[Doctor Stochastic]

For the algebraically inclined: the standard theory of space-time is 3 space + 1 time dimension. The distance between two points: (x1,y1,z1,t1),(x2,y2,z2,t2) is given (for flat spaces) by (x1-x2)²+(y1-y2)²+(z1-z2)²-c²(z1-z2)², where "c" is the speed of light in the system (and the space is flat.)

WALOGIMBAT there are 3 positive and 1 negative terms; the signs could be reversed without changing the physics. This "metric" or "distance formula" produces a hyperbolic geometry. Were there a fourth spatial dimension (call it w), then the distance between (w1,x1,y1,z1,t1) and (w2,x2,y2,z2,t2) would be given by (w1-w2)²+(x1-x2)²+(y1-y2)²+(z1-z2)²-c²(z1-z2)². This describes a different geometrical structure. According to physics (I think), there should be measurable differences between the two models. (I don't know what these would be.)

I have ignored interactions between the various dimensional coordinates. For those interested, see things like quadratic forms or metric spaces, etc.

One need not actually see the dimensions with one's eyes, but only find a method of measuring the effects of such dimensions's existence. This is already done with the quantum variables called "spin" for example.

[Ready2go]

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[freedumb2003]

Math can deal with n dimensions as well as zero and null dimensions. It can also have arrays with null and n axes.

It is fun but conjuring the numbers doesn't get the job done.

[MHGinTN]

That looks downright Lorentzian ... how transforming!

[Quark2005]

According to physics (I think), there should be measurable differences between the two models. (I don't know what these would be.)

Most everything you said looked right to me, except you used z in place of t in the c² term (I assume that's a typo) - and the metrics you use all describe flat spacial geometries. Different geometries emerge when non-diagonal components appear in the metric tensor/matrix in the presence of a gravitational field (things start getting complicated then) - Einstein's gravitational equation dictates how this occurs. Assuming that this 4th spacial dimension has a localized geometry, I'm making an educated guess that the measurable effects would be that gravity becomes much stronger at super-short distances. (What distance, I don't know - experiment rules out anything greater than a cm or so, I think.)

For those interested, see things like quadratic forms or metric spaces, etc.

Most people will be deterred once they actually find out what goes into basic general relativity physics - not because it's too difficult in principle, but because it's, well, boring. Riemannian geometry is what it's all about, and unless you have a vested interest in learning it, it's all a lot of tedious algebra.

WALOGIMBAT

Now this has me confused!