New readers may start here.
Good question. I define it as follows. We have 3 dimensions we all know about – call them North-South, East-West and up-down. They’re all at right angles to each other. The fourth dimension is a direction at right angles to all three of them.
That is a direction in which, as the late and much-missed math writer and puzzle expert Martin Gardner put it, we humans “cannot even point”, however hard we try. So it’s not easy to imagine. The best way to start to picture it, I suggest, is to imagine that you’re a pondskater. You live on the pond surface and you can skate as much as you want N-S, or E-W, or anywhere between those two. But you’re stuck on the surface of your pond and you have no concept at all of “up” (or indeed of down, come to that). You've no reason to think that 'up' or 'down' even exists.
You’re an intelligent pondskater however and you take math lessons. Your teacher (from another world above or below the pond we assume) teaches you about Pythagoras. Pythagoras, of course, works perfectly well on the pond surface. 3 squared + 4 squared = 5 squared just as much there as anywhere else, the teacher explains. Then he or she says something like this.“You, being a pondskater, can’t imagine ‘up’, which is a direction at right angles to N-S and E-W. But just suppose, in your imagination, that there was such a direction. Pythagoras would work just as well in 3 dimensions as he does in 2. Say you started at point X and you found you could get to a mysterious point Y by skating 2 metres North from X, then 3 metres East, then mysteriously jumping 6 metres ‘up’. Now you could find how far away you were at your new point Y, from X where you started, by adding 2 squared + 3 squared + 6 squared (= 4 + 9 + 36) and taking the square root. The square root of 49 is 7 so you’d know that at Y you’d be exactly 7 metres away from X.”
(In parenthesis: we all know about Pythagorean triples such as 3, 4 and 5 where 3 squared + 4 squared = 5 squared. What about Pythagorean quadruples? I’ve just found one by accident, namely 2, 3, 6 and 7 where all the numbers are integers – how many more are there?)
We humans are lucky enough to live in 3 dimensions so we can see that the pondskater is mistaken in thinking that “up” doesn’t exist. He/she can’t see it, or imagine it, or even point in that direction. It’s there nevertheless. And what I’m saying is that we ought to be able to think of a 4th dimension which is analogous to the pondskater’s third. It’s at right angles to N-S and E-W and up-down. We can’t point to it, but we can surely imagine it. If the pondskater can imagine “up”, surely we ought to be able to imagine “fourwaurds” (or whatever we’re going to call it)?
Or suppose you were an animal warden, in charge of cats and dogs. You could count them and keep a 2D chart of their numbers, cat numbers along the horizontal axis and dog numbers along the vertical. Maybe your job would be to make sure by careful breeding that the plot of the total number of animals (cats plus dogs) fell on some arbitrary line on such a chart. Cats plus dogs = 100 for instance. Add elks and you could keep a 3D chart of the numbers. Add foxes and, in theory at least, a 4D chart of the numbers must exist. Come to that you could add goats and get a 5D chart, horses and get a 6D version. And so on.
To suggest that the 4th dimension doesn’t exist is equivalent to saying that foxes don’t exist. And clearly they do. Ask any chicken farmer.
New readers:
- At this point you may like to click here. A new window will open where you can browse for a simple explanation of a 4D cube ... how it relates to a point, a line, a square and an ordinary cube ... and how to draw it ... and how it rolls ... all in the context of trying to understand the curious rotating graphic at the top of this blog.
(Another parenthesis. What about Pythagorean quintuples? Are there any? Suppose you’re living in 4 dimensions and you travel A metres North, then B metres East, then C metres up, then D metres “fourwaurds” such that you find yourself, let us say, E metres from where you started. To work out what E is, you add the squares of A, B, C, and D and then take the square root. Question: are there any values of A to E such that all five quantities are integers? And if the answer is yes, is there an infinite number of these Pythagorean quintuples (as there is with Pythagorean triples)? And what about, in 5 dimensions, Pythagorean sextuples. Are there any? I leave the thought with you.)
This has been drawn to my attention by a colleague:
ReplyDeleteHere are a couple more Pythagorean quadruples. First 1,2,2 and 3. Second 1,12,12,17. Both of these have the curious feature that two of the numbers are equal. It is clear that you cannot have a Pythagorean quadruple with all three numbers equal, since sqrt 3 is irrational. But sqrt 4 = 2, so there are infinitely many Pythagorean quintuples of the form x, x, x, x, 2x. A quick example of a Pythagorean sextuple is 1, 1, 1, 2, 3, 4.
I noted one quadruple 2, 3, 6, 7; and my colleague noted 1, 2, 2, 3. Trying to generalise, I looked at this one: x; x + 1; x(x + 1); and x(x + 1) + 1. Surprise, surprise, these are a Pythagorean quadruple. So (if you had nothing better to do with your time) you could start with, say, a million and get: a million; a million and one; a million and one times a million (1,000,001,000,000); and the latter plus one (1,000,001,000,001). Perhaps more amusingly, you could have: 999,999; 1,000,000; 999,999,000,000; and 999,999,000,001. Somewhat surprising quadruples, I think.
DeleteThe 1, 12, 12, 17 quadruple is amusing and I have been quickly trying to generalise from that, without success so far. Since 1, 2, 2, 3 and 1, 12, 12, 17 both work, I wondered if 1, 22, 22, ? might. It doesn't.
My colleague notes that his knowledge of Pythagorean quadruples comes from using "Pythagoras' Theorem in 3-d", i.e. working out the longest knitting needle that could fit into a cuboid box of dimensions a x b x c. the answer is sqrt of (a^2+b^2+c^2). Examples use a,b,c = 1,2,2 or 2,3,6 or 1,12,12, or multiples thereof. All Pythagorean triples are of the form a^2 - b^2, 2ab, a^2 + b^2 where a,b,c are integers such that a is bigger than b which is bigger than one. A favourite triple is 20,21,29 which is the nearest you get to an impossible Pythagorean isosceles right-angled triangle, unless the numbers are huge. Impossible because sqrt(2) is irrational.
DeleteI can give you infinitely many Pythagorean n-tuples. It's very simple if you think about it. You just gotta use the hyper sphere and scale them up. I also know of one example in quadrilaterals that is part of the 3d sphere but not generated combinatorically. I think it's kind of special. 200 bucks and I'll do a youtube video of it some time. It's not a bad investment. How many people pay guides thousands of dollars to go see wild life?
ReplyDelete