Engineering Designs in Planiverse
The easiest form of transportation in Arde is walking. However, even walking presents difficulties. For example, if two Ardeans need to pass each other, they cannot walk by each other because there is no third dimension. Therefore, in order to pass each other, one of the Ardeans must lie down on the ground while the other walks over them. However, in a city with many more Ardeans, it would be very difficult if someone had to lie down each time two people passed each other. Therefore, they created a traffic system in which there are pits in the ground covered by a rope which the westward and eastward bound Ardeans alternate going into while the others can pass above.
Another form of transportation in Arde involves balloons. The balloons are formed by a rope that is filled with a lighter gas, which allows it to operate like a helium balloon on Earth. In this particular transportation, the balloon is attached to a train which does not go on tracks like on Earth, but flies to its destinations instead. There are two compartments so that people can enter from either the left or the right side.
The reason that the train doesn't go on tracks is because it is very difficult to use wheels in a two dimensional world. The diagram on the right shows three failed ways of making a vehicle using wheels. The main difficulty in using wheels is that the lack of a third dimension makes it impossible to use an axel. The wheels on the first vehicle on the left don't rotate because of friction. The second vehicle has smaller wheels between the vehicle and the large wheels, but those would get stuck as well. The third design would work for a little bit, but then one of the small wheels would fall out and thereby render it useless like the first two. The vehicle shown below is a working vehicle that uses wheels by surrounding the entire vehicle with wheels. However, this would be very difficult to use as it would require ventilation and hooking up the wheels each time that the vehicle was used. Therefore rolling vehicles are not commonly used in Arde.
While most housing on earth is above ground, all of the housing on Arde is underground. If houses were built above ground in two dimensions, as one couldn't walk around it, anytime one passed a house, one would have to either go through it or climb over it. With the underground housing, the only obstacle in passing a house is to get past the entrance. Another key aspect of the Ardean housing is the swing stair. It is constructed by attaching the stair to the house with a spring and a hinge, as seen in the diagram. This allows the stair to be out of the way when not in use, and to connect two floors when in use (that is, when the weight of an Ardean is on it or it is pulled down from the lower floor by a rope). There are two key uses for this type of stair. First of all, if none of the stairs could be moved, each floor would need to be to the left or the right of the floor above it. If the first staircase went down to the right, each of the next staircases would need to go to the right. In this case, there would be no point in making multiple stories as one would be using the same amount of lateral space for a tunnel of the same size. Another useful factor of the swing stair is that it allows one to enter and exit a house from either side as well as pass by a house relatively easily. To pass the house in this diagram, one would simply go down a few steps and then up a few steps. However, if traveling to the left, one would need to pull the steps down before climbing.
To solve this problem, the Ardeans created a door, as seen on the right. The bottom piece of this door is a wedge that holds the bar going up and down in place. The hinge on the top of that bar allows it to swing back and forth if the bottom piece is not wedged in. Therefore, when the wedge is in place, this door supports the hallway, and when it is not in place, an Ardean can pass by merely pushing it out of the way. However the main problem is how to get the wedge in and out. With this door, the wedge needs to be pulled out to the left. This is where the two levers come into play. From the left, one can push the left lever to remove the wedge, and from the right one can pull the right lever to remove the wedge. After the wedge is removed, the door can be pushed (from either side) to pass through. Then, to replace the wedge, one either pushes the right lever or pulls the left lever.
There is still one more problem with the door. It will support the building while the wedge is in place, but what happens when the wedge is moved? What supports the building then? The simple answer to this is that instead of using one door, the doors always come in pairs so that while one is being opened, the other can support the ground above it.
Logical Circuits in Planiverse
Computers in the planiverse are doable. The primary difficulty in creating computers is making logical circuits without wires crossing; wires cannot cross separate from each other in a 2-d world.
The solution is the NAND gate. Corresponding to the logical NOT-AND operation, it can be constructed without any crossing-over of wires, and from combinations of NAND gates, any logical circuit can be made.
The NAND gates can be combined to form an XOR gate:
From this, computers can be made.
Original Design for Planiverse
Of course, even 2-D creatures want to have fun, so it's an interesting thought problem to come up with some games for the planiverse dwellers to play. Of course, it's hard to apply normal, three-dimensional world games, so 2-D analogues must suffice.
One such example, described in Gardner's Sciam article, is two-dimensional checkers (seen in figure a). Using a 1x8 (as opposed to 8x8) board, checkers pieces move forward by one, and are mandated--as in --> --normal checkers--to take a piece if possible. In fact, the game is identical to a game of checkers played only along the center diagonal. This game isn't actually terribly exciting--the player who --> --moves first will, in rational play, --> --always lose; after all, there aren't --> --that many possible combinations of --> --moves (chess, in theory, could also be --> --solved, except that there are so many --> --possible moves to consider). Of --> --course, when the board is longer --> --(e.g. 11x1) and there are more pieces, --> --it's harder to figure out who will --> --win.
There are, of course, limitless other variations. Gardner's article notes a chess-like game that can be played (see figure b) with a king, rook, and pawn on each side. Gardner doesn't explain exactly how it works (e.g., is there the normal process of checkmate, or does one just need to actually capture the king, as in checkers).
Other possibilities exist, too, such as a linear form of the game G.
Though it might at first seem hard to design normal, 3-d world chess (a 2-D game, not the 1-D planiverse version), theoretically it is trivial. After all, chess moves can all be recorded using a notation (e.g. "e4" means move the pawn to the square designated by coordinates 'e' and '4'). So, if everything relevant to the game of chess was relayed in such information, it could be transmitted just through a matter of bits. But this is not surprising; if one developed a 3-D version of chess (that is, 8x8x8), even though it would be very hard to play it in the real world, moves could be explained as moving the piece on one set of coordinates to somewhere else.
So, we set out to design a version of normal chess that could be easily playable in the planiverse world. We made eight sets of rows, each attached on the righ to a structural support. To allow the planiverse creatures to see all the pieces, the rows were fanned out at angles. Because the creatures can see color, we used that to differentiate the different pieces.
Of course, the creatures will have to climb through the chess set themselves to move the pieces. For that reason, we added little sticks between each square on to which the planiverse creatures can clutch.
Finally, it turns out that, even though it's hard for us to visualize what they would see as our normal chess board, the planiverse creatures actually turn that into a form of "depth" perception, so the board looks quite normal to them.
It should be noted that the 2-d world makes engineering issues very different. In trying to come up with designs, we found that everything either seemed very intuitive and trivial in a sense, or else incredibly difficult. Even things that seemed complex--eg, making a calculator--are easily decomposed into simple-to-accomplish tasks. On the other hand, some tasks--like making wheel-based devices--are incredibly difficult.
If you're interested in some of the more conceptual issues, look at GRAPH THEORY! I looked through a bit of it, and it raises some important stuff. Of course it's kind of dense, but yeah have fun.