Describing a room you cannot see: a grid as a shared language for position

0:00Here is a wardrobe, and here is a loop of wool laid round its edge. Lift the wool away and you are holding the wardrobe's shape. You can post it to somebody. They can feel it, measure it, draw round it. Now try to post them where the wardrobe is. You cannot, and the reason is not that it is difficult. Shape belongs to the object. Position does not. Position is a relation between the object and something else, and until you have agreed what that something else is, there is nothing to send.

0:38Someone is describing their new room to a person who will never see the drawing, and who will read the whole thing with their hands. So they build it. A board, and thread stretched between pins, one loop for each thing in the room. Run your fingers round a loop and you learn a shape, and a size, and nothing else at all. Is it by the door? Is it under the window? The loop is silent, and it will stay silent however carefully you feel it.

1:09Worse: build a second loop for a second object of the same size, put them anywhere you like, and the two feel identical. Everything that distinguishes them has been left on the table. So the board underneath is not blank. It is ruled into a grid, evenly spaced, crossing at right angles. Now a loop is not floating. It sits at particular crossings, and a finger can count its way there from the corner.

1:42That is the whole repair, and it is worth being precise about what it added. It did not make the wardrobe any more definite. The wardrobe was always exactly where it was. It made the wardrobe's position sayable. One more thing has to be written down, and it is the thing people skip. One centimetre on the board stands for one foot of real room. Without that sentence the model is only a shape again, at some unknown size.

2:15With it, every length in the model is a claim you can check. The bedroom's twelve foot wall is twelve centimetres of thread. The wardrobe's four foot face is four centimetres. And it cuts the other way too. Push a pin two millimetres off and you have moved something two point four inches across a real room. Getting the scale wrong is not a small error. It is a different building.

2:44Read that same twelve centimetres as twelve metres and the wall is over thirty nine feet long. Not twelve feet. Thirty nine. The thread has not moved. Only the sentence agreeing what it means. And nothing inside the model would give the mistake away. Every length is still in perfect proportion to every other. The model is only wrong about the world. Here is the room the model is describing, drawn flat, as if you were looking straight down at the floor.

3:19A bathroom on the left, six feet by nine. A bedroom beside it, twelve feet by ten. Multiply and you get the floors. Fifty four square feet, and a hundred and twenty. A hundred and seventy four square feet altogether, and not one of those three numbers is written anywhere on the drawing. They are all things you work out. Against the bottom wall of the bedroom, a wardrobe, four feet by two.

3:50That wardrobe is eight square feet of floor, which is exactly one fifteenth of the bedroom. Now slide it along the wall, one foot at a time. There are nine places it can stand on a twelve foot wall. Its shape is identical in every one of them. Four feet by two, nine times over. Nine different positions, one single shape. If you only sent the shape, you sent one ninth of what you knew.

4:21Two doorways, and each one is drawn with a faint arc sweeping through the room. That arc is the door leaf swinging open. It is worth noticing what has just been recorded, because it is not an object. It is the floor that has to stay clear. Nothing can stand inside that arc, not because the drawing says so, but because the door would hit it. A plan can carry movement, and not only furniture. That arc is a rule about the future, drawn on a picture of the present.

4:58Now the question this drawing cannot answer, and the answer is more interesting than the drawing. Where are the windows? They are certainly somewhere. A window has a perfectly definite position. But there is no place on this page to put it. A real room has three independent measurements. Length, breadth and height. A plan keeps two of them and throws the third away, and the one it throws away is height.

5:28That is why the doors survive and the windows do not. A door reaches the floor. A window is a hole partway up a wall, and its height above the floor is precisely the measurement this drawing no longer has. Give the same room a nine foot ceiling instead of an eight foot one and it gains a hundred and twenty cubic feet, and this plan cannot tell the two rooms apart.

5:57There is a way of giving positions that needs no grid at all, and everybody uses it. Against the left wall. In the far corner. Next to the door. So it is worth asking exactly how much those phrases leave out. Mark the bedroom floor at every foot, both ways. That is a hundred and forty three marked points. Against the left wall fits eleven of them. At three feet along the left wall fits exactly one.

6:28Now count every point that any wall name can reach at all, all four walls together. Forty four. Which leaves ninety nine points in the middle of the room that no wall name reaches, ever. A pair of numbers reaches all hundred and forty three, and reaches each one once. One warning, for anyone who copies a plan rather than drawing it. Suppose the same room is drawn twice, and in one drawing a boundary runs flat across at six feet, and in the other the same boundary slopes, starting a foot lower and rising to meet the wall.

7:09Both drawings look right. They are not the same room. The strip between those two lines is a triangle, and over a three foot run it is one and a half square feet, which is one thirty sixth of that bathroom floor. Small, and completely real. A drawing is a claim, and two drawings can make different claims about one room. So redraw from the thing itself, never from the other drawing.

7:39So here is everything two people have to settle before either of them can say where anything is. A corner to count from. Two directions to count along. A unit to count in. Three agreements, and the model quietly makes all three. The corner of the board. The two edges. The centimetre standing for a foot. Notice that shape needed none of them. You can hand somebody a shape across a table with no agreement of any kind. Every single one of these three exists to make position sayable.

8:18The person reading this room with their fingers is not being given a special adaptation. They are being given the general case, made touchable. A grid, a fixed corner, a stated unit. That is exactly what a graph is, and the only difference here is which sense you read it with. Position was never a property of the wardrobe. It was always an agreement between two people about where to start counting.