LAB Science Fiction - file 02
The Mathematical Romance – “Flatland”
(sponsored by Amazon)
The BEST and ONLY mathematical romance ever. But also a comical and poignantly cynical critique of Victorian society. “Flatland: A Romance of Many Dimensions” (1884) by Edwin Abbott also challenges our narrow fixed ideas and prejudices through various dimensions.
“Flatland” explores higher-dimensional geometry through a 2D narrator, a Square, who was born in the 2D World, Flatland, and discovers the 1D World, Lineland, and the 3D World, Spaceland.
“Flatland” uses geometric shapes to explain dimensional slicing, perspective, and the concept of 4th and higher dimensions, at the same time, satirizing Victorian social hierarchy.
“I (Square): I must indeed confess that I do not in the least understand your Lordship. When we in Flatland see a Line, we see length and brightness. If the brightness disappears, the Line is extinguished, and, as you say, ceases to occupy Space. But am I to suppose that your Lordship gives the brightness the title of a Dimension, and that what we call “bright” you call “high”?
Stranger (Sphere): No, indeed. By “height” I mean a Dimension like your length: only, with you, “height” is not so easily perceptible, being extremely small.” (§16. How the Stranger vainly endeavoured to reveal to me in words the mysteries of Spaceland, Flatland)
Dimensionality
“Flatland” describes the two-dimensional (2D) world in a comical tone, populated by geometric shapes, from triangles, squares, to polygons, and circles. They have ONLY length and width, but NO height, even NO notion of height.
Inhabitants of Flatland (2D) live on a flat, paper-like surface in the 2D environment, moving ONLY in forward/backward or left/right directions. They perceive ONLY 1D lines, viewing others as lines that shrink or grow, bright or disappear.
Residents of Flatland (2D) are NOT able to comprehend what height, the 3rd dimension (3D), or up/down is. They see others as lines, distinguishing them ONLY by brightness, which indicates the sharpness of corners, acute angles, or obtuse angles.
Our protagonist Square is a well-educated lawyer in Flatland (2D). Even intelligent Square struggled to perceive higher dimensions from the lower-dimension perspective.
A stranger Sphere from Spaceland (3D) passing through the 2D plane of Flatland appears as ONLY a changing circle as a 2D slice. Likewise, Square from Flatland (2D) would appear as a 1D line to inhabitants of Lineland (1D). It shows different perspectives by dimension.
A Sphere from Spaceland (3D) visiting Flatland (2D) is perceived by the inhabitants as a point initially, then expanding into a circle and again contracts, just like being sliced by the CT (Computed Tomography) scan. It's similar to how a 4D hyper-sphere would appear to 3D humans.
“Flatland” suggests a mathematical thought exploration, using dimensional analogy to explain 3D, 4D, or higher dimensions by reducing them to 2D or lower-dimensional perspectives.
“Sphere: Now stretch your imagination a little, and conceive a Square in Flatland, moving parallel to itself upward.
I (Square): What? Northward?
Sphere: No, not Northward; upward; out of Flatland altogether. If it moved Northward, the Southern points in the Square would have to move through the positions previously occupied by the Northern points. But that is not my meaning.” (§16. How the Stranger vainly endeavoured to reveal to me in words the mysteries of Spaceland, Flatland)
“Upward, not Northward” – This symbolic phrase was taught to Square by Sphere. This phrase highlights the limitation of 2D thinking to introduce a NEW perpendicular dimension ‘height’, and the difficulty in comprehending higher dimensions. Our Square has repeatedly reminded this slogan to evangelize his family and inhabitants in Flatland (2D).
Geometric Social Hierarchy of Flatland
The society of Flatland (2D) is constituted by a strict, rigid hierarchy based on geometry. The social status is determined by the number of sides or the complexity of a shape. Generally, the more sides a male figure has, the higher his social standing.
1: Women – Straight Lines
Women in Flatland (2D) are depicted as simple straight lines. By view angle, a woman is seen shorter, longer, or even as a point.
Women are considered the lowest class in the social hierarchy of Flatland (2D) because they are a straight line with NO sides.
Women are portrayed as dangerous, volatile beings due to their very sharp ends. So Women MUST follow strict social rules to prevent stabbing accidents.
As Women in Flatland (2D) can be nearly invisible by view point like a needle, this is Edwin Abbott’s black humor about the scary side of women in the view of men.
“The dangers to which we are exposed from our Women must now be manifest to the meanest capacity of Spaceland. If even the angle of a respectable Triangle in the middle class is not without its dangers; if to run against a Working Man involves a gash; if collision with an Officer of the military class necessitates a serious wound; if a mere touch from the vertex of a Private Soldier brings with it danger of death;—what can it be to run against a woman, except absolute and immediate destruction? And when a Woman is invisible, or visible only as a dim sub-lustrous point, how difficult must it be, even for the most cautious, always to avoid collision!” (§4. Concerning the Women, Flatland)
2: Soldiers and Workmen – Isosceles Triangles
Soldiers and Workmen in Flatland (2D) are Triangles with two equal, thin sides and one very small side. So very sharp, acute Isosceles Triangles.
Soldiers and Workmen are the lowest class of males in Flatland (2D). They are seen as unskilled and unintelligent beings in the 2D world.
“Our Soldiers and Lowest Class of Workmen are Triangles with two equal sides, each about eleven inches long, and a base or third side so short (often not exceeding half an inch) that they form at their vertices a very sharp and formidable angle. Indeed when their bases are of the most degraded type (not more than the eighth part of an inch in size), they can hardly be distinguished from Straight lines or Women; so extremely pointed are their vertices. With us, as with you, these Triangles are distinguished from others by being called Isosceles…” (§3. Concerning the Inhabitants of Flatland, Flatland)
3: Middle Class and Professionals – Equilateral Triangles, Squares, Pentagons
Middle class, tradesmen, and professionals in Flatland (2D) are regular polygons with equal sides and equal interior angles. Their regularity looks more beautiful than that of the lower classes.
The middle-class inhabitants have more stability and social standing than the Isosceles Triangles. The protagonist, our Square, belongs to this group.
4: Nobility – Hexagons to Many-Sided Polygons
Nobilities in Flatland (2D) are Polygons with six or more sides. They are high-ranking members of the 2D society.
As a figure adds more and more sides, the Polygons become closer and closer in appearance to a Circle, which means more beautiful and more honorable in the Flatland 2D society.
“Next above these come the Nobility, of whom there are several degrees, beginning at Six-Sided Figures, or Hexagons, and from thence rising in the number of their sides till they receive the honourable title of Polygonal, or many-Sided. Finally when the number of the sides becomes so numerous, and the sides themselves so small, that the figure cannot be distinguished from a circle, he is included in the Circular or Priestly order; and this is the highest class of all.” (§3. Concerning the Inhabitants of Flatland, Flatland)
5: Priests / The Ruling Class – Circles
Finally, Priests and the ruling class in Flatland (2D) are Polygons with so many sides to perfect Circles. They are the highest echelon in the hierarchy of Flatland 2D society.
If they have tens or hundreds of sides, the Polygons appear to be Circles to the naked eye. And the closer the perfect Circle, the more beautiful and more honorable they are.
Circles are viewed as divine, infallible, and hold ALL religious and political authorities.
“It is a Law of Nature with us that a male child shall have one more side than his father, so that each generation shall rise (as a rule) one step in the scale of development and nobility. Thus the son of a Square is a Pentagon; the son of a Pentagon, a Hexagon; and so on.” (§3. Concerning the Inhabitants of Flatland, Flatland)
Social Evolution of Flatland
By the Law of Nature in Flatland (2D), male children are born with one more side than their fathers. For example, a Square’s son is born a Pentagon, a Pentagon’s son is born a Hexagon, in turn. This evolution allows for limited upward social mobility over generations.
Soldiers and workmen in Flatland (2D) are acute Isosceles Triangles. Evolution for them involves the smallest angle gaining 30 arc minutes ( 30’ = 1/2 ° = 0.5° ) each generation. So, Isosceles Triangles are becoming more obtuse, closer to the equilateral triangle generation by generation.
As the angle increases, Isosceles Triangles become Equilateral Triangles (craftsmen), then increasing the sides, squares/pentagons (gentlemen), and evolve to Polygons of many sides (nobility).
And the top of the hierarchy of the Flatland 2D society, the highest class, Circles (Priests), are perceived as having an infinite number of sides. They are also the rulers of Flatland (2D), achieved by generations of polygonal refinement.
While the Law of Nature promotes upward movement, it is NOT altogether guaranteed. However, a parent can hope his son becomes a one-more-sided being to belong to the upper class. Parents, particularly those on the cusp of nobility, often submit their children to surgeries to increase sides at the Neo-Therapeutic Gymnasium to gain upper status.
Edwin Abbott cynically likens the rigid class system of Victorian society, in which a citizen's status was determined by birth rather than talent or merit, to the social hierarchy of Flatland (2D).
Eugenics in Flatland – Irregularity
The 2D society of Flatland is strictly stratified by birth, based on the number of sides. Circles holding the highest authorities and Women, straight lines, the lowest.
There is an obsession with maintaining the purity of shape, mirroring eugenics, keeping bloodline intact and class unchangeable. It inhibits the improvement of the rigid class system in Flatland (2D).
The lower classes, specifically Isosceles Triangles, attempt to improve their status, but the establishment of the perfect hierarchy of Flatland (2D) is too rigid and invincible.
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| Eugenics Society exhibit in 1930s |
Irregularity, caused by natural deformity or poor birth from an unfit pedigree, contrasts vividly with the regularity of spotless, beautiful shapes by birth, such as Equilateral Triangles, Squares, Pentagons, and Circles, which make up the orthodoxy and authority of Flatland (2D).
Irregular figures, such as non-equilateral triangles and irregular polygons, do NOT conform to the required geometric regularity of Flatland (2D). These inhabitants are deemed lower-class and face severe discrimination in the 2D society.
Moreover, in Flatland (2D), an irregular figure, even an irregular polygon with unequal sides, is treated as a criminal, social outcast, or a dangerous, degenerate character.
In terms of the doctrine of regular configuration under the rule of elite Circles, irregular figures are viewed as severe threats to social order. So they should be harshly punished, including exile, life imprisonment, or painless extinction, leading to the destruction of the individual.
As Irregularity is viewed as a disease that MUST be cured, the State also attempts to correct Irregularity through forced surgeries at the Neo-Therapeutic Gymnasium or Regular Hospitals. But if in the cases hopelessly irregular, it's a shame that the irregular figures should be sent to the State Prison or the State Executioner.
“The art of healing also has achieved some of its most glorious triumphs in the compressions, extensions, trepannings, colligations, and other surgical or dietetic operations by which Irregularity has been partly or wholly cured.” (§7. Concerning Irregular Figures, Flatland)
“Flatland” is NOT only a popular introduction to mathematical dimension theory, but also a poignant satire on the rigid class system associated with real Victorian social hierarchy.
Social evolution in Flatland (2D) is an artificial, state-mandated system using the guise of the Law of Nature to maintain a strict, oppressive hierarchy. Through the social hierarchy of Flatland (2D), Edwin Abbott cynically criticizes classism and sexism in his contemporary Victorian society.
Through the concept of Irregularity of Figure, Edwin Abbott suggests the social issues of discrimination, racism, and unfair judgments based on disability and physical appearances.
The Neo-Therapeutic Gymnasium and the Law of Nature of Flatland (2D) are plain references to Eugenics prevalent in the late Victorian era in the 19th century.
Although published in 1884, just shortly after Francis Galton (1822-1911) coined the term Eugenics in “Inquiries into Human Faculty and Its Development” (1883),
“Flatland” is the first literary work in history, addressing the eugenic theme that would later become realized and controversial as the genetic control of social and human evolution.
Further reading (sponsored by Amazon):
● Rudolf Rucker (2012). Geometry, Relativity and the Fourth Dimension (Dover Books on Mathematics). 159 pages. Dover Publications.
(sponsored by Amazon)
“Geometry, Relativity and the Fourth Dimension” is a highly readable, popular exposition of the 4th dimension and the structure of the Universe! In “Geometry, Relativity and the Fourth Dimension,” a remarkable pictorial discussion of the curved space-time we call home, it achieves even greater impact through the use of 141 excellent illustrations!
Finding a perfect analogy in the situation of the geometrical characters in “Flatland,” Professor Rudolf Rucker continues the adventures of the 2-dimensional world visited by a 3-dimensional being to explain our 3-dimensional world in terms of the 4th dimension in “Geometry, Relativity and the Fourth Dimension”!
Table of Contents
Preface
1: The Fourth Dimension
2: Non-Euclidean Geometry
3: Curved Space
4: Time as a Higher Dimension
5: Special Relativity
6: Time Travel
7: The Shape of Space-Time
8: Conclusion
Annotated Bibliography
