: : : There can be no five-sided squares yet undiscovered because by man-made definition they would no longer be squares! : : Only according to the Euclidean definition; which is a helpful approximation rather than The Ultimate Truth.
: : There might indeed be a five-sided square; but it would require non-standard definitions of the terms 'square' or 'five' or 'side'.
: You got me. I should have specified which geometry I was employing. Of course you can understand that I used the most popular form of geometry to make my point. Then again, it serves me right for doing so with a rocket scientist, of all people.
This is a real problem, actually...
Have you ever seen a square; something with four *exactly* equal sides, four *exactly* equal angles of 90.000000...00° and infinite thinness?
Nope. Neither you or I have ever seen these shapes because they just don't exist; even if a square was 1 atom thick, it would have a measureable thickness.
Everything in nature is either a prism, a rough sphere, a Platonic solid or something irregular; or a combination of one or more instances of one or more forms. As such, the figures we do our maths on are just abstractions; they don't apply to 'real life' any more than a cone does to a mountain.
And everything theoretical is open to challenge; 2D geometry is a pretty solid theory and has lasted incredibly well, but when it comes up against non-Euclidean space-time as per Einstein's general theory of relativity, it breaks down.
In the end, the best theory is the one that most simply and completely explains the observed phenomena; and the theory you use depends entirely on the accuracy you need.
So, for example, if firing an artillery shell or Apollo 8, Newton's Laws will suffice adequately; Apollo 8 was a few seconds late getting to the Moon. However, if looking at a quasar, Newtonian gravitation won't give you a good answer, since the quasar is exhibiting relativistic properties; and the light will probably be curved by an intervening object causing a curvature of space-time and deflecting the photons.
You use the theory that best (and most simply) describes the scale you're examining.
: : Taking my triangular example above, one person could say 'the internal angles of a triangle add up to 180°' - another could say 'aha! but the internal angles of *this* triangle add up to 270°' - and they'd both be absolutely correct; they have merely failed to state the caveats and implicit assumptions in their model; the spatial geometry that they were using.
: Maybe so, but if the debate is taking place in a pub, the second guy will find his audience slowly backing away from him with reassuring grins while the barkeep hastily calls the local asylum.
True; but more because of an accident of education; most people are never taught anything apart from Euclidean geometry.
: : The statements 'there are five-sided squares' and 'there are no five-sided squares' fall into exactly the same category; they both fail to put the qualifying assumptions.
: : If you included, say, space-time, then a cube would have four dimensions.
: Is that the reigning scientific consesus re: the 4th dimension, that it's time? I suppose that is a reasonable conclusion, but if the Fifth Dimension is a hokey singing group, I'd expect the fourth to be a bit more concrete.
It really depends; there's not a lot of firm evidence either way. If you subscribe to string theory, then there could be ten dimensions - most of which are folded up on each other where we can't see them. I'm not convinced.
(In addition, one of the (odder) interpretations of Pauli's exclusion principle is that multiple electrons can stay in orbitals due to existing in different sets of dimensions; so a twelve-electron orbital could exist over 36 dimensions!)
: :Unless you say this specifically, anyone you ask will say quite firmly that a cube has three; length, breadth and height.
: : (Hardened physicists will probably say a number between 4 and 'possibly infinite' for the number of dimensions a cube has...)
: Aha! Couldn't there be 1 or 2 or 3 sided cubes according to some other definition? I believe the noted physicist Dr. A. Square is hard at work on such a theory.
Yup. Although most people will start from the Euclidean geometry as a nice simple place to start.
: : As such, my original statement to Stuart was considered; I've never seen a five-sided square; my physical perceptions and model of the Universe specifically prohibit five-sided squares. But I can't in all integrity say that they do not exist; merely that the physical evidence I've seen makes them impossible to perceive physically.
: Makes sense.
: : Which is where we come down to reasonable belief; it is more reasonable to most people to agree with something that is backed up by all the physical evidence but not made certain either way by experimentation (e.g. that there are dodos left alive somewhere) than it is to believe something that flies in the face of all our physical evidence (that five-sided squares exist and frolic in fields of clover) - because the only evidence that we have to support or refute such an idea is our own physical perception.
: : I know it's pretty abstruse and overly picky, but it's best to be clear about this; I can no more say 'God does (or doesn't) exist' than I can say 'Five-sided squares exist (or don't exist)'. However, in the absence of any firm data either way, I go with what I've got; my physical perceptions and models of the Universe; which is why I feel that my position is reasonable; it is at least founded on something I can demonstrate and replicate under experimentation.
: Thank you. You've given me newfound appreciation for the unparalleled clarity of the Code of Federal Regulations.
No problems ;)
Farinata.
