Which coordinate goes to zero on which axis or plane

0:00A zero looks like an absence. It looks like the number you write when you have nothing to write. Inside a triple of coordinates it is the exact opposite. Each of the three entries is a distance. It says how far the point sits from one of three flat surfaces that meet each other at right angles. Any other value leaves you floating somewhere in a whole surface's worth of places at that distance.

0:29A zero says one thing and one thing only. You are on it. Not near it, not missing, not unknown. On it. That makes a zero the most definite value an entry can take, and once you have seen that, counting the zeros in a triple tells you what the point is standing on before you read a single one of the other numbers. Start with one entry gone. Say it is the middle one.

1:00The middle entry measures the distance from one particular surface, the one that leaves the middle direction out. Zero says that surface holds the point. And that is all it says, because the other two entries are still completely free. Let me put a grid of places out in space so this can be counted. Forty-nine of those places lie on that surface. Across all forty-nine, the number whose middle entry is anything other than nothing is zero.

1:32Meanwhile the first entry takes seven different values across them, and the third takes seven as well. One entry pinned, two entries free. That is the price of a surface. Now take two entries gone. The first being nothing means one surface holds the point. The second being nothing means a second surface holds it too. A point on both is on whatever the two surfaces have in common, and two flat surfaces meeting at a right angle have a line in common.

2:08So two zeros put the point on a line. Which line, though, is the part worth slowing down for. Ask the surfaces directly, rather than reading it off the names. Of the three surfaces, the number that hold every single place of that line is two, and they are exactly the two whose entries went to nothing along it. Run that for all three lines and the answer never wavers.

2:35The number of lines held by some number of surfaces other than two is zero. Here is where the names bite back, and this is the single most common mistake in the whole subject. A point sits on the first line. What is its first entry? The name says first, so the tempting answer is that the first entry has gone to nothing. It is exactly backwards. The first entry is the one that survives.

3:07Along that line the first entry takes seven different values on the grid, and it is the only thing still moving. The second and third are nothing at every place there is. And of the seven places on that line, the number whose first entry is nothing is one, the place where all three lines cross. One out of seven. The entry that shares the line's name is the one thing that does not vanish, and the two that do vanish are named after the directions you have just left behind.

3:44Three nothings, then. All three surfaces hold the point at once, and only one place in all of space can manage that. It is the crossing, where the three lines meet. But there is a question hiding here that is worth asking out loud. If two zeros already pin you down so hard, why is the third one not spare? Because two zeros do not pin you to a place.

4:10They pin you to a line, and a line is not a place. On the grid, the number of places whose first two entries are both nothing is seven, a whole line's worth of them. Exactly one of those seven is the crossing. The other six are somewhere else entirely, strung out along the direction that is still free. Two nothings tell you what you are standing on. Three tell you where you are.

4:39So the count of zeros is a readout, and now it can be checked properly. Take the grid: seven values along each of three directions, three hundred and forty-three places in all. For every one of them, count how many entries are nothing. Two hundred and sixteen have none at all. A hundred and eight have exactly one. Eighteen have two. One has three. Those four add back to three hundred and forty-three, so nothing has been dropped and nothing counted twice.

5:13Now go round again and ask a completely different question of the same places: what is this place, decided only by which surfaces and which lines hold it, never by looking at its entries. Four counts come out of the first question, four kinds out of the second. The number of counts that match more than one kind is zero, and the number of kinds showing more than one count is also zero.

5:42Neither reading was ever shown the other's answer, and they line up perfectly. Let me build something you can hold all of at once. Put one corner of a box at the crossing, put the opposite corner at some chosen point out in space, and run its edges along the three directions. Eight corners. Count the nothings at each one. One corner has none: the chosen point itself, hanging in open space clear of everything.

6:15Three have exactly one, and those three lie on a surface. Three have exactly two, and those three lie on a line. One has three, and that is the crossing. One, three, three, one, adding to eight. Every pattern the readout allows turns up in this one box, and the middle two turn up three times each, once for every direction you can drop. The count of places on the grid that this box shares is two, so those four numbers are a second measurement and not a slice of the first.

6:51Now turn the readout around and use it to find a corner instead of checking one. Which corner of the box lies on the middle surface? The wrong move is to hunt for it in the drawing. The right move is to ask which measurement vanishes. Four of the eight corners sit on that surface, and every one of the four has a middle entry of nothing. But three of those four carry more than one nothing, which means they have slipped onto a line, or all the way down to the crossing.

7:24The number carrying exactly one nothing is one. That is the corner, and its entries come out as two, nothing, five. Notice what did the work there. Not the picture, and not the letters on it. The zero count found the corner, and the other three were ruled out by having too many zeros rather than by lying somewhere else. Two questions that look like opposites are really the same question asked from two ends.

7:55First: a point sits on a line, so what can you say about its entries? Two of them are nothing, because that line lies on two of the surfaces at the same time. Second: a point sits on a surface, so what can you say? One entry is nothing, and about the other two you can say absolutely nothing at all. The asymmetry is not a quirk of the wording, it is the geometry.

8:23Seven places of the grid lie on that line; forty-nine lie on that surface. Count the entries each one holds still: a line pins two of them, a surface pins one. So a line frees one entry and a surface frees two. Being told you are on a line is far more information than being told you are on a surface, and the zero count is exactly how much more.

8:51One more trap, and this one is nothing but spelling. The same surface gets written two ways, with its two directions named in either order, and it is easy to start believing they are two surfaces. So build it twice. Take the two directions one way round, build the surface, and keep its normal. Take them the other way round and build it again. The two normals come out pointing exactly opposite to each other, so these really are two different arrows.

9:25Then ask them, place by place across the whole grid, which places they hold. The number of places the two disagree about is zero. Forty-nine places each, and the same forty-nine. And the entry that goes to nothing on each of them is the same entry. Two names, opposite normals, one surface, and the order you say the letters in changes nothing whatsoever about which measurement vanishes. There is a short list people tend to memorise here, and it is worth knowing what it leaves out.

10:02Count the general forms there are: one for each of the three lines, and one for each of the three surfaces. Six of them. The list that usually gets memorised carries three, and they are the three lines. One more, a single surface, tends to arrive separately as a remark off to the side. That leaves two forms that neither one hands you, and they are the two remaining surfaces.

10:31Nothing is wrong with the list; it is just short. And this is the difference between carrying a rule and carrying a list. If you have the list, those two are simply missing. If you have the rule, you write them down from the count. So here is the whole thing, in the order you would actually use it. Count the nothings in a triple. None, and you are out in open space, in one of eight regions.

11:00One, and you are on a surface. Two, and you are on a line. Three, and you are at the crossing, and there is only the one of those. That is not a fourth rule to memorise alongside the others. It falls straight out of what an entry is. Each entry is a distance from a surface, and a distance of nothing means that surface is holding you. Count how many surfaces are holding you and you have counted how much freedom you have left: three loose directions, then two, then one, then none.

11:36A zero is not the entry you could not work out. It is the most definite thing a coordinate is able to say.