Even if adventurers stay closer to home, the two-dimensional properties of Flatland can still be surprising. Consider, for example, fire. In our world (and on the rim of coinworlds) if something is set on fire the combustion products rise with the air heated by the fire. More air replaces it from the sides, so that the fire continues to be supplied with oxygen. In Flatland there’s no “up,” so even though the hot gasses are less dense than the cold, they simply expand rather than going away; the fire is soon surrounded by warm carbon dioxide (or some Flatland equivalent) and smothered. It’s going to be difficult to keep a fire lit for any length of time, and fire may be a comparatively late invention. Certainly it isn’t needed for illumination…
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.
Flatland (Ch. 2)
“It’s probably quantum.”
Terry Pratchett, Pyramids
THE omnipresent sourceless white light of Flatland is a problem. The question isn’t why there’s light – there are many possibilities, most of them assuming that Flatland is lit by an external source in a three-dimensional universe - it’s how Flatlanders can see when their eyes are continually flooded with white light.
A Square tells us that there’s white light everywhere. This is repeated several times. This surely implies that there’s light illuminating every particle in Flatland, including the cells of his eyes (or whatever they’re made of). That means that if he looks at something in the distance there will be a little light reaching his eye from that source, and a comparatively huge amount hitting it from the interior of his body, the fluid in his eye, and light impacting directly on his optic nerves.
A Square also tells us that living creatures are a little brighter than their surroundings and inanimate objects. That’s an important clue.
One of the reasons for believing that Flatland is a genuine two-dimensional space is that Flatlanders can see at all under these conditions. The explanation is a little complicated.
If Flatland were a thin 3D world objects would reflect photons in all directions. Nearly all of them would escape from the plane of Flatland, since Flatland would be much thinner than the wavelength of light. Only the minuscule fraction of photons that were radiated at exactly the right angle and polarization to stay entirely within the plane of Flatland would make it to an observer’s eye. It’s implausible that any object would reflect enough light with exactly these properties to be perceptible at any distance, even the limited range described by Abbott.
What happens to the rest of the light? Most of it passes through the plane of Flatland without striking anything, but a proportion ends up inside Flatlanders, the walls of their homes, etc. Far more hits the interiors of their bodies than the thin edges! Most of it will pass through without interacting with their bodies, but a proportion of photons will hit something and release energy. Obviously something has to be done with all this energy; one possibility is that they use some of it to power their bodies (see below), another is that it somehow released again, as heat, light, and other forms of radiation.
When a photon excites any atom, there’s a quantum effect in which the energy is released as a more energetic photon. If this is a genuine 2D space , excited 2D atoms will be forced to emit photons exactly in the plane of the space, and they will have to be polarized to stay in that space. There’s nowhere else for them to go. Since the photons emitted by Flatland objects will be more energetic than those coming in from outside they will be a slightly different colour, polarized , and readily distinguished from the omnipresent whiteness.
Or will they?
This sounds plausible, but if all Flatland atoms are excited they will all radiate polarized light. This unfortunately applies to the atoms of flesh, nerves, etc. that make up a Flatlander’s eyes. Our Flatlander additionally needs to be VERY sensitive to the direction from which photons come.
One way around this is to assume that only the material that makes up the dense outermost layer of a Flatlander’s body – the rigid lines – emits photons in this way. Elsewhere in the body the energy is used for other things such as photosynthesis, or passes through tissue that is so transparent that few or no photons are stopped. If true, there would be relatively low levels of this specially polarized light inside the eye.
A complication is the fact that eyes use a quantum transformation for vision; in humans the visual purple takes in photons and uses them for a chemical reaction which triggers the optic nerves. A similar effect is likely to happen if a 3D photon strikes a Flatlander’s retina (or whatever is used for vision), so that the material used for vision is the most likely part to be swamped with random energy.
One way around the problem is suggested by the antennae used in GPS systems and many other radio applications; if Flatland “eyes” are in fact an array of light sensors, sensitive to polarization, phase, and timing, most of the external light and random photons could be tuned out. There’d still be a scattering of random flashes when two or more sensors were triggered – this could explain the omnipresent mist that limits visibility to a few feet – but overall there would be excellent vision at short range. One consequence of the level of timing and computation required to make this process work might be the ability to judge angles very accurately, as displayed by all Flatlanders, but it might also be the reason why Flatlanders are so obsessed with geometry.
 We’ll leave the question of how 3D photons interact with 2D matter for the advanced class. Let’s assume for now that 3D particles, spheres, etc. can come in from outside, but 2D objects and particles can’t normally escape the plane of Flatland.
 Numerous 3D species, such as bees, have eyes sensitive to polarization. In Flatland this ought to be universal.
 Flatland prisoners are described as surviving for months without food. This seems odd, but it’s plausible if food is used mainly for growth and repairs while photosynthesis provides energy. Photosynthesis also takes away much of the need to breathe; to an extent the body of a Flatlander would be a closed system, with external air used mainly to keep oxygen and carbon dioxide balanced and for speech.
 The colours used for illustrations in this book are an artistic conceit; if they existed at all they would only be detectable by the use of cross-polarized filters and image enhancement – a one-atom thick body is simply too thin to be seen by any normal means
What do you think?