…and, at the word, I grasped a moveable Square, which was lying at hand
Flatland (Ch. 21)
HOW DOES A Square grasp and move that “movable square?” For that matter, how does any Flatlander move any object, including himself? It may seem an odd question, but Abbott made it a very difficult one to answer. His illustrations of Flatlanders do not show limbs, or any means of propulsion whatsoever, yet it’s obvious that they live, work, use tools, and otherwise manipulate their environment. A Square acknowledges the problem in Chapter 11, quoted earlier, but annoyingly says nothing to answer it.
If Flatland were a thin 3D world this wouldn’t be so much of a problem. We could assume that Flatlanders had hundreds of tiny feet under their bodies, moved like snails, or undulated like rays. Unfortunately it’s a true 2D world, and that means that none of these options are available. Flatlanders can’t have feet or snail-like undersides because there’s nothing for them to move on, and can’t undulate up and down because there is no up or down!
The easiest way to get around this is to assume that there is a strong taboo against any mention of external limbs, much as the Victorians had a taboo against mentioning legs (especially those of women) in polite society; Flatlanders do indeed have arms or tentacles, but would never dream of mentioning them because they are considered vulgar. But if they existed they would make it difficult for any Flatlander to claim true symmetry – unless, of course, they folded or retracted when not in use. The illustrations below show three possible designs for such limbs; a flexible tentacle-like structure (a) which exudes from an opening in the body if needed and is retracted when not in use, a tentacle-like arm with a relatively hard outer surface which folds against the body (b), and a whip-like tongue (c) which is retracted when not in use.
A flaw of version (a) is that it requires a break in the outer shell of the Flatlander, so that the section between the eye and the tentacle opening isn’t connected to the rest of the shell, and might easily be pushed in or otherwise damaged. It’s not going to stand up to the life of a fighting Isosceles! The folding limb (b) is superficially more plausible given a sufficiently flexible material making up the outer shell, but it’s an unfortunate fact that there is no evidence for it in the text, and a hard-surfaced limb would immensely complicate the Recognition skills Abbot describes, and make it ridiculously easy for the lowliest Isosceles to impersonate a Regular triangle or a Higher Form.
The tongue design is possibly the most plausible, and can be compared to the tongues of frogs and the long cilia of single-celled animals such as Euglena, but has the disadvantage of giving the eye/mouth a lot to do – it’s now a mouth, an eye and a hand, and will often be needed to do all of these things simultaneously. Isosceles would be particularly handicapped, since they wouldn’t be able to handle objects and fight with their points simultaneously. Having said that, a tongue of this sort does have the advantage of being out of sight when it isn’t in use so won’t interfere with the recognition process too badly.
If A Square was telling the literal truth, Flatlanders genuinely have no external limbs. So what does a Flatlander use as a replacement?
One simple answer is suction. If Flatland is only one atom thick then it’s very easy to make a hermetic seal; in fact, as Dewdney has pointed out  it’s actually difficult to avoid doing so. Any two points linked by a solid barrier are effectively sealed and airtight. Reduce the air pressure very slightly behind the seal and you have something that works like a suction cup. If Flatlanders have even rudimentary control of the outer layer of their bodies, and can do the equivalent of tensing a muscle for a few seconds, they can easily stick any solid object to their bodies.
The illustration below illustrates this process in four stages. At (w) an Isosceles is moving towards a gun. As he approaches it (x) he pushes out a layer of his outer surface slightly smaller than the “(“ shaped base of the stock of the gun. Once the gun is in contact with his body (y) he relaxes this bulge, creating a partial vacuum underneath the base. The gun is now stuck to his body, and a fair amount of force will be needed to remove it.
Of course this doesn’t explain how he can light the fuse on his gun… maybe that whip-like tongue is still needed.
Some other possibilities for sticking to and moving objects (but not necessarily using them) include adhesive skin secretions, electrostatic forces, magnetism, Van der Waals forces which attract molecules (as seen in the adhesion of Geckos), Zero-point energy, etc. Since the surfaces joining together will always be one atom thick Flatlanders can take advantage of many phenomena that aren’t readily available to 3D organisms.
There’s no reason to assume that there is only one answer. Maybe Flatlanders hold things by suction, manipulate them with a retractable tentacle, and supplement both with magnetism and sticky slime. Provided all Flatlanders have the same abilities it doesn’t make any difference to the game.
None of this actually explains how Flatlanders move. We know various things about their locomotion; that travelling somewhere involves some form of effort, that it is harder to travel North than in other directions, especially for women, and that females “undulate” as they travel, although this is more of a safety measure than a necessity.
A mundane explanation, but one that causes a few problems, is the use of very short cilia-like mobile hairs for propulsion. This assumes that Flatland air is extremely dense, of course. If the hairs were short enough and spent most of their time flattened against the body they wouldn’t affect the process of angular recognition, and their motion might only be perceptible as a slight rippling in the brightness of a moving flatlander. Streamlining would be an advantage, so Isosceles and women would move faster than anyone else. A snag is that anything carried (such as the pouches shown in illustrations of some of the characters) would obstruct the hairs and slow Flatlanders down. This idea also doesn’t explain why women should find it difficult to go North.
Perhaps they use light pressure. Flatland appears to get its light from 3D space, which ought to mean that it packs a vast amount of energy compared to most other forces in Flatland. If Flatlanders have somehow evolved an ability to harness it, their movement could be the effect of organs roughly equivalent to solar sails inside their bodies. This won’t work very well if the light is exactly perpendicular to the plane of Flatland, but more or less works if light radiates from a source slightly to the North and above or below the plane of Flatland. As already noted, most light would normally pass through Flatlanders without being absorbed or reflected, but if Flatlanders have an internal organ or organs with varying reflectivity, like the elements of an LCD panel, they could be “switched on” to reflect light, or “switched off” to allow light to pass through the body. When light is reflected it exerts a microscopic amount of pressure; if that pressure is exerted at an angle it should produce a little thrust. Flatlanders might not know how they moved, but they would move. For fine control it would be useful to have multiple reflective organs, switched on or off to control rotation and possibly even allow tacking towards the light source, to the North. In the illustration a sixteen-sided aristocrat is using these organs (shown black) to move forward and slightly to the left; a second smaller reflective patch to the right of the centre line adds a little off-centre pressure, so that he is slowly rotating anticlockwise as he advances. The yellow arrows show the primary and secondary thrust.
This could explain why women have trouble going North; since they have very thin bodies they would have comparatively poor directional control and might find it difficult to tack. Some form of energy would be needed to power the process, hence feelings of tiredness after a long “walk”.
All of this requires an immense amount of hand-waving, especially in explaining how Flatlanders absorb enough energy to move without frying themselves, and referees may prefer to tackle the problem by refusing to explain it. After all, it’s what Abbott did…
 The Planiverse, various chapters.
Comments, as always, are very welcome.