Dimensional Coordinates

you are hereThe 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.

Your GPS device provides your current location in terms of longitude and latitude. Those numbers link your physical location with a specific point on any globe or map of the Earth.

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.

3DThe 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.

string straightLet’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.

string curledOne 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.)

map coordsIf 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.

paper wadWhich brings us to the mathematical weirdness that a “flat” space  can be crumbled, folded or curved.

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.]

long and latFinally 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.)

apartment BAnd 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?

prime meridianNah, 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°) 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° 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.

polarOne 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 takes 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.

400 S HopeWhen 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!

Calabi-Yau Manifold - a six-dimensional object!

Calabi-Yau Manifold – a six-dimensional object!

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.

About Wyrd Smythe

The canonical fool on the hill watching the sunset and the rotation of the planet and thinking what he imagines are large thoughts. View all posts by Wyrd Smythe

17 responses to “Dimensional Coordinates

  • cftc10

    Reblogged this on cftc10.

  • bronxboy55

    I had no trouble following your discussion of dimensions and spacetime. But your very first sentence made me a little nervous. Those maps that say “You are here” confuse the life out of me. If you’re ever at the mall and see a guy wandering aimlessly, please come over and say hello.

  • It's only P!

    And now back to work!

    During my breakfast reading, in Night train to Lisbon, a Portuguese book was referred to: ‘Um ourives das palavras.’ A goldsmith of words.

    I thought of you. 🙂

  • It's only P!

    An afterthought: If only we could measure exactly where we are standing with people!

    • Wyrd Smythe

      Oh, yes. Life would be so much simpler.

      In fact, I’m laying the groundwork to point out exactly why that’s so challenging (and perhaps impossible). The short answer is that people have an infinite number of dimensions, so you need an infinite number of measurements to specify them (or yourself (or the difference/relationship between you)).

      Occasionally in science fiction an author describes a race that (for one reason or another) has a society based on complete honesty with each other. In one case I recall, the race had fur that that moved and revealed their every emotion. In several cases, the races were telepathic, so their thoughts were open to all.

      I’ve also seen a couple movies based on the idea of humans who can’t lie (did you see that one about the invention of lying? silly/stupid on some levels, but thought-provoking in others). And there was a House, M.D. episode where a man had a disease that removed his social filtering (with ugly results).

      It’s interesting to think how such a reality might play out… politics and corporate life would sure be different (for the better, I can’t help but think).

      • It's only P!

        Nope, didn’t see the invention of lying, but I saw Liar Liar. 😉 By nature I have no filter but people don’t like honesty, (no matter how diplomatically presented it is) so these days I shut up or concoct white lies, but this really is bad for my health. 😛

        Two things came to mind after reading your thoughts: 1) Sybil, and 2) Subpersonalities. I read the first and a friend told me about the second one. We may all have a number of subpersonalities. Some of us are not aware of these (schizo’s) and others are, but still can’t change the reactions and responses of the subs. In the book it is suggested that you can give your subs names, ha ha! (Subpersonalities: The People Inside Us – John Rowan. I didn’t read it.)

        In the Night train to Lisbon, a philosophical book, the following quotation could be meaningful for your groundwork. (original in French by Michel de Montaigne). The English translation I found online is very rough so I translated it from the Dutch.

        Each of us merely comprises pied shreds which are so loosely connected with one another that every single one of them is continually fluttering as it pleases; therefore, within ourselves as many differences are present as between ourselves and others.

        That only took 45 minutes. 🙂 I love philosophy that much.

      • Wyrd Smythe

        Liar, Liar was cute; I’d say The Invention of Lying is worth a viewing. It’s a bit convoluted to make the premise work, but it’s fun to see what they do with it (and such a clever idea).

        The hard part, I find, is figuring out when to spare your health and spare the white lies. Like you, I incline towards the truth and really struggle with half-truths. The words, “It’ll all be okay,” tend to stick in my throat when I know there is no rational basis for believing things will be okay. I understand that it’s emotionally supportive and even wishful thinking, but to me it’s also a falsehood, and it bothers me.

        I didn’t really have our sub-personalities (I’ve heard them called “actors”) in mind thinking about all the dimensions of a person, but that would all be part of it. I’ve read that MPD (actually, DID now), assuming it is real, is when those actors become distinct and non-integrated. We all have them, but most of us integrate them as facets of our single personality. When they become fragmented and distinct, that’s DID.

        Your quote seems to support that! Perhaps the key to a well-integrated personality is being in touch with, and in some control of, all those fluttering shreds.

        Way ahead on the naming. Guido is my (wanna-be) tough guy, Poindexter (of course) is my science guy,…

      • It's only P!

        Rational beliefs are also subjective, just as ‘reality’ is. When doctors told Morris E. Goodman that he would never walk again he refused to believe it and set himself a goal to walk out of hospital. And he did. It’s a well known motivational video based on a true story. People were given up by doctors in hospitals but recovered after they consulted a witch doctor (a.k.a. homeopath 🙂 ). Isn’t it better, therefore, to say that things will be okay?

        This leads me to the differences within ourselves, or contradictions. ‘Better a harsh truth, than a false hope.’ But what is the truth? No-one knows the future. A therapist I consulted when my son was three said that she could see that I still had love for my husband and that we could work on reconciliation. This was her truth. My truth was that I wanted to leave him, which I thought she could help me with. Instead she became our marriage counsellor. Two marriage counsellors later and my son now being seven I finally instigated that divorce, and have always wished that I had done it sooner. It is therefore tricky to go along with other people’s truths (perceived reality).

        The quote which I translated has really made me think. With all the differences/contradictions within ourselves and the differences between us and others (plus their own), the human race is a maze within a maze (a riddle, wrapped in a mystery, inside an enigma; Churchill said of Russia). Impossible to fathom. So many undercurrents. Why we even try…

        Churchill continued: but perhaps there is a key. That key is Russian national interest.

        So if we want to understand, reach and influence others, we could study Dale Carnegie’s How to win friends and influence people, right? I read it, and practised it sometimes. All of life then becomes a game, a contrived affair, within which I no doubt would lose touch with my own core. How then, can one strike the golden mean?

      • Wyrd Smythe

        My reply got long (surprise!), so I’m excusing myself out to the top level.

  • Wyrd Smythe

    > Rational beliefs are also subjective, just as ‘reality’ is.

    It may not surprise you that I’m going to disagree with the first part of that. For me, it comes close to saying, “Math and logic are also subjective.”

    Rational beliefs, if they are truly rational, should be based on reason and logic alone, and would be the opposite of subjective in my book. Now, of course, if we leave off the word “rational,” the whole picture changes. And I do agree that reality is (effectively) subjective. (That, in fact, is one of Kant’s key contributions to philosophy, as I suspect you already know.)

    But that may be quibbling over a definition of terms that doesn’t really apply to your point.

    > When doctors told Morris E. Goodman…

    I have a strong immunity to such “inspirational” stories (to be honest, they kind of piss me off). I’ve known too many where the doctor was right. I’ve lost three in the last decade, all of whom pegged the needle on being positive, life-affirming people adored by all who knew them.

    The fact that one guy got lucky and wrote a book about it… just makes me mad.

    > Isn’t it better, therefore, to say that things will be okay?

    The words stick in my throat and I cannot offer them genuinely. You may well be right, at the least about offering positive support, but it’s just not in me. (And, perhaps sadly, the older I get, the less willing I am to playact for the benefit of others. I am increasingly weary of having to meet everyone else on their grounds.)

    > ‘Better a harsh truth, than a false hope.’

    That is more aligned with my view of the world. Knowing a harsh truth (“you will almost certainly never walk again”), then you can set your sights on defying it, accepting it, dealing with it, trying to fix it, whatever.

    It may amount to simply having the pedant’s lust for truth. A single mom I dated a while back had a very bright daughter, and one time the school sent home a page of “high IQ kid” traits (I can’t recall why… parent awareness?). One key trait is over-adherence to rules and laws. I immediately flashed back to an oft-told family story.

    We were on vacation, by car, and had stopped in a small town for lunch. Upon returning to the car, there was a ticket on the windshield (meter had expired). The little Poindexter blurts out in dismay, “Daddy, that’s the third time you broke the law!”

    Poor dad was trying to think of what the first two times were! (I used to remember; don’t now.)

    Ironically, with regard to the law and rules, I kinda ended up going the other way for a while (might be the Preacher’s Kid genes kicking in), and later returned to a more normal honesty. (Cheating on taxes: no, Post-It notes from work, yeah, sure.)

    > But what is the truth?

    “Is truth unchanging law? We both have truths; are mine the same as yours?”

    Sorry, was channeling Jesus Christ, Superstar there for a moment. (That’s from the meeting with P.Pilot.)

    > This was her truth. My truth was that I wanted to leave him,…

    And I have a hard time labeling her perception as a truth. In the most casual sense of the term (“It is a truth that I love coffee ice cream.”) it more or less equates to perception or opinion. On the other end of the spectrum, a truth is a kind of knowledge.

    In the philosophy of epistemology (study of knowledge), the phrase is “justified truth belief,” and there are criteria towards establishing what qualifies. Something must actually be true (it wasn’t in this case), there must be justification for believing it to be true (doesn’t sound like there was), and there must be the belief (okay, one outta three).

    She had an opinion, not a truth (in my (pedantic) book).

    > With all the differences/contradictions within ourselves and the differences between us and others…

    On this we agree completely, and it’s been the key thread of this discussion.

    > So if we want to understand, reach and influence others, we could study Dale Carnegie…

    No; not for me. I hate “managing” people, and I really hate being so managed.

    In high school I figured out you could get with many girls just through being persistent. You could get with even more if you were willing to be manipulative. In fact, it’s amazing what you can do if you’re willing to be manipulative.

    At some point early on (still high school) I had a huge counter-reaction to the whole idea, and swore off that sort of thing forever. From then on, it was openness, honesty, and meeting halfway. Less successful, but I like the person in the mirror better this way.

    I picked up a book many years ago that was about rhetoric, which you can easily assume I loved. The thing is, the book was quite clear that it was strictly about the rhetoric regardless of content or correctness of that content. Once I fully realized that, the book become unreadable to me. I never got but about 1/3 through it.

    Even those articles about how to have a successful blog don’t speak to me. I don’t want a successful blog; I want my blog as a part of me and who I am. If that brings success, that’s great, won’t turn it down, but that is the only acceptable terms.

    Winning to win has never interested me. I never sought grades, I never had any interest in cheating on exams. Success, in and of itself, is meaningless to me. Success as a byproduct or acknowledgement of my work is another matter.

    > All of life then becomes a game, a contrived affair, within which I no doubt would lose touch with my own core.


    > How then, can one strike the golden mean?

    It’s all a process of self-education and experimentation, as far as I can tell.

  • SF or Fantasy, Pick One? | Logos con carne

    […] of degrees of freedom. An orthogonal property, an axis or a dimension, is a degree of freedom. (See Dimensional Coordinates for […]

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