Reading a triple as three perpendicular distances, one per plane

0:00A point somewhere in a room, and three numbers. Everyone knows those two things go together. But going together is not one claim, it is two, and they pull in opposite directions. The first says every point of space hands you a triple. The second says every triple hands you back a point. Neither one gives you the other for free, and a rule that only runs one way is worth very little.

0:29What is remarkable is that a single construction settles both, because you can walk it forwards and then walk it backwards. So watch the construction, and watch it twice. Start with three flat surfaces meeting at one corner, each square to the other two, and a point P floating somewhere off them. The first move is the only one that needs any thought. Drop a perpendicular from P straight down to the flat surface, and call where it lands M.

1:02That word perpendicular is doing real work, so it gets tested rather than assumed. Two hundred and twenty-four different lines were drawn across that surface through M, going every whole-number direction, and the segment from M up to P was set against each of them in turn. The number that came out at anything other than a right angle was zero. That is what makes M the foot and not merely a point underneath.

1:34The second move repeats the first one dimension lower. From M, drop a perpendicular onto the first of the three lines, and call where that lands L. Now, is M really the only place the first perpendicular could have landed? Six hundred and twenty-five places of that flat surface were walked through and each was asked whether the drop from P could end there. One survived. The same six hundred and twenty-five were then measured for distance from P, and again exactly one was nearest, and it was the same place.

2:12So the foot is unique because a search found one, not because someone said so. Three segments now exist, and they are the answer. From the corner out to L is the first number, from L across to M is the second, and from M up to P is the third. For the point being drawn here, those come out two, four and five. And notice what got built on the way.

2:40M is not a scratch mark, it is a genuine point of the flat surface, and it has a triple of its own: two, four, nothing. Across sixteen different points run through this construction, the number whose foot disagreed with them in the first two entries was zero, and the number whose foot had anything but nothing in the third was also zero. Every point drags a shadow, and the shadow keeps the first two numbers and loses the last.

3:13Now run the whole thing in reverse, and start from three numbers instead. Go two along the first line, and stop. From there go four parallel to the second, and stop. From there go five parallel to the third. The two stops on the way are exactly the two points the forward construction produced, in the opposite order: two, nothing, nothing, and then two, four, nothing. Of the sixteen triples put through, the number the return trip sent somewhere other than where they came from was zero, and the number that did not come back unchanged after going out and returning was zero.

3:56Every move is forced by the one before it, which is why the point a triple names is not merely available but unavoidable. There is a second way to reach the same triple, and it is not a repeat. Leave the perpendiculars alone and instead cut space with three flat surfaces through P, each one parallel to one of the three you started with. Of those three, the number failing to pass through P is zero, and the number meeting one of the original surfaces at anything other than a right angle is also zero.

4:36Solve for where all three cross and you get P back, which is the least the construction owes you. Ask instead where two copies of one surface and a second surface cross, and the solving refuses: three surfaces sharing a line share no single place. So this is a real calculation and not a picture with an answer written underneath it. Six flat surfaces now bound a box, three through the corner and three through P.

5:09Take every way of choosing one surface from each facing pair and solve; eight places come out, and the number of them standing on another is zero. Read each one through the construction and every triple is built from nothing and from P's own three numbers, with no exceptions. One corner carries three nothings, three corners carry two, three carry one, and one carries none, which is the box counted by how much of P each corner has kept.

5:43Three of its faces get named by their four corners, and each name turns out to be exactly the corners that reach the full distance in one direction: the number of names that did not match was zero. All three of those faces meet in exactly one corner, and that corner is P. The box itself is scaffolding, though. Rebuilt around three different points it took three different shapes, and the number of those points whose triple changed because of it was zero.

6:19Here is what the second construction buys, and it is the whole point of the topic. Each of the three lengths in the box is a distance from one of the three flat surfaces. Not a distance along a line, a distance from a surface. So there are two readings of the same triple now: three lengths measured along three lines, and three distances measured from three surfaces. Run both on all sixteen points and the number they disagree about is zero.

6:52They agree because each surface cuts its own line square, at the corner the whole thing was built from. That is worth saying aloud, because otherwise the two constructions look like two unrelated definitions that happen to give the same answer. And now the trap, which almost everyone walks into. The first number is not the distance from the first line. It is the distance from the one surface that first line does not lie in.

7:25Ask which of the three surfaces fails to contain the first line and exactly one does, and it is the surface the other two lines span. Each line is left out by exactly one surface, and the three surfaces doing the leaving out are all different. Take the point two, four, five: its squared distance from that surface is four, and its squared distance from the first line is forty-one.

7:55Those are not the same measurement and they are not close. Across all sixteen points, the number where the distance from the line happened to match the first number anyway was zero, and it takes a deliberately chosen point like three, nothing, three to make them agree. The distance from a line mixes the other two numbers together; the distance from a surface does not touch them. Which raises the obvious objection: a distance cannot be negative, so how can a coordinate be?

8:32Put a point and its mirror image through the middle surface and compare them. Of the sixteen, the number whose mirror differed anywhere except the second entry was zero, and the number whose second entry did not simply turn around was also zero. All sixteen sit exactly as far from that surface as their mirror does. But only three read the same, and those three are precisely the ones already sitting on the surface, where there is no side to choose.

9:06So a coordinate is doing two jobs at once: it reports how far, and it reports which side. Of these sixteen points only two have all three numbers positive, and between them they reach all eight regions of space, so the all-positive corner is one case out of eight and not the general one. Back to the two claims from the start, and now they can be counted rather than believed.

9:37Three hundred and forty-three triples were built into points. The number of them landing on a point another triple had already reached was zero, the number of points read back as a triple another point had already given was zero, and the number of triples nothing was read as was zero. That is what the pairing actually asserts, and it took both directions to get it. To see why one direction alone would not do, watch the map that only goes one way: sending each of those points down to its foot.

10:14Three hundred and forty-three points, forty-nine distinct feet, and as many as seven points sharing a single one. Perfectly well defined, completely useless for naming anything. And the order is not a mere convention either: cycle which surface is which and the same point reads four, five, two instead of two, four, five, with zero of the sixteen reading the same in both. One more thing, and it is the kind of thing a drawing does to you.

10:48In the picture, the upright stroke from M to P crosses the second line. It really does cross it: solve for the crossing on the board and there is exactly one, seventy-two twenty-fifths along. So it is tempting to letter that crossing and call it nothing, four, nothing. But ask space rather than the board. The number of places the upright stroke through this point shares with the second line is zero, and it only becomes one when the first coordinate is nothing; three of the sixteen points qualify, and this one does not.

11:27The place the crossing is really a picture of is two, four, eighteen twenty-fifths, which is nowhere near nothing, four, nothing. And the drawn length from the corner to that crossing is seventy-two twenty-fifths while the drawn length it would have to equal is four, so the picture does not even claim they match. A crossing in a flat drawing is two strokes passing at different depths. Which leaves one small trap worth naming before you go.

11:59The letter C gets used twice here, for two different points. In the first construction it sits on the second line; in the box it sits on the third, four units out one way against five up the other. They lie on different lines and they are not the same place. Nothing in the drawings tells you that, so carry the letters lightly and the construction firmly. Three numbers, three distances, three surfaces, and each number measured from the surface its own line is missing from.

12:34Run it forwards and you can name any point; run it backwards and you can find any point that gets named. That is the whole of it, and it is why a triple is a place and not just a label.