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Literature Study GuidesFlatlandPart 1 Chapters 5 6 Summary

Flatland | Study Guide

Edwin Abbott Abbott

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Flatland | Part 1, Chapters 5–6 : This World | Summary



Chapter 5: Of Our Methods of Recognizing One Another

The narrator moves on to the subject of how figures in Flatland recognize each other and assess each other's angles when they appear to each other as a straight line. There are three main ways. First, people in Flatland have a more highly developed sense of hearing and can distinguish not only acquaintances, but also different social classes through voice recognition. The upper classes, however, are not very good with voice recognition. Thus for them there is a danger of being fooled by a clever Triangle who has trained himself to sound like a more complex Polygon.

Second, among women and the lower classes, most people introduce themselves by feeling, and the most common introduction is some variation on "permit me to ask you to feel and be felt by my friend Mr. So-and-so." The people of Flatland, through natural inclination and training, are able to discern shapes quickly by touching a single angle. The only exception is the subtle angles of a many-sided polygon, which may differ by only a few degrees and confound even the highly educated. With women and acute Isosceles, feeling is potentially dangerous if people do not hold relatively still. This happened with one of A Square's own triangular ancestors, and it set the family back generations.

Square takes a moment to answer readers' potential question about how Flatlanders can understand angles without being able to see them. Square points out that they can feel them and are trained to do so. They also keep specimens of very acute Triangles chained in classrooms to help children learn to feel angles. In some places these "educational specimens" are fed and can live for several years, but in "better regulated" regions, they found it is more efficient to starve them across the course of a month and then replace them. This is more expensive, but Square opines that it means that the children have fresher angles to learn from. It also helps control the Triangle population.

Chapter 6: Of Recognition by Sight

The higher classes of Flatland recognize one another and assess shape by a third measure: sight. They are able to do this by extensive training and because many parts of Flatland have a persistent fog that lends a slight atmospheric fade to a figure. With diagrams, Square demonstrates how if one is looking at a shape straight on from a view that bisects one of its angles, the amount of obscurity the fog gives to the two visible sides on either side of that angle will let the viewer determine the angle, and thus the shape, of the figure.

Obviously, in motion and with other figures at various angles, this all becomes very difficult, even for a mathematician such as Square. Square believes that this completely justifies all the money that is endowed to universities that teach sight recognition to the elites. He also believes that only Polygons can truly appreciate polygonal society. Upper-class Polygons use sight exclusively, as they consider feeling both vulgar and counterproductive to learning sight recognition.

A few noble youths are unable to pass their final test at the university, and they are in Square's opinion "truly pitiable." They are not fit for the society of the upper classes and are not permitted to do the work of the lower classes. They also have trouble marrying, and their offspring are, according to Square, also unfortunate and sometimes even irregular. These unfortunate nobles have often been at the head of revolutions, and some statesmen have begun to argue that any nobles who cannot pass their exams should be imprisoned for life or painlessly euthanized.


Abbott's grasp of geometry shows in these chapters. The interior angles of any Polygon follow the expression (n – 2)*180/n. This means all angles in an Equilateral Triangle will be 60 degrees ((3 – 2)*180/3). In a Square all angles will be 90 degrees ((4 – 2)*180/4), and in a Pentagon they will be 108 degrees ((5 – 2)*180/5). This means that the difference in angles between a middle-class Triangle and a Square will be a whopping 30 degrees, and between a Square and a Pentagon 18. However, a 20-sided Polygon will have internal angles of 162 degrees, and a 24-sided one will have internal angles of 165 degrees. That is a mere 3-degree difference, which, as Square points out, is easy to confuse, especially when both figures are in motion.

This question of angles also acts as pointed social commentary. A difference of 58.5 degrees between two Triangles does not materially affect their position; they are still servants, soldiers, and serfs. But the difference between a 59-degree and a 60-degree Triangle affects the entire lives of the individuals involved. The gap in prestige between an Equilateral Triangle and his Square son is likewise immense. But the difference between increasingly multisided nobles is so slight as to be imperceptible even to many educated people, despite it being the subject of fierce pride and intense training. Though Square talks about all these things as if they are "Natural Law," in many ways they are bizarre and arbitrary.

Training nobles in sight recognition is, according to Square, incredibly expensive, though he claims that the expense is justified by the loftiness of the "Art." He also sees no problem starving at least one Triangle a month in each school to help children learn to feel angles, which, while not lofty, is a practical concern. In fact he views the deaths as a positive control on the Triangle population. Once again, the reader sees Square's and Flatland's class prejudices at work. While there is some debate about this practice, Square implies that the questions are more economic than moral. However, even these "economic" decisions are made based on assumptions about what is valuable. Triangles, clearly, are not.

The story about Square's ancestor's accident setting his family back generations in terms of angles illustrates the Victorian conception of criminality as evolutionary regression, even though in this case it was an accident. Square's contradictory view is that the increase of angles is at once Natural Law and something that must be carefully cultivated and guarded.

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