1 The Author desires me to add, that the misconception of some of his critics on this matter has induced him to insert in his dialogue with the Sphere, certain remarks which have a bearing on the point in question, and which he had previously omitted as being tedious and unnecessary.