Ditching the Fifth Axiom - Numberphile

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Featuring Dr Caleb Ashley. Abandoning "The Fifth Axiom" - Euclid's so-called Parallel Postulate - gave birth to a whole new world... the world of Hyperbolic Geometry.
More links & stuff in full description below ↓↓↓
Extra footage from this interview: • Fifth Axiom (extra foo...
Filmed at MSRI. More hyperbolic geometry videos soon.
bit.ly/HyperbolicGeometry
Euclid's Postulates: mathworld.wolfram.com/EuclidsP...
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Videos by Brady Haran
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Опубликовано:

9 апр 2015

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Комментарии : 560

@FinetalPies 9 лет назад

"This circle is a straight line" "That's a bit of an exaggeration don't you think" Hyperbolic geometry

@decearing-egg 4 года назад

Hyperbolic geometry is basically exaggeration incarnate :p

@green0563 4 года назад

That's why it's hyperbolic. It's hyperbole.

@Sci0927 3 года назад

hahahahahaha

@themobiusfunction 2 года назад

But in spherical geometry, lines are actually circles

@mattk8440 9 лет назад

I have a really hard time drawing straight/parallel lines in maths class. Now i can just tell my teacher they're in hyperbolic space

@numberphile 9 лет назад

Kitty Kat nice idea :)

@HiAdrian 9 лет назад

Kitty Kat I think it helps to keep your hand stiff and use your large joints (shoulder & elbow).

@fossilfighters101 7 лет назад

@AdityaKumar-ij5ok 6 лет назад

Ha ha 😂😂

@Triantalex 9 месяцев назад

@nerdbot4446 9 лет назад

@Cosine_Wave 9 лет назад

Patrick Star U and I have much in common... in hyperbolic space...

@flexico64 9 лет назад

Roshan Sharma I love you people~

@IvanIvan1974 9 лет назад

Patrick Star +Roshan Sharam U2 have done a great job!

@NicolasGaillard 9 лет назад

Roshan Sharma This is gold

@Cosine_Wave 9 лет назад

Dinushka Herath It's not about getting physical, if an intersection is what you're in for, you'll split shortly after.

@gingermatchstick 9 лет назад

I like this guy

@CanariasCanariass 9 лет назад

Me too

@defaultbrowser1 9 лет назад

Matt G Me three

@JohnSmith-ph1hx 9 лет назад

swifterik Was that supposed to be pi?

@jaiyeko 9 лет назад

Me 69

@magicandmagik 9 лет назад

Ginger Matchstick me too

@SpeakShibboleth 9 лет назад

This guy, he should do more videos for you. He explains things so well. Thank you, Dr Ashley.

@Hugh.Manatee 9 лет назад

I really love his voice too.

@aesthetic1950 9 лет назад

***** Agreed

@DisappointedBuddha 9 лет назад

Yeah, full agreement. Especially the extended. Awesome stuff

@Dpaq13 9 лет назад

***** Keep in mind that it is a very simple topic

@SpeakShibboleth 9 лет назад

Paquin13 Yes, yes... You are very smart. Here, have a cookie and go sit in the corner.

@stegwise 9 лет назад

can't wait for the next one. great job Dr Ashley and Brady.

@numberphile 9 лет назад

Withstand thank you

@LogicraftRedstone 9 лет назад

Love his voice. If only he read bed time stories on the side of being a mathematician!

@tiborpejic2341 9 лет назад

Please do more non-euclidean geometry. Elliptic geometry, differential geometries, projective geometry, perhaps something more on higher dimensions and tessellation.

@ahcripes7651 7 лет назад

this guy is great, he talks so informally yet communicates his ideas perfectly. he sounds like just an average guy but hes smarter than everyone i interacted with today combined.

@Majoofi 9 лет назад

This is missing definitions of Hyperbolic, and Straightness.

@flexico64 9 лет назад

Majoofi I look forward to more info to come~

@nudel500 9 лет назад

Flexi Co Majoofi the extra footage is on their second chanel, i don't know why they didn't show it in this video

@flexico64 9 лет назад

I saw that -- still so much more to cover though! ^^

@geneticallyinferior1 9 лет назад

Majoofi the definition of hyperbolic = the reverse of hypobolic. the definition of straightness = the opposite of gayness

@geneticallyinferior1 9 лет назад

Flexi Co kinda hard to make a youtube video about undergraduate level and above geometry

@Erik-yw9kj 9 лет назад

Today I learned the Euclid spoke in Comic Sans. =X

@Erik-yw9kj 8 лет назад

***** Oh I love cheese, yes thanks! =)

@deffinatalee7699 4 года назад

He’s speaking the language of the gods

@calebgeballe2724 6 лет назад

Euclid could have just said "Two non parallel lines intersect at exactly one point."

@TimothyReeves 3 года назад

But parallel must be defined.

@seytanuakbar3022 3 года назад

@@TimothyReeves Two lines are parallel to each other if they intersect in infinity.

@NateROCKS112 3 года назад

@@seytanuakbar3022 no, it'd just be if and only if they never intersect. They don't intersect at infinity.

@seytanuakbar3022 3 года назад

@@NateROCKS112 Read the definition of parallel lines in math books.

@gabrielbrunser6401 3 года назад

Parallel lines do intersect at infinity

@lawrencedoliveiro9104 5 лет назад

5:57 Presumably it’s actually a “geodesic”, i.e. the shortest distance between those two points.

@lightspeed9762 4 года назад

Geodesics is line that will be left behind on a surface by a really tiny car moving at constant speed and not turning left or right. Its not always the shortest distance between two points

@theleastcreative 9 лет назад

Dr. Ashley is great! I'd love to hear more from him in the future!

@BrutishLearner4 9 лет назад

This topic is very fascinating, thank you Brady!! Absolutely can't wait for more!

@LemonWedge123 9 лет назад

This guy makes complicated mathematics look cool. Not that the other mathematicians you guys have on aren't cool or anything, but I'd definitely have stayed awake in high school math classes if somebody like Dr. Ashley had been teaching me.

@WTZubul 9 лет назад

Really excited for this video series!

@RomanNumural9 9 лет назад

Could you make a video taking more about the hyperbolic plane?

@numberphile 9 лет назад

Josh McGillivray we have more coming, here and on Numberphile2 - this was just a little snippet! :)

@baronlz 9 лет назад

Numberphile please talk about spherical geometry/trigonometry as well

@screwhalunderhill885 7 лет назад

I just realized I smiled throughout the whole video. This guy's enthusiasm rubs off on me.

@gigglysamentz2021 9 лет назад

Dr Ashley's voice is a treat ! I think I'm gonna watch this video before going to sleep x)

@gusamarante 9 лет назад

Hey Brady! I would really love to see these guys insights on Pascal's Triangle. Keep up with the great videos! :)

@barbietripping 2 года назад

6:23 - "yes... we said parallel lines are lines that dont intersect... they dont intersect" that smile right after.

@ragnkja 9 лет назад

I've crocheted a hyperbolic plane. It looks like kale.

@lawrencedoliveiro9104 5 лет назад

Don’t believe the hype. Rbola.

@lambdabaryon 9 лет назад

Thank you sooo much for this video, I really hoped that you'd make one about non euclidian geometry.

@torresfan1143 9 лет назад

Very proud I follow this channel from at least 1 and a half years ago . Great Job

@tiziobelo 2 месяца назад

Do you still follow this?

@torresfan1143 2 месяца назад

@@tiziobelo Mate I'm a theoretical physicist myself now😅a big part of how I got into this stuff will always be channels like these

@MostlyLoveOfMusic 9 лет назад

If that's a hyperbolic straight line, then what does a hyperbolic curved line look like?

@jaguarfacedman1365 7 лет назад

dgmcl3284 more or less curved and I mean that in the most literal way.

@AdityaKumar-ij5ok 6 лет назад

MostlyLoveOfMusic I don't think they exist in hyperbolic geometry

@albertwood8836 6 лет назад

Anything that's not a vertical line, or a circle centred on the x axis, is a 'curved line'.

@shrirammaiya9867 6 лет назад

Anything that's not straight

@kartikkalia01 4 года назад

@@shrirammaiya9867 just like me :)

@tonberrytoby 9 лет назад

I have always wondered why the first example non-euclidean geometry is always the hyperbolic geometry, even though the elliptic geometry seems to be a more obvious alternative.

@Djorgal 9 лет назад

tonberrytoby Because that's historically the first and that's conceptually the simplest in term of maths.Elliptic geometry is more understandable because you're already used to its model of drawing onto a sphere, but geometry exists outside any particular model. You can do geometry with only your five axioms without ever drawing anything.

@tehlaser 9 лет назад

tonberrytoby That was Brady's editing. In the extra footage Dr. Ashley did describe elliptic geometry first. You can see the sphere he drew to the right at around 6:20

@DavidB5501 6 лет назад

Euclid's axioms, without the parallels one, were often interpreted as implying that there is only one straight line between any two points. This excludes Riemannian geometries.

@joshuakostynuik 5 лет назад

tehlaser that wouldve been such a better way to edit it

@EamonBurke 9 лет назад

Dr. Ashley is great! I'd have loved to have him teach a course or two that I sat through.

@FishKungfu 9 лет назад

All this hyperbolic talk made my head explode!

@gui1521 9 лет назад

Oh look, a straight line -> o

@josephlarsen9414 9 лет назад

Flandre Scarlet and yet ∞ isn't. ):

@suzystar3 6 лет назад

Naah, only in elliptical space.

@gattabat 6 лет назад

'o look, a straight line -> o'

@Sci0927 3 года назад

parallel lines : ) l l ( nonparalell : /\ \/ >

@tedchirvasiu 9 лет назад

i approve this guy. more of him, please

@alexandercampbell9178 9 лет назад

That is not the meaning of the second Postulate.

@AlarmClock65 9 лет назад

Alexander Campbell Right. It deals with extending a line segment as far as one wishes, in the direction of the line.

@Wafflical 9 лет назад

AlarmClock65 Couldn't you just treat it as two points and use the first?

@rkr1410 9 лет назад

Alexander Campbell The first one is worng, too.

@johnhilbert7640 9 лет назад

rkr1410 How is it wrong?

@cottonycloud- 9 лет назад

Alexander Campbell Extending all segments within an area should produce a plane, but I'm not sure if this is what he refers to. I agree with the extension of a segment into a line.

@lukasmiller8531 9 лет назад

Love this guy!

@sitearm 9 лет назад

I like it! It's taken the last 5-10 years for me to catch up from university maths to the stuff Einstein used for special and general relativity not to mention Riemann yadda yadda. I love the key phrase moving to multiple worlds.

@astrofox2409 9 лет назад

Numberphile, could you compile a list of about all the functions, including some of the more complex ones, as well as how to graph such functions?

@spider0151 9 лет назад

Best handwriting i've ever seen from a Numberphile video. :)

@matthowell6562 9 лет назад

Please do something about the idea of parallel lines meeting at infinity. Great videos.

@Rukalin 9 лет назад

what a coincidence! today in physics-class I explained non-euclidean geometry to a friend of mine :D

@falnica 9 лет назад

God dam it!, upload the two together so I don't have to wait please!

@BigParadox Месяц назад

I have seen so many times people think they can object to what Euclud claimed, by suddenly taking the claims out of context and applying them to some kind of curved space, and then expect people to say "wow, you are smarter than Euclid, who must have been a very primitive thinker".

@RollyBM 9 лет назад

Very nice. More about projective planes please.

@jwt242 9 лет назад

This guy is awesome!

@cansomeonehelpmeout 7 лет назад

At 0:35, doesn't the 2. axiom mean that the length of a line segment can be extended, and not the "width"?

@vorpalneko 9 лет назад

The second axiom is presented incorrectly. Extending a finite line segment in a straight line means that you can make a line segment bigger as much as you like. It's not about breadth or planes, but about extending a line segment into a bigger line segment. ... Remember that the Greeks had a lot of trouble with the concept of infinity, so (unlike modern mathematicians) they didn't acknowledge the line as having infinite extent. Instead, they thought of them as finite lines (i.e., what we now call line segments) that you could arbitrarily make bigger if you chose to, a "potential infinity" rather than an "actual infinity".

@MathCuriousity 7 месяцев назад

Exactly what I was thinking!!! Thank you for vindicating my thought!!! This is extremely irresponsible and lazy of them to not fix this mistake! Selfish lazy and immoral!!!

@ruifernandes3524 9 лет назад

His handwriting is majestic

@geneticallyinferior1 9 лет назад

dr. ashley is amazing!!!!!!!!!

@cheongziyong8871 9 лет назад

Isn't number 2 the fact that you can extend a line segment into a ray/line?

@AlarmClock65 9 лет назад

Cheong Ziyong Yup.

@martixy2 9 лет назад

Computerphile did a whole slew of videos on this exact same thing, but exploring the concept from a different perspective.

@DisappointedBuddha 9 лет назад

If you want definitions of straightness and hyperbolic geometry, watch the extended on Numberphile 2. It's really good and everything makes sense.

@ScowlieMeerkat 9 лет назад

"No fair! That's like totally cheating!" Warm regards, Euclid and Scowlie

@DOSTalks 9 лет назад

How do you find such great speakers all the time?

@PwnDealer 9 лет назад

I went back and listened to him say "Let the following be postulated" at least 100 times.

@Richard_is_cool 9 лет назад

He is so chilled... It's nice to just sit and listen.

@alexgheorghiu6523 9 лет назад

Woah, that's my kind of crazy maths!

@poiewhfopiewhf 9 лет назад

Finally, a brother on this channel

@RedBar3D 9 лет назад

More videos with him!

@nathanialblower9216 7 лет назад

What happens if you deny one or more of the other postulates? How many can you deny and still call it geometry?

@Aassymcass 9 лет назад

Mind = Blown

@pickupsomemilk 9 лет назад

The second axiom was explained wrong, it simply means that you can extend a line segment into an infinite straight line. Also sphericaly geometry might have been a simpler place to start than hyperbolic :P

@newkid9807 4 года назад

No you’re wrong I just checked elements by Euclid lol

@boriecua 4 года назад

No this actually was Euclid's way of extending to a plane. Currently reading Elements.

@Xzcouter 3 года назад

@@newkid9807 You are wrong though. Without this axiom, you will not be able to extend any line. You can just check Proposition 5 in the book or wherever we use Postulate 2 to extend a line.

@TommyRobinsond20 9 лет назад

Looks like I've stumbled onto the weird side of maths again. *slowly backs away*

@FoamySoaps 9 лет назад

Neat coincidence, I have a midterm on this stuff next week

@matchboxp 7 лет назад

My maths textbook has a question literally asking us to simplify Euclid's 5th axiom.

@taesheren 9 лет назад

Very interesting. Non-euclidean geometry is amazing stuff.

@RishabhsinghQuides 7 лет назад

make some more on this topic

@mimzim7141 4 года назад

i thought the second just meant one can make a segment longer as we wish. Not that we can transform lines into planes?

@Ujellyfish 9 лет назад

I know you've looked at a few series related subjects but can you do a video on Maclaurin and Taylor expansions?

@MrSpaniard41 9 лет назад

trippy as fuck

@ximecreature 9 лет назад

Did I give you the idea of this video with my contribution to ISpyMath? That would be an honour.

@chrisg3030 8 лет назад

Now that we can have various curves in that hyperbolic plane which are all perpendicular to the boundary we can proceed to "ditching" the 4th axiom as well: not all right angles are equal to each other. A simple example is when you draw a straight line from the centre of a circle to intersect it at the circumference. Clearly the two right angles formed outside the circle's boundary are bigger than the two inside. It's just that we follow the convention of reconceiving and measuring an angle formed by curves with each other (or with straight lines) as if they were really formed by the straight line tangents to those curves. Only then do all four right angles become equal. In general I suggest that all the axioms are questionable and of eventually limited validity, which is precisely why they needed to be put.

@BeCurieUs 9 лет назад

One man deserves the credit, One man deserves the blame, And Nicolai Ivanovich Lobachevsky is his name. Hi! Nicolai Ivanovich Lobach- I am never forget the day I first meet the great Lobachevsky...

@markmandel6487 5 лет назад

Tom Lehrer, R.I.P.

@alastairbateman6365 8 лет назад

THE NUMBERPHILE SONG: 'I'll hit you with my FIDDLING STICK, hit you, hit you. One plus one plus two makes six, hit you, hit you, HIT YOU!

@LordNethesis 9 лет назад

I love the shift in the videos !

@alejorabirog1679 9 лет назад

Is there gonna be more of this?

@dudewhoisnotfunny 9 лет назад

best way to understand non euclidean space is to draw a triangle on a ball and see that you can make a triangle with 3 right angles

@OlavRH 5 лет назад

What a cool guy

@asdasdasdasd7483 9 лет назад

woah... that's some crazy magic stuff 2spooky4me

@ferretyluv 2 года назад

There’s a simpler more intuitive demonstration of hyperbolic space: lines of longitude on a sphere are parallel and will always intersect. There are infinitely many lines on a sphere that will intersect.

@navneetkumar9516 2 года назад

Any CBSE fellows here? I had to learn this thing in 9th class and after 3 years I still remember it WORD TO WORD!!! and scare all the children going from 8th to 9th by saying you have to learn that...

@mmatt314 9 лет назад

Gives me a warm fuzzy feeling when people get so worked up over maths in the comments!

@bejoysen4468 9 лет назад

Right angles don't have to be equal, even in Euclidean. "Equal" in geometry means the same. "Congruent" means identical in proportion and form. A right angle is the union of two perpendicular rays. All right angles are congruent but don't share the same two rays as sides. Therefore, they are not equal unless they are the same angle.

@giabao576 3 года назад

I was confused as heck when I read the 5th axiom a day or two ago

@DrEMichaelJones 9 месяцев назад

"When viewed with hyperbolic eyes." Lol

@SuperFinGuy 9 лет назад

What a concept, lines can bend... Pre-Aristotelian philosophers knew this before Euclid came up with his reductionist geometry.

@williamshanks255 9 лет назад

Why is the text in Numberphile speech bubbles Comic Sans? Why?

@numberphile 9 лет назад

William Shanks irony

@KrazyPlonk 9 лет назад

Numberphile I do that...

@Yupppi Год назад

My immediate question with these axioms is "how do you define right angle" as 90° is way too arbitrary. The angle between two perpendicular lines? But what makes them perpendicular? That random point along (set distance) the line is at the shortest distance from the other line?

@vsm1456 Год назад

Euclid's Elements, in addition to 5 axioms, also includes a lot of definitions, it's there: "Definition 10: When a straight line standing on a straight line makes the adjacent angles equal to one another, each of the equal angles is right, and the straight line standing on the other is called a perpendicular to that on which it stands."

@jdgrahamo 9 лет назад

I don't see how it can be a straight line if the surface it is drawn on is curved. Is that not cheating?

@craigruchman7007 4 года назад

What it means to me is Euclid’s 5th is just a special case in a bigger picture. So....is space perfectly flat? If not, even by the smallest amount, things get strange.

@rewrose2838 5 лет назад

A curved line with a radius-> infinity will 'look' like a straight line and that's what the line on the left is.

@torokati44 9 лет назад

Your pronounciation of Bolyai made me smile a little bit. Don't get me wrong, I don't mean to offend you, in fact I'm glad you even mentioned him; but just in case you wanted to hear how it's really meant to be said, click the speaker icon in the bottom left corner here: translate.google.com/#hu/en/Bolyai%20J%C3%A1nos

@monkeymath1 9 лет назад

I think there should be another one that says for any 2 non-parallel straight lines, there can only be one intersection point.

@marinekjb1611 9 лет назад

Was that Richard Canary at the very end of the video?

@sephyxer5187 9 лет назад

What are the uses of hyperbolic geometry? Of course, if there are.

@vasilivanich3842 9 лет назад

Fun fact: Lobachevsky designed an easy (theoretically) experiment to prove the actual world of ours has a curvature. All you needed to do was to imagine a triangle with one vertex on Earth, one on the Sun, one on a distant star - and then measure the sum of its angles. Simple as that, if the sum is less than 180, then we live in a hyperbolic space. Unfortunately the precision wasn't there at the time and the experiment failed.

@AlarmClock65 9 лет назад

Vasil Ivanich Nice! Were they ever able to do it?

@vasilivanich3842 9 лет назад

AlarmClock65 they did something along these lines but the latest data implies with rather large confidence level that we live in a flat universe.

@vixthesnarky2885 9 лет назад

Out of 4 line segments that are parallel. These can be of any length though they have to be straight. Make a circle.

@boumbh 9 лет назад

Reminds me about the representation of the Earth going in a straigth line in a space that is distorced by the Sun’s gravitational field.

@JTheoryScience Год назад

hey this guy is pretty cool, he should have higher production for his first expose

@Wifi_Cable 9 лет назад

I just learned a new party trick.

@keatonlarson4657 9 лет назад

Ciccarello Hopefully your at a party where people understand axioms and hyperbolic functions. I had a freind who watched the Infamous -1/12 video and every one just said he was stupid because he did not know anything else.

@titanarmy4116 8 лет назад

Is numberphile just the videofication of the book "Journey through genius"?

@TheSpacecraftX 9 лет назад

Aaargh. My brain!

@mspeir 9 лет назад

Ignoring the hyperbolic geometry, the axiom is flawed as it assumes that both lines must reside on the same plane. The example at 2:35 falls apart if one of the blue lines lies on a different plane. They may not be parallel, but they'll never meet.