What's the Deal with Euclid's Vth Postulate? (Part 2) 500 Subscribers Special!

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In his book, the Elements written in 300 BCE, Euclid described planar geometry using five postulates or axioms. One of these axioms was considerably more complicated than the others and seemed to some to be likely provable as a theorem from the other four more natural postulates. But no ancient Greek was ever able to prove Euclid's fifth postulate and this question persisted for over 2000 years until it was finally resolved in the 19th century CE. It took 2000 years to confirm that indeed you cannot prove Euclid's fifth postulate from the other four, but this proof resulted in us learning of another type of plane geometry that is completely consistent and satisfies the first four of Euclid's postulates but not the fifth. This type of geometry is called hyperbolic geometry and weird stuff happens there: triangles have less than 180 degrees, similar triangles are congruent, and rectangles are nowhere to be found. This discovery inspired further advancements in geometry including Riemannian and semi-Riemannian geometry which is the mathematical framework that Albert Einstein used to develop general relativity and advance our understanding of gravity and the universe.
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[@bluemarblescience]

Well done Alan. I'm sure a certain group of Euclidiots we know will be somewhat upset by this - they might even make a few dozen videos to ensure they didn't understand a word of it!

[@scholarsauce]

Thanks, Blue! I'm glad you enjoyed the video. It was a lot of fun to make!

[@LakotahWind]

@scholarsauce Come on Alan, do you know there are a LOT of people coming forward speaking on your behalf? Can you? Are you allowed to? Speak for yourself with these people you have been set up to have an adversarial position with? Was this your intention or is it something that has blind-sided you into an awkward position? Is Blue Marble your chosen advocate to represent you in this matter?

[@user-gx1rk8yw6l]

If I am not mistaken, then the up-to-then consistent-MISSING of the target of rockets into space by using "old-style" Euclidean geometry changed to consistent-HITTING the target when using the updated 5th axiom/postulate. I remember reading about this some decades ago, but probably misremember exact details.

[@ElectroNeutrino]

The bit with the two pieces of paper gets really close to affine spaces.

[@williamjones3313]

I think that Euclid was not thinking about parallel lines but about triangles. He was making an axiomatic statement about triangles. Triangles are the heart and soul of geometry. Unique parallel lines are just an afterthought. Possibly, Euclid reasoned; "Suppose I have two parallel lines, intersected by a third? Then the 'triangle' formed has one angle of zero degrees. So the sum of the three angles is the sum of two right angles. Now tilt the parallel lines toward each other. As the lines move 'inward. and the apex angle increases, and the sum of the base angles decrease; but the sum of the three angles remains constant. Therefore, the sum of the three angles is always equal to the sum of two right angles. I have another problem If two lines pass through the same point and are both parallel to a third line; are they not parallel to each other? Then, by all reckoning, they are essentially the same line. Why does this apparently axiomatic fact need to be proven?

[@scholarsauce]

Thanks for watching my video and for commenting. These are great questions. There are multiple ways to think about what Euclid's fifth postulate was getting at, for sure. I think the parallelism thing ended up being what we landed on because with that it's easier to see what the possible differences are. To answer your other questions, first, we have to be careful by what mean by "axiomatic". The colloquial usage of the word axiom is not the way mathematicians use the term. Colloquially, the word axiom is used to indicate something that seems self-evident. However, in mathematics, we use the term axiom to refer to a statement that is a baseline logical assumption about the topic we are going to discuss. Then all other facts about that topic must be proven logically from those baseline assumptions. We reject the notion that something can be proven by being seemingly obvious; instead, as I said, every fact about that topic must have a explanation of how it follows logically from the assumed baseline assumptions or axioms. We also have a general desire to know what the minimal baseline assumptions are that logically force a phenomenon that we want to understand. So, when Euclid is trying to describe the geometry of a flat plane, that's his target, and he wanted to identify a minimal set of assumptions that would guarantee the geometry of a flat plane. His best attempt of a minimal set of assumption were his five postulates (i.e. axioms). Because of the complexity of the fifth axiom, the question was raised of whether flat plane geometry could be logically guaranteed by assuming just the first four. Thus, another characterization of the question was this: What parts of geometry does just the first four of Euclid's axioms guarantee? It seems that in reference to your description of the transitivity of parallelism (if line L is parallel to line M, and line M is parallel to line N, then line L is also parallel to line N) that you have used the term "apparently axiomatic fact" in the colloquial sense of the term "axiom". But Euclid did not include that statement in his list of axioms, but rather proved that it logically followed from his five axioms. Since transitivity of parallelism isn't one of the baseline assumptions or axioms, then that fact has to be proven to follow from the axioms in order to be accepted as a fact of that geometry. It turns out, though, that the transitivity of parallelism is a logically equivalent statement to Euclid's fifth postulate in the context of Euclid's first four postulates. That is, you could have used it in place of the Euclid's fifth postulate and produced the same geometry. However, that statement, being equivalent to Euclid's fifth postulate, turns out to not be guaranteed by the first four of Euclid's postulates. Thus, in hyperbolic geometry, parallelism is NOT transitive and so you don't get the effect that you mentioned about two lines through the same point being parallel to a third forces them to be the same line. In fact, the situation ends up being a little more interesting. In hyperbolic geometry, where we use Euclid's first four postulates but also assume the hyperbolic parallel postulate, there are two different classes of parallel lines. These are those parallel lines that admit a common perpendicular and those parallel lines that are asymptotically parallel. Parallel lines in this second class of lines get progressively closer together as you approach infinity in one direction, but never touch, hence the name. The properties of Euclidean parallel lines end up getting sort of split between these classes. For example, the first class has common perpendiculars just like Euclidean parallel lines do, but the second class satisfies transitivity of parallelism in some sense. That is, in hyperbolic geometry, it is true that if line L is asymptotically parallel to line M and line M is asymptotically parallel to line N and they get closer together in the same direction as line L and M, then line L is asymptotically parallel to line N in that same direction. Several other properties of Euclidean lines end up getting split between these two classes, which I think is pretty curious. I hope that that answers your question. If you're interested in all the other weird types of things that parallel lines do in hyperbolic geometry, I recommend checking out Gerard Venema's book Foundations of Geometry. It's very well written and includes a lot of amazing facts about the hyperbolic plane in the second half of the book.

[@user-gx1rk8yw6l]

Simply BECAUSE Euclidean geometry makes NO statements about the SHAPE of a plane, I am trying to picture the consequences of the quadrilateral V1-A1-K1-U1 at 15:10 having 4 right angles when drawn on the surface of a globe (if lines CD & FG are not-intersecting parallels AND lines M-U1 & I-K1 are intersecting parallels--ie points M & I are the same point and ditto N & J at their other end). Latitude & longitude lines, in other words. The size of said quadrilateral leads to interesting things regarding said right angles, for the quadrilateral can become a triangle when M=I, & even 2 intersecting lines when also N=J. ..

[@scholarsauce]

This is an interesting question, but there's a slight problem with your set up. Not all latitude lines are "straight" on the globe. Only great circles are. The equator is the only latitude line that is a great circle. If you try to connect two points on the same latitude besides the equator, the shortest distance between them will not follow the latitude line, but instead the great circle that actually goes a little closer toward the pole (north or south, whichever it's closer too). On the other hand all longitude lines are great circles, so you can use them as lines. So if you try to make a Saccheri quadrilateral with the base on the equator and the two sides going up two different longitude lines to the same latitude line, the fourth line will follow something more towards the pole than the latitude line. However, since the latitude line makes a 90 degree angle with the longitude one, the geodesic between the two points that pulls more towards the pole will make an obtuse angle there and you'll get two congruent obtuse summit angles. You do rightly point out though that this quadrilateral collapses into a triangle if you extend the longitude lines to the pole. As an aside, all quadrilaterals on a sphere have strictly greater than 360 degrees similar to how the triangles have strictly greater than 180.

[@user-gx1rk8yw6l]

@@scholarsauce Re "straight" & great circles on a globe: True. But any latitude line can always (on said globe) be connected by a line at right angles to it to any other latitude line & there intersect it at a right angle. So every point on any latitude line is LOCALLY parallel to 1 point on any other latitude line. (Said intersecting lines are all on great circles.) But the 5th postulate no longer holds for the point right beside the original point because the 2nd line intersects the original line. (Am here ignoring the "ridiculous(?)" idea of a no-dimensional point being BESIDE a ditto other point.) Anyway, yes, I understand that on a globe only great circles are "straight" lines AKA geodesics. And yes, the GEODESIC of the not-equatorial latitude line causes a Saccheri quadrilateral with 2 right-angle corners & 2 obtuse corners. But all 4 corners between the latitude lines & the longitude lines are 90 degrees. As to your aside, I can see "not less than 360 degrees" for all quadrilaterals, but NOT "strictly greater than 360 degrees". Because all PAIR-WISE intersecting great circles enclose at least 2 quadrilaterals each with 4 exactly right-angle corners. Ditto the "strictly" for triangles.

[@user-gx1rk8yw6l]

@scholarsauce The "devil's advocate" approach has a problem: namely the assumption that not-True always equals False (& not-False always equals True). The system must be 2-valued for the assumption to be valid. This has the consequence for the axioms of Euclidean geometry that a PRE-axiom (if you will) is required for ANY of its axioms to hold.

[@user-gx1rk8yw6l]

Umm... what might the angle-sum of a triangle be on the surface of a toroid?😵‍💫😵🤓

[@scholarsauce]

Now this scenario is a bit weirder because it depends on what you mean by the torus. Unlike the sphere and hyperbolic plane which have constant positive and negative curvature, respectively, the torus doesn't have a standard metric on it. That is, depending on how you set up the torus can affect how you measure distances and angle, even if the tori are the same topologically. For example, the flat torus, made by using a flat rectangle like a piece of paper and identifying the top edge with the bottom and the right edge with the left, exhibits Euclidean geometry behavior in a patch on its surface. So any triangle on that surface would have 180 degrees. This is similar to drawing a triangle on that piece of paper and then rolling it up into a cylinder. Since no stretching was involved, the geometry of that triangle stays the same and it still has 180 degrees in it. However, the flat torus can only be embedded into R^4 or higher dimensions. If you try to use the more familiar torus as it's embedded into R^3, then this one isn't intrinsically flat anymore nor is its curvature constant. This makes the shortest paths (aka geodesics) act differently depending on where you draw them. Triangles on the outside edge where the curvature is positive will produce triangles with more than 180 degrees, while triangles draw in the donut hole as it were, where the curvature is negative will produce triangles with less than 180 degrees. By the way, positive curvature at a point just means that all directions from that point bend away in the same direction, like a sphere. Negative curvature at a point means that some directions from that point bend one way, while others bend a different way, like a saddle point. The positive and negativeness of the curvature gets their name from the fact that the Gaussian curvature is positive or negative there and the Gaussian curvature is the product of the two principal curvatures (for a 2D surface) which are the directions of greatest and least curvature (so most positive and most negative). If the principle curvatures have the same sign at a point, then everything is pointing in the same direction from that point and the product will be positive. If they have different signs, then the product is negative. If one of them is zero, then surface is actually flat intrinsically. This detects for example the flatness of the cylinder. I bring this all up because it can give you a way of determining when the angle sums of a triangle will be with respect to 180 degrees. For positively curved regions, the angle sums will be greater than 180 degrees. For negatively curved regions, the angle sums will be less than 180 degrees. For flat regions, the angle sums will equal 180 degrees. This is a consequence of something called the Gauss-Bonnet Theorem. If you have a triangle covering a region that includes regions of different signs of curvature, the angle sum would be governed by the Gauss-Bonnet theorem and could be greater, equal, or less than 180 degrees. Thanks a lot for this comment! I think I might actually make a video on this idea and what the Gauss-Bonnet theorem says about all this!

[@user-gx1rk8yw6l]

@@scholarsauce You are so welcome! 🙂 If/when you make a video on this subject, please include figures overlapping themselves... Including figures that are "larger" than the torus. Also of course the self-intersecting torus (AKA donut with negative hole?), the Möbius strip, & figures on their surface(s). The concept of a "flat torus" might be useful with Flat-Earthers. (Not that I am even slightly willing to engage in any discussion with any such people...)

[@GonFishin-uk8jb]

Thank you for confirming we don't live on a globe. You da man

[@scholarsauce]

Nothing in this video commented on the shape of the Earth whatsoever, so how did you come to that conclusion?

[@GonFishin-uk8jb]

@@scholarsauce I just finished watching you on some clown named blue marble science. Def not science, def not globe. I wonder why you didn't post that "interview" with blue on your page

[@iridium1118]

Are you okay?

[@sphaera2520]

Op self reported that it was never about the evidence, logic, or anything. Being so wrapped up in flat earth, that’s all they are. Now someone could say, on a hyperbolic surface triangles have

[@bluemarblescience]

@@GonFishin-uk8jb Oh look - a free range Bevarian sloth. We don't see many of these out away from the Try Thinking swamp.

[@LakotahWind]

You never said anything before about the Rectangle lie the kids have been told!! Jeez! When do the lies ever end??

[@user-gx1rk8yw6l]

Maybe reading the 600-page book "Lies my teacher told me" (1995, by James W. Loewen) would be enlightening. Is withholding *anything* from the-WHOLE-truth to be considered a Lie?

[@sphaera2520]

@@user-gx1rk8yw6l these guys are just fishing for excuses to validate their feelings. They’re upset at the world for being outcasts, for being left behind, and so they project that own incompetence onto everyone else and try to find reasons why they’re not the problem. Most recognize that the general approach to education is learn the basics, then add complexity as you grow older and have more capacity to digest nuance and non-intuitive ideas.

[@user-gx1rk8yw6l]

@@sphaera2520 Learning the basics & adding complexity is a GoodThing, yes. But said intuitiveness is learned from a TOO-MUCH-encompassing set of basics. So that what should have been simply an ADDITION to learned intuition must now be taught/learned as a CHANGE, because it then *contradicts* intuition. This means that curriculum-designers are taking the easy way out by projecting onto students the *designers'* inability to design truly-good curricula. Example: my youngest son learned sets BEFORE addition, so that he could CORRECTLY add apples & oranges as FRUIT. This method skips any need for the "you cannot add different things" lie. Said "incompetence" is more-likely a curriculum-caused problem than student-inability.

[@sphaera2520]

@@user-gx1rk8yw6l maybe intuition was not the best word to use there, more abstract/intricate/advanced probably conveys my meaning better. Regardless, I’m not sure I necessarily agree with the example in your response. You can’t add different things isn’t a lie, your own example shows this. You switched from adding apples and oranges (different things) to adding fruit and fruit (same thing). In the end, you still didn’t add different things, you just taught your kid in some situations it can be convenient to relabel objects with a class they share so the addition now makes sense. This is clearer when there is no obvious way to relabel them, eg 1 kg + 3 m/s leads nowhere.

[@user-gx1rk8yw6l]

@@sphaera2520 True that "You can’t add different things" is not a real lie. But the need to COUNTER one's learned response by restarting & RE-labelling things is a not-necessary complication. I think that "intuition" IS actually the better word. Though "abstract/intricate/advanced" are ALSO applicable, they shift the focus away from ADDITION-to-already-learned-stuff towards the not-necessary REPLACEMENT-of-learned-stuff. The difference is CONTINUING on a path via learned habit (ie STRENGTHENING intuition) versus BREAKING that habit & RESTARTING that path. If the student STARTS with super-sets (here "fruit"), then ADDING subsets ("apples" & "oranges") CONTINUES on the sense-making path. This building-without-breaking method is very useful in ALL fields. For instance, all this hoopla about the Big-Bang theory being WRONG (their capitalization, not mine) would not exist, for one already knows that things come in sets, & that subsets are possible (& probable) most-anywhere.