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Recently seen by a colleague as a True/False question in homework, I'd be interested in your answers. Please respond to the poll, express opinions in replies, and boost for reach.

A line is parallel to itself:

I say "True" provided the definition is the slope is the same. The other concept that the lines don't touch each other is very weak and does not hold true for separate parallel lines in non Euclidean geometries.

@olligobber @ColinTheMathmo I had an instinctive preference for 'true' and was trying to think why, and the mention of equivalence relations crystallized it for me. A line paralleling itself feels of a kind with the idea that an angle is congruent to itself.

(But I don't do serious geometry and there may be good reasons to not want parallelism to be an equivalence relation.)

@11011110 The question then is whether that's a good definition. It means that if A is parallel to B, and B is parallel to C, you can't necessarily say that A is parallel to C.

That feels ... unfortunate.

CC: @nebusj @olligobber

@ColinTheMathmo @nebusj @olligobber But that's completely typical in hyperbolic geometry.

@11011110 But Euclidean and Hyperbolic geometries are different, and in Euclidean geometry it's "most true". So allowing it to be "always" true just solidifies a difference that's already there, and creates a consistency.

CC: @nebusj @olligobber

In another direction, in 3(+) dimensions, I think it's common for "parallel" to mean "in the same direction", in contrast to "intersecting". Now those two pieces are completely independent, so there are four types of lines: skew, intersecting but not identical, parallel but not identical, identical.

@bmreiniger That's true. The question remains as to whether the best/cleanest/elegantest underlying idea "parallel including coincident", or is the "best" underlying idea "parallel, not including coincident".

In some sense it really doesn't matter, but the thought processes in this non-important example may then prove to be useful and important in developing the concept of "taste" in definitions, etc.

My understanding is that you need other line to compare to first to say if they are parallel or not. Statement just doesn't sound good to me, maybe I'm wrong

@ColinTheMathmo @grainloom I’d say true if you define it as parallel at distance=0

It's interesting to see the majority of people from the fediverse I can see getting this wrong

@ColinTheMathmo Definition #1: Lines with the same slope are parallel. True.

Definition #2: Lines that never intersect when extended infinitely are parallel. False.

@vertigo Your definition is not the only option.

It's interesting that this was set as a homework question for a child, it's created an interesting discussion among several working mathematicians.

@ColinTheMathmo To be clearn, I didn't say it was. You asked for an answer to the question, and a rationale behind the answer. I provided one such answer to each.

This is why I clarified with the question being meaningless as queried.

It's always fun to have a go at Euclid, but did the working mathematicians say anything interesting? (In my working definition, a thing might be interesting because of the presence or absence of novelty, aesthetics, or utility)

@yaaps There was quite a lot of interesting stuff, yes. I'm gestating a blog post to try to sum up the interaction.

Diagrams here:

https://www.solipsys.co.uk/Chitter/ParallelLines.svg

https://www.solipsys.co.uk/Chartodon/ParallelLines.svg

CC: @vertigo

@ColinTheMathmo @yaaps Would love to read it. :)

@vertigo @ColinTheMathmo that was my reasoning as well, but i was never strong on relations

!@#$% lazy question, perhaps with no pedagogical function.

@seachanged As stated, in that context, agreed. It has, however, sparked a fantastic conversation elsewhere with potentially huge pedagogical value.

Here:

, but, my intuition is still on the fence about it

@ColinTheMathmo

Mu.

@ColinTheMathmo that depends of the truthness of zero being a even number

@dani Really? How so?

@dani To claim that zero is neither odd nor even is definitely not mainstream. To help me understand your thinking, let me ask a few questions:

* Is zero a number? (Some people say zero isn't a number at all).

* Is 6 an even number?

* Is 2 an even number?

* Is -2 an even number?

* What is your definition of an even number?

Maintaining the 2-fold, the zero for me, is: out off the 2 classes; or gives an error; or is impossible to answer.

Answering your questions:

* That is a great question. In the decimal system, which is in scope, zero is a number

* 6 is even

* 2 is even

* -2 is even

* even number is a number which the modulus on the 2 ( n % 2 ), gives zero

@dani In every other instance, adding two even numbers gives and even number. On that basis, (-2) plus (2) should give an even number, which implies that 0 would be even.

And consider (n%2) when n=0 ... the answer is 0. That implies that 0 is even.

0 is between two odd numbers, that implies that 0 is even.

Do you still think that 0 is neither even nor odd? If it's *not* even then you can have two even numbers which add to a result that is *not* even. That seems unhelpful.

@ColinTheMathmo you are right. Makes sense

Dual of Zoë🍵@zoec@mathstodon.xyz@ColinTheMathmo True, which makes “being parallel” an equivalence relation among lines?