Three perpendicular planes, and the eight regions they cut space into

0:00A lamp hangs from the ceiling of a room, and you want to say where the bottom of it is. Walk to the point on the floor directly beneath it and measure two distances - so far along one wall, so far out from the other. Two numbers, and you have a point on the floor. But the lamp is not on the floor. Every lamp hanging anywhere above that same spot gets the same two numbers. Something is missing, and what is missing is the height.

0:32So position in a room takes three numbers, not two. The room itself already shows you how. Its floor and two walls that meet that floor are three flat surfaces, and they all meet at one corner. Here is the only decision anybody makes. Fix three flat surfaces through a common point, each one at a right angle to the other two. That is it. Everything else is forced by that sentence, and the parts that are not forced are small and worth pointing at when we reach them.

1:09Notice what a flat surface is before we use it. It is a set of places, and the only question we will ever ask it is which of its two sides a place is on. Not how far. Just which side. One yes-or-no answer per surface. That is the whole instrument. Everything counted from here is counted with it. Now, where do the three lines we measure along come from?

1:38They are not chosen. Two of the surfaces meet along a line, and that line is one of them. Take the other two pairs and you have the other two. Worked from the three surfaces alone, the three lines come out along the three obvious directions, one for each pair. Two things were checked about them. Of the three lines found that way, the number that fail to lie inside both surfaces that made them is zero, and the number whose direction is not square to both of those surfaces is zero.

2:14And the three lines are at right angles to each other: three right-angled pairs among them, which is all the pairs there are. So the axes are a consequence, not an ingredient. Hand somebody the three surfaces and the lines follow; there is no further say in the matter. Each surface has a name, and the name is just the two lines lying inside it. Every one of the three surfaces holds exactly two of the three lines, and every one leaves out exactly one. Three surfaces, three lines left out, and all three are different.

2:55That last count is the point. It is what stops two of the surfaces being the same surface with two names. And it gives you a check that needs no memory: name a surface by the pair inside it, and the line it omits is whatever is left. Ask a pair of surfaces facing the same way for their shared line and you get a refusal instead: two surfaces facing the same way share no line. A pair that genuinely crosses is never refused.

3:30Two things remain that really are choices. The first is which side of each surface counts as the positive one. Above rather than below, right rather than left, towards you rather than away. Three separate choices with two options each, so eight ways it could have been set up - and every one gives a different reading of the same fixed place. Eight distinct readings, one of which is the arrangement we use.

4:00But here is the part that matters. However the sides are chosen, the number of regions they leave is the same: across all eight ways, the distinct region counts number one. So the choice changes what a place is called and not how many things there are to call. The second thing is the corner itself. Sweeping a window of four thousand nine hundred and thirteen places, the number that all three surfaces hold is one, and it is the point where they meet.

4:35Now the count everybody quotes. It is not being quoted here, it is being measured - because the usual argument writes down two times two times two and calls that a proof. Here is the honest version. Sweep that same window of places. Ask each place the only question we have - which side of each surface - and collect the distinct answers that come back. Whatever that collection has in it is the number of regions. Nothing else has been assumed.

5:08The collection has eight things in it. Not one of them was found twice, and a coarser sweep of the same space finds exactly the same eight. That is the count, and now it is a measurement. The sentence says three surfaces at right angles. It is worth finding out what happens if you drop that. Take three surfaces that all share one line, like the pages of an open book. Same instrument, same window. Six regions, not eight.

5:43Take three parallel ones. Four. Take a set where one surface is repeated, so there are really only two. Four again. And here is the honest one, the one that stops this being a slogan. Take three surfaces that are not at right angles at all - only one right-angled pair among the three - but which genuinely point in three independent directions. Eight. The slanted triple reaches it too.

6:13So the right angles are not what forces eight. Of those four other arrangements, the number reaching eight is one, and it is the slanted one. What the right angles buy is something else: they make the three answers readable as three separate distances along three separate lines. The count comes from the three directions being independent; the right angles are how you guarantee that without checking. And it is not that the failures lack right angles. Two of the three arrangements that miss eight have right-angled pairs of their own.

6:54Eight regions, so eight names. Each line splits into two rays at the corner. A region is fenced in by exactly one ray from each of the three lines, and the name is just that list, written around the corner point. Eight names come out of that rule. None of them repeats, and compared with the eight names people are asked to learn, the number that disagree is zero. So there is nothing to learn. There is a rule, and the list is what it prints.

7:29There is also a grid of twenty-four marks - three rows of eight - and it is the thing people actually memorise. It has a shape, and two sentences rebuild all twenty-four cells. The third row is one block of four pluses and then one block of four minuses. Underneath it, the four ordinary plane quadrants are stamped twice, in their usual order. That is the rule. Run it and compare it against the grid, cell by cell: twenty-four cells built, twenty-four cells to check against, and the number that disagree is zero.

8:09The check is a real one. Flip a single cell of the grid and the comparison finds exactly one disagreement. And it lines up with the sweep. The eight answers the grid names and the eight regions the sweep found are the same eight - none the grid names that the sweep missed, and none the sweep found that the grid does not name. So the first four are the four quadrants raised above the flat surface, and the last four are those same four dropped below it. Of the first four, the number whose third mark is not a plus is zero. Of the last four, the number whose third mark is not a minus is zero. And the number of places where the two halves differ in their first two marks is zero.

9:00It is very tempting to think this is just counting in twos. It is not, and that is the commonest way to get an answer wrong here. Build the obvious order - plus before minus, first mark slowest, third mark fastest - and lay it beside the order actually in use. Of the eight, the number the two orders agree on is two. Two out of eight. Take the reading that is plus, plus, minus. In the order in use it sits fifth. In the obvious binary order it sits second.

9:36The two orders hold the same eight readings, so the disagreement is about order and nothing else. But if you count in binary you will be wrong six times out of eight, and the two-line rule is what to use instead. Going the other way takes no work at all: ask the three surfaces, get three marks, find the column. Here is a pair worth slowing down for, because they differ in one mark only. Minus three, one, two - and minus three, one, minus two.

10:11Of their three marks, two are shared and one differs. They come out in regions two and six, and in letters, one is ex-prime, oh, why, zed, and the other is the same with a prime on the zed. So changing one sign moves you to a different region and changes exactly one letter. A place lying on one of the surfaces gets a refusal rather than an answer: a place on one of the surfaces is in no region. All three surfaces have such places in the swept window.

10:48Now eight places offered as practice. Read one at a time, they land in regions one, four, eight, five, six, two, three and seven. Eight answers, eight distinct, and the number of regions no offered place reaches is zero. All eight, each hit exactly once, in a scrambled order - only one of them was already in its numerical place. That turns a drill into a check that the eight really do cover everything.

11:22One last thing, because the picture people carry is wrong. A region is not a corner of a box. It runs out forever in three directions. Take each region and walk out along it to a distance of ten, then a thousand, then a million. Twenty-four places tried that way, and the number where the region no longer holds the place is zero. The nearest of those is ten out and the farthest a million, and the number of them lying inside the window we swept is zero.

11:57Draw a box out from the corner instead. Eight corners, and the number of them outside the single region the box sits in is zero. The box sits inside region one. But the box is not the region. A place a hundred units out in each direction is in that same region and nowhere near the box. One closing thought, because none of this had to happen. The plane version of these ideas was published in sixteen thirty-seven. The three-surface version, in the form now taught, arrives in a letter written in seventeen fifteen - seventy-eight years later.

12:40It appears in print as a full treatment in seventeen forty-eight, a hundred and eleven years after the plane work, and thirty-three years after the letter. Of the four dates that mark this out, only two fall inside the century after the plane work. Nothing was missing. The idea was available the whole time and simply was not carried through - worth remembering while looking at three surfaces in a corner and thinking it obvious.