Abstract Science, part 2 – Flatland

For part two of this series I have to delve into the world of fiction. But I’m not going to say why until the third post. The fiction I am going to discuss is an 1884 Victorian satire called Flatland: A romance of many dimensions, by Edwin Abbot. It is set in a two dimensional Victorian England; and England without height.

All of the characters are two-dimensional, geometric shapes: triangles, squares, pentagons, hexagons etc. Their infrastructure is much the same. The book is fantastic, and I won’t spoil it for you in this post. It deals with status; the more sides you have the higher your status is. So a square is above a triangle and a hexagon above them both. Eventually people are so-many-sided that they are indistinguishable from circles, and these people are high-powered religious figures. But these are not the quirks I wish to deal with.

I wish to deal with the quirks of vision in a two-dimensional world. As three-dimensional figures you and I see in two dimensions; we see a flat picture projected onto the retina of the eye. We produce the three-dimensional image in the brain by combining the two two-dimensional images we have in each eye. The two-dimensional characters in Flatland (called Flatlanders) see in one dimension; they only see lines, and these lines are shining lines (i.e. they glow). The illusion of the second dimension of vision is created by each line appearing more intense, or brighter, the closer it is to the observer, and less intense, or less bright, the further it is.

A Flatlander looking at a Triangle Flatlander

This picture explains how they see in Flatland. The Triangle at the left of the picture is a Flatlander. The eye on the right is another Flatlander’s eye. The dashed-line describes the direction of vision of the viewing Flatlander. The bar in the middle is what the viewing Flatlander sees of the Triangle. It is a straight line and changes in brightness hightligh that the Triangle has a corner; that the corner is closer to the observer means it is brighter. Our view of an actual two-dimensional triangle is granted to us because we can move in three dimensions and therefore move “up”* above the plane Flatland exists in.

* “Up” is something a Flatlander cannot conceive of. A Flatlander can move left, right, forwards or back, but “up” and “down” are not options.

A Flatlander looking at a Hexagon Flatlander.

In this image the observer is viewing a Hexagon. The bright bit of the line is longer than it was viewing a Triangle, so even with a one-dimensional view a Flatlander can distinguish between shapes (and therefore status).

Take a moment to imagine walking around Flatland as a Flatlander. Each person and building appears to you as a line, and the brightness of that line tells you how far or close they are. What would it look like if a Sphere, a three-dimensional shape, moved through Flatland? Would you see any of it? Could you see all of it?

As a sphere moves up through Flatland, in Flatland it appears as a circle changing in size.

The fact is that as a Flatlander you would see a cross-section of the Sphere; you would see a circular cross-section where the Sphere was in Flatland. The rest of the Sphere is not in Flatland. This is the story of Flatland: a three-dimensional shape, a shape from a higher dimension, enters Flatland and takes a Square on explores different dimensions, starting with a one-dimensional world and a none-dimensional world and then the third-dimensional world.

The one-dimensional world was inhabited by lines, and each line could only see its neighbour, and each neighbour appeared only as a point. Each line knew it was not a point, but was a line. But the stuff that made up the line was its insides, and its points were its outsides that could be seen. When the Flatlander described to the King of the one-dimensional world (Lineland) that he could see that each person was a line the King dismissed him as crazy.

The one-dimensional world, Lineland. The Linelanders see their neighbouring Linelanders as dot, but the Flatlander from outside Lineland sees lines.

In the none-dimensional world there was just one character, an actual dot. No one could convince the dot of higher dimensions because the dot could not conceive of something else talking to it, so when it heard a voice it considered that a thought in its own head and congratulated itself on challenging itself in that way.

In the three-dimensional world the Sphere and the Square looked down at Flatland, and the Square saw straight through houses and saw the insides of people.

From within Flatland you see lines with varying brightness making up Flatland. From above Flatland you see full shapes.

The Square saw Flatland from “above”, literally a view from another dimension. Houses that should have appeared concealed were open from above. People whose inside should have been hidden had them entirely on display from above. And thinking about it, the Square demanded that this knowledgeable three-dimensional shape, that had brought it such perspective and knowledge, should show it more; the Square demanded to see the fourth-dimension. The Sphere said that there was no such thing, don’t be ridiculous.


4 thoughts on “Abstract Science, part 2 – Flatland”

      1. I wanted to get the book – is it about the flat world that is used to explain multiple dimensions? I think it might be the same or similar to how Sagan used flatland to explain dimensions… though possibly more complex in detail.

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