When a genius 16 year old Pascal discovered a geometry pattern

Pascal discovered this amazing geometry result when he was only 16. The book "The Art of the Infinite" by Robert Kaplan and Ellen Kaplan has a wonderful introduction to projective geometry and a proof this this theorem.
Proof of Pascal's Theorem for the Circle (which also proves any conic section under projective geometry)
mathgardenblog.blogspot.com/20...
Proof of Pappus Theorem
mathgardenblog.blogspot.com/20...
Algebraic proof
www.ias.ac.in/mathsci/vol120/n...
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[@SuperNolane]

Scaling is not a proof of this conjecture because it is affine transformation which means all lines stay lines.

[@nivpearlman6514]

this was an intuitive explanation for why it might be true it works *because* scaling keeps lines as lines actually the whole idea becomes kinda trivial if you consider projective transformations(not sure if that's how it's called in english) because it keeps conic sections as conic sections and lines as lines

[@nikhilnirmal7772]

Денис Мирзоев must see channel Nikhil Nirmal Geometry Theorum of 18°72°90°

[@TheBaggyT]

"Scaling is not a proof of this conjecture because it is affine transformation which means all lines stay lines." My thoughts exactly. Would have been much better if the points were moved, swapped, or have different ratios between them (e.g. in the last example, the ratio of AB:BC looks the same as A'B':B'C').

[@mienzillaz]

Exactly.. i was like seriously..? That's your proof..? Little lolz

[@emilyemily9953]

@@mienzillaz it reminded me of that tim & eric video about space where they talk about the big dipster and the little dipster

[@ophello]

Stretching the diagram doesn’t really prove anything. Move a point on the diagram to a different part of the figure to illustrate the properties.

[@DN-qp4tc]

Not really..

[@sunnychauhan4525]

Stretching the figure will not change the ration between them

[@mathalysisworld6693]

exactly. Scaling the figure didnot matter. He should have instead moved a point.

[@keshavkasat9465]

Squishing was so dumb...

[@kwinvdv]

Keshav Kasat Yes, of course a linear transformation will keep a straight line straight...

[@Nebula_ya]

Yh, the squishing changes nothing.

[@FreddyFrazzer]

I literally came to the comments to vent about the same thing. Moving the points on an individual basis would have been a lot more enlightening #squishingwasdumb

[@BN-fi9wi]

YES THANK YOU!

[@nicolastroncoso1791]

I don't see any proof or demostration to this, should I call this geek bait? a good content but without demostration is totally unnaceptable. We want things correctly proofed, in the only way math can go. I'm not saying that this is incorrect i'm just saying that this is a horrible way to introduce such a magestic idea.

[@expiringphilosophy7605]

A 394 year old discovered this AMAZING theorem

[@chrisg3030]

There's hope for me yet.

[@tangohotellima1895]

🤣🤣

[@RubyPiec]

@@damontan4749 r/woooosh

[@weckar]

Err, you realize that if you deform an image by scaling any line will stay a line, right?

[@sarthakgirdhar2833]

😂true

[@ojasdeshpande7296]

Line will stay line...but it may not pass thro those 3 pts

[@Borthralla]

This can be used to show that any set of 5 points define a unique conic section.

[@kwinvdv]

Philip M You are referring to fact the any sixth point (and thus the conic section) can be found such that the intersections lie on a straight line?

[@alphacore4332]

Great video, but I don't think there is any scenario where deformation of a line by stretching an entire axis will result in anything other than a line, so using that as a demonstration of the theorem isn't appropriate.

[@padphak2021]

yeah the scaling is really useless

[@SendyTheEndless]

That was great. I've not studied geometry formally but I felt good about being able to guess what you were going to say before you said it (though in all fairness, some of that could be deduced simply by the way you were presenting the information!). There's something pleasing and comforting about how intuitive euclidean geometry can be.

[@aljuvialle]

Thank you for this fact. It's great to have an explanation of the problem here and a few links to the proofs in the description. When I opened the video I thought you'll provide a proofs as well. It would be great if you'll record a video with a proof as well.

[@fCauneau]

Pappus theorem seems to result from Pascal theorem, as straight lines can be defined as a peculiar case of circles (infinite radii)

[@glendariorodriguezrafael7305]

Yes, but you have 2 lines (2 "circles", not one circle) ... I think it is most appropiate to visualize the 2 lines as a degenerate hyperbola

[@skylerbehunin7777]

am i the only one who grew up around yu-gi-oh that realized he unintentionally made the seal of orichalcos

[@garywatson3343]

skyler behunin that's why yugi was LIFE

[@spartandude0117]

The power of the pharaoh will be his

[@DamienZshadow]

Before that Yu-Gi-Oh! arc, Alister Crowley made it infamous in the world of the occult.

[@DamienZshadow]

@Kosem72 I am such a sucker for geometric designs such as these that I am now noticing them also on variations within logos used by so many bands that I am into right now. Mindless Self Indulgence, Bring Me The Horizon and so many others use this exact pattern. It's just so cool looking.

[@Pixiesfairiedust]

A man of culture I see.

[@xxxprawn8374]

when you scale everything in a general direction, you're also scaling the line, therefore preserving it's "lineness"

[@andyiswonderful]

Here's another example, proved by Archimedes, and by me. Five pages of algebra. Draw any three circles of unequal circumference on a plane. Call them A, B and C. Take A and B and draw the two mutual tangents. The lines will intersect in a point. Do the same for AC and BC. The three points of intersection will always be co-linear.

[@arkomisra7563]

😂

[@jacksainthill8974]

Did you figure it out?

[@ffggddss]

+ MindYourDecisions I'd like to point out that Pappus' Theorem is a special case of Pascal's, because the case of a pair of intersecting lines, is a special case of a conic section, in which the plane making the section, passes through the apex of the cone. A pair of parallel lines, however, is not a special case of a conic section, although it is a limiting case. Also, the stretching and squishing operations you're doing, to show that the collinearity persists, are just single-axis dilations in the plane, and so, *must* preserve collinearity. "Parallelogram," or affine transformations would also do that; but, being symmetric linear transformations, those are always just a composition of two single-axis dilations along perpendicular axes; or equivalently, one single-axis dilation together with a whole-plane dilation, so you've already covered that with what you were doing. More convincing, though, would be to move one of the points along its conic section line, to see how the two intersection points affected by that, change, and how they remain collinear with the third intersection point. Neat result, and thought-provoking! This theorem must have roots in projective geometry. Did Pascal go on to develop anything else along those lines (pun intended ;-)?

[@rzvn104]

:)

[@anastasiaklyuch2746]

This is fun and interesting, but stretching simply changes plane dimensions, it never changes ANYTHING but angles. The better demonstration would be if you moved one or more of the points, changing where intersection points lay on the intersection line (though I understand it is harder to animate, but simple slides work too).

[@imix.icomonlinetutorial9626]

In 3:31, you can see that if you extend the lines(outside the ellipse), little bit further, points may not be collinear but coplanar.

[@forestpepper3621]

I had learned Pappus' Theorem for lines, but had not known Pascal's generalization to conics. I wonder if either of these theorems can be generalized to higher dimensions.

[@wildbobentertainment]

Could you do a follow up on why it matters and any applications to such a theorem. Cheers.

[@dinopad10]

Will this work on a 3D plane with (x, y, z) coordinates? I'm not sure how I'd approach this except perhaps to use intersecting planes, rather than lines...?

[@JNCressey]

Maybe not. For example, take the vertices of a triangular prism with square and equilateral triangle faces. One triangle is ABC, the other triangle is A'B'C', and the other 3 edges are AA', BB', and CC'. The three intersections then happen at the centres of the square faces; and they're not in a line, (although they are in a plane). And you could line up a sphere or two parallel planes to the points ABCA'B'C', so you can take this to be an example for either of those shapes. Although, if you just meant dropping the parallel lines or circles from the video into 3D space, then sure, that'd work; you could add as many extra dimensions to the space as you want and it wouldn't affect the shape.

[@chrisg3030]

I'm guessing or extrapolating that if in 2D we have at least 3 collinear linear intersections, then in 3D we get at least 4 coplanar linear intersections. Similarly in 1D, a single line, we get at least 2 copuntal linear intersections. The only way I can give the last proposition meaning is that two intersections, line a with line b and line b with line c, are copuntal if a, b, and c all intersect at the same point. If a, b, and c are the same single line you get at least 2 such intersections. Trivial perhaps, but then most edge cases seem to be.

[@massimookissed1023]

Two non-parallel lines are also a conic section. (Through the tip of the cone.)

[@sadhlife]

yeah, but the last theorem was mainly for the 2 parallel lines case, because it works there as well.

[@Vandalfoe]

I came here to say that.

[@laxmipapney7182]

yes it is a degenerated case of a Hyperbola

[@Felhek]

Unicursal hexagram used by Aleister Crowley. Btw this particular figure by some people represent the Water element. I think Air should be between Fire and Water. We need to study these things.

[@hidgik]

I discovered a proof of this construction when i was three days old, but the margin was too small to hold the proof. Margin problem. Insoluble problem.

[@Firefly256]

Fermat is that you?

[@hidgik]

@@Firefly256 Yes. Its Fermat. But these days I prefer to call myself Format. In line with the computer revolution.

[@djbudsellers]

great video but you might want to make a correction at 4:32 (the explanation) shouldn't the pairs be AB'/BA' & AC'/CA' & BC'/CB'?

[@MindYourDecisions]

You're right, thanks for telling me and sorry for the typos. I have added an annotation for the correct pairs.

[@timothybexon6171]

djbudsellers that is exactly what I thought when I saw it. Glad someone else noticed otherwise I would have thought I was going crazy.

[@SmileyMPV]

Any two non-parallel lines are also a conic section. Just make sure to have your section intersect with the tip of the cone.

[@lagduck2209]

Actually any two parallel lines are also a conic section if you make sure your cone is a cylinder, which is usual cone with tip at infinity.

[@saumyahadani640]

At 3:30 u said that we extend the lines to get them to intersect. What if 2 lines which have to intersect are actually parallel? Thank You.

[@MindYourDecisions]

Read about Pascal's theorem when the lines are parallel: en.wikipedia.org/wiki/Pascal%27s_theorem

[@hyh2012]

+Saumya Hadani One way to treat parallel lines is to imagine them intersect at infinity

[@saopy]

I like how you just did a screen shot at the beginning so the squiggly lines are underneath the title

[@emilyx33x]

I remember doing this in year 8 and being amazed :D

[@SKarea51]

I suppose the theorem should use AB'/A'B.. etc. You show AB'/AB'...

[@antoniofruttaldo]

4:44 aren't AB' and B'A the same line? Aswell as AC' and C'A, and BC' and C'B?

[@skilz8098]

It all comes down to the simplest of all mathematical expressions: (1+1). It may not seem apparent but it is embedded within this expression! Here's my conjecture... Every single concept of mathematics, all branches, and levels are all embedded in the simple expression (1+1) ... Everything is derived or integrated from it. Just the act of adding 1 to itself, the application of applying the operation of addition which is a linear transformation, translation to be exact defines the unit circle. There is perfect symmetry, reflection, and a 180 degree or PI radians rotation embedded within it. It isn't directly obvious at first, but take a piece of paper and mark a point on it and draw a line segment of an arbitrary distance towards your right. Now label the starting point 0 and the ending point 1. To add 1 to this line segment or unit vector is to take the total length or its magnitude and translate it along the same line in the same direction. The tail of the new vector will be at the head of the original and the head of the new vector will be pointing at 2. By doing this the total distance is from the initial point of 0 to the new location will be 2. This turns the expression of (1+1) into the equation (1+1) = 2. This equation is actually the definition of both the Pythagorean Theorem and the Equation of the Unit Circle that is positioned at the origin (0,0). You see, we went to the right from the starting point and labeled that 1. We could of went to the left and labeled it 1 as well. However, they are opposing directions, and vectors have two parts, magnitude its length, and its sign or direction or angle of rotation. So, we can label the point to the left with -1. Here the starting point of 0 is the point of reflection, point of symmetry and the point of rotation. If we rotate the point 1 to the point -1 with respect to the initial point 0, you will make an arc that is PI radians which also takes you from 1D space into 2D space. Now we have the Y coordinates as well. This is simply due to 1+1 = 2 which is also 1x2 = 2. And this is evident because we know that a circle with a radius of one has a diameter of 2, its Circumference is 2*PI and its Area is PI units^2. We know that the Pythagorean Theorem is C^2 = A^2 + B^2. We also know that the equation of a circle centered at the origin is r^2 = x^2 + y^2... They are the same exact equation. When we look at the general equation of a line in the form of y = mx+b we know that m is the slope between two points and b is the y-intercept. The slope is m = (y2-y1)/(x2-x1). We can let m = 1, and b = 0 and this gives us y = x. A diagonal line that goes through the origin. This line has an angle of 45 degrees or PI/4 radians above the X-axis. We know by definition that the slope is rise/run. We also see that it is (y2-y1)/(x2-x1) which is also dy/dx, change in y over change in x. If we look closer we can see that dy/dx is also sin(t)/cos(t) where (t) is the angle above the x-axis. This is also tan(t). The rotation of the point 1 to the point -1 that generates a 180-degree or PI radians rotation is also the definition of the angle between any two points on the same line. When you use the dot product between these two points to find the scalar value that is in regards to their angle of separation it will generate -1 for angles of 180 degrees. This is also a definition of a line. With the process of using the dot product between two given points, there are 3 things you can determine from it, the lines are perpendicular, they are parallel or neither. When the result is 0 and you apply that to the arccos you will end up with 90 degrees, when you take the arccos of -1 you will end up with 180 degrees. This shows that -1 and 1 are on the same line. Knowing that the addition of all the interior angles within a right triangle adds to give you 180 degrees, consider this, the angle of two different points on the same line has the same exact value. This is just something else to think about! When you look at the original equation of the line y = mx+b we can see this as f(x) = tan(t)x + b. This all comes from 1+1 = 2! Every polynomial, every geometrical shape, including vectors and matrices, their operations, even concepts in calculus such as derivatives and integrals are rooted in (1+1)! Just by performing the addition operator of 1 to itself you are implicitly performing multiplication and a higher degree of functions as well which takes a 0D point at 1 with a unit length vector of 1 which is 1D and by translating it, you inadvertently are moving it through 2D, 3D, 4D, 5D, ... ND space. This is so because of the equivalence of the following expressions: (1+1) = 2 === (1*2) = 2 === (2^1) = 2. So just by adding you are doing multiplication, exponentiation, and integration as well. The act of performing a linear transformation of translation itself will take you from 0D space which is simply plotting an arbitrary point in arbitrary space to 1D space that gives you a line segment or a vector. This is without direction that isn't relative to any other line segment or vector. It will also take it to 2D space that does include angles between 2 vectors, implied direction, and that will generate an area of the geometrical polynomials from triangles to circles. It will also take you to 3D space since we can rotate the unit vector in many different orientations of 180-degrees within different planes giving us volume, and higher degrees or the integrations of volume. This gives us an indication of the algebraic polynomials: y=x, y = x^2, y=x^3 and the magnitude of the term with the highest degree of exponent or power of a polynomial determines which dimensional space we are working in. My thoughts, "In truth and even in physics, they don't tell you this: they claim that time is the 4th dimension of space and that there are only 11 known speculated dimensions," I beg to differ... It is my claim that there are an infinite amount of dimensions! For each polynomial term in this sequence, x^0, x^1, x^2, x^3, x^4 the power tells us which dimension of space we are in absent of time. Time is only a factor that scales that dimension of space, time is present in all dimensions except for the x^0 dimension. There is nothing stopping us from having a polynomial in the form of: lim n->inf ax^n. If we have a term such as x^1,420 then we would be in the 1,420th dimension of space! That was just for positive integers greater than or equal to 0. There are also partial and negative dimensions. Don't think so, check this out... partial dimensions x^1/2 which is the same as the square root of x or 1/x^2. How about negative dimensions? That's simple it just a complete reflection of every + integral, fractional and irrational dimension! This can be defined by (N/M)D = lim N -> (+/-)inf, M -> (+/-0)inf X^(M/N) including the indeterminate forms of 0/0 and (+/-)inf/(+/-)inf! How do you think the Complex plane exists in the first place? I just love how everything within mathematics is all connected and related! I'm sure most of you know all of these basic concepts but just wanted to illustrate all of their interconnections and how everything we know about mathematics is rooted in (1+1). Yes, I take a Physical approach to mathematical induction! Why? Because without physics, or the ability to move, or translate, then the operation of addition would have no application or meaning! You can not even add 1 to itself in a scalar manner without treating them first as vector quantities. The number 1 itself is the unit vector, and the unit vector is the number 1 itself. You need physics, namely motion to perform the operation of addition! And as soon as you have motion, you have, limits, derivatives, and integrals! Time is only a byproduct of the act of physical motion or displacement of an object, it is not a dimension on its own. Time is just that; time is time yet it is still completely relevant. Because it answers the question of how long, but it doesn't answer the question of how far! Space is how far, distance and direction are how far and which way, motion itself which is energy is time and that is how long or when!

[@kurzackd]

Why is "Pappus theorem" separate instead of just being an addendum to Pascal's theorem?

[@sallyxu4668]

cuz pascal didnt discover it 🤣

[@kurzackd]

@@sallyxu4668 lol.

[@siggiarabi]

when you changed the picture to show that the points stay on the line, it was kinda not well done since it was a still image.

[@AlexanderQ689]

The dots are not centred on the intersections. I recommend using Geogebra or Geometer's Sketch Pad

[@PacoOtis]

Thanks for the video! But, how do we use this?

[@villageblunder4787]

You can do it at the intersection of any two pairs of two points on any two shapes too. .

[@Jivvi]

Why doesn't Pascal's theorem include the example with two non-parallel lines? That's a conic section too.

[@ramiel555]

I'm glad I'm not the only one that realized how pointless the scaling was >.

I'm too

[@dyvel]

Deforming any image will always keep fearues as alignment. What's the point with that first resizing? Would be more interresting to move the points around.

[@newton6664272]

Can we say that straight lines is a particular case of curves with the centre (or focuses) placed at infinity?

[@HahNullMuehr]

Little flaw in the text at the end: It reads 《AB'/B'A》, what is actually the same. Should read 《 AB'/BA' 》to describe a pair of connections. Also 《BC'/C'B》 and 《AC'/C'A》 do not make sense.

[@jordantorres9892]

description repeats the word 'this' twice

[@ezhaanali169]

Oooh that is why this is called collinear!! Thnx this helped me in my pure maths coordinate geometry.

[@ExpIohd]

Two things: 1. You should have applied a shear to the shapes for a more dramatic demonstration. 2. Your presentation skills have improved so much since you made this video.

[@Baer1990]

Wish you showed what the green line does when you move a point. But I figured it out

[@rhishisakthivel4637]

The problem in the ellipse where the primes and the naturals are not collinear does not work when a natural and a prime is parallel to its inverse So for example if the line A prime to B was parallel to the line of B prime to A then there would be no intersection outside the elipse, thus it's fake for that case.

[@dimanarinull9122]

just want to say Columbus's egg; it works both ways. after a thing is discovered it's easier to recreate it from scratch. but also it becomes "trivial" and without adding a bit of a bg story on pascal this entire video is like hearing my math teacher from 4th grade explaining to me why I can't solve 4 dimensional equations without her teaching when I show her a solution for something like this: w + 12 = x^2 + z - 2y z - 3x = w^2 + 4y and so on... I used to do it with 2-3 equations intersecting each other so i can have a simple function as a solution(i was a 4th grader then - so don't be harsh about those equations being easy) reminder in 4th grade we all learned things like the multiplication table and adding and subtracting numbers with 2-4 digits, useless if you already solving linear equations for the last two years and write things in a scientific notation. this theorem is not really that interesting itself without the background of pascal and how he realized it without a story how can people relate to the mindset that led to this discovery? without the apple story newtons law of gravity is completely obvious - even for kids in kindergarten and before(even if the apple actually fell to his side and not on his head, and he was working on gravity for a long time before that moment and after) the story makes it funny, engaging, relate-able and most importantly memorable. I study program engineering and I am a very proficient learner, but the only reason I can learn so good is cause I search for the stories behind the thing I learn. if you can find the story of the theorem/discovery/equation for your next video it'll be much more fun to watch with the story then without. good job on making the video it would've been fun if i haven't already learned this things several years ago.

[@spencertaylor6910]

Dang I wish I could be as smart as you

[@areox27]

What about triangles? Or 3D shapes?

[@danielbishop982]

Bruh, you literally just scaled everything the same amount on the same axis, I could achieve the same effect by tilting my screen.

[@jibirijibiri8581]

Two lines are also a conic section.

[@billhill897]

With respect to your final explanation of the theorem you quoted non-parallel lines as AB’/B’A, but isn’t AB’ and B’A the same line and likewise for the other two pairs. Shouldn’t it be AB’/A’B and likewise for the other two pairs of non-parallel lines?

[@rodriguevoirol1869]

It would have been interesting to move for exemple point A and see the lines move with.

[@olechuga2]

It must be because I am dense in trying to "understand" these games; ... I see, by your examples, that you didn't mention, the other lines that also "crossed" --> no green dots or straight lines.

[@pwmiles56]

It works with parallel lines as well. If the hexagon is ABCA'B'C' and its opposite sides i.e.AB, A'B'; BC, B'C; CA', C'A, are parallel, their intercepts all fall on the line at infinity so they are collinear, the hexagon can be inscribed in a conic e.g ellipse or hyberbola

[@climbJackbean]

I think the description about Pascal theorem on the last slide should be (AB'/BA' and AC'/CA' and BC'/CB'). Am I right? Or did I misunderstand the idea? Thanks for the help.

[@SOJACjac]

Squeeze any pic and the relative are still relative. Also, I'm no mathematician, but in the Archon, this is known. What I mean is it's what happens when you have predetermined rules in 6 points relative

[@ahmedbaig7279]

The Pascal Law was taught in Physics and it is related with fluids. The common flush may be derived on that Law. I am not familiar with the Pascal Theorem. I will dig it out.

[@DidarOrazaly]

I'm never hear pappus theorem. Actualy this video good explain about point and simmetry. I'm don't full understud english language but i like this type visual explanation. Thank's a lot for you education work.

[@saadatsaadat6120]

If i have 1 apple 2nd person has 2 ,3rd has 4 ,5th one has 8 ,6th one 16..... What will be the no. Of apple that an nth person has . Will it be equal to {2^n-1}

[@granadierfc1953]

Is it Pascal's theorem or pappus theorem

[@13th_hero]

Instantly I'm drawn to the idea of lei-lines. Vortex and energy centers...

[@SafetyBoater]

Similar to this, given 3 circles of distinct radii the intersections of the pairwise tangent lines are colinear. This is a fun proof.

[@parrydm01]

Does this apply to concentric circles? I'll have to think about this.

[@sanketvaria9734]

but where does this comes into play in real life?

[@pooransinghsikarwar732]

How we proof geometrically

[@viniciusfernandes2303]

Thanks for the video!!!

[@johnhasen845]

Is your final written description in error? AB' is the same line as B'A. Did you not mean to write AB'/A'B? Or have I missed something?

[@dhimanbhowmick9558]

Really, this is like magic, wow, great Pascale.

[@travisbaskerfield]

Seems to work if the two lines are perpendicular as well.

[@ThinkTank255]

Very misleading. This is was discovered by Pascal in 1639 at the age of 16... *but those were very different times.* 16 was not that young back then. Pascal was born wealthy. Okay, now, let's see. You are born wealthy, the year is 1639, there is nothing to do, you have private tutors and a library full of books. What the hell else are you going to do other than mathematics?

[@alexandrehuat773]

Nowadays, there are also 16 yo kids that find out new theorems.

[@QuantumPhyZ]

Fight wars for glory! Edit: Don't remove credit when is due. You can make the same argument about Maxwell, and we all know that isn't true. Sure thing that helped but it was his mind who was able to put this things on paper.

[@jimhagler3716]

@@QuantumPhyZ and he was pointed in the right direction by a poor uneducated guy named Faraday.

[@julioandrewssp]

What if AB' and BA' are parallel lines?

[@GlorifiedTruth]

Would it also work on a superbola?

[@thehumanflute]

Isn't a pair of straight lines also a conic section ?

[@aymbaut9361]

WOAH, WHEN I MOVE THE PICTURE AROUND IN MS PAINT THE PICTURE DOESNT CHANGE WOAH

[@mladenmatic731]

This is sth i've learned in primary school 15 years ago and it is called focal points.

[@mikerutecky7531]

focal point is also a turm in art the point in witch ur eyes are focused on example would be a blury immage with the focal point being a high clearity spot in some point of the pic ..iv dun art with that concept in mind shit around the focal point isent as good as the spot i ment for u to focuse on !!

[@OganySupreme]

People are talking about how stretching the diagram doesn't prove anything. They say to move a specific point around to prove this phenomenon. Still, the result proves this to be true. It's all about how he showed the proof.

[@padphak2021]

will it also work with 4 or 5 pairs of dots ?

[@sergiokorochinsky49]

it has to. choose 3 points and get the line. now change the first point by the fourth one... the other two points ensure it is still the same line.

[@imusthegreat]

@Sergio, but changing 1 starting point has influence on 2 of the 3 new points. My intuition says it will not work for more than 3 pairs of dots.

[@Jivvi]

@Tom yeah, you're right. Moving a point changes _two_ of the intersecting lines, and moves _two_ of the points of intersection in such a way that they are still on a line, but it will not be the same line as before.

[@clickaccept]

There is a natural generalisation of the Pascal construction (with single Pascal line). It woks for 6,10, 16, 27, 126, or 2160 points, each figure generalising the previous one. Beyond that, the figure becomes infinite. The construction also generalises to a figure with multiple Pascal lines. arxiv.org/abs/1703.03271

[@mathsolutions9083]

Hi presh, i am from nepal and i need the proof of this pascal's theorem from you. I hope you will provide me the solution

[@patrickdriscoll9962]

How about points on a sphere

[@JJCUBER]

I'm pretty sure that it was pointless to "try and deform the picture" since you are resizing the circle or original shape with it so if you did that with any photo I think it would work....

[@mikerutecky7531]

it works with tryangle also .........every thing is lines and shades " everything" i do art and for last few years i been upsesed with free handing hypercubes tetraheadrons exc idk y

[@JustGotALife]

Where is this theorem applicable in real life?

[@ankurbiswas987]

Where is the proof of this theorem?

[@grahamward7952]

will this work on a 3d model.... example inside of a sphere? if you reverse design this with star constellations, do we get any additional meaning of the stars?

[@chrisg3030]

I'm guessing that a 3d model would result in at least 4 coplanar intersections, a 2d model in at least 3 collinear intersections (the present case), and a 1d model - a single line - in at least 2 copuntal intersections. Presumably this means that such a line, call it AA', intersects with itself at a minimum of 2 points. The last one might be trivial if it means anything at all, but then edge cases often seem to be. (I'll think about your other question)

[@0megazeero]

Hey I am 14 and I found out a pattern in numbers ^2 Basically, the pattern is that 1^2 is 1. So, to get to the next number ^2, you only need to sum the next odd number Like 2^2 = 4 (1 + 3) 3^3 = 9 (4 + 5) 4^4 = 16 (9 + 7)

[@AlgyCuber]

what if the line pairs are parellelelel?

[@thomasolson7447]

Sounds like something to try on a sphere.

[@sakshamverma10a51]

Nice theorem sir

[@odaibm]

How to prove it?

[@theflamecuber637]

Just look at the auto generated captions at 0:05

[@adib7787]

The tumbnail is just bmth logo flipped sideways

[@Knuckles2761]

Love old videos, before elevator music and pauses for self-ads.

[@hanstorbeyns8498]

Yes, Some time ago

[@olufpaden3352]

Pretty intuitive i think. I am not even suprised.

[@AmitKumarKumar-ug9pc]

Name of thorem

[@iustinianconstantinescu5498]

Proof!!!

[@taitywaity1836]

this is the same way you work out the cross product of two vectors

[@sahilkumarsahilkumar7681]

What imoprtant this dicovery..

[@lowriderrookie]

Does it work with 3d shapes?

[@chrisg3030]

I'm guessing or extrapolating that if in 2d we have at least 3 collinear linear intersections, then in 3d we get at least 4 coplanar linear intersections. Similarly in 1d, a single line, we get at least 2 copuntal linear intersections. The only way I can give the last proposition meaning is that two intersections, line a with line b and line b with line c, are copuntal if a, b, and c all intersect at the same point. If a, b, and c are the same single line you get at least 2 such intersections. Trivial perhaps, but then most edge cases seem to be.

[@jessstuart7495]

Any introductory projective geometry book recommendations?

[@bascostbudde7614]

If you read German (and you should! :) ) Albrecht Beutelspacher: Einführung in die endliche Geometrie. I. Blockpläne. B.I. Wissenschaftsverlag, Mannheim/Wien/Zürich 1982, ISBN 3-411-01632-9.

[@bascostbudde7614]

Lehmer is a name that comes to mind too. (querying) Derrick Lehmer. not sure his work is in print but it must be digitally findable in a university library.

[@mohammedal-haddad2652]

What software did he use?