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Last semester, I took a drawing class at my university. Every day, the professor would show us some examples of what we would learn, by pinning several pictures to the enormous bulletin board at the back of the room. During the third week, she tacked up some examples of images with perspective and without. The perspective drawings all looked very realistic, of course, but I thought one of those without perspective was by far the most interesting as a work of art. The picture was of the interior of a room; a kitchen, I think. The walls were drawn with convincing angles, and were lined with shelves. A window opened on a bright flower-garden. The floor was a bright red-and-white checked tile pattern. But instead of shrinking towards the back of the room, the tiles were identical perfect squares from the front of the room to the back. The artist had drawn them by making parallel crosshatch lines with a ruler. The tiles lay flat on the vertical surface of the paper; they didn’t recede from view. This created extreme, almost uncomfortable contrast between the relatively realistic shape and detail of the rest of the room and the starkly unrealistic floor. Especially at the back of the room, where the floor would normally be a subdued element of the background, the two aspects of the picture were in such extreme disagreement that they couldn’t be reconciled. It almost didn’t look like a room anymore. For me, this curious quality made the picture a far greater artistic success than it would have been with the addition of perspective.

Later on, the picture gave me an idea. The shock of a floor that appeared to be vertical would be much greater if it were to appear in real life than it would be in a drawn image, or even in a photograph. Though I haven’t constructed any models, given a large enough room I do think it would be possible to make a “square”-tiled floor that is configured to compensate for perspective from the entrance. The tiles would lengthen and widen gradually, so that to a person standing at a particular place, they would appear to all be the same size, rising up vertically rather than receding toward the back wall.

I’ve visualized it in my head. I would begin by establishing the size and position of the entrance of the room, perhaps a doorway about six feet wide. Then, I would set up two straight lines of tape or rope from that doorway, as long as I wanted the room to be. It would have to be long enough for perspective to take dramatic effect, so perhaps about 150-200 feet. I would then carefully measure the apparent angles of these ropes. Perspective would mean that the ropes would appear to slope towards one another and away from 90°, if viewed from one end. So, if the ropes appeared to slope away from 90° by, say, 25°, then I would make the ropes slope away from one another by 25°, ending much farther apart than they began. This would make them look as though they stretched away from the doorway at an perfect right angle. It would also mean that the tiles would have to be slightly trapezoidal, rather than square, so that they could fan out around the doorway and fit all the way up to the widening walls. But given the length of the room, the angle would most likely be slight enough for the tiles to appear square at the doorway.

As well as becoming progressively wider, the tiles would also need to be longer at the end of the room opposite the doorway. If they were all the same length, then from the perspective of the doorway, the nearest tiles would appear to be eight-inch squares, while the farthest ones would appear to be eight inches wide (thanks to the widening of the room) but much less deep. So there must be a spot, at some distance from the door, at which the tile appears to be a totally one-dimensional line, or even a speck. That spot is referred to as the “vanishing point”. I would walk out from the doorway, find that spot, and measure its distance from the door. I’ll give that distance the arbitrary value of 600 feet. So “apparent length of tiles” is 8 at the door and 0 at 600 feet, meaning that we can express the “apparent length of eight-inch tiles” as y=8-(8x/600) where y is “apparent length” and x is “distance from door”. From this point, we can develop an equation relating length of tile to variable apparent distance: y=z-(xz/600), where y is apparent length, z is real length, and x is distance from the door. Using y=8 and x=200, we find that in order to appear to be an eight inch square from a distance of 200 feet, a tile would have to be 12 inches long. The next tiles would be 11.97 inches long, using 199 for x “distance from door” instead of 200, because the last tiles were 12 inches long. Continuing from here, the tiles would continuously get shorter until they would be exactly eight inches long and wide at the door.

Adjusting for both length and width in this way would result in a long, gradually widening hallway, with a floor that would appear to rise straight up in front of the doorway. The tiles would not look square from every angle, but that would only make it more exciting to walk down the hallway and notice the very gradual lengthening of the tiles. I think a hallway like this would be extraordinary as the nave of some kind of modern cathedral.

I really, really want to build this. I wish I had a few million dollars.

(All of the real numbers here (the 25°, the 200 feet, the 600 feet, and the 8 inches) are arbitrary. I haven’t made any observations on which to base them, and they could be totally ridiculous. Real numbers based on observations can be substituted in.)

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