Euclid's Big Problem - Numberphile
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- Опубліковано 11 гру 2014
- Trisecting angles and calculating cube roots was a big problem for Euclid and his cohorts. Discussed by Zsuzsanna Dancso at MSRI.
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TRISECT WITH ORIGAMI: • How to Trisect an Angl...
CIRCLE THE SQUARE: • Squaring the Circle - ...
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Revolutionize math.
Turn 19.
Die in a duel.
What a life story.
His life is pretty sad honestly.
He was a great mind that wasn’t recognized by authorities of the time and he died very early.
Quentin Styger History disagrees.
@Quentin Styger Yeah they were the dominate land power in Europe for around 1000 years
Revolutionise. Maths. Please stop destroying English, America.
@@shweatypalms4423 lol
exactly 7 years later and this video still bangs. What a fantastic bit of teaching
Sorry, I ruined your 7 likes 🫢.
I had a professor who insisted on taking a few weeks to teach us all of this, and I really didn't get why it was such a big deal until we continued on throughout the semester. Turns out, using a straight edge and compass is a great way to not only understand geometry, but to also to become aware of just how many assumptions we never knew we were making about mathematics when we are taught it.
same but my teacher looks like walter white
@@IDontKnow-dl3lq cookin math
Message from Zsuzsanna (in the video)
Hello Everyone,
Everett Sass and Fenrakk101 (possibly others as well) posted correct
solutions on constructing a segment of length square root 3.
[Spoiler Alert! If you read them, you can discover that the idea behind
both is the same: to construct a right triangle with hypotenuse 2 and one
leg of length 1, so the other leg will be length sqrt(3).]
This solves the puzzle of "tripling a square", that is, you can now
construct a square of area 3. As some of you point out, this is not the
same as doubling the cube, which comes down to constructing a segment of
length cube root of 2, which is impossible by Euclidean means.
Some of you mention the idea of trisecting an angle by constructing an
isosceles triangle and trisecting the base. This does not work: if you
look closely, the middle part will be a bigger angle than the two side
parts. In other words, the three thin triangles are not congruent. As one
person points out, what you'd need to do is trisect a circle arc, not the
base of the triangle, and that is impossible by straight edge and compass.
Of course you can trisect _some_ angles, like a right angle or a 45 degree
angle, but there is no general procedure using straight edge and compass
that will trisect any arbitrary angle.
Sorry about staining that nice straight edge with the marker! I felt bad.
Hello! I was wondering if you could maybe talk a little bit about The Martingale System and The Elliot Waves Theory.
Kindest regardings,
Orionar Johnattan.
Hey guys so it's pretty difficult to get noticed on youtube but I was really hoping that you would give my channel a shot. I make educational videos in hopes that it will help students with their classes. You don't even have to subscribe if you don't want to but it would mean a lot to me if you took a look! Thanks so much in advance!
So I cannot trisect most angles with these 2 things, except for a 45 or 90 degree triangle? Has this been proven impossible to do, and what do I win if I trisect it? Also Numberphile what happened to that guy with the Richard Hammond accent?
Numberphile I think I'm missing something.
You can trisect a line, but not an angle using only a compass and a straight line.
So... Why can't you take your angle, measure points at equal distance from the intersection of the angle, then draw a line between those two points (forming an isosceles triangle).
Now all you need to do is trisect that line as per the previous example and connect the points to the angle and bingo, one trisected angle of any size using just a straight line and a compass.
You say this is impossible, so I'm gonna verge on the side that I'm missing something here rather than I've solved an impossible problem!
James92453 that is already addressed in the comment; the three new angles that you derive from that process will not be equal, the middle angle will be larger.
if you use the new unit you derived from constructing and trisecting your new line and extended it to infinity, and connected all of your new points to the one point at the angle, you'll find that the angles you are constructing will start approaching 0. in other words, dividing an angle into more than 2 equal parts is impossible with this method.
Ths one is really great. She explained points very well, she created tension in the storytelling ("before I tell you the answer" and goes onto another segment) and generally was clear and fun.
"...and then people thought about it for 2000 years..." : my favorite line of the video.
I really enjoined this video; with my limited knowledge of math, this is one of the few videos I can fully understand without a single question. She had shown many examples and clear, understood proof. I really enjoined her!
Euclidean geometry uses a straight edge and compass in two dimensions.
Origami does not allow drawing of circles within the paper (within those two dimensions) ... but ...
Folding is equivalent to the use of a compass, in a third dimension!
And utilizing three dimensions allows third roots. Marvelous!
*****
how do you 'fold' in N dimensions.
robkim55
You have to use mathematics to "simulate" physical folding in higher dimensions, but that's merely *our* limitation as three-dimensional beings.
The principle of geometric calculation by straight edge and compass is an expression of the ultimately *physical* nature of all phenomena.
Except that Einstein demonstrated to us in 1915 that we live in a non-Euclidean world.
@@JonWilsonPhysics we do?
@@ntf5211 but it's relatively flat besides the effects of gravity, as far as we are aware
This video was awesome!
jmasterX thanks
+Tyler Durden check out the rock paper scissors video
it sounds like galois was a legend. revolutionising a fundamental part of mathematics before he turned 19 and then fighting a duel. i feel so useless right now
+12345DJay I can't believe he joined that duel. I don't know how the same person can be such a genius and such an idiot. Well, I know, but I just can't believe it.
+12345DJay We have got to re-establish dueling to solve this cubed root business
+Trail Guy Challenge extended, challenge accepted.
yes, a major branch of algebra is named after him. (maybe because they couldn't come up with a better name. for example we don't call set theory "cantor theory" :D )
12:58 Brady not only cracks this classic dad joke but then actively chooses to include it in the edit. Decisions like these are key to the channel’s success
There is something so beautiful about these simple geometric ideas. Love this content!
Good video. I was spellbound by the simple complexity and complex simplicity underlying the ruler-and-compass-constructions. Brady's questions are always to the point and he says insightful and poetic things like "so the cube root is the point you can't reach".
well! that smile trisected my heart
Huh? Show me a sharp angle of your heart.
what an interesting subject and such a soft spoken guest.
thank you for this vid.
I always had his question.
Lets say you are the first to prove something in math. What do you do after? Do you contact someone? :/
Tap Studios
Check it. Check it. Check it, and then check it. And if you truly find that it's right, put it on the Arxiv.
+Tap Studios Put it on Reddit. Duh.
lol. you try to publish before anyone else.
+TAP. Studios lol, the obvious one is don't email that to professors, since tons of cranks do that often, it's vexing.
show it to you high school teacher
Her voice is absolutely mesmerizing.
More like mASMRizing
Lots of vocal fry going on (not an insult, just an observation.)
kom-pahhhs
It’s sexy.
Vodkacannon that is the stupidest thing I’ve ever heard
Everything Euclid couldn't. People thought about it for 2000 years.
No one thought about it for 2000 years
@@readingRoom100 I thought about it for 1000 years owned
Yeah I watched the video too
@@readingRoom100 Imagine it's 536 and some dork is like "can you trisect an angle?" and you're like, "sir, I'm a farmer and the harvest has failed."
What do you mean wjat couldn't Euclid do??
This video is odly calming and relaxing, probably from the way and rate she speaks with. And most of all, it was intresting.
Trisection of angles? It is certainly possible. I have discovered a truly marvelous proof of this, which this comment box allows too few characters to contain.
Fermat..?
Fermat was either a genius or a troll... I'd like to think he was both.
Validifyed is that Thermat's Last Feorem?
Validifyed ujuik
Dale S actually google changed the comment box, so I think you can post it :)
I think the secret to Brady's success is the paper. Lots of mathematicians in different Numberphile videos have asked if they could have more of the "wonderful paper".
This video brings back nostalgic memories of doing geometry in primary school, with compasses and straight edges and protractors, long before I took high school geometry and learned about postulates and theorems and etc. The Greeks were really on to something when they thought everything boiled down to geometry, that geometry was pure and everything else revolved around it. The Fundamental Theorem of Algebra video showed how algebra is connected to geometry.
What a math try hard
I can unisect an angle.
WITCH!
+NoriMori looool
NO, REALLY? (SARCASM)
Unisect? Monosect? Sect? English and its prefixes...
Sectumsempra!
I really enjoyed this video. Thank you for sharing!
Thank you Zsuzsanna for some refresher examples.
Awesome video! Would have happily sat here and watched another 4 hours of this.
She is great! Super understandable. I we get to see more of her in future videos. ^_^
This was lovely 😍. I find myself exploring videos of classic Math that can be relevant today, and this is one of them.
Wow. I've always had a piecemeal understanding of the relation between algebra and geometry. You've presented that relationship completely and elegantly. Thank you!
This was a very relaxing video to watch.
Amazing video. And I love Zsuzsanna's accent!
This is a wonderful video! It brings back many memories.
Zsuzsannas eyes shine so brightly when explaining. I had to smile through all of the video. Awsome girl!
You can say Pierre got to the... root of the problem.
Prophet
i recommend you tri a different angle
A radical assertion, indeed!
This whole thing is irrational.
It's funny in Portuguese "Pierre" is pronounced exactly as "πR", and you can really "square πR"
Is This a joke about Gregor Mendel?
I miss geometry class, drawing all these angles and circles and stuff was so fun
Great video. I've been wondering about Galois theory for a while now and how it was implicated in the unsolvability of the quintic equation and trisecting an angle. Thanks!
This is the best episode I've ever seen. At the very end I was even, for the first time, to begin to glimpse the relationship between Galois and these questions. What he's looking at when he's extending Fields. Good!
That seemed like a lot of extra steps to construct a square with length root 2.
Once you have the perpendicular lines, all you need to do is draw a circle with radius 1 around the intersection of the perpendicular lines and those 4 points are your vertices for a square with side length root 2.
It works because the distance between the points on the same line is going to be the diameter of the circle: 2. this is going to be the diagonal of the new square, so if you have a square with diagonal length 2, you necessarily have a square with side length root 2.
Or just draw a diagonal on your unit square and copy it
Nah prolly not true
@@VikeingBlade In the Socratic Dialogue "Meno," Socrates talks a slave boy through using this method. First they construct a unit square, they then construct three more unit square to form a larger square with area 4. Then they connect the corners of the unit squares that are at the midpoints of the larger square with area 4. We can see by inspection that the area of the inscribed square is exactly twice the area of the original unit square because the original unit square contains exactly two right isosceles triangle and the inscribed square contains four of these triangles.
There is a game called euclidea with stuff just like this
A beautiful explanation, thank you.
I'm a completely halfwit when it comes to maths, yet i still do find these Numberphile videos so entertaining. I'm puzzled. But great to watch while recovering from knee surgery and way to much time indoor for the next couple of months.
I absolutely love her English. omg...
+AirIUnderwater
It is so cute >_
you like mumbling?
Magyar akcentussal beszél, mert magyar
That how you get qualified into MIT.
@@tamassimon5888 Hát valahogy úgy, de azért voltak akadozások
0:25 Oh, just like in computer science where ideal Turing machines have infinite memory and time, and in Physics where we tie frictionless masses together with massless strings. Gotta love the world of the abstract! ;)
Infinite straightedges and compasses are only to generalize the theorems. You could easily say that they are finite in length, but then there would be nothing saying that anything you showed would still be true if someone went and grabbed a larger straightedge or compass. Basically, it could be reworded to say an arbitrarily large straightedge and compass and would still hold true.
"Infinite" generally has two different meanings. The one we generally know and love, actual infinity, is aleph0, which is the cardinality of N. The infinities in your examples and the videos just mean, "no matter how long a straight edge we need, we could eventually construct one." Those are potential infinities, things that have no bound to how great a distance from 0 they can get.
NowhereManForever Nope. In Euclidean geometry a line is per definition infinite long.
Menea6587 This has nothing to do with his comment or the video.
NowhereManForever
"Infinite memory and time"
"Massless string and frictionless surface"
That's just amazing! Brilliant!
My favorite numberphile video. Mind boggling for sure
He died at 19 in a duel LOL ... manliest mathematician ever
+Roxas99Yami true but he could be manlier if he'd won it...
+2CSST2 just participating in a duel makes you manly.
Source? Because I call bs
Roxas99Yami Confusing Euclid with Galois?
Watch the video, they talk about Galois
I was reading about classical constructions the other day, and there's something I don't understand. One of the restrictions of compass-and-straightedge constructions is that the compass is assumed to collapse when lifted from the plane, making it impossible to directly transfer lengths. But the compass equivalence theorem means that this is ultimately an immaterial restriction, since lengths can still be transferred indirectly (albeit in a complicated fashion). So what I don't get is, why does that restriction even exist if it doesn't make any practical difference?
Do you mean theoretical difference? It makes a practical one. It is a practical problem. You mean. .. Why not build better compasses?
+zee anemone No. I mean exactly what I said.
NoriMori, you are essentially correct. A compass should be used to construct a circle through one known point with its centre at another known point. And it does make a difference. There is a construction to trisect the angle with a
marked straight-edge and compass. If one can directly transfer lengths off the paper with a compass, one can effectively have a marked straight-edge. (Note that a marked straight-edge is forbidden; a straight-edge joins two
known points by an infinitely long straight line.)
Really great video, I love how applicable geometry is to the physical world. With regards trisecting an arbitrary angle; does the number of dimensions you are working in play into it? The fact you can use origami to do it would imply so, I think. When folding the paper you are moving the 2d section through 3d space. Isn't there something about needing one more dimension to someone roots the higher up the root number line you go?
As for trisecting the angle... If you draw a circle with the centre at the meeting point of both lines. Then you draw a line through both points where the circle intersects with the lines of the angle. Next you trisect this line. Now you have the two points you are looking for to trisect the angle.
Another great video! I am loving these, is anyone else too?
i too am loving it
Beautiful talk. Beautiful teacher,Thank you very much
wow, i had never truly seen geometry like this, this is enlightening! it is way more interesting than what i previously thought!
extraordinary!
Yessssssss this is the kind of videos I would like to see here
YOU DEFILED THAT STRAIGHTEDGE WITH A MARKER.
Edit: I now see from +Numberphile's comment that Zsuzsanna apologized for this. XD
14:34 But… I didn’t mean to cause you trouble… _sobs and leaves_
Clearly a horticultural joke.
Did you just create a channel to write this?
Nice to see a hungarian mathematician on Numberphile. She speaks english like everyone else here in Hungary. Funny to hear the hungarian accent your video.
Amúgy ha olvasod ezt Zsuzsa grat hogy kijutottál Amerikába és most ott kutatsz! Pacsi!
I remember an old Mathematical Games article by Martin Gardner where he described a device for trisecting angles. It was something like a compass, with two more legs between the outer ones. As you expanded the outer legs to a specific angle, the two inside legs always maintained a division of 1/3 - 1/3 - 1/3 of the distance (or angle, actually) between the outer legs.
+losthor1zon
Yes, I remember something of that - I seem to recall another kind of instrument, passive, rather than the 'active' one you're describing - that was formed with some kind of special curve as its edge. I can't recall how it was used to do the trisection.
poor ruler at 2:57 hurts to watch
I thought I was the only one that got upset about that
just a ruler? *just* a ruler?!?! honestly how can you say somthing so obtuse? that isnt *just* a ruler... that... no your right its just a ruler. funny note: my phone auto corrected a misspelled "ruler" as "euler" HA math jokes
In my day it was called a 'rule' not a ruler, which is more, er, regal. To 'rule it out' was to take your rule and draw a line through the words you didn't want. To this day, the expression 'rule it in' makes me wince.
mackexr
Indeed..
Lol man some peoples OCD is off-the-charts obscure, rofl.
the most thing that i like in Numberphile is that almost every matematician its from a different country
+Omar St Same!
Universities are like sports teams you try to get the best regardless of where they are from, and then you hope another university doesn't get their hooks in to them and entice them away.
This was a really good video. Thank you!
I don’t know why, but geometry has always been the ‘prettiest’ branch of maths to me. I like a formula or some interesting arithmetic, but there’s something so satisfying and aesthetic about things like this video
1:10 Wait a second! So now we're talking about modern compasses instead of Euklidian compasses?
Well if you were in a 4D world and had a 3D surface to mess about on cube roots should be possible right?
why would we need 4D if we can access 3D from 3D
Because it'd quite difficult to draw in 3D accurately without a computer and if you had a computer why bother using geometry to do cube roots
No. Even in arbitrarily many dimensions the distance norm is still in terms of squares.
I really would like to write on 3D paper
This is great! Zsuzsanna. Dancso is a wonderful teacher.
doing a bit of oscillation, and mid way through, had an epiphany, on how to triple the square! love this difficulty of question for the viewer to do! Feed me more problems
men i nschool , this is what they also should have shown us.... might be really usefull for CAD programs
You've hit on a very good point. When we learned draftsmanship at highschool, the school was just starting to transitioning from paper to CAD (1986). We learned all these classic techniques for geometric construction using compass and straight ruler. For any one working in the engineering or technical field, this knowledge is very practical.
12:13 impossible in Greek is: αδύνατον, not άδυνατον. In ancient Greek it would be: ἀδύνατον. I'm just correcting something that I saw it's s wrong. Generally I really liked this video a lot.
είναι πνεύμα ψιλή απο τα αρχαία ελληνικά και όχι τόνος του μονοτονικού συστήματος που έχουμε σήμερα οπότε δεν είναι λάθος. Το μόνο λάθος είναι οτι δεν έχει μπεί η οξεία στο υ.
@@ItzCrisonFTWWoohoo! I'm patting myself on the back for reading this comment just from having taught myself Κοινή in college-I never took Modern and I only had to look up λάθος (a little embarrasing since it comes straight from the aorist stem of λανθάνω via a little sliding of the meaning).
Thank you so much for this entertaining and beautiful video. So Euclid was right all along. Loved the teases along the way.
Great job, Zsuzanna Dancso your lecture was interesting
but....
if it's possible to divide a line segment in 3, couldn't this be done to trisect an angle: cut the 2 lines from the angle in same length; close an isosceles triagle; divide this new segment in 3; connect the dividing points to the original corner... ?
The resulting angles aren't all equal... the center one will be different than the order ones.
"older" should be "outer"
Slarti Bartfast yeah... it's true... :/
Slarti Bartfast
What would happen if we would divide the line infinetley , we could by knowing how much we divide know what each point is worth , so if we would to add all of the points to find out where the spots are equally different .
I think it wouldn't be possible to infinitley pinpoint but idk ...
(If it were i would think the angles would be equal !)
This would mean that the inability to pinpoint it would make the angles different by just a slight .
But then again we can pinpoint a line in it's 1/3 so i guess this theory would stand .
Btw +LordSatoh , i thought the exactly same thing ...
P.S. i think you meant to 4-sect the line .
This could be a proof that you can't cut a line in 3 equal sections ...
Slarti Bartfast I thought the same thing as LordSatoh, but had my doubts that such a simple, intuitive solution could have gone so long without being figured out. Thanks for pointing out the error in our intuition :)
Yeeey a fellow Hungarian :D She has the same family name as one of the most famous comdeian in Hungary :D
Was wondering if you could construct a right angle with only a compass.
Thanks for showing it so I don't have to go through the Elements to find it. :)
Also I think I'm in love with this approach to maths, it's so practical and 'simple'. Straightforward.
As for trisecting an angle, I'm not sure if this is allowed, but it's mathematically valid...
- Use the compass to draw an arc connecting the two lines.
- Move and rotate a markable edge (straight edge, piece of paper, etc) from one end of the arc to the other, marking it's arc length as a straight line (this can be interpreted as 1 unit, if it makes you any happier).
- Trisect the straight line that is the arc length.
- Retrace the arc length with your trisected line, marking 3 equal arc segments.
- Connect the vertex to the marks on the arc.
The problem, of course, is knowing how precise the measurement and retrace of the arc length and segments are. If it were a free body as opposed to a drawing, you could just roll the arc over a straight line and measure that way.
Sweet, pretty, nice and mathematician..what a combination..!
At around 3:10, shouldn´t it have said "perpEndicular"? I´m really unsure now.
Didn't she? I'm not sure what you heard there
You're right, it should have. Just a simple spelling mistake though, the maths is sound.
Rochester Oliveira it wasn't what you heard, it was the caption on the diagram :)
per- + _pend_ + -icular
It's nice to see a video that's more my speed. Reeeeeally slow and simple :D
This video made me nostalgic because half of the things done in this video were things that I learned how to do when getting my certificate in mechanical/architectural engineering: bisecting and abritrary angle, finding the perpendicular bisector of an arbitrary line segment, finding parallel lines, trisecting an arbitrary line segment.
However, I never was taught that you could double an arbitrary square. And one thing that was not talked about in the video was finding the center of an arbitrary triangle.
Boy, I used the word "arbitrary" a lot in this comment.
Nice comment
Isn't it entirely possible to make a root-3 segment from a triangle? Take your unit length, double it, use the new line as the base of an equilateral triangle. If you draw a line from an angle to a side (bisecting that side) the length of that line would be root 3.
That seems to make sense, I can't see why that wouldn't work, actually. That's weird, we need someone smarter to explain this.
(Square) Root three segments, yes. Third root segments, no.
i was thinking the same thing. it looks like it works, can't figure out a flaw in it.
Root three is completely different to a cube root.
Constructing a Cube with sides of the square root of 3 does not create a cube with volume of 2. The sides of that cube have an area of 3 so the volume of the cube is about 5.2. The cube root of 2 which would be the side length for a cube with volume 2 is about 1.26.
She's Hungarian!!!
I had two Hungarian teachers, they were very well educated, nice people and good teachers.
Nice explanation!
this stuff if so cool! thank you for sharing these math insights.
PLEASE HELP!! I don't understand why this is difficult...! Can't you just
- Draw a circle at angle A, making points B and C equidistant from A;
- Connect BC, trisecting the new line at D and E
- Connect AD and AE, and then you're done??
I mean, I understand from a recent video that all triangles can be represented as an Equilateral triangle viewed in 3d space
Nobody here is smart enough t answer you apparently, but the basis of mathematics and science says keep going until you’ve been proven wrong or you can prove it correct
Is that really called "a compas" in english? It's really strange, because I think of "a compass" every time I hear that word. In Danish, they are called (If translated directly) "a school-follower". It's a really strange name, but I guess it's because it helps you follow school? Or something? o.O
***** Oh okay. Thanks for the info :)
Well, I guess school-follower is a bit weird, because it's not about following the actual school building itself, but about making sure you don't fall behind with homework and such. That you listen in class etc. It's that kind of "following". Can't for the life of me think of a better term in english, at this moment, lol.
Edit: I actually just looked it up, and apparently the danish word for it, which is "Passer" comes from (old) german. A word identical to it, which meant to measure or adjust. It might also be related to the French word "compas".
TL:DR Danish is weird! lol
In Poland "kompas" means a device used to show north in field. The circle-creating compass is called "cyrkiel" (sounds almost exactly like circle)
I have the opposite problem in Portuguese, because the device to make a circle is called "um compasso", while the device that points to north is called "uma bússola". So every time I pick up a compass in a Zelda game, the first thing that comes to mind is this stuff! lol
It really is called a compass. Just to make it even more confusing, a compass is also a device that indicates north. I'm not sure why these two things have the exact same name, but its probably because you can use them both for navigation. We also have something called a sextant, but that is specifically for navigation, and you wouldn't be drawing circles for fun with it.
In Slovakia, we call that "kružidlo" which would translate roughly into something like "circler" - maker of circles :-)
Excellent teacher so bright and both charming and very pretty too! Lucky students😊
If people are interested in straightedge and compass constructions, there is an app called Euclidea that basically poses these construction problems as puzzles. If you like thinking about this stuff you will love the app, it can be really tricky and fun to find the most efficient solutions.
For example, an especially tough one: given a circle and a point on that circle, can you construct a tangent line through that point using only 3 lines/circles?
So you have to go one dimension up to crack cube roots. What about fourth roots? O_o; Do we go up one more dimension? :P
fourth root is square root of square root, which isn't a problem
That seems so wrong being unable to trisect an angle with Euclidian tools. It just seems too benign to be impossible. I'm going to waste so much time now trying to get it to work even though I was just told it's been proven impossible.
How is that going for you?
K463178 Can confirm, it cannot be done.
TheJaredtheJaredlong I think i may have found a way to do it
K463178 I found a way that worked for acute angles, but I couldn't get it to work on obtuse angles. I think my method for the acute angles was also technically wrong, and the error is so small it _looks_ right, but when I tried it on some obtuse angles it was very clearly wrong.
TheJaredtheJaredlong bisect an angle, create a perpendicular line, and divide that line in 3 equal parts, obtaining so 2 points. Connect then the points to the origin of the angle, and erase the bisect line and the perpendicular line.. why not?
Euclid's compass was a collapsing compass. You can not use it to transfer a distance from one starting point in the plain to another. As soon as you lift the compass from the plain, it collapses! If you want to construct a 20 cm line from a given 1 cm line you must use Book I Proposition 2 of Euclid's Elements.
Plane! D'oh!
i dont get it...
+Brendan Ward Compass equivalence theorem.
Also, you could always just place the compass on a line and then keep constructing half-circles down the line without lifting it from the plane. :D
if my math teachers had been so kind and friendly, I might have learned to love math... I'm certainly in love with math now that I found this channel...
Even a slaveboy could figure out how to double a square!
-Socrates
haha....Plato reference.
I believe Socrates claimed that the slave boy was not "figuring out how to double a square" but only "recollecting" what his immortal soul already knew. (Meno)
Adidas = Trisector :0
Regarding the angle /3:
I did it the following:
1.) choose any radius and construct the circle with die middle being at the angle itself.
2.) mark the point where the circle crosses the "angle lines"
3.) connect these crossing points
4.) trisect the line
5.) draw lines from the angle through the trisected line.
DId I do something wrong or did I not even understand the problem?
Beautiful!
>>> HEY BRADY!!
Hungarians unite!
userful1 persze angolul írd le xd
Legalább más is megérti
van bojler elado!
bojler eladó
Christobanistan No
Could you solve to n roots if you used n dimensional paper? (Such as finding a cube root using 3 dimensional paper)
This geometry makes sense. Numbers are a human construct of measure, so the Eucilean seams more of a anchoring reality in how we understand our surrounding. This is rather amazing, and I am shocked I never learned this basic method to geometry....sure some overlap, but not based in. This should be in schools from perhaps 2nd or 3rd grade on and into the complexity in high school and university.
Brady's big problem: Getting Zsuzsanna into his bed
Line of 20cm? A line is infinitely long, a line with restrictions such as 20 centimeters is by definition known as a line "segment".
+Enlightenment Not in Euclid.
Question couldn't you trisect an angle by trisecting the the two line segments that make up the angle then connect the corresponding points and trisect that line segment then draw a line between those points and the angle
Talking about this particular case of constructing a square with an area twice that of a unit square, where the only problem is to construct a length of root 2: this could've been achieved from the unit square itself, without having drawn another straight line and constructing a group of right triangles.
Note that root 2 is also the length of the diagonal of the given unit square, so you could've directly joined the two diagonally opposite vertices of the square and measured the length of that line segment using your compass, and you're done.
I know that the method was aiming at the demonstration of constructing lengths of positive irrational values, but just wanted to point out a quicker solution to * that particular question *.