@@blazebluebass if there are two equivalents shots fire from either side of a perfects sphere, such that their paths run parallel to the outside of a second sphere...
And it involves Numberphile again lmao. Still remember when that -1/12 vid they posted years ago garnered so much attention, so many Math youtubers went ahead and made a video to discuss the same thing.
@@leif1075 it's toxic when you don't approach the subject with humility. Being completely clueless and thinking you know the solution just places you in the peak of Mount Stupid in the Dunning-Kruger Effect.
@@Zkflames because he's referring to an appearance of Stanford professor Tadashi Tokieda on the numberfile channel, implying that his explanation to the given problem was not quite on point. So we are enthusiastically anticipating the math equivalent of the Drake VS Kendrick beef 😄
@@Zkflames *Revelation 3:20* Behold, I stand at the door, and knock: if any man hear my voice, and open the door, I will come in to him, and will sup with him, and he with me. HEY THERE 🤗 JESUS IS CALLING YOU TODAY. Turn away from your sins, confess, forsake them and live the victorious life. God bless. *Revelation 22:12-14* And, behold, I come quickly; and my reward is with me, to give every man according as his work shall be. I am Alpha and Omega, the beginning and the end, the first and the last. Blessed are they that do his commandments, that they may have right to the tree of life, and may enter in through the gates into the city.
There are two planes tangent to three spheres. What you mean is the plane cutting all spehers in half by going through all three midpoints (of which there is only one)
If you invert that, you can start with a line and 3 points. Then trace 6 lines, 2 from each point, and find the circles... Also, it seems from the video that there are some other spots from the intersection of the lines that are possible to inscribe a circle. I dont know why, but the idea of starting with a line and 3 points to generate 3 circles seems so exotic...
That would be great. But, this is why logical reasoning is paramount in proving theorems. B=>A is not equivalent to A=>B. In fact, when both A=>B and B=>A is true, we invoke the special condition "if and only if".
@@AnkhArcRod No, not like that. You draw a line with 2 dots and one circle beside. When you project 4 tangent lines from dots to the circle, you get areas where 2nd and 3rd circle must be. Then you have to work out proportional math, why the lines that are touching those 2 circles (wherever in their areas they may be) have to intersect on the 'primordial' line.
@@capsey_I have a simple proof of impossiblity for it to be any arbitrary 3 circles. If 2 of the circles are the same size, you won't get an outside point from the two double tangent lines - they're parallel, and this don't intersect
Learnt about this in my technical drawing class. I think it was called radical axis and is the basis for a lot of methods for manually finding tangential relations without knowing the radius
radical axis is different? I think it is the locus of the point where the power of tangents to the two circles are equal. And the radical axis for two different circles not touching each other lies between them and we get it by S - S' = 0 where S and S' are the equations of two circles
Only works when circles have different sizes. The result is always a triangle, and it's the same calculation for perspective. We have 3 imaginary lines, going from the center of the spheres, and meet in the cross section. When we move a ball, or change it's size, 2 out of the 3 horizons adjust always keeping the relation that origins the line. There are several relations that we can set to figure out the problem, and the result should be a line equation with variable angle, that depends on ball position and size.
A special case of this theorem (iirc if one circle is a point) was part of the solution to ARML 2024 Individual Round problem 10, one of the hardest problems on one of the hardest high school math competitions in the world.
Something you didn't mention in your setup - the three circles must be all different diameters or the lines will never converge and you will not be able to get a crossing. This restriction may feed into the answer somehow.
Here is another fun way of looking at the problem: Imagine the three different sized circles in 2d are actually the result of taking a picture of three equally sized spheres in 3d space with different distances to the camera. If you add the plane spanned by the centers of the spheres to the picture, this plane defines sort of a horizon line infinitely far away from the camera. My gut says this horizon line should be the same as the line constucted via the tangents.
I think I know a cool geometric explanation. Think of these circles not as circles of different sizes but as circles that are at different distances from you. E. e. the picture you see is a projection of a 3d scene with circles (not spheres!) all parallel to the viewport. The circle that apperas smaller is just farther from you in 3d space than the others. Then the tangent lines to a pair os circles on the projection may be thought of as parallel lines in 3d space. Also all those lines are parallel to the plane that goes through the centers of the circles (easily proven by drawing a line through a pair of centers). So all the "intersections" of thase parallel lines are at the apparent horison of that plane. Since the horizon of a plane is a straight line, the theorem is proven.
I watched this video without sound. It took me several tries to read the captions correctly because my eyes kept jumping to the white text instead of the yellow. Color theory is as important to communication as number theory.
Here is another "fun" geometry pizzle, since you like the third dimension a lot, you have n spheres, generalize a formula for a point that lays inside all spheres at the same time, assuming there is one
Probably the only 3B1B video where I wasn't satisfied with the expalanation. While he usually goes into rigorous proofing which I happily have to research and find out more about, he just said that there is a plane that sits on all three of these points and it happens to go through the tangent points exactly, without going into the "why".
I am a high schooler with interest in mathematics, however, my biggest issue is that while I can solve "difficult" (long ques or those ques which need a new way to think) easily, I STRUGGLE with basic calculations. My parents insult me a lot due to this. How can I solve this issue? Thank you.
Btw, by "new way of thinking", I mean questions that don't really follow any kind of convention and u just have to figure them out on your own, their might be multiple or even just one way.
Basic math calculations are pure memorization and practice - lots of _perfect_ practice. Additions, subtractions, multiplications, divisions, etc. Once you do a lot of them, speed and accuracy will come. Pen and paper, not apps. Write down, say them out loud, randomize the tables, etc. Then get a computer program to generate random pairs of numbers and solve it on paper. You can also use a calculator to generate numbers: tan(1) tan(2), etc, then get pairs of numbers and do operations on then. Them multiply them, get tan(5) and sin(8), write them down, multiply, then check the answer. Do not optimize for speed, always optimize for right answers. Take your time and speed will come. Calculations, abstract thinking, modeling, geometry, are all different skills areas. Maybe you have a talent for some of those, and not others. Practice what you are bad at.
Basic calculation skills only come with practice. Practice so much that you can do it in your head, then you’ll be ready. Start with 2 digit by 2 digit multiplication, move onto algebra and geometry textbooks, and remember to regularly review everything that you've learned and practiced
I remember this problem appeared in a gifted student math contest exam in my country that I participated. Needless to say, I didn't even get a commendation.
I don't even understand why this is a issue.. the points are self evident - or is it that you need ot prove it mathematically? That's just a definitional issue
The short: here’s a 2D geometry puzzle The solution: we can solve it by stepping into 3D. See linked video! The linked video: Time for 4D geometry! We’re going to be busting out the nth dimensional tesseract at this rate
I know people joke about there being drama, but yeah science people love discussing this stuff and correcting mistakes to prevent them from propagating !
Drama in the youtube math community
Shots fired! Quick, count the bullet 😂
@@blazebluebass if there are two equivalents shots fire from either side of a perfects sphere, such that their paths run parallel to the outside of a second sphere...
And it involves Numberphile again lmao. Still remember when that -1/12 vid they posted years ago garnered so much attention, so many Math youtubers went ahead and made a video to discuss the same thing.
@@fluffybear46LMFAOO I remember that 😂 what a time
Though they also did the Parker Square which has been absolutely funny as a running joke
They did a collab before, I doubt it’s against the channel as a whole 😂
My toxic trait is thinking that i can prove this geometrically
EDIT: guys i watched the video and it actually is a simple geometrical proof
Hence the opening "geometry puzzle"
@@michaelloew522why did you think that was toxic??
It's not really that difficult tbh. I proved it right after I watched this short. My proof wasn't anywhere as simple as theirs tho
@@leif1075 it's toxic when you don't approach the subject with humility. Being completely clueless and thinking you know the solution just places you in the peak of Mount Stupid in the Dunning-Kruger Effect.
Proof for why these points exist on a line: they just do fam, look
I layed my finger across the points, and it lined up. That concludes the proof
Proof: trust me bro, why would I lie
Ork ahh proof
The proof is by magic.
bro you gotta start solvin these math conjectures for us
Shots fired. Whole Stanford campus waiting for a tokieda 3b1b diss track 😂
can you explain whats happening
@@Zkflames because he's referring to an appearance of Stanford professor Tadashi Tokieda on the numberfile channel, implying that his explanation to the given problem was not quite on point. So we are enthusiastically anticipating the math equivalent of the Drake VS Kendrick beef 😄
@@Zkflames
*Revelation 3:20*
Behold, I stand at the door, and knock: if any man hear my voice, and open the door, I will come in to him, and will sup with him, and he with me.
HEY THERE 🤗 JESUS IS CALLING YOU TODAY. Turn away from your sins, confess, forsake them and live the victorious life. God bless.
*Revelation 22:12-14*
And, behold, I come quickly; and my reward is with me, to give every man according as his work shall be.
I am Alpha and Omega, the beginning and the end, the first and the last.
Blessed are they that do his commandments, that they may have right to the tree of life, and may enter in through the gates into the city.
There exist one tangent plane to three spheres. The line is its intersection with plane through their centeres.
Another plane "underneath", to be pendantic..
There are two planes tangent to three spheres. What you mean is the plane cutting all spehers in half by going through all three midpoints (of which there is only one)
This doesn't work if the smaller sphere is located between the two larger ones such that there is no plane tangential to all three.
@@worrisomeDeveloperthis is the flaw in numberphile's proof
I figured this already.
You, 3 Blue 1 Brown straight throwin SHADE. Betta gat up crew!
can you explain whats happening
@@Zkflames
Jesus is the way, the truth and the life. Turn to him and repent from your sins today ❤️
If you invert that, you can start with a line and 3 points. Then trace 6 lines, 2 from each point, and find the circles... Also, it seems from the video that there are some other spots from the intersection of the lines that are possible to inscribe a circle. I dont know why, but the idea of starting with a line and 3 points to generate 3 circles seems so exotic...
Maybe better start with line and 2 dots.
Don't forget to prove you can construct any three circles with this method otherwise your proof only works for circles that you can make
That would be great. But, this is why logical reasoning is paramount in proving theorems. B=>A is not equivalent to A=>B. In fact, when both A=>B and B=>A is true, we invoke the special condition "if and only if".
@@AnkhArcRod No, not like that. You draw a line with 2 dots and one circle beside. When you project 4 tangent lines from dots to the circle, you get areas where 2nd and 3rd circle must be. Then you have to work out proportional math, why the lines that are touching those 2 circles (wherever in their areas they may be) have to intersect on the 'primordial' line.
@@capsey_I have a simple proof of impossiblity for it to be any arbitrary 3 circles. If 2 of the circles are the same size, you won't get an outside point from the two double tangent lines - they're parallel, and this don't intersect
Learnt about this in my technical drawing class. I think it was called radical axis and is the basis for a lot of methods for manually finding tangential relations without knowing the radius
radical axis is different? I think it is the locus of the point where the power of tangents to the two circles are equal. And the radical axis for two different circles not touching each other lies between them and we get it by S - S' = 0 where S and S' are the equations of two circles
Only works when circles have different sizes.
The result is always a triangle, and it's the same calculation for perspective. We have 3 imaginary lines, going from the center of the spheres, and meet in the cross section.
When we move a ball, or change it's size, 2 out of the 3 horizons adjust always keeping the relation that origins the line.
There are several relations that we can set to figure out the problem, and the result should be a line equation with variable angle, that depends on ball position and size.
"Only works when circles have different sizes." otherwise the lines are parallel 😂😆🤣
And not aligned, and smaller not in the middle ...
A special case of this theorem (iirc if one circle is a point) was part of the solution to ARML 2024 Individual Round problem 10, one of the hardest problems on one of the hardest high school math competitions in the world.
The circle situation just got even crazier
I proved this using geometry and algebra when I was in grad school (chemical engineering). Ten pages of calculations later, it was done. Whew!
It’s a three leg stool!
MathTube is 'boutta have a throwdown.
Grant, you are ever-inspiring! Thank you!
It's 3 points of directions of the xyz axis, and the line just intersects all 3 of them
Something you didn't mention in your setup - the three circles must be all different diameters or the lines will never converge and you will not be able to get a crossing. This restriction may feed into the answer somehow.
Can you explain a specific Problem?
How many circles fit in another?
Love to See a Video✌️
Here is another fun way of looking at the problem: Imagine the three different sized circles in 2d are actually the result of taking a picture of three equally sized spheres in 3d space with different distances to the camera. If you add the plane spanned by the centers of the spheres to the picture, this plane defines sort of a horizon line infinitely far away from the camera. My gut says this horizon line should be the same as the line constucted via the tangents.
its because the exterior lines will make a triangle
Thank you for this, Grant, I'd love to hear about the type of math/science media that you consume both in audio/video and book format :)
Geometry? Third dimension? Oh come on, Sheeple!
^^" You called out Numberphile.
But in math, we welcome people who criticize us and fix our mistakes.
🗿
Now I want to know what the mistake was.
you sound like someone who doesn't welcome people who criticize you
They did a collab before, I doubt it’s malicious in intent 😊
You know what else we could imagine?
3blue1brown vs numberphile beef is crazy
A rook and a bishop are placed at random. What's the probability that they stand on the same line
Truly remarkable, thank you!
What if two circles were the same size…
If you move blue circle far enough there will be no straight line
something about a shadow of a straight line, must also be a straight line?
Grant is wonderful! I love these vids.
What application is this?
If 2 circles are the same size, their tangent lines will not intersect.
I wonder if the same is true for inner tangents
What an animation! Fantastic.
"Full solution" sounds like "false illusion".
I think I know a cool geometric explanation. Think of these circles not as circles of different sizes but as circles that are at different distances from you. E. e. the picture you see is a projection of a 3d scene with circles (not spheres!) all parallel to the viewport. The circle that apperas smaller is just farther from you in 3d space than the others. Then the tangent lines to a pair os circles on the projection may be thought of as parallel lines in 3d space. Also all those lines are parallel to the plane that goes through the centers of the circles (easily proven by drawing a line through a pair of centers). So all the "intersections" of thase parallel lines are at the apparent horison of that plane. Since the horizon of a plane is a straight line, the theorem is proven.
With the 3D hint it becomes pretty obvious.
:( I draw two of the circles with the same size and I cannot find the point. Help.
I watched this video without sound. It took me several tries to read the captions correctly because my eyes kept jumping to the white text instead of the yellow. Color theory is as important to communication as number theory.
But what if two circles are the same size? Creating parallel lines? Doesn't that break thsi puzle?
Menelaus theorem goes brrr
Homotety trivial
"I have been drawn as the cool blue pi and you as the confused brown pi"
Aah guy
like Desargues theorem?
Corect me if im wrong. Is it the same thing as horizon line when drawing perspective?
Here is another "fun" geometry pizzle, since you like the third dimension a lot, you have n spheres, generalize a formula for a point that lays inside all spheres at the same time, assuming there is one
That’s definitely the anodyne logo at first
Can you prove with projective geometry?
Things heating up in the math fandom
Aren't every circles's tangents external?
Probably the only 3B1B video where I wasn't satisfied with the expalanation. While he usually goes into rigorous proofing which I happily have to research and find out more about, he just said that there is a plane that sits on all three of these points and it happens to go through the tangent points exactly, without going into the "why".
What program did you use to visualise this?
Manim, a program he made himself.
not a program, but a python library
wanted to add that the code for his videos is open source on his github
Numberphile is like the Veritaseum of maths.
Reminds me of the “fourth side of a triangle” idea.
Oh my god, I thought for a moment this was tightly related to the Euler Line
The lesson: Here's some circles
The exam: prove triangle has 180° using Monge's Theorem
it's easy to prove just draw every possible circle configuration and use a ruler
Waiting for diss track from numberphile 😞
Using Menelaus’s theorem for triangle O1O2O3
What is applicability of this problem?
What happens with four circles?
Usually impossible
Oh no my circles are of same diameter.
Uuuuh, a plane resting on the three spheres intersects with the initial plane along the line?
Eigenvector?
Numberphile was WRONG?
It would be great if there was a link I could click to go to the video
im pretty sure its the play button above the title press that
@@Zkflames that worked. Thanks!
Horizon line in perspective
I am a high schooler with interest in mathematics, however, my biggest issue is that while I can solve "difficult" (long ques or those ques which need a new way to think) easily, I STRUGGLE with basic calculations. My parents insult me a lot due to this. How can I solve this issue? Thank you.
Btw, by "new way of thinking", I mean questions that don't really follow any kind of convention and u just have to figure them out on your own, their might be multiple or even just one way.
Study and practice allow solving of basic calculations. Your natural talent can only take you so far.
Basic math calculations are pure memorization and practice - lots of _perfect_ practice. Additions, subtractions, multiplications, divisions, etc. Once you do a lot of them, speed and accuracy will come. Pen and paper, not apps. Write down, say them out loud, randomize the tables, etc. Then get a computer program to generate random pairs of numbers and solve it on paper. You can also use a calculator to generate numbers: tan(1) tan(2), etc, then get pairs of numbers and do operations on then. Them multiply them, get tan(5) and sin(8), write them down, multiply, then check the answer. Do not optimize for speed, always optimize for right answers. Take your time and speed will come. Calculations, abstract thinking, modeling, geometry, are all different skills areas. Maybe you have a talent for some of those, and not others. Practice what you are bad at.
@@enzoferber3982 I never use calculators, just my own brain. Sometimes, my brain's thinking is so far ahead that I cannot even calculate properly.
Basic calculation skills only come with practice. Practice so much that you can do it in your head, then you’ll be ready. Start with 2 digit by 2 digit multiplication, move onto algebra and geometry textbooks, and remember to regularly review everything that you've learned and practiced
I remember this problem appeared in a gifted student math contest exam in my country that I participated.
Needless to say, I didn't even get a commendation.
A mistake in a Numberphile video? Imagine muh shock.. 😂
well its simple, is this case.
3 is the magic number
I don't know the solution, but I'm going to assume it has something to do with squares
My short and sweet proof:
Get a ruler to see if those three points indeed lie on a straight line -- done, end of story
This kind of reminds me of Desargues' Theorem
same!!
I don't even understand why this is a issue.. the points are self evident - or is it that you need ot prove it mathematically? That's just a definitional issue
The numberphile Burn 😂😂😂
The short: here’s a 2D geometry puzzle
The solution: we can solve it by stepping into 3D. See linked video!
The linked video: Time for 4D geometry!
We’re going to be busting out the nth dimensional tesseract at this rate
It can be proved by menelaus theorem
The proof is "bro, look at it"
Okay isn't this the same argument as there's always a way to cut a sandwich in half, even if the ingredients are wildly set apart?
No. They are different types of statements and I don't see a connection between the arguments.
A 2D solution requiring a 3D idea? Without knowing anything at all about this, I'm gonna guess and say it requires vectors doesn't it?
fun is getting a headache....
I feel like in artistic call it's "Horizontal lines" 🤔
The proof is trivial, and is left as an exercise for the reader.
caviat that they all have different radius.
Menelaus theorem easy solution
Only on two axis
I think that was implied that they’re using 2D geometry
The answer is 42
projective geometry much? :3
I knew I would find a osc member here after that intro
@@imnotapartygoer 🤸
1000 configs?!
looks like triangle to me so yeah
pfft more like NumberFail am I right?
The proof is by magic.
If you bevel the edges it can be used as a hat!
Isn't this just perspective? That line is the horizon line, and so when the points are moved, so does the line.
that's one way of seeing it
@AntonioLasoGonzalez would you say it is a matter of perspective?
I know people joke about there being drama, but yeah science people love discussing this stuff and correcting mistakes to prevent them from propagating !
using menelaus will solve this easily :u
Beautiful
numberphile
they're cones