📅 2010-Nov-07 ⬩ ✍️ Ashwin Nanjappa ⬩ 🏷️ geometry, topology ⬩ 📚 Archive

**Rating:** 4/4 (Takes the reader on an inter-dimensional journey!)

If you dip your toes into any general reading involving dimensions, space or topology, Flatland is the book that comes heavily recommended. **Flatland: A Romance of Many Dimensions** was written by **Edwin Abbott Abbott** more than a hundred years ago in 1884. It is a fictional story narrated by a resident of Flatland. As its name might imply, **Flatland** is a 2-dimensional world, whose residents are straight lines and regular polygons. The book is seemingly written by **A Square**, a square Flatland resident, who now finds himself behind bars for proselytising the existence of a higher dimension (3D). The first half of the book introduces the layout and beings of Flatland. A strict **class system** is in force in this land, based on the number of **sides** of the **polygonal** residents. **Straight lines** (line segments to be precise) are women, the lowest class, they receive no education, have no careers and exist mostly for procreation. **Isosceles Triangles** are soldiers and workers. The more sides the polygon has, the higher his class. The highest class are the **Circles**, who are actually polygons with innumerable tiny sides. All irregular sided polygons are either fixed or killed to prevent their spread. Flatland residents perceive each other by sight and touch (feeling). The most interesting event in Flatland's history was a **Color Revolt** which threatened the existence of the entire class system. This revolt, ironically, would have given residents the freedom to color (paint) themselves as they wished to. However, this revolt was cleverly put down by the Circles whose authority it undermined.

More interesting events occur in the second half of the book. One night A Square has a dream where he visits a 1-dimensional world called **LineLand**. Residents of LineLand are line segments of different lengths, they all live and die on a single straight line, which is their entire world. A Square shocks the king of LineLand when he is able to not only enter and leave their world at will, but can see their entire world and their insides too. Soon after this dream, A Square is visited by a **Sphere**, a being of **SpaceLand**, a 3-dimensional world. He takes A Square to his higher dimensional world and introduces him to the existence of solids. Being a 2-dimensional being, A Square initially finds it very hard to perceive this 3D world. But, by using analogy Sphere is able to convince A Square. For example, A Square already knew that a Point when moved along a direction formed a Line and a line moved parallel to itself formed a Square. Using such analogies, Sphere educates him on the existence of polyhedrons such as himself and **Cubes**. Together they visit the 0-dimensional world of **PointLand**, where a single **Point** is the sole happy resident. Unsurprisingly, the Point is found to be quite full of himself! 😊 A Square's story though ends in tragedy. On his return to Flatland, he tries to spread the **Gospel of 3 Dimensions**, only to be punished with life imprisonment for this heresy.

Flatland is a tiny book and has undergone lots of editions over the years. I read the **Princeton Science Library** (1991) edition, with an introduction by **Thomas Banchoff**. In my opinion, it is better to read this introduction after reading the novel. Banchoff introduces the life and times of Abbott. He lived in Victorian England, when education was not yet provided to women and the class system was in play. Both of these are well satirized in Flatland, by representing women as lower-dimensional (1D) entities and by the rigid polygonal class system. The eradication of all irregular polygons is also a representation of Abbott's times, when anyone who deviated in form or thought was labeled as a freak.

Flatland is above all a dimension-travel book. It is surprisingly easy and light to read. By anthropomorphizing polygons, Abbott skillfully makes all dimensions easy to understand. After reading his book, the reader cannot but believe that higher dimensions must exist! Even if seeing such a higher dimension is beyond us, Flatland provides us the signs and hints of how such a higher dimensional being will appear and affect our world. For example, a 4D being could see *inside* us and could appear and disappear at different places at will. It is no secret that Abbott might believe that God is such a higher dimensional being. The book does not deal with **space-time**, where time is the 4th dimension. This is not surprising since space-time became popular only after Einstein. But, even then, I believe time would be (or is) a different kind of dimension, not a strict topological higher dimension. Flatland is a highly recommended read that is sure to take the reader's mind on an inter-dimensional journey! 😊

A few quotes from the book:

Sphere: "I had hoped to find in you -- as being a man of sense and an accomplished mathematician -- a fit apostle for the Gospel of the Three Dimensions, which I am allowed to preach once only in a thousand years: but now I know not how to convince you."

I: "[...] take his servant on a second journey into the blessed region of the Fourth Dimension, where I shall look down with him once more upon this land of Three Dimensions, and see the inside of every three-dimensioned house, the secrets of the solid earth, the treasures of the mines of Spaceland, and the intestines of every solid living creature, even the noble and adorable Spheres."

"That Point is a Being like ourselves, but confined to the non-dimensional Gulf. He is himself his own World, his own Universe; of any other than himself he can form no conception; he knows not Length, nor Breadth, nor Height, for he has had no experience of them; he has no cognizance even of the number Two; nor has he a thought of Plurality; for he is himself his One and All, being really Nothing. Yet mark his perfect self-contentment, and hence learn his lesson, that to be self-contented is to be vile and ignorant, and that to aspire is better than to be blindly and impotently happy."

"It fills all space [...] and what It fills, It is. What It thinks, that It utters; and what It utters, that It hears; and It itself is Thinker, Utterer, Hearer, Thought, Word, Audition; it is the One, and yet the All in All. Ah, the happiness, ah, the happiness of Being!"