Mentor
The title is admittedly clickbait. Or a joke. Or a provocation. It depends on with whom you speak, or who reads it with which expectation. Well, I cannot influence any of that. I can only tell how I mean it, namely as an entertaining collection of simple truths which later on turn out to be not as simple as thought if considered from a scientific point of view. Some of them could even lead to heavy debates within the scientific community, so maybe I should say: from my point of view. So before you get excited or even angry about what is to come, please keep in mind to take it with a big grain of salt and try to feel entertained, not schooled.

#10. The sum of all angles in a triangle is 180°.
We all live on a globe, a giant ball. The angles of a triangle on this ball add up to a number greater than ##180°##.
And the amount by which the sum extends ##180°## isn’t even constant. It depends...

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Janosh89, Dale, Wrichik Basu and 3 others

## Answers and Replies

stevendaryl
Staff Emeritus
Science Advisor
I have one nitpick about the discussion of rationals. In the foundations of mathematics, what's done is this:
1. We start with the natural numbers, 0, 1, 2 ... (some people start with 1 instead of 0).
2. Then we can define the integers as equivalence classes of pairs of naturals: ##(a,b) \approx (c,d)## if ##a+d = b + c##. So ##8 - 12## is defined to be the equivalence class ##(8, 12)##.
3. Then we can define the rationals as equivalence classes of pairs ##(n,m)## where ##(n,m) \approx (p,q)## if ##n \times q = m \times p##.
4. Then we can define the reals as equivalence classes of convergent sequences of rationals.
However, having developed the reals, we can then, for most purposes, just consider a real to be a "black box" with certain closure properties, and then consider rationals and integers and naturals as particular subsets of the reals. Then under this reinterpretation, an expression such as ##2/3## means the real number resulting from dividing the real number 2 by the real number 3. So ##2/3 = 4/6## is an equality.

pbuk
AndreasC
Gold Member
Hmmm idk... Most of these are really minor nitpicks. #10 and #8 are both true when talking about the Euclidean plane, and that's what students learn about. Actually the discussion of #8 is kind of confusing. What do you mean they intersect at the horizon, they clearly don't. It just looks like they do. I'm not sure what perspective has to do with the rest of the discussion.

#6 is kind of overcomplicated. I see the central point but the way you have presented it is not how it is usually discussed in schools because it's more complicated than it needs to be.

#5 is bound to confuse more than it might help. 12/12 does equal 1. It's a different representation of the same thing. In analysis they are treated as a subset of the reals anyways, not as equivalence classes, so I don't really see how this is beneficial.

#4 is taught that way in schools as far as I know.

#1 is also weird because I'm not sure what you mean by "correct" definition. The most common definition is, a number which can't be decomposed into a product of two smaller natural numbers. I don't see why this definition is "wrong".

symbolipoint
Homework Helper
Education Advisor
Gold Member
The angles of the triangle being 180 degrees SPECIFICALLY tells of those as INTERIOR angles.

Infrared
Science Advisor
Gold Member
Actually the discussion of #8 is kind of confusing. What do you mean they intersect at the horizon, they clearly don't. It just looks like they do. I'm not sure what perspective has to do with the rest of the discussion.

This is pretty standard, you might want to look into projective geometry. You can embed ##\mathbb{R}^2## into the projective plane via ##(x,y)\mapsto (x:y:1).## If you have two parallel lines in the plane, say ##x+y=0## and ##x+y+1=0## for specificity, their projective closures are given by the equations ##x+y=0## and ##x+y+z=0##, which intersect at the "point at infinity" ##(1:-1:0).##

The "horizon" is given by ##z=0## here.

Although, I would summarize this by saying that there aren't parallel lines in projective geometry, not that parallels intersect.

AndreasC
Gold Member
This is pretty standard, you might want to look into projective geometry. You can embed ##\mathbb{R}^2## into the projective plane via ##(x,y)\mapsto (x:y:1).## If you have two parallel lines in the plane, say ##x+y=0## and ##x+y+1=0## for specificity, their projective closures are given by the equations ##x+y=0## and ##x+y+z=0##, which intersect at the "point at infinity" ##(1:-1:0).##

The "horizon" is given by ##z=0## here.

Although, I would summarize this by saying that there aren't parallel lines in projective geometry, not that parallels intersect.
Sure, but why does projective geometry have to enter this?

About half of the article is interesting, especially the stuff about tangents etc. But the rest is kind of like trying very hard to find enough wrong things to go to 10. I hope fresh_42 doesn't take it personally though, he has some other good insights (also good problems)!

martinbn
Science Advisor
Sure, but why does projective geometry have to enter this?
It doesn't. Of course the "myth" is not a myth, in Euclidean geometry of the plane the sum of the angles in a triangle do add up to two right angles. Nothing wrong about this.

AndreasC
Mentor
Although, I would summarize this by saying that there aren't parallel lines in projective geometry, not that parallels intersect.
Yes, I saw this weakness, too. But I decided to live with it since a detour to projective geometry would have been off-topic and too long.

DaveC426913
Gold Member
The angles of the triangle being 180 degrees SPECIFICALLY tells of those as INTERIOR angles.
Yes. The marked interior angles of that triangle add up to about 300 degrees.

symbolipoint
Homework Helper
Education Advisor
Gold Member
Yes. The marked interior angles of that triangle add up to about 300 degrees.
I am a little confused but not worried as maybe I should.

We maybe do have some things taught or learned wrong in school. Somethings are a little beyond what teachers know how to say or to present. I did see at one period something definitely incorrect but could do very little to change it because that was somebody else's job and not mine, so I was not really allowed.

symbolipoint
Homework Helper
Education Advisor
Gold Member
Yes. The marked interior angles of that triangle add up to about 300 degrees.
I THOUGHT it was about a triangle but I did not handle the picture properly.

DaveC426913
Gold Member
I am a little confused but not worried as maybe I should.
I am a little confused too. I assumed your post #4 about angles of a triangle was trying to point out something wrong with the diagram in post #1, but I don't see anything wrong. I may not be giving you enough credit.

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symbolipoint
Homework Helper
Education Advisor
Gold Member
I am a little confused too. I assumed your post #4 about angles of a triangle was trying to point out something wrong with the diagram in post #1, but I don't see anything wrong. I may not be giving you enough credit.
#10. The sum of all angles in a triangle is 180°.
We all live on a globe, a giant ball. The angles of a triangle on this ball add up to a number greater than 180°.

And the amount by which the sum extends 180° isn’t even constant. It dep
I did not read the whole thing carefully the first time, misunderstoood the picture, and then rechecked just the wording above the diagram. The one sentence, "The sum of all angles in a triangle is 180 degrees", is wrong because it is imprecise. Needed, say "INTERNAL ANGLES", or is it "interior angles"? Now, I forgot something that I learned but the basic fact I do know. (But for the triangle - the planar figure).

DaveC426913
Gold Member
"The sum of all angles in a triangle is 180 degrees", is wrong ... Needed, say "INTERNAL ANGLES",
In its defense it does say "in a triangle", not "outside a triangle".

Tom.G
Science Advisor
OK, down to nit-picking:

But that "triangle" in fig. 1 presented on a plane, is a hexagon as presented.

bhobba
Mentor
All true. But even the concept of point (except when using Hilbert's Axioms) is problematical. It is supposed to have a position but no size. Such, of course, do not exist. We all carry around assumptions we are not aware of. Generally, they cause no problems. But occasionally, they rear their ugly head and need to be challenged. An example is the good old 1+2+3+4...... = Infinity. But as we all know from studying complex analysis by the 'trick' of regularisation, it is -1/12. We normally take for granted that when talking about natural numbers, they are not considered a subset of the complex plane. However, in some calculations, such as the Casimir force, if we wish to avoid infinity, that is exactly what we must do. It is related to the epistemological position of Model Dependent Realism:
https://en.wikipedia.org/wiki/Model-dependent_realism

Thanks
Bill

For 400 degrees one, that is Gradian, and practically no one use it, they prefer only degrees or radians

trees and plants
All true. But even the concept of point (except when using Hilbert's Axioms) is problematical. It is supposed to have a position but no size. Such, of course, do not exist. We all carry around assumptions we are not aware of. Generally, they cause no problems.
Personally i thought of that too. We do visualise them but their definition differs from the visualisation we make, but visualisation helps us work.

About the straight line,that it is breadthless length as Euclid said? How could we visualise the breadthless part?

But we make use of visualisations of them to help us.