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A definition in classical geometry (for example, Birkhoff's formulation, but I suppose it could be all of them) is that a line is always considered to be parallel to itself. I understand this is probably for convenience, but in my mind since two distinct lines are parallel if they have no points in common and a line has infinitely many points in common with itself. Perhaps the idea is to ease the definition that two (non-parallel) lines intersect at one and only one point?

Q: What's the purpose/what inconvenience would be caused if we didn't have that definition?

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    $\begingroup$ Do you know what an equivalence relation is? $\endgroup$
    – saulspatz
    Apr 17, 2018 at 22:29
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    $\begingroup$ If you consider a line to be parallel to itself, then you can say that any two lines in the plane with the same slope are parallel. Under the definition of "have no points in common" that statement is false, since you need to add an exception. $\endgroup$
    – Joshua Ruiter
    Apr 17, 2018 at 22:29
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    $\begingroup$ A big reason is to make "is parallel to" an equivalence relation. Equivalence relations are used a lot in math, and it's convenient to just be able to use standard notions about equivalent relations, without having to make exceptions. You don't even have to know the use cases in advance; it's just going to make life easier eventually. $\endgroup$
    – saulspatz
    Apr 17, 2018 at 22:33
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    $\begingroup$ @saulspatz Oh! I see. The definition in question basically gives us a ~ a. Thanks. $\endgroup$
    – Ryan
    Apr 17, 2018 at 22:36
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    $\begingroup$ It's for transitivity: if $a\parallel b$ and $b\parallel c$, then $a\parallel c$. And we would not like to add 'except if $a=c$', would we...? $\endgroup$
    – CiaPan
    Apr 18, 2018 at 6:47

7 Answers 7

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The idea is that you want "parallel" to define equivalence classes (called "pencils", cf. Coxeter, Projective Geometry, and Artin, Geometric Algebra), which require the defining relationship to be an equivalence relationship: reflexive, symmetric, and transitive. Those classes then have some nifty uses, like defining projective space by adding a point at infinity for each pencil (which is the considered to be on each of those lines) and a line at infinity for each class of parallel planes (this line containing all the points at infinity corresponding to the pencils of lines in that class of planes).

Also, you were already going to have to rethink the definition of "parallel" as having no points in common, if you are going to do solid geometry. Parallel lines also need to be coplanar...i.e., there need to be two other lines that intersect each other and that each intersect the parallel lines (five distinct points of intersection). Lines that are not coplanar are called "skew" not "parallel".

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    $\begingroup$ citation needed for 'often called "pencils"' $\endgroup$
    – OrangeDog
    Apr 18, 2018 at 12:27
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    $\begingroup$ en.wikipedia.org/wiki/Pencil_(mathematics) $\endgroup$
    – Andrei Sipoș
    Apr 18, 2018 at 13:33
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    $\begingroup$ Wiki is more a citation for "called pencils long ago"... it's a cute name but I wouldn't use it and expect people to know what you are talking about... $\endgroup$
    – djechlin
    Apr 18, 2018 at 16:27
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    $\begingroup$ @djechlin: “pencils” is still absolutely current, although (as with many terms) it’s been generalised way beyond the original meaning. For instance, here’s a 2008 paper in Advances in Mathematics: Pereira and Yudvinsky, Completely reducible hypersurfaces in a pencil, which studies pencils of hypersurfaces in projective space. $\endgroup$
    – Peter LeFanu Lumsdaine
    Apr 18, 2018 at 23:29
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    $\begingroup$ any book on projective geometry will define and use pencils - e.g., here in Coxeter see chapter 11. $\endgroup$
    – davidbak
    Apr 19, 2018 at 3:04
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If two lines are both parallel to a third line, they should be parallel to each other, even if they are the same line.

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    $\begingroup$ @MauroALLEGRANZA Of course it depends, and this is an (or the) answer to the question "Why is the definition like this?" $\endgroup$
    – JiK
    Apr 18, 2018 at 10:11
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    $\begingroup$ @MauroALLEGRANZA I think the point wnoise is trying to make is that it makes stating properties easier. Like how we say that $1$ is not prime. We could say that $1$ is prime, and then write it as an exception to basically every number theory theorem out there. It's the same for parallelism – if you need to end every statement about parallel lines with "[...] are parallel or equal", it's probably easier to just say that parallel includes equal. $\endgroup$
    – Najib Idrissi
    Apr 18, 2018 at 14:56
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    $\begingroup$ This isn't really an explanation because you can replace "even" with "unless" and it's exactly as reasonable. $\endgroup$
    – djechlin
    Apr 18, 2018 at 16:29
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    $\begingroup$ @djechlin not really. If you just say “If two lines are both parallel to a third line, they should be parallel to each other”, then the “even if they are the same line” is already implied and already holds true if you don't explicitly contradict it with an “unless”. $\endgroup$
    – leftaroundabout
    Apr 19, 2018 at 8:02
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    $\begingroup$ @djechin: This answer on its face is is indeed just an assertion, but the form of this assertion ("even if") is an implicit explanation: statements are nicer with this choice, and only a modicum of mathematical maturity is required to read that into the answer. The example given is A || B && B || C => A || C is "obviously" a nicer, more composable theorem than what you would get for excluding the A = C case. $\endgroup$
    – wnoise
    Apr 19, 2018 at 20:41
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This is also quite natural when starting from the definition in a metric space:

Two lines $a$ and $b$ are parallel if for every point $p\in a$, the distance from $p$ to $b$ is the same.

Then, $a$ is parallel to itself because for every point $p\in a$, the distance to $a$ is zero.

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  • $\begingroup$ I think this is ultimately the correct answer. Various English lexicons give the main definition of "parallel" as "being an equal distance apart everywhere". Note how we can talk about parallel rows of trees, which have nothing to do with equivalence classes. $\endgroup$
    – user21820
    Apr 19, 2018 at 5:58
  • $\begingroup$ @user21820 I would not say it is the correct answer. The axiomatic definition going back to Euclid is arguably more elegant than anything referring to distances. $\endgroup$
    – leftaroundabout
    Apr 19, 2018 at 7:57
  • $\begingroup$ By "correct" I meant "the intuitive notion which we are interested in". After all, we all have the intuitive idea that train tracks are parallel even when the tracks curve. $\endgroup$
    – user21820
    Apr 19, 2018 at 8:27
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It is a (useful ?) convention.

According to Euclid's original definition two parallel lines must be different:

Definition 23. Parallel straight lines are straight lines which, being in the same plane and being produced indefinitely in both directions, do not meet one another in either direction.

Thus parallelism is not reflexive.

The same with Hilbert's version :

Axiom III. In a plane $\alpha$ there can be drawn through any point $A$, lying outside of a straight line $a$, one and only one straight line which does not intersect the line $a$. This straight line is called the parallel to $a$ through the given point $A$.

We may compare with Birkhoff's definition :

In consequence of Postulate II, any two distinct lines $l, m$ have either one point in common or none. In the first case they are said to intersect in their common point; in the second case, they are said to be parallel; a line $l$ is always regarded as parallel to itself.

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    $\begingroup$ I would be very interesting to understand why Birkhoff specifically includes that a line is always parallel to itself when that fact does not follow from the definition. $\endgroup$
    – Todd Wilcox
    Apr 18, 2018 at 13:20
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    $\begingroup$ @ToddWilcox - I think that C Monsour's answer is the "reasonable" one: to define an equiv relation and the "generalization" to space. Regarding equivalence, I think that we can handle it in the same way as $<$: we have that not-$n \lt n$ and thus "$\lt$" is not reflexive. Thus, we introduce $\le$. In the same way, we can add a new relation: "paralell-or-equal". $\endgroup$
    – Mauro ALLEGRANZA
    Apr 18, 2018 at 13:27
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In .. "parallel" to the answers already given, defining that a line is parallel to itself has the advantage to match, on the algebraic side, the distinction of linear systems based onto the coefficient matrix and the total matrix.

So we define parallel the lines for which the coefficient matrix has rank $1$ ( thus they have parallel normal vectors), and then we distinguish them as "distant" (no solution) or "same"(infinite sol.) according to the rank of the total.

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The definitions of parallel vectors:

Two vectors $u$ and $v$ are parallel if their cross product is zero, i.e., $u\times v=0$. Two vectors are parallel if they are scalar multiples of one another.

Similarly parallel lines can be characterized as:

Two lines are parallel if they are scalar additions of one another.

Obviously, in this sense the equal vectors as well as the same lines are parallel.

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When comparing different functions, often points of interest are where the slope of the graphs is the same: parallel.
It would be quite inconvenient if two slopes would be considered "not parallel" just because they're on the same line.

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