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February 8

Euclidean distance

Computing the euclidean distance (our common length of things), I was never happy about the sums of square roots of sums of squares. So now, I am wondering what makes the Euclidean distance so special. In other words, what are the minimal defining properties?.

Obviously, the length of a line segment between two points is translation invariant, so we can restrict ourselves to consider the length of vectors, or line segments between one point and the origin.

The second obvious property is that it is scalable be non-negative scalars, thus $l(\lambda {\vec {v}})=\lambda l({\vec {v}})$ .

The third is that it is rotation invariant, thus for a unitary matrix $l(U{\vec {v}})=l({\vec {v}})$ . (This implies $l(-{\vec {v}})=l({\vec {v}})$ .)

I'm not sure if the triangle equation $l({\vec {v}}+{\vec {w}})\leq l({\vec {v}})+l({\vec {w}})$ already follows from the above properties, or if it is required at all.

OK, and there must be a vector for which $l({\vec {v}})>0$ .

Are these properties enough to define the euclidean distance uniquely (up to a positive scale factor), and if not, what else is required? 95.115.236.56 (talk) 13:15, 8 February 2015 (UTC)[reply]

Yes, these characterize the Euclidean distance (up to scale). The most important assumption is that the distance be invariant under rotations. If you drop this (but still assume l(-v)=l(v)), then you get a norm (mathematics). Depending on the situation, the assumption of rotation invariance might be thought of as begging the question, though, because rotations are defined in terms of the Euclidean distance! A way out is to suppose instead that some subgroup of the general linear group acts transitively on the unit sphere { v | l(v) = 1 }. (So the unit sphere is "linearly isotropic".) Of course, one can characterize this property in other ways as well. Sławomir Biały (talk) 14:28, 8 February 2015 (UTC)[reply]

Thanks. Do you have a link or some keywords for further reading? 95.115.236.56 (talk) 14:55, 8 February 2015 (UTC)[reply]

So, from my perspective, the general problem of characterizing properties of norms "geometrically" in terms of their unit balls is closely related to the study of convex bodies (of which unit balls are an example). There is a theory due to Hermann Minkowski (see Minkowski functional). But this is all rather more general than studying Euclidean spaces. For Euclidean space, you can find a proof in volume 1 of Berger's "Geometry". He shows, for instance, that any compact subgroup of the general linear group is contained in an orthogonal group. (And gives three different proofs, using rather dramatically different sets of mathematical ideas.) It's probably worth a look. Sławomir Biały (talk) 16:49, 8 February 2015 (UTC)[reply]

If you look more generally on vector spaces with the p-norm

\|\mathbf {x} \|_{p}:={\bigg (}\sum _{i=1}^{n}|x_{i}|^{p}{\bigg )}^{1/p},

what characterizes

p=2

is that it is the only one that induces an inner product, and hence, angles and geometry.

But I think the real answer to "what makes the Euclidean distance so special" is that our physical universe is, in appropriate scales and to a very good approximation, Euclidean. This is equivalent to saying that rotations and angles are a thing in our universe.

I guess it could be a fun exercise to come up with physical laws for other p-norms, and figure out whether they make possible the creation of life that would ask these questions. -- Meni Rosenfeld (talk) 23:41, 8 February 2015 (UTC)[reply]

FWIW, Conway's Game of Life is AFAIK Turing-complete, and can be thought of as using a combination of

L^{1}

and

L^{\infty }

metrics (the local rules rely on discrete

L^{\infty }

, but the fastest spaceships go at c/2 in one direction or diagonally at c/4 in each direction, so the global geometry can be thought of as

L^{1}

. -- Meni Rosenfeld (talk) 23:56, 8 February 2015 (UTC)[reply]

A notion of "distance" is mathematically a "metric" - so you might want to read up a bit on Metric_(mathematics), and metric spaces in general. SemanticMantis (talk) 16:27, 9 February 2015 (UTC)[reply]

To build up on Sławomir's comment "rotations are defined in terms of the Euclidean distance" - a unitary matrix is defined by

{\bar {x}}^{T}y=1

if x and y are the same column, 0 if these are different columns. The first condition guarantees that the standard basis vectors, which have norm 1, will be mapped to vectors of norm 1. The second condition guarantees that the standard basis vectors, which are orthogonal to each other, are mapped to orthogonal vectors. But

{\bar {x}}^{T}x

gives the squared norm of a vector only if we assume Euclidean distance - so without this assumption, there is no reason to think of unitary matrices as rotations. -- Meni Rosenfeld (talk) 23:18, 9 February 2015 (UTC)[reply]

Is there a 1:1 equivalence between algebra and geometry?

Can every concept in algebra be represented using geometry? Like representing the natural numbers as a line, or the roots of a quadratic equation with a parabel. Is any field of mathematics speciallized in comparing algebra dn geometry?--Senteni (talk) 19:19, 8 February 2015 (UTC)[reply]

Well, when you get to algebra equations with a dozen variables, it becomes rather difficult to represent those geometrically, since they would require higher dimensions. StuRat (talk) 16:40, 9 February 2015 (UTC)[reply]

No, there is no 1-1 bijection. Things like a group action can often be described geometrically, but it's inherently an algebraic thing. Likewise for finite fields and many other algebraic structure. I mean sure, I can say that this set of lines (|-|-|||--|) represents the classification of finite simple groups, but that would be facile and silly :) - that is a purely algebraic statement about purely algebraic objects - the fact that we can draw a diagram to illustrate the Klein four group doesn't make that a geometric object.

There's a whole field known as algebraic geometry, but it's a little hard to see the connections to classical Euclidean geometry and basic algebra. In the modern day, it's about using tools from abstract algebra to solve problems in geometry.

There's also a thing called a Geometric_algebra - but that's almost certainly not what you're after. SemanticMantis (talk) 17:23, 9 February 2015 (UTC)[reply]