It seems kind of obvious that is what will happen because when new circles are drawn on the uppermost space, they are arbitrarily placed precisely arranged in such a way that they will touch and not overlap. Because the rectangle with fixed area is acting kind of like a panto router it is merely translating this pattern to a smaller scale. The reason they are nested circles is because one corner of this rectangle is arbitrarily fixed and used to create the originating point of a radius. So it's kind of like, if I draw a bunch of circles that just barely touch and translate them into a rotational map of the same space, I will get a circular space of touching circles. Which is kind of like, yeah, of course you will.
@@EamonBurke What do you think arbitrarily means? Doesn't seem to me that you are using the word right, even the two words "arbitrarily" and "precisely" seem kind of a oxymoron when put together. The circles are not arbitrarily placed and neither is the fixation of the corner of rectangle. I would even argue that is the exact opposite. Both the circle and the rectangle follow a strict set of rules or deliberate thought in placement.
The animation just begs to draw the next horizontal line, one unit up. It should touch all the circles from both rows, and the 'origin', so you can imagine how it must be drawn fairly easily. I think it would look neat though.
I believe you'd just get smaller or bigger circles, whose touching on the right hand side of that original circle? That is, all parallel lines above the "original line" would be a smaller circle sharing one point with the "original circle" and bigger circles for the parallel lines below the original.
I love this "style" of episode - 2-4 similar but unrelated topics in one longer vid. It's like the Neil Sloane "amazing graphs" series of vids from this channel. Great stuff!
What does the circle on the right map to, the one formed as a limit of all the increasingly smaller circles? And what happens when you go below the line? Do you just get a mirror image of the pattern within the circle?
Below the line would be outside the main circle. The circle formed as a limit of the other circles is the sides of the triangle that is formed from the circles drawn above the line
Below the line you'll get mostly another copy of the circle but going to the right instead of to the left of the origin. I'm not exactly sure what the two rows of circles just below the line will map to. I should make this picture.
Even though the picture at 8:01 makes it look like at the limit there’s a big empty circle on the right, that’s just where the simulation stopped. If you think about it though, if the lower right corner of the rectangle is the origin (0,0), then there are circles above it touching all points of the form (0, 1 + (2k+1)/2) for k = 1,2,3,…. So the height of those rectangles is 1 + (2k+1)/2, which makes the width the reciprocal, and as k grows to infinity that width approaches 0. So as more and more circles are added you are getting circles that intersect the x-axis at points closer and closer to the origin, which means there isn’t a big empty gap on the x-axis, it’s filled with an infinite string of circles approaching that pivot point.
But our schools is teaches maths in a way like they are sst , Because of there teaching some people hate maths either maths is a subject no one can hatee
Apollonian gaskets are cool. They have a connection to Ford circles, which I think have been covered in another video, and thence to Farey sequences and the Stern-Brocot tree (ditto).
Well, my Ford does need a new set of head gaskets. Where can I find that brand? :D I'd say that I'm sorry and I couldn't help it, but I won't lie to you.
Yeah, but seeing it in motion it's hard to not see! That is what I love about these animations: they give me a much better intuition of concepts the textbooks or my unfortunate tutors every could!
THIS is the beauty and simplicity of Math. Our numbers give reasoning and make it complex, But behind those smoke and mirrors of numbers and variables, is geometric beauty. Amazing work Matt Henderson. And discoveries/explanations like these is why Numberphile is an OG of the youtube Math community
Here's one way of seeing why the intersection of the circles and lines is a parabola: Say the center point has a circle of radius r intersecting one of the lines. Then that intersection point is r units from the center. But since the circles and lines are moving right at the same rate, the intersection point is also r units left of a certain line. That line is the directrix, and the center point is the focus.
Would this map into 3d? Would fixing a corner on a constant volume cuboid and drawing a plane with another produce a 3 sphere drawn by a third? And would similarly drawing an array of 3 spheres above the plane propagate the volume of the one below the plane with infinitely many touching 3 spheres?
Time is the worst dimension IMO... We can only perceive the past, why can't we just turn around and see what's in the future?! It works just fine for the 3 spatial dimensions, you just rotate and look. With time, there's not even a way to rotate. Something's deeply wrong, and no-one even seems to care... AWFUL dimension
How many circles can you stack on top of the line for a circle R=1m, such that the last small circle you project on the bottom vertex has radius of planck length
Seeing 18000 views, 1800 likes and 18 dislikes makes this moment even more mathematical to me. Or not. I dunno, just a coincidence. 180 comments would make it mind-blowing though...
at roughly 5:40 they go "and what this really is is circle inversion" and throw up the other videos they've mentioned circle inversion in before, including epic circles
I'm really mesmerized by the "Infinitely Many Touching Circles" part - very, very beautiful pattern. I'm really curious what it would look like if you instead used one of the corners of the fixed-area rectangle to inscribe circles on a non-square grid (triangular or hexagonal?). Would you get the same pattern? I wish I could try this out myself, but I'm not a programmer... sigh.
It's not the same pattern, naturally, but i don't think the difference would be visible without it being pointed out: Consider that every time circles touch, they do so in both "worlds". A triangle pattern above means you have each circle below touching 6 others, rather than 4 as shown. An hexagonal pattern is just the triangular pattern with some gaps, unless i misunderstood you.
I believe it gives a rotated, translated, and possibly mirrored inversion. In ordinary inversion, points drawn on the circle of inversion map to themselves.
Yep. Circular inversion was my point in life where I realized that the world as we see is simply subjective and depending on other perspectives thing may look different for an identical object.
