# Talk:Affine space

## Definition

I liked the definition -- a lot more intuitive than other more formal ones.

Thank you. I put it there. It's not my original work; it's "folk mathematics", but I like to think I can claim some credit for realizing it pedagogical superiority and putting it here. Michael Hardy 23:58, 27 Nov 2004 (UTC)

## Removing the story

I vote to put back the informal description which MarSch removed. Oleg Alexandrov 19:11, 7 Apr 2005 (UTC)

Wikipedia has a reputation for articles only mathematicians can understand. And removing this kind of stories, and also striving for more general approaches makes an article maybe better for mathematicians (which I am not sure) and, I think, worse for everybody else. That the story is duplicated is no big deal, it has a useful purpose in both articles. Oleg Alexandrov 19:17, 7 Apr 2005 (UTC)

I agree that the story is better as an informal description than anything else I've seen in this article. And it is misleading to say that physics sees physical space as an affine space, because the affine structure fails to indicate, for example, which lines are at right angles to each other. Michael Hardy 21:13, 7 Apr 2005 (UTC) ... and I would add that John Baez, the mathematical physicist, loved the story. Michael Hardy 21:15, 7 Apr 2005 (UTC)

I have been thinking about the physical space and you have a point: it is an affine space with a metric and conformal structure(angles). But it is still also an affine space and the only motivation thusfar that this article has ever seen. Thus I think we should put this back in. MarSch 13:11, 12 Apr 2005 (UTC)

I wish the story would concentrate on subtraction of points to vectors as per the definition. Now it focuses on adding points in an affine way which I have put into formal form in the second example. I find this a bit confusing. Further I disagree with the use of a formula for one side of the story and not the other, making it more difficult to compare. Mentioning John Baez serves no purpose whatsoever and saying that the proof is a routine exercise is also garbage. cancelling the origin yields the result. Okay so John and Smith know how to "add" two points. Does that have any meaning? Well yes, it is the halfway point. And what is that? It is the point p such that a-p=p-b. Or (a-b)/2=a-p. And why did you put _every_ tex formula on a separate line? That is really ugly and difficult to read. -MarSch 14:29, 8 Apr 2005 (UTC)

Tex formulas have size and font that are different from the regular text. I think that putting them on the same line is ugly.--69.107.121.247 01:48, 2 October 2006 (UTC)

## "Affine module"?

Is there an affine equivalent of a module (mathematics), i.e. a module that has forgotten its origin? —Ashley Y 09:48, July 20, 2005 (UTC)

## Transitive vector space action

That's not enough. The real line acts transitively on a circle. Can we have a better definition back? Charles Matthews 22:09, 12 October 2005 (UTC)

Hmm, that definition has been here since the inception. Would it suffice to define it as a faithful transitive vector space action? that would rule out the circle, at any rate. -lethe talk 23:54, 31 December 2005 (UTC)
Yes, it has to be a principal homogeneous space. Actually that's the real definition ... Charles Matthews 08:53, 1 January 2006 (UTC)

## Affine space before vector space

Since an affine space is like a vector space which has forgot its origin, one should regard affine spaces as a more basic mathematical object than vector spaces. Hence, I would like a precise definition of an affine space without any reference to a vector space. Afterwards one can define a vector space as an affine space together with a choice of origin.

--Toreak 18:37, 20 October 2005 (UTC)

You need the vector space for the displacements, but an affine space is more general than a vector space because it need not be a vector space itself.--Patrick 21:42, 20 October 2005 (UTC)
The problem may be resolved, except for affine spaces over the field {0,1,2}. See the discussion below on the axiomatic definition of affine spaces.--Mark, 17 May 2006
The case for the field {0,1,2} requires special treatment, when adopting the algebraic approach described below, as does the case of affine spaces over {0,1}.
Another route that may be later discussed here is to note that an affine space may also be thought of as a projective geometry in which a specific "subspace at infinity" has been designated. The advantage of this second approach is that (1) projective geometry only requires 3-5 axioms, depending on whether you want to include Desargues' Theorem and the Theorem of Pappus, and (2) the underlying field is internally constructible and does not need to be assumed at the outset. In dimensions more than 2, Desargues' is provable, and Pappus' Theorem is only required to show that products in the underlying field commute.
The most likely route to this approach is to split the "point" concept in Projective Geometry to "point" and "direction"; and split the "line" concept to "line" and "horizon". Then each of the axioms needs to be divided into the various special cases (some becoming redundant or reducible to less generalized forms). For instance, the axiom that asserts that every line has 3 points becomes 2 axioms: (a) every line has 2 points and a direction; (b) every horizon has 3 directions on it. The axiom that asserts (c) two points uniquely determine a line now ramifies into special cases, (d) given a point and a direction, there is a unique line containing the point lying in the given direction, and (e) given two directions, there is a unique horizon which these directions lie on.
This may be brought into more conventional terms by replacing the "direction" concept by that of the "parallelism" equivalence relation; and replacing the "horizon" concept by "plane". The only drawback to this general approach is that Projective Geometry normally assumes its spaces are of 2 or more dimensions, so you lose the 1-dimensional Affine Spaces with this approach (as well as the 0-dimensional space).
All you're really doing is writing down postulates for Descriptive Geometry and then exploiting its capability of internally defining a field to construct an affine space structure on top of it. -- Mark, 19 May 2006

I think there's something to be said for the approach Toreak proposes, but in this kind of forum we have to live with the fact that everyone's already familiar with the concept of vector space. At least that's my gut reaction; I'll say more later.... Michael Hardy 21:44, 20 October 2005 (UTC)

I propose to define affine spaces as sets where you can evaluate affine combinations. See this note about affine spaces. --Toreak 15:40, 21 October 2005 (UTC)

If you do this, please try to be sure to put a more-or-less intuitive explanation of affine combinations before the technicalities. Michael Hardy 23:11, 21 October 2005 (UTC)
If it is an alternative approach and the definition in the current version of the article is common, the alternative can be added but should not replace the existing one.--Patrick 09:03, 22 October 2005 (UTC)

## p-affine space?

The article states:

If the sum of the coefficients in a linear combination is 1, then Smith and Jones will agree on the answer!

or more formally:

An affine combination of vectors x1, ..., xn is a linear combination
$\sum _{i=1}^{n}{\alpha _{i}\cdot x_{i}}=\alpha _{1}x_{1}+\alpha _{2}x_{2}+\cdots +\alpha _{n}x_{n}$ in which the sum of the coefficients is 1, thus:
$\sum _{i=1}^{n}{\alpha _{i}}=1$ .

My question is this: is there a generic name for a combination where

$\sum _{i=1}^{n}{\Vert \alpha _{i}\Vert ^{p}}=1$ .

and would such a thing be called an "affine p-space", or what? Or is it the case that only the p=1 space has properties which are disappear for p not one? (It should be clear that the case p=2 shows up in a zillion different areas of math; I haven't yet seen it be given a distinct name of its own.). linas 15:59, 5 November 2005 (UTC)

Put the norm in and you have a totally different ball game. Charles Matthews 18:22, 5 November 2005 (UTC)
You're applying a norm to a scalar? Where did you get this "p-affine space" idea? 190.224.94.57 (talk) 00:37, 11 June 2009 (UTC)

## Getting carried away with symbolic logic

If ever there was a page that needed 'or in words' after its symbolic logic, this is it. What's that square ended right hand arrow? I believe that Wikipedia is intended to provide readable explanations, given a fair background. I have a fair background, but the assumption of the writers is that the symbols of set theory, as compared to logic as compared to this relatively esoteric area is well established and understood by everyone. It isn't. Centroyd

This is common notation for a function (mathematics), although that article currently does not explain it, because it is undergoing a re-write :-( Very briefly $f:X\to Y:x\mapsto y$ means that the function f maps the space X to the space Y, carrying the individual element x to the element y. In old-fashioned terms, it means that f(x)=y, although the old-fashioned notation doesn't allow X and Y to be described easily. (which is why its not much used). linas 06:13, 31 December 2005 (UTC)

## Theta:SxS to V such that (a,b) goes to a-b

In the current article (Jan 7,2006) the formal defintion says S is a set and Theta sends (a,b) in SxS to a-b in V. Then S is meant to be a subset of V? It doesn't say so, and I'm confused. Please explain, I want to learn this! -Richard Peterson

S is not a subset of V: S contains points, V contains vectors, these are not considered the same.--Patrick 11:55, 8 January 2006 (UTC)
 For a precise definition of affine space, the set S should be described.-Richard Peterson


RP has good point. The minus sign usually is shorthand for a function X x X -> X. In an affine space, it's used in a very nonstandard way, as it takes points from one space (the affine space) to a different space (the vector space). Saying that Theta(a,b) = a-b is simply a tautology; "a-b" has absolutely no meaning other than being equal to Theta(a,b). Presenting subtraction as if it is somehow an intrinsic property of the affine space separate from Theta is deceptive.--69.107.121.247 01:55, 2 October 2006 (UTC)

I added the cleanup tag. The faithfully homogeneousgobbledy gook space stuff is a whitewash of that it's not understood, learned authority in place of reason. The "Theta:SxS to..." business doesn't make sense. This has been unsatisfactory for more than a year. If the writers do know how to fix it, fine. Otherwise let's put on an experts needed tag.Rich 18:45, 16 April 2007 (UTC)
Yes well placed. You might find this pdf slideshow of help. It explains things very well. --Salix alba (talk) 22:49, 16 April 2007 (UTC)

## Axiomatic Definition of Affine Spaces

Interestingly, there is a simple intrinsic characterization of Affine Spaces over fields other than the trivial field {0,1} and the field {0,1,2}. The case of Affine Spaces over {0,1} is separately characteized as abelian group with idempotent elements. To arrive at an axiomatization for these, one needs an analogous concept of "Affine Group", which is discussed below. The case for {0,1,2} also requires special handling.

The operation defined by [A,r,B] = (1-r)A + rB may be completely characterized by the following properties

• [A,0,B] = A
• [A,1,B] = B
• [A,rt(1-t),[B,s,C]] = [[A,rt(1-s),B],t,[A,rs(1-t),C]].

One recovers the vector space operations by designating a point, O, as the zero vector and then defining

• rA = [O,r,A]
• A+B = [[O,1/(1-t),A],t,[O,1/t,B]]

The latter operation requires that t be an element of the field other than 0 or 1.

This was initially developed in 1996 in

## Generalization to "Affine" Groups

Just as an affine space may be thought of as a vector space in which the zero vector is no longer distinguished, one has an analogous concept for groups, where the identity element is no longer distinguished. Unfortunately, the name "affine group" is not possible, since that term is already being used for something else; hence the quotes.

An "affine" group is defined by a ternary operation a/b.c, subject to the axioms

• a/a.b = b
• a/b.b = a
• a/b.(c/d.e) = (a/b.c)/d.e

It is abelian if, in addition,

• a/b.c = c/b.a.

For abelian "affine" groups, the first two axioms are equivalent, but the third remains independent as can readily be seen by a suitable counter-example for a 2-element "affine" group.

The ternary corresponds to the operation a/b.c = ab^{-1}c. Here, the group operations are recovered by first selecting an element E to call the identity and then defining ab = a/E.b and a^{-1} = E/a.E. This defines the group "localized at E".

The group 'associated' with an "affine" group may be recovered by the equivalence relation (a,b/c.d) ~ (c/b.a,d). The corresponding equivalence class [(a,b)] defines an operation a\b which thus satisfies the identity a\(b/c.d) = (c/b.a)\d. One may then define the product by (a\b)(c\d) = a\(b/c.d) and prove that a\a is the identity and the inverse of a\b is b\a. The resulting group is isomorphic to the group localized at any element E.

An affine space over the field {0,1} is an "affine" group in which a/b.a = b.

These sorts of algebraic generalizations of affine spaces have a distinguished history. See
Baer, R., Zur Einführung des Scharbegriffs, J. Reine Angew. Math. 160 (1929), 199-207
Certaine, J., The ternary operation (abc)=ab-1c of a group, Bull. Amer. Math. Soc. 49 (1943), 869-877. Michael Kinyon 18:13, 5 September 2006 (UTC)

## Intrinsic Definition section

I am bothered that the section entitled Intrinsic definition of affine spaces seems to be original research based on some (obviously unrefereed) posting to the sci.math.research newsgroup. However, I have no objection to a section giving an intrinsic definition or description of an affine space, and in fact, there is a published source upon which one could draw, namely

Bertram, Wolfgang, From linear algebra via affine algebra to projective algebra, Linear Algebra Appl. 378 (2004), 109-134.

A preprint version under a different title is also available at the Jordan Theory Preprint Archive. The axioms in Bertram's paper seem more intuitive (to me, at least) than the ones in this article. However, I hesitate to be bold, because the section was added last May without anyone objecting, so maybe my concerns are misplaced. Any thoughts on this? Michael Kinyon 23:41, 3 September 2006 (UTC)

I've tagged that section for cleanup because of the notation. For example, we see this:
[A,m,B,s,C = [C,1-ms,B,(1-m)/(1-ms),A if ms is not 1.
But we should see something more like this:
[A, m, B, s, C] = [C, 1 − ms, B, (1 − m)/(1 − ms), A] if ms is not 1.
Among the differences: proper spacing, italicization of variables (but not digits and not brackets, etc.) proper minus signs instead of stubby littly hyphens, right brackets to match left brackets. This sort of stuff runs through the whole section. Michael Hardy 19:14, 18 December 2006 (UTC)
I've put most of the eqns inside [itex] tags, which fixes some formatting problems, eqn 3' is suspect, but I don't know enough on the topic to know what it should be. The whole section could do with a lot of cleanup so tags still remain. --Salix alba (talk) 15:15, 24 January 2007 (UTC)

## Affine n-space

I was redirected here from the algebraic variety article when I wondered what affine n-space was. Either the link from the aforementioned article was to the wrong article (what would the correct link target be?) or perhaps there should be a sentence or two about that notion? 128.135.100.161 16:09, 9 December 2006 (UTC)

## Another characterization (?)

Does anyone else think this section should be deleted? The "axioms" describe real affine 3-space, so not only is the construction strange (and assumes we know what point, line and plane are--might risk circular logic) but it is very weak, and probably belongs more to a full course on classical geometry. MotherFunctor 07:51, 29 April 2007 (UTC)

That was my immediate reaction when I saw it, and in fact I came to this page meaning to make the same proposal, so was happy to find yours. The reference  it cites is to p.192 of Coxeter, which is the second page of Chapter 13, "Affine Geometry". There is no mention of these axioms or of David Kay that I could find anywhere in the book, so the reference is bogus. My guess is that it should cite David Kay's 1969 text "College Geometry". More importantly than the incorrect citation however is that this is an embarrassingly badly organized account of synthetic affine geometry as the flip side of analytic affine geometry. The entire article should divide at the root into analytic (Cartesian) and synthetic approaches to affine geometry. Since the dominant mode of affine geometry today is analytic (in part because it handles d dimensions as gracefully as two and three), the synthetic portion need not be as comprehensive as the analytic. However the division should be made up front, and the synthetic approach be given its due as appropriate for an encylopedia article. Even if 3D affine geometry is sufficient for the synthetic approach, it should consist of more than a mere listing of axioms of questionable interpretation and completeness. For example do Kay's axioms imply that any three out of four noncoplanar points are noncollinear or is the reader supposed to take that for granted, etc. etc.? --Vaughan Pratt 21:24, 11 September 2007 (UTC)

## Confusing section 'Informal descriptions'

Hi - I found it very hard to understand the section 'Informal descriptions'. I think it is confuses points and vectors, and that, when the distinction beween the two is one of the important aspects of affine spaces. In fact I had to go elsewhere to study 'affine spaces' before I was able to go back and understand staht section - and even then I think it only barely passes as meaningfull.

Anyway, for my own record I wrote up an informal description, which you are free to use as you like:

An affine space distinguishes itself from a vector space by not

requiring a notion of absolute position of points (and thus, no notion of an origin). Instead one can speak of the relative position of one point to another. The difference between two points is a vector and such a vector (in general) is something different from a point. In fact, it is an element of some vector space associated with the affine space. A point and any vector may be added to produce a new point. Addition of two points, however, is generally not defined on an

affine space.

Any vector space is an affine space with itself as the associated

vector space, but not all affine spaces are vector spaces, one example is the real numbers, R, shifted by some unknown but fixed value k, call this space K. Points are then elements on the form (x+k) where x is a real number. Since (a+k) + (b+k) = a+b+2k cannot be expressed as (x+k) without knowing the value of k, addition is not defined and thus, K is not a vector space (unless one is willing to consider more exotic definitions of addition.) It is, however, an affine space with R as the associated vector space, since (a+k) - (b+k) = a-b is a real number and (a+k) + b = ((a+b)+k) is on the form (x+k), and therefore a

point in K.

Even though one cannot add points in an affine space, one may choose

some point O as a pro forma origin and then define addition of points Pi, as O + SUM(Pi-O), but of course the result will depend on the choice of O. On the other hand, if one instead considers linear combinations as in O + SUM(ci(Pi-O)) then the result will be independent of O if and only if SUM(ci) = 1. In general any linear combination where the sum of the coefficients is 1 is called an affine

combination.

Please excuse my ASCII math ;-) Kristian 00:13, 19 July 2007 (UTC)

## "An affine space is a vector space that's forgotten its origin"

I think this is wrong, or at least confusing. Vector spaces are the spaces in which vectors live, in a pure vector space, there is no definition of the concepts of points, let alone origins. Even the wikipedia article on vector spaces has no mention of points or origins. Just because the most common vector spaces we use are affine spaces in an affine frame, doesn't mean that a vector space itself has the concept of a frame. 128.16.9.29 09:26, 10 October 2007 (UTC)

Except that when you call a set with a structure on it a "space", you often call its members points, and people speak of "a point in a vector space", meaning of course a vector. The zero element of a vector space is often called the origin. Michael Hardy 22:43, 10 October 2007 (UTC)

## "The smith and jones story in the informal section is just wrong"

The a and b described in the story are not vectors in the first place- position 'vectors' are not vectors at all- they don't transform in the right way. —Preceding unsigned comment added by 128.230.72.213 (talk) 22:10, 12 April 2009 (UTC)

In effect they become vectors if you choose one point to be the origin. The point is that although they are not vectors, since there's no privileged point serving in the role of an origin, nonetheless not all of the structure of the vector space is lost: you can still find combinations of points in which the sum of the coefficients is–1. Michael Hardy (talk) 00:22, 13 April 2009 (UTC)

## Better variable names

 $V\times A\to A$ , written as (a,b) → a + b


wouldn't it be much clearer to use

 $V\times A\to A$ , written as (v,a) → v + a?


ThinkerFeeler (talk) 06:39, 9 June 2009 (UTC)

## Informal tone

Don't think the tone is up to scratch, I understand this is mainly the informal section but generally this means not rigorous rather than just non-encyclopedic section. I don't understand the subject well enough to do too much however.Wolfmankurd (talk) 11:34, 9 July 2010 (UTC)

This page only makes sense to mathematicians. —Preceding unsigned comment added by 72.66.226.35 (talk) 06:05, 18 March 2011 (UTC)

You cannot explain an affine space simply, just like you cannot explain quantum spin simply. It's just a fact of the universe that some things cannot be dumbed down. It's like explaining color to someone blind from birth. —Preceding unsigned comment added by 67.11.202.195 (talk) 05:31, 26 March 2011 (UTC)
I don't think that can be true, otherwise none of us would know what an affine space is. The concept is somewhat abstract, but it is not impossible to explain it simply - it is all a matter of using the right examples. Gandalf61 (talk) 09:59, 26 March 2011 (UTC)
It's not abstract at all. Take a look at Weyl’s marvellous book “Space, Time, Matter” which makes affine geometry very understandable in terms of addition of successive localized vectors (just what we did at school in fact!). The article quotes Weyl but makes it all very abstract and incomprehensible. Learn from the masters! JFB80 (talk) 15:40, 26 May 2013 (UTC)

## Wrong signs in informal description?

A recent edit changed some signs in the informal description. The version before the edit makes sense to me, the current one doesn't. I'm hardly an expert; any thoughts on this? 84.73.177.141 (talk) 15:10, 9 June 2012 (UTC)

Yes, you are right. I undo that wrong edit. Thank you. Boris Tsirelson (talk) 16:23, 9 June 2012 (UTC)
Thank you, too! 84.73.177.141 (talk) 14:03, 10 June 2012 (UTC)

## Number Line Example

In the examples section, I don't see how children doing math on a number line relates to affine spaces. If someone explains it to me I'll update the article to use that example but be more clear. Nick Garvey (talk) 15:32, 22 February 2013 (UTC)