The maps you find in some buildings and malls have a little marker flag that says, “You are here!” The marker connects the physical reality of where you are standing at that moment with a specific point on a little flat map.
But to fully represent our location, longitude and latitude are not quite enough. (We might be high overhead in a hot air balloon!) To fully represent our position, we need a little more ‘tude, but in this case that’s altitude, not attitude.
We need three (and only three) coordinates to completely represent our location in space. This post is about why.
The short answer is, “Because we live in three-dimensional (3D) space.” Three dimensions, three coordinates. Simple.
And it is exactly that simple. What “three-dimensional” means is that it requires three (and only three) coordinates to fully specify a location. There is a one-to-one correlation between a “dimension” and a “coordinate.” Specifically, a coordinate is a measurement along a dimension.
Let’s break it down by starting with one dimension. Think of a piece of string. (How long is a piece of string?) For the moment, let’s stretch it out alongside a ruler, and let’s imagine the string and the ruler are the same length. It should be easy to see that every point along the string has matching point along the rule.
This just formalizes a very simple concept. A piece of string has a (total) length and a measurement along every point of that length. The important thing here is that the measurement is just one number.
What the number means depends on your ruler; it could be inches or centimeters or something more exotic, like points, pica or pixels. The key is that it is just one number; remember that as we continue.
One other important point before we move on. The measurement (let’s start calling it a coordinate) along the string has the same value regardless of whether the string is stretched out straight, wound around a pencil or crumbled in a ball. One inch along the string is always one inch of string.
The string is a one-dimensional reality. You’ve heard about someone being “one-dimensional.” It means their character exists along only one axis; they have no breadth or depth. Let’s add a dimension.
Most maps are flat, and as mentioned above, longitude and latitude will locate you on a map of the Earth. Some maps have a grid on them, and the grid squares are often numbered like spreadsheets: rows are letters and columns are numbers. (It can just as easily be vice-versa.)
If you think of the map as an “X-Y” graph, then a point on the map has an X and Y coordinate.
Regardless of whether it’s longitude & latitude or letters & numbers or X & Y, it always requires two (and only two) numbers (coordinates!) to specify a point on a flat surface.
Now, by “flat” I don’t mean smoothed out on a table. Remember the string. Likewise the coordinates on a map don’t change if the map is spread out, folded up or crumbled in a wad. By “flat” I mean that the wad could be smoothed out and the flatness made obvious again. A wad of paper still consists of a flat sheet of paper.
Which brings us to the mathematical weirdness that a “flat” space can be crumbled, folded or curved. And by this definition, the surface of the Earth is (just like they used to think) flat. It’s a curved flat surface. It’s also infinite and bounded. Life is filled with irony.
[In fact, you might think the surface of the Earth is fairly bumpy, what with all those mountains and deep ocean trenches. In fact, if you shrunk the Earth to the size of a billiard ball, it would have a smoother surface than the billiard ball. When viewed from a distance, Mt. Everest amounts to damn near nothing.]
Finally we return (at least for the moment) to three-dimensional space. If we lived strictly on the (flat) surface of the Earth, longitude and latitude would suffice.
But, no, we like things like hot air balloons, airplanes, submarines and caves (not to mention the more ordinary tall buildings and sub-parking levels). Because we persist in using all that three-dimensional space, we need three coordinates to fully place ourselves.
As with the string (inches, etc.) or the flat surface (X & Y, etc.), there are various ways to come up with three numbers.
One obvious one is the traditional X-Y-Z. The “Z” being added, of course, to extend the idea of the X-Y graph from 2D to 3D. (What can drive one a little crazy is that there is a lack of agreement on which direction those point. Especially inconsistent is which one is “up”. One might think that, since “up/down” is the new direction, it would get the “Z,” but that is not always the case.)
And while it might not seem like a riff on “X-Y-Z,” your address number, street name and floor are exactly that. Home addresses just assume a single family, so no need for floors, but those who live in apartments do need some third coordinate (apartment “B” or “203”).
Some coordinate forms are unusual. Imagine specifying location relative to the center of the Earth as an elevation angle, a rotation angle and distance. So, for example, I might say my location has a 45-degree positive elevation, a 94-degree negative rotation and a distance of roughly 6,600 (km). Weird, right?
Nah. I just described latitude, longitude and altitude in a very slightly different way. Latitude is literally your angle of elevation relative to the center of the Earth. Think of standing in the center looking out at the Equator. Straight up (90 degrees) is the North Pole, straight down, the South. Likewise longitude is the angle of rotation (about the polar axis) with Greenwich, England being the 0-degree point.
Granted, we tend to measure altitude from the Earth’s surface rather than from the center, but trifles, trifles.
The real point is that there are a variety of ways to “map” a space into coordinates, but you always need as many coordinates as you have dimensions. Two very common ways of doing this are Cartesian (where you divide space into regular “square” units and assign numbers or letters to the columns, rows and so forth. The coordinates there are a list of which column, which row, etc.
One can also do angular mappings where position is specified as an angle of some kind (I’m 45° north and 93° west) plus a distance from the angle-measuring point. (If you’ve ever heard of polar coordinates, that’s what we’re talking about here.)
So remember this: when we talk about “three dimensions” what we mean is that it requires three numbers to specify your position.
And the logic does extend when you hear talk of four or five dimensions. If you do have a 4D or 5D space, it does take four or five coordinates to specify position. In fact, in physics, we actually live in a kind of 4D space. The fourth dimension being time.
When time is a factor, it requires four coordinates to specify where you are. For example, “I’ll be on the eighth floor (1) of the building at 400 (2) Hope Street (3) at 10:30 (4).” The building has many floors; a floor is required. There are many addresses on that street; an address is required. There are many streets in the city; a street is required. There are many hours in a day; a time is required.
If we try to think about four or more physical dimensions our brains are seriously stretched. We exist in a 3D world, so it’s generally impossible for us to visualize having a fourth (or more!) direction we could move without changing our position in the first three!
A term for a dimension is “degree of freedom (of movement).” For a moment, let’s go back to two dimensions and think about a map.
Let’s say you’re at 45° north. You can move east or west without changing that. The east/west direction is a degree of freedom you have that doesn’t affect (at all) the north/south one.
Likewise, you can move north or south without changing your east/west coordinate. This is a key distinction of dimensions. You move along a given dimension without changing your position in any others.
This has gotten long. Let’s leave it here for now so you can absorb this. (As always, feel free to ask questions.) I’ll resume this topic another time, no doubt.