Hinton's Discworld

In 1907, the book “The Episode in Flatland,” written by a mathematics teacher at Princeton and then the University of Minnesota, Charles Howard Hinton, was published in London. In it, Hinton made an attempt to handle the flat world.

In the two-dimensional world, created from Hinton’s imagination, the same flat planets revolve around the flat-as-a-pancake Sun, on one of which, called by the author Astria, and events will take place,

written off in the novel. You can easily imagine this two-dimensional world by placing a few coins on the table. Let one of them, say, a metal ruble, represent the Sun, while the other, smaller coins, represent the planets rotating around it. The force of gravity in this two-dimensional world behaves differently than in our space. The only difference is that it changes in inverse proportion to the distance between the bodies, and not the square of the distance, as in the world we are accustomed to.

Astria rotates around an axis, like any planet. For example, in which it rotates is called east, the opposite is called west. Hinton seemed to place his flat actriyans (inhabitants of Astria) on the “rim” of the planet. They can mix and remain in the plane of the Astria disk. Trees grow and houses stand in the same plane. Therefore, when encountering a tree, the actor must either climb over it or surround it. In order for one to pass the other, two inhabitants of the planet must move either one under the other, or jump over each other, as two acrobats would do on a tight rope.

To see something behind him, the actor must bend down and stand on his head or use a mirror. Since the second method is more

convenient, then not a single resident of Astria leaves home without a mirror.

In addition to the fact that each Astriin carries a mirror with him, all their houses are also equipped with mirrors. The structure of houses communicates with the two-dimensional world. They have windows and doors through which you can enter and exit, but to prevent the house from collapsing, only one door or window can be opened at a time. If the west door is open, the east door and windows must be closed, otherwise the top of the house will collapse. For the same reason, houses in Astria do not have pipes. After all, there is no way to connect the ends of the pipe so that the connecting ends are not closed.

The Astrians’ bodies have a complex structure, but to avoid unnecessary detail, Hinton renders them as rectangular triangles with arms, legs and one eye. All Astriin men are born with their faces facing the east, and all women are born with their faces facing the west. This orientation remains with them until death, since for flat creatures (two-dimensional people) there is no way to turn around without going into the third dimension. It’s easy to kiss a jen or daughter of an astrian. But in order to eat his son, the father comes to turn him over and hold him upside down.

Cargo in Astria is transported by ships and planes. The propeller is excluded (as it rotates, it enters the third dimension), so all planes have flapping wings.

Ropes in Astria cannot be tied into knots. This is only possible in a three-dimensional world. It is possible to strictly prove mathematically that ropes (one-dimensional bodies) can be tied into knots only in three-dimensional space, but we will not give the proof of this theorem (and even its exact formulation) here.

In a flat world, wheels and axles will be completely eliminated. Heavy bodies can be transported in Astria using the method of rolling over circles, just as three-dimensional things can be moved on cylindrical rollers placed under them.

This is the mathematical background of Hinton’s novel. But that’s not all there is to the 181 pages of “The Episode in Flatland.” There is love, and war, and an impending disaster (the approach of another planet, which insisted on changing the orbit of Astria, so that life there would become impossible), and a happy ending. But this is presented much less interesting than everything related to mathematics and mechanics.

In conclusion, we suggest solving a problem that will help you test your perception of the two-dimensional world.