David Eisenbud Numberphile Playlist: bit.ly/Eisenbud_Videos David Eisenbud: math.berkeley.edu/people/faculty/david-eisenbud 3264 and All That: A Second Course in Algebraic Geometry: amzn.to/3lQfyKR David Eisenbud author page on Amazon: amzn.to/3G2VJ9S

@Syncrotron9001

Жыл бұрын

When you make it above 9000 let me know

@jenspettersen7837

Жыл бұрын

Cool video! I am very excited about the recent einstein tiling discovery, I hope you'll manage to make a video about that!

@averagelizard2489

Жыл бұрын

Can you please do SSCG(3) next?

Eisenbud's uncertainty principle: when drawing circles, the location of the pen cannot be known with any accuracy whatsoever.

I love his D. Eisenbud's cadence and soft spoken-ness. It's always a pleasure when he's on Numberphile : )

@subnormality5854

Жыл бұрын

I miss the old days of the Eisenbud 17-gon

@wheatdaddy_9629

Жыл бұрын

Woah, phrasing, pal

@Sad_bumper_sticker.

Жыл бұрын

Indeed, his way of speaking creates a truly unique cogni-feast ambience, I could listen to him teaching for hours without losing focus.

@codycast

Жыл бұрын

Some of you guys are strange.

@kostoffj

Жыл бұрын

Math ASMR

Parker Squares and Eisenbud Circles. Can't wait to see what approximate value gets created on Numberphile next.

@WhereNothingOnceWas

Жыл бұрын

Numberphile Cinematic Universe lore

Joke reference for those unfamiliar with English-language culture: "1066 And All That" is a parody of History teaching in British primary schools, published in 1930.

@NoNameAtAll2

Жыл бұрын

can you give a timestamp as well?

@jorgechavesfilho

Жыл бұрын

This 1930 book inspired several other history books with similar titles and in the same sarcastic tone.

@DukeBG

Жыл бұрын

Oh, I was definitely not familiar with this fact! Is there something special about the year 1066 in that title?

@neilmasson3609

Жыл бұрын

​@@DukeBG That was the year that England was invaded by the Normans. It was supposed to be one of the only two dates taught in history which people actually remembered. I think that the other one was nineteen fourteen eighteen.

@NoNameAtAll2

Жыл бұрын

@@DukeBG it's important for english history - french,english and norman leaders battled over who would rule brittain

"The point at infinity" - greatly appreciate the rigor! always surprised that other mathematicians on numberphile dont state it that way.

"if you may remember i did a video of the fundamental theorem of algebra", yes it was 8 years ago and one of my favourite numberphile videos ever

Oh my, it's been so long since we had an Eisenbud video! So great to see him back

@TippyHippy

Жыл бұрын

l put my hamster in a sock and slammed it against the furniture.

Lovely topic! I can't help but feel that a little more explanation could be given to statements like _"A circle in the space of line,"_ not everyone watching numberphile is going to immediately see a parameter space as a geometric object.

@viliml2763

Жыл бұрын

You mean "a line in the space of circles"? I thought that was a weird choice of phrase too but I guess that just means a one-parameter continuous family of circles

@QuantumHistorian

Жыл бұрын

@@viliml2763 Yep, that's what I meant. And yes, that's exactly what it means. But to someone not used to abstracting things via geometry it's not obvious.

@lonestarr1490

Жыл бұрын

@@QuantumHistorian And I think this non-obviousness is well placed in that context. It makes people stop and think, "wait. What _is_ a line really?"

@QuantumHistorian

Жыл бұрын

@@lonestarr1490Yes, exactly, except that because the speaker *doesn't* stop and instead continues the stream of new ideas, the viewer doesn't have the time to stop and think unless they manually pause (or, if they do, they'll fall behind the rest of the video). That's precisely why the speaker should spend a little bit of time clarifying what he means by that in order to give the viewer time to digest that alternative way of thinking about things.

when you combine complex numbers with conics on Numberphile you get an iConic video

11:50 What about the xy term? I don't understand why we're not dealing with a 6-dimensional space when conics are free to rotate.

Off topic, but is a video about the singular aperiodic tiling (an "einstein") that was JUST discovered a few weeks ago in the pipelines? If so, I'm really looking forward to it!

@osmia

Жыл бұрын

+

@nosuchthing8

Жыл бұрын

Here, here! And the trig functions that prove the Pythagorian theorem too.

@SaveSoilSaveSoil

Жыл бұрын

I can't wait to see Prof. Kaplan on Numberphile!

@oz_jones

Жыл бұрын

@@nosuchthing8 *hear, hear

i just happened to watch the video on matrix factorization a couple hours ago, i always misread the title as matrix multiplication and thought it wouldn't be particularly interesting since i already understand all that.. lovely to see another one with david eisenbud already! saw a video too, on projective geometry, which is related to this video's topic somewhat too! definitely one of the most odd sub box moments ive had like that before, most i can think of right now

Wow! This is fascinating! As a photographer, I know the long side of an 8 MP picture has 3264 pixels, so that's somewhere else that number shows up! 😀👍🏼

I am actually impressed not by the video and theory but by how he drew those circles neatly

7776 is called "weremeke" in Arammba, also "wärämäká" and "wermeke" in other Yam languages.

Loved this - such a clear explanation of ideas in Algebraic Geometry!

The 1066 and all that reference is very much in keeping with the Eisenbud style.

Collatz Conjecture Guy returns

@user-jc2lz6jb2e

Жыл бұрын

"Collatz Conjecture guy" 🙄 This is David fucking Eisenbud

@asheep7797

Жыл бұрын

@@user-jc2lz6jb2e donald knuth goldbach?

@RobotProctor

Жыл бұрын

@@user-jc2lz6jb2e it's supposed to be tongue and cheek; it's my ignorance not his. I bet he would get a kick out of the comment, not feel anger or anything. If I thought he would be upset by it I wouldn't have made the comment.

@emirates4321

Жыл бұрын

​@@user-jc2lz6jb2e nobody cares

@NotoriousSRG

Жыл бұрын

Dang his mom really named him specifically

Man, the Numberphile animations have come a long way...

8:53 should be (y - b), not (x - b)

@ravi12346

Жыл бұрын

Also at 11:40, there should be an xy term as well. That *would* mean there are six parameters instead of five, but really we should think of all of these equations up to a multiplicative scalar (e.g. x^2 - y = 0 is the same conic as 2x^2 - 2y = 0), which drops us back down to five independent parameters.

@emilianol203

Жыл бұрын

@@ravi12346 x^2-y=0 is the parabola y=x^2. But without the rectangular term xy u can't make a hyperbola

I once fell into a rabbithole of 3, 7, 9, 11, 13, and 37. I wasn’t disappointed.

@junerae

Жыл бұрын

intriguing, could you say more?

Y’all should make a video on how those two students proved the Pythagorean theorem using trigonometry, everyone is talking about how cool it is but I haven’t been able to find anyone actually explaining the proof

At 15:32, "Sottile" turns to "Sotille" for a few frames!

Equation of a circle at 8.56 should be (x-a)^2 + (y-b)^2 =r^2

15:57 "Circles are circles. You know what they are. They're round." Wow! There's my profundity for the night. My brain is full! 🤣

This video was incredibly interesting, and I feel like the title and thumbnail don't do it justice - I could care less about a random integer sequence, but a story about quadratics and the fundamental theorem of algebra? That's definitely something I want to watch!

@xenmaifirebringer552

Жыл бұрын

Yeah, conic curves and anything with a graphical representation feel more appealing to me than random integers

I always enjoy an Eisenbud appearance.

When will there be a video on the new mathematical proof by two US students, Calcea Johnson and Ne’Kiya Jackson? They proved the Indian sum of squares theory (x^2 + y^2 = z^2) using triogonmetry, the first people to do it. They're in high school.

15:13 I was in a program co-hosted by Dr. Sotille (Dr. "I can make all 3264 conics have real values for a, b, c, d, *and* e") a few years back. I still remember that half an hour into each session of the program, when middle-school and high school students would split up, he'd just yell "HEY!" to catch everyone's attention. Seriously- I can still imagine him doing that, which is the first thing I thought of after you mentioned him! (and after I realized he was it- that made me do a double-take!)

equidecient spheroidal points - love it!

Fascinating: like a route map through space in the microcosm and the microcosm

I love this guy.

He mentions at the end that the "theory of excess intersections" plays a role in physics, anyone know where that is?

Does that ironing board in his office have a story behind it?

@asheep7797

Жыл бұрын

Yes, it was used for ironing.

@mtranchi

Жыл бұрын

@@asheep7797 Lol, standard slapstick humor. Kudos :)

@colinwood9717

Жыл бұрын

I couldn't stop staring at it!

Fascinating!

All the books in highschool that had conics gave them as solution to ax^2 + by^2 + cxy + dx +ey + f = 0. That is six coefficients. The cross term xy was missing in this and my understanding is that it gives the tilt of the non symmetrical conics. Eg. you can alwas using just rotation of the axis change the co-ordinates so that the cross term disappears. But if you are talking about crossing points of conics I think you need to have it in there. Or can you actually write equations of two conics that have non-parallel axis in a co-ordinates where both equations do not have the cross term?

Never clicked on a video so fast in my life! Saw the title and I just knew it would be David Eisenbud speaking, because of his book "3264 and All That" Side note, maybe a very small mistake: I computed the intersection of the hyperbola xy=1 and the degenerate conic xy=0 mentioned in 4:24 in projective space for fun, and I got 2 distinct points: [0:1:0] and [1:0:0]. So I don't think there's a tangency at infinity. Also there is a projective line at infinity, not a single point at infinity. Might have been a momentary confusion with the Riemann sphere which is usually used to compactify C^1, whereas the real projective plane is used to compactify R^2.

@issoroloap

Жыл бұрын

Hi! The point is that each of them is a double point (xy=0 and xy=z^2 gives xy=0 and z^2 =0, so the second equation gives twice the line at infinity). After all, as Eisenbud explained, you should expect 4 solutions in total, for the intersection of two conics. This is like a circle and an ellipse meeting at 4 points, or being tangent at 2 double points.

@TheIcy001

Жыл бұрын

@@issoroloap You’re right! I neglected the fact that there are 4 points of intersection of two conics counted with multiplicity. Furthermore I realized that I can see the tangency by the fact that the slope of the line joining the origin and the point on the curve (this is y/x) doesn’t change sign as you wrap around. However, I can still make the point that merely having slopes converge is not a sufficient geometric interpretation of tangency. That’s just saying they intersect at infinity. Maybe physical distance approaching 0 on the other hand is sufficient for tangency (for algebraic curves), but I haven’t worked that out yet

Why 5? Where did the xy term go?

@michaeltajfel

Жыл бұрын

Yes, the xy term should be included, but you can multiply all six constants by the same number, and you get the same conic. Thus there are really five constants to determine a conic.

@aceman0000099

Жыл бұрын

xy isn't a constant, so it can't be altered in the same way

@EebstertheGreat

Жыл бұрын

@@michaeltajfel But what if the xy term has a coefficient of 0?

@ipudisciple

Жыл бұрын

The right answer is that there are 6 parameters but only up to scaling. The equation is axx+byy+cxy+dx+ey+f=0, but replacing (a,b,c,d,e,f) by (ga,gb,gc,gd,ge,gf) has the same solution set, so we get 6-1=5 parameters. They simplified for presentation and you caught it.

@EebstertheGreat

Жыл бұрын

@@ipudisciple That's still only true for nonzero g.

More eisenbud videos!!

Facinating !! Thanks :)

Great video.

Hi! Beautiful channel 👍

Is there a reason that the general equation shown at 12:40 doesn't have an xy term? (ax^2 + by^2 + cx +dy + e) I have a vague memory that the xy term can be eliminated by rotating the coordinate plane, but it has been too long since I last looked at this stuff to remember for sure.

@landsgevaer

Жыл бұрын

I think it is a mistake, or perhaps intentionally sweeping it under the rug, although the conclusion that there are 5 degrees of freedom in the parameters is still correct. Generally, ax²+bxy+cy²+dx+ey+f = 0 seems to have six, but if we multiply the entire thing by a (nonzero) number we get a *different* equation for the *same* conic, so that is overcounting one degree of freedom. Your own argument that you could rotate away the xy seems correct, but I would object that you then would get a different (rotated!) conic. Maybe somebody else has a better justification for the choice in the video...

@diniaadil6154

Жыл бұрын

xy can be rewritten as 1/4 * [(x+y)^2- (x-y)^2]

@landsgevaer

Жыл бұрын

@@diniaadil6154 Yeah, so that is a transformation to variables v=x+y and w=x-y, but if you do that, the x² and y² terms are going to reintroduce v*w terms, so you haven't (generally) lost the product term... Unfortunately, I think it is a bit more tricky than that.

@stevenmellemans7215

Жыл бұрын

I also spotted it and I have no clue other than it is a mistake.

@DukeBG

Жыл бұрын

It can be eliminated by rotation, yes. Not just v=x+y and w=x-y, but a more generic-looking matrix

8:56 is meant to have (y - b)^2 for anyone wondering

Sometimes I have no idea what you’re talking bout but when that happens I still know more.

How come there is no x*y hyperbola term in the general conic equation at 11:48?

At 9:00 the video reads (x-a)^2 + (x-b)^2 = r^2, I think this is typo - of its circle it should say (x-a)^2 + (y-b)^2 = r^2, no?

Very cool but here is an idea. Are there points in the plane that are not on a tangent to the n conics and is there a way to determine if you are on such a point?

i'm a simple man, if i see numberphile posting a video of the bob ross of mathematicians, i watch the video and like it.

I need to go back and finish my Masters in Mathematics

Great video

This professor speaks so melodiously that I wonder whether he has a joint appointment at the music department.

The conic stuff will always conjure up Cliff Stoll's bread in my mind. What's going on with that ironing board?

A voice that is a cross between Tommy Chong and the Ren & Stimpy singer of Happy, Happy, Joy, Joy. 'All the little creatures, maaaan.'

"well i wanna tell you about some numbers" fantastic! i love numbers

8:52 It should be y-b, not x-b.

Hey guys can you please update the viewcount of the 301 video

Somewhere at about 3:37 the equations become weird. I think the guy means that (1 + a^2)x^2 + 2abx + b^2 -1 = 0 has either two complex solutions or one complex solution of multiplicity two.

Algebraic geometry looks very difficult. Is it really so? Or if I study step by step, can I smoothly follow the subject?

@moose9002

Жыл бұрын

I think "step-by-step" is the key word here. Algebraic geometry requires fairly broad background to really get into, however, this background material is often best understood knowing how it used in algebraic geometry! Commutative algebra is certainly the biggest culprit in this regard. From my experience of learning the subject (which is admittedly not so much, but this is consistent with what more experienced people have told me) what really helps is the willingness to revisit things with the new perspectives you gained. I don't think there is any need to be intimidated, just read what you find interesting, and fill in the background as you need it. Eventually you'll learn a lot!

is a tilted cut through a cone really a perfect ellipse? I mean, at the higher end of the cut, the curvature of the cone´s surface narrower and at the lower end whider.

@razielhamalakh9813

Жыл бұрын

You'd think, right? That's actually a common question. Turns out, because the cut approaches the wider part of the cone at a shallower angle, the section is in fact a perfectly symmetric ellipse. It is counterintuitive, I'll grant you.

... that's the best use of an ironing board I've ever seen ...

why there is no xy term in quadratic formula?

I love this guys voice so much, but I do wish he'd articulate his words a little more d:

Quadratic from Latin for 'square', not Greek for 'two', though.

Yes, but why the ironing board?

please make a video on the recently discovered aperiodic hat tiling

Woof. Is there any way to visualize the complex solutions that makes any sense? It's difficult to imagine tangency between two curves in the real plane that involves the complex plane. Do you need a three-dimensional projection to show such a thing?

@GGoAwayy

Жыл бұрын

Turn the paper 90 degrees so you can't see the front or the back and are looking at it edge on... the circle and the line are overlapping in that dimension?

(@10:07): First, we had Parker Squares -- now, we have Eisenbud Circles!

19:00 bit of a typo/misspoken fact? (x-a)²+(y-b)² not (x-b)², right?

at 1:44 the ellipse doesn't match with the formula below it. if x² is multiplied by 2, it should get skinnier, not flatten.

I would love to know if there's a simple explanation for where the number 3,264 comes from? We know 2^3 = 8 and 6^5 = 7,776 but how do we get to 3,264?

@theflaggeddragon9472

Жыл бұрын

Intersection theory on moduli spaces. There's not a simple explanation

@soyokou.2810

Жыл бұрын

​@@theflaggeddragon9472 What is the expression of the number given by that theory?

@theflaggeddragon9472

Жыл бұрын

​@@soyokou.2810 I am no expert in enumerative geometry (I barely know basic algebraic geometry), but after skimming the relevant section, here's what I can say. The space of plane conics (as Eisenbud indicated in the video) is 5-dimensional; in fact it is P^5 (projective 5-space). This means the parameters are unrestricted and have no nontrivial relations (no two equations ax^2 + by^2 + cx+dy + e with different coefficients are isomorphic. Given a plane conic C, it's _dual_ C^* is the set of tangent lines, a smooth conic in the dual projective plane (space of lines in projective plane). The difficulty in narrowing 6^5 = 7776 to 3264 comes down to casting out "degenerations" of conics (double lines and such, as Eisenbud mentioned). This is technical and requires working on a _compactified_ moduli space of conics. This is the closure of the space of usual conics C in P^5. In fact, we take pairs (C,C^*) in P^5 x (P^5)^* and take the closure in there. We call this space X. Fix five general place conics C_,i, i = 1,...,5. The space of tangent conics in X is a hypersurface of degree 6. As mentioned in the video, taking a naive intersection gives a count of 6^5. The issue is the degenerate intersections occurring on the boundary of X. Now there is an object called the "Chow ring" A(X) = direct sum A^i(X), essentially formal integral sums of subvarieties of X modulo an equivalence relation. The A^i(X) encodes varieties of codimension i in X. Algebra in this ring allows us to compute intersections and many other things in algebraic geometry. On the open subset of smooth conics U in X, the hypersurface Z of conics tangent to a given conic has degree 6 (mentioned in the video). Let a,b in A^1(X) be pullbacks to X in P^5 x (P^5)^* of hyperplane classes on P^5 x (P^5)^*, and c,d in A^4(X) be classes of curves that are pulled back from general lines in (P^5)x(P^5)^*. One cna show that A^1(X) is generated by a,b over the integers. Take an equivalence class [Z] containing the hypersurfaces we want to intersect. The degree of its 5th power tells us the number of intersections (this is the point of working with Chow rings). Hypersurfaces lie in A^1(X), and it's free part has rank 2, so you can write [Z] = pa + qb for some p,q in Q and x,y forming a basis for A^1(X) (tensor Q). From basic properties of degree, you can show that [Z] = 2a + 2b in fact. Hence deg[Z]^5 = 32 deg(a+b)^5. So its enough to calculate the degree of a^ib^(5-i) for each i = 0 ,..,5. By symmetry, enough to do for i = 0,1,2. The calculations with explanation are on page 307-308 of 3264 and all that. All in all, you get deg([Z]^5) = 2^5deg(a+b)^5 = 2^5(5C0 + 2(5C1) + 4(5C2) + 4(5C3) + 2(5C4) + 5C5) = 2^5 * 102 = 3264. Reply

@adamcetinkent

Жыл бұрын

​@@theflaggeddragon9472 Blimey. That's what we get for asking questions! 😂

@viliml2763

Жыл бұрын

​@@theflaggeddragon9472 "no two equations ax^2 + by^2 + cx+dy + e with different coefficients are isomorphic" ax^2 + by^2 + cx+dy + e and k(ax^2 + by^2 + cx+dy + e) have different coefficients but are isomorphic the issue is forgetting the sixth, xy term

At what point does slicing a cone give me a parabola? :P There's a place where it it is an ellipse and then the other point when there's a clear parabola, but in between those two limits there's an area where the slice looks like an ellipse whose tip has been cut. Is there a name for that position where the slice starts resembling a proper parabola?

@embryonicsuperfemme

Жыл бұрын

This would be easier to explain with a picture, but I'll try. He mentioned two degenerate conics: a single point x^2 = 0 and two lines (cut the cone in half). There is a third which is one line. If you take any point but the tip and draw a line to the tip you will have a line that runs along the side of the cone. You can imagine the cutting plane as just touching that line, like you were preparing to wrap the cone with it. Any plane parallel to that plane by pushing inwards will intersect with a parabola. Any deviation from this angle would either tip to ellipse or hyperbola.

@Lattamonsteri

Жыл бұрын

@@embryonicsuperfemme i think i got it now, even without the picture :D thank you for your reply/explanation!

10:17 - Funny, I always thought they were Parker Circles.

Can someone further explain why a line tangent to a circle goes through two points?

@MeOnStuff

Жыл бұрын

It's not that it goes through two points (it only intersects the circle at a single point). It's that if you solve the underlying equations you get a repeated root. It's the same as, say, the equation x^2 = 0. This has one solution (x=0), but through the fundamental theorem of algebra we know every degree n polynomial has n roots: we can write it as a*(x-c_1)(x-c_2)...(x-c_n) = 0, where the c_i are complex numbers. So for x^2 = 0 this gives (x - 0)(x - 0) = 0. The c_i are, collectively, the roots, so in this example we have the roots 0 and 0 i.e. a repeated root at x = 0. The solutions are just the roots listed without repeats (without multiplicity, to use the maths term for it). Hope that helps.

And another interesting thing about 3264: 32=2^5 64=2^6

So… What’s the deal with the decorated ironing board???

I'm not a mathematician, and I really don't get this.. How will y=x² meet up again and be tangent at infinity? That would mean that the square root of y would be 0 at high enough values for y, wouldn't it? At least *sqrt(+inf)* = 0..? Thinking about it a bit more, thinking of the parabola as a conic section, if the cone has a bottom plane, they would indeed meet up again - as straight lines, so those would be tangent to the bottom plane, i.e. "infinity", though it would also "work" with a finite sized cone, it just have to be "closed". The way it's drawn at 4:46 would imply the cone has a convexly curved "bottom" though, with a "tangent transition" between the cone and the curve part. Otherwise there would be a "corner".. Anyway, I can't get the math to work with just y=x², but maybe it works with the ax² + by² + cx + dy + e formula..

8:55 Shouldn't that be (y-b)^2 ?

It's kind of wild how abstract concepts can make a book and for a family 😊

Translation to written words scroll is not helping full screen view. Pls do something

So many things went over my head that I now have a new haircut.

Interesting

1:37 my brain came to a crashing halt when he showed that graph and said xy =1. The graph is incorrect and the equation is not a quadratic. I’m sure it’s a trivial error. Can someone tell me what the equation is supposed to be?

@michaelfahie4228

Жыл бұрын

I kept watching and realize that it’s the graph that was weird, not the equation

Professor Eisenbud's speech patterns remind me of Floyd the barber from _The Andy Griffith Show._

Does this include x^2 + y^2 = 0?

@willnewman9783

Жыл бұрын

Yes, but it should be thought of as being over the complex numbers, so it is more than just the origin.

my grandson's grandson is gonna find this trivial

2^6 * 3 * 17. Seems like a pretty quick journey to me.

Could this be the first video by numberphile on actual algebraic geometry?

@stephenbeck7222

Жыл бұрын

What do you define as algebraic geometry and what is just regular function graphings or like high school analytic geometry in the coordinate plane stuff? Eisenbud’s first numberphile video years ago was on the graph characteristics of odd functions.

@soyokou.2810

Жыл бұрын

@@stephenbeck7222 Algebraic geometry is the geometric study of multivariable polynomials like the curves in this video. Classically, it uses lots of projective geometry like in this video, but modern algebraic geometry uses a lot of commutative algebra. Eisenbud is himself a famous algebraic geometer as the author of the book Commutative Algebra With a View Towards Algebraic Geometry.

11:39 why don't you invite to the party the term xy?

Slice a cone on the diagonal = Oval. Slice a cylinder on the diagonal = Ellipse.

Someone should formalize those shaving methods. There is no way that they worked so consistently without something going on there.

Employer: shows me the graph of the salary growth Me: but it never reaches the value we agreed on Employer: oh it does, it does! but the point is imaginary

How can he just say that two things which don’t intersect are intersecting, or that one point is two?

Cut through the top, and you get a triangle, or sharp corner, a special hyperbola

6:15 my brain: "boobies" Edit: 11:06 and the vid editor said "and I took that personally"

Cool

32 x 102 and 32 x 243

it's not necessary for the slice to be parallel to the central axis to get a hyperbola. if it were it would mean that there's another type of conic section between the parabola and hyperbola. so it's curious that absolutely nobody postulates the existence of such a thing, and yet most people assert that a hyperbola arises when the cut is taken parallel to the axis of the cone. these are the actual conditions for getting the conic sections: circle - if the slice is perpendicular to the axis of the cone ellipse - if the slice is between perpendicular to the axis and parallel w/ the wall of the cone parabola - if the slice is parallel with the wall hyperbola - if the slice is between parallel with the wall and parallel with the axis you should learn to say things correctly, since it reduces the amount of gaslighting that students have to deal with to figure out wtf you're trying to communicate.

@sumdumbmick

Жыл бұрын

also, a proper 'cone' has six lobes. most of the ones you depict have 1, and only for the hyperbola do you finally show the classic 2. but if you use the correct number, 6, then your slices correctly illustrate all of the relationships which occur between conic sections. it's quite nice. and it also demonstrates very elegantly how thoroughly fubar modern philosophy of mathematics really is.

@sumdumbmick

Жыл бұрын

@3:54 the 2 imaginary solutions here are on the hyperbola. they're imaginary here because the lobe of the cone that the hyperbola slices is on an axis perpendicular to the lobe sliced by the circle. why wouldn't you just mention that? or did you just not know about this?

@sumdumbmick

Жыл бұрын

this isn't controversial, either. it's literally how Special Relativity works. the curve of the relationship between t and t' is circular for vc. this is why tachyons would require energy inputs not to accelerate.

@sumdumbmick

Жыл бұрын

the other closure of the parabola occurs at the same exact spot as the vertex you have. it just looks like a mirrored copy of the parabola you drew. there is nothing happening at infinity. that's nonsense. for the hyperbola you showed, there is no contact between the y-axis and the hyperola at y = +inf. the hyperbola reaches a height of +inf when x is the successor to 0, which contrary to Peano, is not 1. but this successor is a value that we use all the time without understanding it, since it's absolutely required for evaluating limits. when you do something like: lim x->0+ 1/x = +inf you obviously can't evaluate at x=0, because division by 0 is undefinable. further, we know this function is discontinuous, since for x0 we get a positive branch that grows in magnitude as we approach 0. so, when we take this limit and say that it gives us positive infinity, what we did is we evaluated it at the successor to 0. which I will notate as L(0). now, you can trivially see that L(0), 0 and -L(0) are completely distinct values, because 1/L(0) = +inf, 1/0 is undefinable, and 1/-L(0) = -inf. your hyperbola reaches L(0), and when it does its height is the largest possible infinity that exists, but it does not reach 0. and thus it is simply nonsense to claim that it touches the vertical asymptote, x=0, at y = infinity.

@sumdumbmick

Жыл бұрын

it's hilarious to hear someone speak of rigor in mathematics when it's been known for 92 years that modern mathematics cannot possibly be rigorous. Incompleteness demonstrates that your assumptions about how mathematics works are wrong, and yet you just carry on acting as if that never happened... ok, but when that's the choice you make, you don't get to speak of rigor.

By the way: Gauß was awesome, but his "proof" of the fundamental theorem of algebra contained a gap. Only Jean-Robert Argand gave a complete proof. (Gauß did find the Tukey--Cooley algorithm 150 years before Tukey and Cooley. In general, some mathematicians are rightly considered great mathematicians, but their achievements are being confused in rather arbitrary fashion.)

@adrianf.5847

8 ай бұрын

Actually, the proof by Argand seems to use an incomplete infinite descent argument, which I believe would need some sort of Weierstrass theorem or ODE method to work. So I don't even know who first proved this theorem.