I have a question about the whole 7 x 13= 28 I saw this on UA-cam and I think I figured out what happens I'd thought I ask if you could make a video about it
Brady, I need to thank you for keeping the parker memes alive for more than a year without ruining them. I got my parker square shirt already and will see how many parker circles I can fit in it.
Actually, the term "spiky" perfectly describes Parker Dimensions, which are similar to normal mathematical dimensions except that there's something a little off about them; personally, I think they're adequate.
@@seDrakonkill Actually, the nature of the dimension is irrelevant to the maths involved. 4+ dimensional graphs are useful. Being able to compute geometry in higher dimensions is a great toolset for data analysis.
Student:"How do imagibe things in the nth dimension???!!!" Teacher:"Easy! You just imagine the 4th dimension and let 4=n" Or Student:"How do imagibe things in the nth dimension???!!!" Teacher:"Easy! You just imagine the Yth dimension and let Y=N"
I wish I had thought to say that to any one of my math teachers in school. "You need to show your work or you won't get credit!" "Well, professor, the trick here is to not worry about it."
It's not the higher dimension spheres that are "spiky". It's the higher dimension cubes that are spiky. The corner n-spheres have radius 1 in all directions and all dimensions. For the cubes, the distance from the center to the corners increases without bound with the number of dimensions, while the distance from the center to the faces remains constant. That's a spiky n-cube. The central sphere can grow endlessly, since the corner spheres are "close" to the corners, which are further from the center as the number of dimensions increases.
At first I was like, "wait, if the sphere is already larger than the side of the box, and it keeps getting larger, shouldn't it eventually go beyond the corners?" but then I remembered, that in higher dimensions, the distance to the corner is multiplied by sqrt(n), so sqrt(4) for 4D, sqrt(5) for 5D and so on, so it's not the sphere that is getting spiky, it's the box!
The sphere in a way could get spiky, say, in dimension 100. Some points in the sphere have a coordinate of 1 somewhere with all the others zero, but there are also points with all coordinates of just 0.1 or -0.1. Only a few points have a high value in one or two coordinates.
@@skrlaviolette The sphere has a larger diameter than the box's side length. However, the box's corners keep getting further away as the # of dimensions rises. In dimension 100, a cube centered at the origin with side length 2 will have corners ten units away from the origin, as opposed to only two units away in dimension 4.
@@coopergates9680 Yes, in 2D, the corners of the cube are sqrt(2) away from the origin, in 3D sqrt 3. If we say the the side length is 4 , it means to me, that the surface of the cube interect with the axis at a distance 2. The sphere intersects with the axis at its radius, which is grater than 2 for n>10, right? Or is it different for higher dimensions, because the surface of a 10D cube is made out of 9D cubes?
I'll note this also works in 1 dimension. By the definition of sphere as "the set of points equidistant from a point", a 1-dimensional sphere would simply be a line segment. The pattern suggests a "largest sphere" with a radius of sqrt(1) - 1, i.e., 0, which is exactly the gap between two line segments with a radius of 1 filling a line segment of length 4.
Being a bit pedantic, but the 1-d sphere isn't a line segment, it's two individual points equidistant to the center. It's like in 2-d, your circle is made of a 1-d line shifted through the 2nd dimension, a 1-d sphere is a 0-d object shifted through the 1st dimension.
7:19 Oh wow, this gets really interesting ! The title could have been: "Spheres packed in a box from 2 to 12 dimensions! You won't believe what happens in the 9th dimension!"
I'm going to become a famous pop star just so I can name my debut album "You won't believe what happens in the 9th dimension!" Don't worry, I'll credit you as a producer.
I don't usually pay much attention to thumbnails, but the thumbnail for this video is absolutely perfect! It tells you what the video's about, works in the joke about the slightly scraggly circles, and, Matt's expression is priceless! Seriously, you did an awesome job designing this one.
By the way, just letting you know, higher dimensional spheres AREN'T spiky by any means, if fact they're still symmetrical in all directions, and are convex just like their lower-dimensional counterparts. You might think now, that this is contradictory to what we've heard in the video. BUT IT'S NOT! Let's take the 10-dimensional case for example. The central sphere in fact is able to go out of the 4x4-box boundry, while still touching the outer spheres. Why? Because the outer spheres don't touch or even get close to any of the axes, they only touch all 9-dimensional hyperspaces formed by each group of 9 axes. They also don't touch: - any of the planes that are determined by groups of 2 axes, - any of the spaces that are determined by groups of 3 axes - any of the hyperspaces that are determined by groups of 4 axes, - any of the hyperspaces that are determined by groups of 5 axes, - any of the hyperspaces that are determined by groups of 6 axes, - any of the hyperspaces that are determined by groups of 7 axes, - any of the hyperspaces that are determined by groups of 8 axes, They only touch: - hyperspaces determined by groups of 9 axes. Basically, they are REALLY far away from any of the axes, and VERY far away from the center, there is plently of space for the central sphere to expand. It's only our 3 dimensional brain that thinks: Hey, all the outer spheres should touch all the sides of the coordiante system, but in reality, they come nowhere near (as explained above). If you're still confused, I recommend you doing some maths to prove yourself wrong: "Let's try to prove, that in the 10 dimensional case, if we take a a sphere who's radius is (sqrt(10)-1), which is larger than 2, then it will have a common region with the outer spheres, let's say for example with a sphere whose center is in (1,1,1,1,1,1,1,1,1,1), and radius is 1 - meaning that they won't just touch, but intersect." This is a very easy example to check, since for a point "to be inside of a sphere" means "to be at most it's radius away from the center", which is really easy to calculate. And you will soon realize that they do actually only touch (and they don't intersect), since the points (1,1,1,1,1,1,1,1,1,1) and (0,0,0,0,0,0,0,0,0,0) are really far away.
I don't like that analogy either. I think it would be more appropriate to say the 3-D shadows (I know there is a term for that ...embedding?) become spikier. It's just that our minds are conditioned to a 3-dimensional world and can't fathom more dimensions or shapes within those dimensions.
sounds like the higher dimensions intersect with the lower dimensions at the exact same point. such as (1,1,1,1,1,1,1,1,1,1). but as soon as one coordinate changes, like (1,1,1,1,1,1,1,1,1,0), the they don't touch and are extremely far away from touching each other. correct me if i am wrong. also i'm not sure why they said spiky, i'm sure they have reason for describing it that way, but i agree with you that they are still spheres by its very definition. we can only think of that box in 3d but have no idea what it is in 5d. 1d is a slice of 2d and 2d is a slice of 3d, 3d is just a slice of 4d. and if 4d is a slice of 5d, i cant imagine what that means for the 4x4 box or its spheres in 5d or going foward.
+Andrew Olesen The 2-D shadow of the 3-D inner sphere is not spikier than the 2-D inner sphere. Why would the 3-D shadow of the 4-D sphere or the sphere in any higher dimension be spikier?
but if you rotate a sphere, the shadow should never be spiky. so a 3d sphere casts a 2d sphere shadow. and a 4d sphere casts a 3d sphere shadow... but still round.. and a 5d sphere should cast a 4d shadow still round. unless "round" is something different altogether in higher dimensions.
I like how that pattern also explains 1-dimensional drawings, square root of 1 - 1 = 0 and how 0 dimension makes that square root of 0 - 1 = -1, which is imaginary/impossible.
+Vampyricon He gave a "solution" to a matematical proposition involving square numbers. So he practically did it but failed from overlooking an important detail and now "parker square" is called to any situation where you almost manage to do something important but you fail miserably.
The way I think of it is not by thinking of higher dimensional spheres as spiky. I actually think that's not the best: I prefer to think of higher dimensional spheres as smooth and to reconcile the pseudo-paradoxically unbounded growth of the central sphere by realizing the higher dimensional space itself (indeed, boxes) are bigger than I think; what I mean is that the space itself grows quickly as you add dimensions, so your intuition about how objects fit together naturally begins to break down a bit.
True. Actually it is the box that is more spiky in higher dimensions, because the the inner hyper angle becomes smaller and smaller relative to the full angle.
It's like the spheres become more and more like the surface of the box, which makes the void bigger and bigger and you can fit a bigger and bigger new sphere in it. At least that's how I think about it.
In 11 dimensional string theory, when you squeeze something down smaller than a plank length, it actually gets bigger. It turns out that the mathematics the describes a string smaller than a plank length is identical to the mathematics that describes basically the inverse string bigger than a plank length (which is why that's the smallest size a string can be).
The moment Matt said "That's adequate" referring to the circle, I nearly screamed at my screen "PARKER CIRCLE!" I was so happy to be vindicated 2 seconds later. Brady is always on his A game.
It should be noted though that to anything used to higher dimensional space the spheres wouldn't seem spiky at all, but rather well... spherical. It is there three (or two) dimensional representation that might be spiky.
@@brcoutme This is spot on, a 2dimensional projection of a 3d sphere to a 2d flatlander would appear spikey in some representations. But a sphere makes as much sense as a circle to us because we live and think in 3 dimensions. A sphere in 8 dimensions projected in some way to us would look spikey because were somehow compacting 8 dimensions in a really skewed way to 3d. But in 8 dimensions the center of a 8d unit sphere still has radius 1 and no matter which surface of the sphere you draw a line to from the center it'll always be distance 1. The sphere growing larger than the box is pretty mindblowing. But it has more to do with the fact that the amount of space inside a box in higher dimensions grows quite large really fast and the spheres are only touching each other at one point. Still hard to wrap your head around the fact that the center sphere can somehow be larger to escape the actual box itself. But i suspect (because intuitively this is the only way it makes sense to me) That is possible to actually have a sphere with a larger radius than the box is long that is still fully contained within the box because of the sheer amount of volume in such a box at higher dimensions. Like a 12 dimensional person wouldn't see the sphere escape the box. The sphere would have a larger radius than the box is long but because of the volume that sphere can be contained entirely inside the box while being longer without leaving the box. I guess the way i think about it is, if you have a 1x1 square you can fit a root2 line in it by turning it diagonally it is longer than the box is wide but doesn't escape the box.
@@spawn142001 The thing is it does escape the box, it was never said our subject sphere doesn't escape the box, just that it is defined by, "kissing" the padding spheres that "kiss" edges of the box. The thing to keep in mind is that the padding spheres always have a radius of 1 (there are more padding spheres in each ascending dimension), therefore at higher dimensions (more than 4) the subject sphere has a larger radius than it's own padding spheres. When we get to much higher dimensions (10 up) the subject sphere is no longer contained by the box. It might be easier to think of this like portals in fantasy or sci-fiction. From a simple direct 2 dimensional view, it may simply appear as if the area between the circles was being filled. On the other hand, their may be angles in our higher dimensions where a 2d snap shot might not show our subject sphere at all.
Loved this video, thank you. My whole life I've felt somewhat annoyed with the perpetual inability to visualize higher-dimensional solids, as though I somehow thought that "if I tried hard enough or tried the right way, I could do it." Of course, it isn't possible for us to really imagine what they would look like, but still like everyone else watching I'm sure, I find myself frustrated by the notion that "in higher dimensions, spheres become spiky." Of course we're all thinking "but what would that look like?" -- as if there was a way to answer this that we could grasp. I'm sure that 'spiky' doesn't exactly describe it, after all by definition all points on a higher-dimensional sphere must be equally distant from its center-- but I guess that it was an imperfect way to help describe certain properties of it. Higher dimensions have fascinated me since practically childhood, I'd love to see more videos on topics like this.
'It's not getting any bigger, it's gaining more directions within it.' As Matt well knows, it's not about how much space you have - it's what you do with it
What if we start at or include the 0th dimension? Would that make the radius equal to -1 ? And what about negative dimensions, do we get complex radii?? The -1st dimension would give a radius of i-1 Or what about imaginary or complex dimensions, does that even make sense??
@@briandiehl9257 LoL The original meme is about "Parker Squares". This video talks about alleged "Parker Circles." And Renshaw here suggests the introduction of "Parker Triangles".
If I remember correctly, in the episode where they meet the tardis put into a person, the doctor says it's an 11D entity... so that might actually be how that works!
TreuloseTomate I was just going to comment this. He has, so far, the best way of intuiting this. Also while you are doing this take time to measure the volume inside the box and subtract the volume of the packing spheres. The difference gives you an idea of just how much extra space there is in higher dimensions. I wish there were 4 spacial dimensions because I am a hoarder. Lol.
veggiet2009 Was it his video that I saw this problem in first? Because I didn't really understand why this happened from that video at all but this video did really make me understand it.
Quintinohthree, are you sure? Because I found this video borderline misleading. Of course hyperspheres aren't spiky. All their points still have the same distance from the center.
The weird thing, for me, is that the distance between a cube whose edge lengths are 1 (a square with sides 1, a cube where each face’s sides are 1 length, a hyper cube with cubic faces whose face’s sides are length 1, etc.) and it’s corner *also* grows without bound for the same reason. A 1d unit cube has a distance of 1 from its corner, 2d unit cube a distance of 1.414…, a 3d cube 1.732…, 4d cube 2, 5d cube 2.236… The higher dimension a unit cube is, the longer it takes to get to the corner.
9:30 "Well, somewhat appropriately, this video about fitting circles and spheres into a square space has been brought to you by ..." RAID: SHADOW LEGENDS?
bro, that's the thing. NO ONE CAN EXACTLY DESCRIBE 4-DIMENSION. We are just the "shadow" of it, a cross-section of it. The same way 2-D drawings are shadows and cross-sections of a 3-D world. 2-D world will never know life moving in the z-axis, the same way we will never know life moving in the fourth axis.
There's a trick you can use in mathematics called not worrying about it. That said, if you can draw a 3-dimensional cube in 2 dimensions (imperfectly), and you can draw a 3d sphere in 2 dimensions (with shading), is there no way to have a 3 dimensional model of a 4d sphere. I know you can model a 4d cube, so I don't see why a sphere would be any more difficult.
In a circle (2-sphere) the boundary curves into 1 extra dimension, in a 3-sphere it curves into 2 extra dimensions and in a 4-sphere it curves into 3 extra dimensions to keep the distance to the center. And a 3-plane(3-dim subspace) in any orientation through the 4-sphere will create a 3-sphere with the radius=sqrt(radius_of_4sphere² -distance_of_plane_from_center²) as a slice in this 3dim-subspace.
@@jamirimaj6880 We can easily exactly describe the 4th dimension, its really rather routine to work in higher dimensions. We cannot ever visualize 4d perfectly of course which is what I think you mean.
It may also be useful to think of higher dimensional spheres at smooth, but higher dimensional boxes as spiky. After all, it is the boxes which have their diameter going towards infinity as the dimension increases. In this mode of thinking, the 'confining' spheres get pushed further and further out into the corners of the box, leaving large amounts of room for the central sphere.
'From 10 dimensions and up, the central sphere is bigger than the box...' Yes, in the sense that its *diameter* is greater than the *side* of the enclosing box; so that it pokes out through the faces. But it's *never* as big as the diameter of the *box,* which is its (main) diagonal. In fact, the box diagonal is always more than twice the center-sphere's diameter: Diameter, D(d) = 2r(d) = 2(√d - 1) . . . box-diagonal = 4√d > 2D(d) = 4(√d - 1) So that central sphere never encloses the box, which really *would* be weird!!
Turns out there are many videos on UA-cam, and from time to time there will be overlap in creators making videos on a particular topic. Sometimes it's not a coincidence; usually it is.
If you keep in mind that the 'space' inside the higher dimension 'boxes' is way bigger and keeps growing, then the 'spheres' don't have to be spiky and the >2 'sphere' can fit in.
I don't think it works that way. The size of spheres are based on measurement of distances; so with a distance of >2 you go over a bigger distance than the size of a side of the 10+D box.
I think the intuition works better if you look at the problem a little backwards, at where the packing spheres are around the middle one. In 1D, the two 'packing spheres (lines) fill the entire box (line) with no room for a middle sphere, so its radius is 0. You see this in the 2D figure by looking at the 1D cross-section of the box across the diameter of the bottom two spheres. Likewise, if you take the 2D cross-section of the 3D box across the diameter of the 'bottom layer' of 4 spheres, you get the 2D figure, with the intuition that the Central Sphere of the 2D cross-section is only an off-center cross-section of the real Central Sphere, and thus smaller. The 3D central sphere has room to 'round out' as you move the cross section higher. Likewise, in 4D, we can take the 3D cross section across the 'bottom layer' of 8 spheres (bottom in terms of the 4th direction), and see that the Central Sphere in the cross-section must just be an off-center cross-section of the 4D Central Sphere - the 4D Central Sphere must 'round out' from it some ways as you move the cross-section upward. Taking this to the logical extreme, then, in each higher dimension, the sphere has more room to round outward from the last off-center cross-section, until in 9D the central sphere fits perfectly within the Box. 9D, essentially, allows for so much 'rounding out' of the Central Sphere that the packing spheres neatly fit within the corners. It isn't so much that they're 'spiky', rather higher dimensional packing spheres leave a lot more space to 'round out' in - the spheres are no less round, it's just that the packing spheres' cross-sections get smaller much faster as you move the cross sectional plane. They 'get out of the way' faster. Another way to think of it is to think of the radius of the Central Sphere as a fraction of the Diagonal of the Box. The Diagonal of the Box of dimension D is equal to 2sqrt(D). The ratio is then (sqrt(D) - 1)/2sqrt(D) = 1/2 - 1/(2sqrt(D)). This approaches 1/2 as D gets larger and larger.So the Central Sphere's radius never becomes larger than half the Diagonal of the Box, as we would intuitively expect. It is interesting that it escape the box's sides, but it never escapes the box's corners, ultimately keeping it from expanding unbounded.
I don't get what "spikiness" has to do with it. The more dimensions you add, the more room you get corner to corner. In a line of 4 units, the length is 4 = sqrt(16). In the 4-unit square, the length of the diagonal is sqrt(32) . In the cube the length of the distance from most opposite vertices is sqrt(48). You are getting longer distances each time you add a dimension because you have to go another 4 units in a direction orthogonal to what you had before adding the new direction.
Woah dude I totally own that book you wrote! :D I want to read it soon but I have 13 books to read just for coursework this semester so it'll probably have to wait until winter break. Also this video blew my mind!
It gets even weirder. I found the equation for the vulme of a n-sphere and the volume of a n-cube, plugged in this formula for the center sphere, and compared the volume of the sphere vs the volume of the cube. For sufficiently high dimensions, the volume of the center sphere is higher Tha the volume of the containing cube. For instance, at 26 dimensions, the cube has a volume of 4.5035 e15, while the sphere has a volume of 1.21 e17.
It's somewhat intuitive if you compare 4 circles in a square and 8 spheres in a cube. Starting at the cubic condition, to get the 2D condition, you would need to cut a section at 1/4 the height of the cube, but the maximum radius occurs at the center of the cube, or 1/2 the height. Picturing this makes it clear that the radius of a sphere in 3D will be larger than the radius of a circle in 2D, and provides a bit of evidence that the radius will increase as the number of dimensions increase.
As the infinite-box stretches in infinite directions and the sphere is in the middle and actually occupies all of those dimensions, it makes sense that the radius of this sphere can (and must) be infinite to reach through all dimensions, and will in fact never stretch outside the cube.
In infinite dimensions, the concept of measuring distance as simply extending the Pythagorean Theorem does not make sense anymore. Measuring distances that way would be similar to measuring volumes in the 3D space: You can find the area of a square, but its volume would be zero. What we can do is change how we measure things. If you're interested, one of the fields that study such infinite-dimensional spaces is Functional Analysis, but it's quite advanced if you haven't studied some other areas (especially Topology, Analysis and Linear Algebra)
I do not agree, extending the pythagorean theorem as n approaches infinity makes perfect sense. How is measuring distance in infinite-dimensions analogous to measuring the volume of a square? Makes no sense to me. Functional Analysis seems very interesting though, will look it up further!
The moment you drew that explanation with square root of 2 was very helpful. Mathematics in school should have examples like this. This really helped me understand your point.
9:27 does that mean that if you were to take a sphere to the 'infinity'-nth dimension, that it would make another sphere? Thus coming full circle? *punpunpun* haha but still a legit question
Weird things happen in infinite dimensions. For example, the box itself would not exist because most of its points would be infinitely far from the origin. The infinite sphere would no longer be compact, and would actually be contractible...
Infinity-nth dimension is probably not a concept, but... If you consider the radius of the sphere is sqrt(d) - 1, where d is the dimension, you could take the limit of the radius R when d goes to infinity. In that case sqrt(d) - 1 -> Infinity too. That would mean the contained sphere tends to occupy the entire d-dimensional hyperspace as the d -> Infinity. Wow... The contained sphere would contain the container spheres and the box. Mathematicians, can we mix high-dimension geometry with analysis like this? I don't think so... It seems so wrong...
First, infinite-dimensional vector spaces are totally a thing. Second, we can't take the formula and extrapolate to infinity naively, because things break in surprising ways. As I said above, the main problem is that the box can't exist because most of its points would be infinitely far from the origin. The proper intuition I think is to imagine the box itself as more spikey as d increases, so that the corners are further and further away while the spheres inside stay a fixed radius. Also other fun things happen, e.g. the sphere becomes contractible but stops being compact. Third, if you like these thought experiments you would probably enjoy learning functional analysis. It's the study of what still works in vector spaces once we have infinitely many dimensions. Pretty wild.
+Joey Beauvais-Feisthauer The box would exist, but it would not have a meaningful measurable size, because that naively constructed infinite dimensional space would have no "inherited" metric. The corners of the box are defined. They're just all the infinituples (x1, x2, ...) for which each xi = +/-1. I believe a metric can be defined for R x R x R x ..., but it would not be the limit of the metrics on R x R x R x ... x R for finitely many copies of R.
An intuitive way to grasp the "spikiness" of high-dimensional spheres is that the Central Limit Theorem starts to apply. The projected mass of the sphere beings to resemble a normal distribution along each dimension, but shrunk down horizontally so that instead of the tails growing longer and longer, the middle instead gets taller and taller and the arms thinner and thinner. This concept is called "Concentration of Measure" in the literature. Nice video!
Parker CIRCLE T-Shirts and Mugs: teespring.com/stores/parker-circle
Numberphile amazing
Best running gag ever.
I have a question about the whole 7 x 13= 28 I saw this on UA-cam and I think I figured out what happens I'd thought I ask if you could make a video about it
+Austin515wolf Abbot and Costello beat him to it. See In the Navy (1941)
HAVE WE GONE TOO FAR? OR HAVE WE NOT GONE FAR ENOUGH?
*"There is a trick that you can use in mathematics called not worrying about it."*
I need that on a shirt
I'll just tell that to my math teacher when I fail.
2017 Xtremum Domini
I heard him say that and I'll probably be laughing about it for a week
I actually had to pause the video and write it down planning to paint it in cubital letters in my office wall
"I'm going to call them spheres no matter the dimensions."
>Continues calling them circles indefinitely
same thing really
Brady, I need to thank you for keeping the parker memes alive for more than a year without ruining them. I got my parker square shirt already and will see how many parker circles I can fit in it.
achu11th Parker circles are "spikier" than the dimension and spiky is a Parker description of higher dimensions :)
Actually, the term "spiky" perfectly describes Parker Dimensions, which are similar to normal mathematical dimensions except that there's something a little off about them; personally, I think they're adequate.
22/7, anyone want some parker pi?
Parker circles are allowed to overlap, so you can fit infinite Parker circles.
P4RK3R C1RCL3
Student: "How do imagine things in the 4th dimension???!!!"
Maths Professor: "Easy! You just imagine the nth dimension and let n=4"
Perhaps the 4th dimension could be time. Imagine a sphere morphing as a function of time
Sebastian Aguiar Generally, In the context of math, the fourth dimension is considered another physical dimension. I believed
@@seDrakonkill Actually, the nature of the dimension is irrelevant to the maths involved. 4+ dimensional graphs are useful. Being able to compute geometry in higher dimensions is a great toolset for data analysis.
Woah what a professional technique
Student:"How do imagibe things in the nth dimension???!!!"
Teacher:"Easy! You just imagine the 4th dimension and let 4=n"
Or
Student:"How do imagibe things in the nth dimension???!!!"
Teacher:"Easy! You just imagine the Yth dimension and let Y=N"
I absolutely lost it at Parker Circle.
There's a t-shirt too
Me too :D
Yeah its absolutely hilarious! XD
Parker triangle
Andrew Kovnat me too I like how he can just make fun of him and they still like it
I love how Brady says, "Parker circle" and Matt's reactions is like, "Ah f*#%" instantly realizing that a new meme was added to the family 😂
There is a trick that you can use in Mathematics : by not worrying about it.- Matt Parker 2017
#parkertrick
Vinit Doke parkertrick
I wish I had thought to say that to any one of my math teachers in school. "You need to show your work or you won't get credit!" "Well, professor, the trick here is to not worry about it."
+
What about the dimension where spheres can be larger than the box containing them? The answer is don't think about it Morty.
You just got Ricked!!!
It's not the higher dimension spheres that are "spiky". It's the higher dimension cubes that are spiky.
The corner n-spheres have radius 1 in all directions and all dimensions.
For the cubes, the distance from the center to the corners increases without bound with the number of dimensions, while the distance from the center to the faces remains constant.
That's a spiky n-cube.
The central sphere can grow endlessly, since the corner spheres are "close" to the corners, which are further from the center as the number of dimensions increases.
Nicolai Sanders Thanks!
This is actually the more intuitive explanation
Thank you I was looking for this.
The idea of a spikey sphere seemed like mental gymnastics.
Thanks man. This makes sense. I was going nuts with the spikey sphere idea.
IT ALL MAKES SENSE NOW
Teacher: "Do you call that a circle?"
Me: "Yes Sir, that's called a Parker Circle."
#ParkerCircle
XD
??
At first I was like, "wait, if the sphere is already larger than the side of the box, and it keeps getting larger, shouldn't it eventually go beyond the corners?" but then I remembered, that in higher dimensions, the distance to the corner is multiplied by sqrt(n), so sqrt(4) for 4D, sqrt(5) for 5D and so on, so it's not the sphere that is getting spiky, it's the box!
The sphere in a way could get spiky, say, in dimension 100. Some points in the sphere have a coordinate of 1 somewhere with all the others zero, but there are also points with all coordinates of just 0.1 or -0.1. Only a few points have a high value in one or two coordinates.
@@coopergates9680 But in these direction the box has even higer values than the middle sphere?
@@skrlaviolette The sphere has a larger diameter than the box's side length. However, the box's corners keep getting further away as the # of dimensions rises. In dimension 100, a cube centered at the origin with side length 2 will have corners ten units away from the origin, as opposed to only two units away in dimension 4.
@@coopergates9680 Yes, in 2D, the corners of the cube are sqrt(2) away from the origin, in 3D sqrt 3.
If we say the the side length is 4 , it means to me, that the surface of the cube interect with the axis at a distance 2. The sphere intersects with the axis at its radius, which is grater than 2 for n>10, right? Or is it different for higher dimensions, because the surface of a 10D cube is made out of 9D cubes?
@@skrlaviolette The boundary of the cube still does intersect the x-axis at x = +- (side length / 2).
*Walks into the fruit section, starts comparing the sphericity of oranges*
Furkell go big or go home. Melons!
#ParkerOrange
The word you want is oblateness. Sphericity is not what you think it is.
I'll note this also works in 1 dimension. By the definition of sphere as "the set of points equidistant from a point", a 1-dimensional sphere would simply be a line segment. The pattern suggests a "largest sphere" with a radius of sqrt(1) - 1, i.e., 0, which is exactly the gap between two line segments with a radius of 1 filling a line segment of length 4.
Being a bit pedantic, but the 1-d sphere isn't a line segment, it's two individual points equidistant to the center. It's like in 2-d, your circle is made of a 1-d line shifted through the 2nd dimension, a 1-d sphere is a 0-d object shifted through the 1st dimension.
@@noahbaden90 You're correct.
“There is a trick you can use in mathematics called… not worrying about it.” - Ah! :-D
The answer is don't think about it Morty.
Eric Eggert
Lots of high school students know this intuitively... a bit too well.
I only use that when I absolutely need to. XD
7:19 Oh wow, this gets really interesting ! The title could have been:
"Spheres packed in a box from 2 to 12 dimensions! You won't believe what happens in the 9th dimension!"
I'm going to become a famous pop star just so I can name my debut album "You won't believe what happens in the 9th dimension!"
Don't worry, I'll credit you as a producer.
@@Dorian_sapiens commenting to see if the album is available.
@gigglysam clickbait
I don't usually pay much attention to thumbnails, but the thumbnail for this video is absolutely perfect! It tells you what the video's about, works in the joke about the slightly scraggly circles, and, Matt's expression is priceless! Seriously, you did an awesome job designing this one.
The Parker shape family is slowly expanding :D
It's expanding to beyond it's bounding box
The field of Parker Geometry is developing quite quickly.
In the end it'll have all the shapes, but not quite. (can't wait to see the parker icosaheadron)
Omg I died laughing.
#parkercircle
+SoulSilverSnorlax Godel proved in 2013 that Parker Geometry is both self-condradicting and incomplete.
"there's a trick you can use in mathematics called not worrying about it".
Yep.
I wonder how many Parker circles would fit in a Parker square
EnGIsNowhere oh shiiiiiiiii
About as many as you'd need to fit, but not quite.
^
1.3
3.14...
By the way, just letting you know, higher dimensional spheres AREN'T spiky by any means, if fact they're still symmetrical in all directions, and are convex just like their lower-dimensional counterparts.
You might think now, that this is contradictory to what we've heard in the video. BUT IT'S NOT!
Let's take the 10-dimensional case for example. The central sphere in fact is able to go out of the 4x4-box boundry, while still touching the outer spheres.
Why? Because the outer spheres don't touch or even get close to any of the axes, they only touch all 9-dimensional hyperspaces formed by each group of 9 axes.
They also don't touch:
- any of the planes that are determined by groups of 2 axes,
- any of the spaces that are determined by groups of 3 axes
- any of the hyperspaces that are determined by groups of 4 axes,
- any of the hyperspaces that are determined by groups of 5 axes,
- any of the hyperspaces that are determined by groups of 6 axes,
- any of the hyperspaces that are determined by groups of 7 axes,
- any of the hyperspaces that are determined by groups of 8 axes,
They only touch:
- hyperspaces determined by groups of 9 axes.
Basically, they are REALLY far away from any of the axes, and VERY far away from the center, there is plently of space for the central sphere to expand.
It's only our 3 dimensional brain that thinks: Hey, all the outer spheres should touch all the sides of the coordiante system, but in reality, they come nowhere near (as explained above).
If you're still confused, I recommend you doing some maths to prove yourself wrong:
"Let's try to prove, that in the 10 dimensional case, if we take a a sphere who's radius is (sqrt(10)-1), which is larger than 2, then it will have a common region with the outer spheres, let's say for example with a sphere whose center is in (1,1,1,1,1,1,1,1,1,1), and radius is 1 - meaning that they won't just touch, but intersect."
This is a very easy example to check, since for a point "to be inside of a sphere" means "to be at most it's radius away from the center", which is really easy to calculate.
And you will soon realize that they do actually only touch (and they don't intersect), since the points (1,1,1,1,1,1,1,1,1,1) and (0,0,0,0,0,0,0,0,0,0) are really far away.
AtricosHU It's an analogy...
I don't like that analogy either. I think it would be more appropriate to say the 3-D shadows (I know there is a term for that ...embedding?) become spikier. It's just that our minds are conditioned to a 3-dimensional world and can't fathom more dimensions or shapes within those dimensions.
sounds like the higher dimensions intersect with the lower dimensions at the exact same point. such as (1,1,1,1,1,1,1,1,1,1). but as soon as one coordinate changes, like (1,1,1,1,1,1,1,1,1,0), the they don't touch and are extremely far away from touching each other. correct me if i am wrong. also i'm not sure why they said spiky, i'm sure they have reason for describing it that way, but i agree with you that they are still spheres by its very definition. we can only think of that box in 3d but have no idea what it is in 5d. 1d is a slice of 2d and 2d is a slice of 3d, 3d is just a slice of 4d. and if 4d is a slice of 5d, i cant imagine what that means for the 4x4 box or its spheres in 5d or going foward.
+Andrew Olesen The 2-D shadow of the 3-D inner sphere is not spikier than the 2-D inner sphere. Why would the 3-D shadow of the 4-D sphere or the sphere in any higher dimension be spikier?
but if you rotate a sphere, the shadow should never be spiky. so a 3d sphere casts a 2d sphere shadow. and a 4d sphere casts a 3d sphere shadow... but still round.. and a 5d sphere should cast a 4d shadow still round. unless "round" is something different altogether in higher dimensions.
"They've gone a bit beyond kissing"
lololol
Someone really needs to make a "Matt Parker out of context" video
I like how that pattern also explains 1-dimensional drawings, square root of 1 - 1 = 0
and how 0 dimension makes that square root of 0 - 1 = -1, which is imaginary/impossible.
-1 isn't imaginary. Root(-1) is. And the root was on the 0, giving root(0) - 1 = -1
#ParkerCircle
What's this Parker Square meme about?
Just search it in youtube.
He'll NEVER live it down...
+Vampyricon
He gave a "solution" to a matematical proposition involving square numbers. So he practically did it but failed from overlooking an important detail and now "parker square" is called to any situation where you almost manage to do something important but you fail miserably.
#Adequate
3:23 I guess you could say those circles are quite into each other :^)
"That's a Parker's circle" ahahhahahahahahahah
haha literally all of the comments are saying this lol!
Because we love Matt Parker
I said that aloud a couple of seconds before he did. 😀
+
The best part was when Matt started to repeat it and then realized what he said.
"Imagine your favourite film with a spiky thing in it. It's a bit like that."
I'm a simple man, I see Matt Parker and a square and I click
I try to click, but don't quite succeed. But I give it a go.
For being simple that sure is oddly specific.
false.
The way I think of it is not by thinking of higher dimensional spheres as spiky. I actually think that's not the best: I prefer to think of higher dimensional spheres as smooth and to reconcile the pseudo-paradoxically unbounded growth of the central sphere by realizing the higher dimensional space itself (indeed, boxes) are bigger than I think; what I mean is that the space itself grows quickly as you add dimensions, so your intuition about how objects fit together naturally begins to break down a bit.
Right, they're not spiky at all; in fact, they're the least spiky things you can have in each number of dimensions.
True. Actually it is the box that is more spiky in higher dimensions, because the the inner hyper angle becomes smaller and smaller relative to the full angle.
Or you could just think of it leaving the box because it is not bound by our three dimensional thinking or model anymore.
It's like the spheres become more and more like the surface of the box, which makes the void bigger and bigger and you can fit a bigger and bigger new sphere in it. At least that's how I think about it.
Steffen Bendel My thoughts exactly. It's the n-cube that gets spiky. As the corners get further from the origin so do the spheres packed into them.
ParkerBox containing ParkerCircles ... brilliant
Moral of the story is: *_”DON’T_* use padding spheres in 10D and up.”.
So the Tardis must be working in a 10+ dimensional space in order to be bigger on the inside
In 11 dimensional string theory, when you squeeze something down smaller than a plank length, it actually gets bigger. It turns out that the mathematics the describes a string smaller than a plank length is identical to the mathematics that describes basically the inverse string bigger than a plank length (which is why that's the smallest size a string can be).
Sam- Fascinating but it's obvious that the Planck length is smaller than the length of a plank.
It's also impossible to make a Planck plank.
Maturkus Not unless you trick the observer, the Planck plank prank.
graphite How many of those have you pulled off? What's your Planck plank prank rank?
Matt Parker is my fav professor from this channel. Reminds me of my own uncle and high school math teacher.
Now we can call every badly drawn circle as a Parker circle
You're username has to be the most intelligent anime character
They gave it a go
3:30 who are we to judge circles’ relationships
First, a Parker square, then a Parker circle. What's next? A Parker triangle!?
3C Kitani my name is parker
Parker Theorem: The Parker square of the hypotenuse of a Parker triangle equals almost but not quite the sum of the Parker squares of the other sides.
talha tariq Oh, sorry. I don't know if there's another "Parker" here.
3C Kitani probs a parker cube.
I prefer the Free Triangle. Made of three right angles. #AH
2:50 I don't know why but i kind of expected you to draw a vertical line in the air there
This video only leaves me with one question - what is the audio waveform on Matt's shirt of?
Maybe it's a rude word...
Donald Trump MAGA
It looks like a pretty long waveform, like a softer piece of music, or a decent amount of speech.
The shirt also had bird doodles (the "flying eyebrows") on it, which I imagine is related. A bird's call maybe?
Probably a waveform for Parker square
1:07 I thought Matt swore in response to the Parker Circle comment and I almost spat my tea out
Yay! Now we have Parker squares and circles!
We need regular Parker polygons of all kind!
I support this motion!
I believe they would work similar to the Eisenbud Heptadecagon.
ua-cam.com/video/87uo2TPrsl8/v-deo.htmlm
The moment Matt said "That's adequate" referring to the circle, I nearly screamed at my screen "PARKER CIRCLE!" I was so happy to be vindicated 2 seconds later. Brady is always on his A game.
Spiky spheres sound like spheres that aren't quite right somehow...Parker Spheres....
It should be noted though that to anything used to higher dimensional space the spheres wouldn't seem spiky at all, but rather well... spherical. It is there three (or two) dimensional representation that might be spiky.
@@brcoutme This is spot on, a 2dimensional projection of a 3d sphere to a 2d flatlander would appear spikey in some representations. But a sphere makes as much sense as a circle to us because we live and think in 3 dimensions. A sphere in 8 dimensions projected in some way to us would look spikey because were somehow compacting 8 dimensions in a really skewed way to 3d. But in 8 dimensions the center of a 8d unit sphere still has radius 1 and no matter which surface of the sphere you draw a line to from the center it'll always be distance 1.
The sphere growing larger than the box is pretty mindblowing. But it has more to do with the fact that the amount of space inside a box in higher dimensions grows quite large really fast and the spheres are only touching each other at one point. Still hard to wrap your head around the fact that the center sphere can somehow be larger to escape the actual box itself. But i suspect (because intuitively this is the only way it makes sense to me) That is possible to actually have a sphere with a larger radius than the box is long that is still fully contained within the box because of the sheer amount of volume in such a box at higher dimensions. Like a 12 dimensional person wouldn't see the sphere escape the box. The sphere would have a larger radius than the box is long but because of the volume that sphere can be contained entirely inside the box while being longer without leaving the box.
I guess the way i think about it is, if you have a 1x1 square you can fit a root2 line in it by turning it diagonally it is longer than the box is wide but doesn't escape the box.
@@spawn142001 The thing is it does escape the box, it was never said our subject sphere doesn't escape the box, just that it is defined by, "kissing" the padding spheres that "kiss" edges of the box. The thing to keep in mind is that the padding spheres always have a radius of 1 (there are more padding spheres in each ascending dimension), therefore at higher dimensions (more than 4) the subject sphere has a larger radius than it's own padding spheres. When we get to much higher dimensions (10 up) the subject sphere is no longer contained by the box. It might be easier to think of this like portals in fantasy or sci-fiction. From a simple direct 2 dimensional view, it may simply appear as if the area between the circles was being filled. On the other hand, their may be angles in our higher dimensions where a 2d snap shot might not show our subject sphere at all.
No, a parker truncated gyroelongated disphenoid.
Loved this video, thank you. My whole life I've felt somewhat annoyed with the perpetual inability to visualize higher-dimensional solids, as though I somehow thought that "if I tried hard enough or tried the right way, I could do it." Of course, it isn't possible for us to really imagine what they would look like, but still like everyone else watching I'm sure, I find myself frustrated by the notion that "in higher dimensions, spheres become spiky." Of course we're all thinking "but what would that look like?" -- as if there was a way to answer this that we could grasp. I'm sure that 'spiky' doesn't exactly describe it, after all by definition all points on a higher-dimensional sphere must be equally distant from its center-- but I guess that it was an imperfect way to help describe certain properties of it. Higher dimensions have fascinated me since practically childhood, I'd love to see more videos on topics like this.
#parkercircle
'It's not getting any bigger, it's gaining more directions within it.'
As Matt well knows, it's not about how much space you have - it's what you do with it
That pun-tastic seque into the sponsors ad at the end was a work of art.
"In 4d lovely stuff happens..." brilliant add placement. One of my absolute favorite books.
8:13 you could have started with 1 dimension. It works too.
It's a lot less intuitive to start with ... specialy because your "sphere in 1D" have r = root(1) - 1 = 0 so it's just a point ^^
What would be the point?
Wait, that's zero dimensions. Never mind.
What if we start at or include the 0th dimension? Would that make the radius equal to -1 ?
And what about negative dimensions, do we get complex radii?? The -1st dimension would give a radius of i-1
Or what about imaginary or complex dimensions, does that even make sense??
Spikey points?
not to mention irrational dimensions
or... complex dimensions lol :P
This video is now 372 days old and it still gets me at 1:16 with "Parker Circle"
Eagerly awaiting the introduction of the #parkertriangle now.
Parker Illuminati confirmed.
@@briandiehl9257 Parkunimatti* confirmed.
Matt Parker
@@pranavlimaye I have no memory of what a Parker triangle is
@@briandiehl9257 LoL
The original meme is about "Parker Squares". This video talks about alleged "Parker Circles." And Renshaw here suggests the introduction of "Parker Triangles".
@@pranavlimaye I see. I don't think i have watched this channel in 3 years
So after 9d we start to make our own Tardis? Cool :-)
If I remember correctly, in the episode where they meet the tardis put into a person, the doctor says it's an 11D entity... so that might actually be how that works!
3Blue1Brown
TreuloseTomate I was just going to comment this. He has, so far, the best way of intuiting this.
Also while you are doing this take time to measure the volume inside the box and subtract the volume of the packing spheres. The difference gives you an idea of just how much extra space there is in higher dimensions.
I wish there were 4 spacial dimensions because I am a hoarder. Lol.
veggiet2009 Was it his video that I saw this problem in first? Because I didn't really understand why this happened from that video at all but this video did really make me understand it.
Quintinohthree z
Quintinohthree, are you sure? Because I found this video borderline misleading. Of course hyperspheres aren't spiky. All their points still have the same distance from the center.
On Infinite Series it was even before 3Blue1Brown
I dont understand why parker didnt bring 10 dimensional oranges
Tesco didn't have any that day.
@@mytube001 😔
I called my dog "PI" because he's infinitely constant.
and irrational?
I dont have a dog and i call it i cuz it is unreal
The 4th dimension has to be so beautiful and symmetrical given the contained sphere being exactly 1
#ParkerCircle
I think Matt will forever be teased with this😂
"The short moral of the story is that high dimensional spheres are really weird." - Probably the best quote of the year.
I have found the pinnacle of entertainment. A grown man taping oranges together.
1:33 I spent 5 minutes trying to answer that, he explained it in 30 seconds :')
"The cheapest spheres I could find in a grocery stores"
I wonder what are the most expensive hyperspheres out there
Parker circles are "spikier" than the dimension theyre in and spiky is a Parker description of higher dimensions :)
The weird thing, for me, is that the distance between a cube whose edge lengths are 1 (a square with sides 1, a cube where each face’s sides are 1 length, a hyper cube with cubic faces whose face’s sides are length 1, etc.) and it’s corner *also* grows without bound for the same reason.
A 1d unit cube has a distance of 1 from its corner, 2d unit cube a distance of 1.414…, a 3d cube 1.732…, 4d cube 2, 5d cube 2.236…
The higher dimension a unit cube is, the longer it takes to get to the corner.
It's in times like these where I quote Rick Sanchez: "Don't think about it!"
Cubik To be fair....
Ok no, I won't.
9:30 "Well, somewhat appropriately, this video about fitting circles and spheres into a square space has been brought to you by ..."
RAID: SHADOW LEGENDS?
Can we get a video where someone tries to describe the geometry of a sphere in 4 dimensions? I’ve looked into it, and it’s really weird and confusing
What's confusing about it lol? All the points that are a certain distance from the origin, looks pretty simple to me man.
bro, that's the thing. NO ONE CAN EXACTLY DESCRIBE 4-DIMENSION. We are just the "shadow" of it, a cross-section of it. The same way 2-D drawings are shadows and cross-sections of a 3-D world. 2-D world will never know life moving in the z-axis, the same way we will never know life moving in the fourth axis.
There's a trick you can use in mathematics called not worrying about it. That said, if you can draw a 3-dimensional cube in 2 dimensions (imperfectly), and you can draw a 3d sphere in 2 dimensions (with shading), is there no way to have a 3 dimensional model of a 4d sphere. I know you can model a 4d cube, so I don't see why a sphere would be any more difficult.
In a circle (2-sphere) the boundary curves into 1 extra dimension, in a 3-sphere it curves into 2 extra dimensions and in a 4-sphere it curves into 3 extra dimensions to keep the distance to the center.
And a 3-plane(3-dim subspace) in any orientation through the 4-sphere will create a 3-sphere with the radius=sqrt(radius_of_4sphere² -distance_of_plane_from_center²) as a slice in this 3dim-subspace.
@@jamirimaj6880 We can easily exactly describe the 4th dimension, its really rather routine to work in higher dimensions. We cannot ever visualize 4d perfectly of course which is what I think you mean.
2:18 Matt Parker grocery shopping: "Can you direct me to the aisle where I might find your cheapest spheres good sir?"
Matt, don't listen to the haters- just Parker Square
It may also be useful to think of higher dimensional spheres at smooth, but higher dimensional boxes as spiky. After all, it is the boxes which have their diameter going towards infinity as the dimension increases. In this mode of thinking, the 'confining' spheres get pushed further and further out into the corners of the box, leaving large amounts of room for the central sphere.
1:01 These circles look very good actually, I challenge Brady to draw better ones XD
'From 10 dimensions and up, the central sphere is bigger than the box...'
Yes, in the sense that its *diameter* is greater than the *side* of the enclosing box; so that it pokes out through the faces.
But it's *never* as big as the diameter of the *box,* which is its (main) diagonal. In fact, the box diagonal is always more than twice the center-sphere's diameter:
Diameter, D(d) = 2r(d) = 2(√d - 1) . . . box-diagonal = 4√d > 2D(d) = 4(√d - 1)
So that central sphere never encloses the box, which really *would* be weird!!
Excellent work - I was thinking exactly this, and I'm glad someone put it better than I could.
3blue1brown just did this a
few videos ago.
That must be more than just a
coincidence.
I was thinking PBS infinite series.
3Blue1Brown is the best
Turns out there are many videos on UA-cam, and from time to time there will be overlap in creators making videos on a particular topic. Sometimes it's not a coincidence; usually it is.
Also this is in Matt's book so...
Pure class!! I’ll never tire of watching Numb/Compphile vids, or rewatching old ones. Especially those with with Matt in them :)
If you keep in mind that the 'space' inside the higher dimension 'boxes' is way bigger and keeps growing, then the 'spheres' don't have to be spiky and the >2 'sphere' can fit in.
I don't think it works that way. The size of spheres are based on measurement of distances; so with a distance of >2 you go over a bigger distance than the size of a side of the 10+D box.
The Matt Parker memes are literally the best thing ever, I love this guy.
Let's get ahead of him before he makes a video: Parker Platonics (tetrahedron, cube, octahedron...)
Nothing more
I think the intuition works better if you look at the problem a little backwards, at where the packing spheres are around the middle one. In 1D, the two 'packing spheres (lines) fill the entire box (line) with no room for a middle sphere, so its radius is 0. You see this in the 2D figure by looking at the 1D cross-section of the box across the diameter of the bottom two spheres. Likewise, if you take the 2D cross-section of the 3D box across the diameter of the 'bottom layer' of 4 spheres, you get the 2D figure, with the intuition that the Central Sphere of the 2D cross-section is only an off-center cross-section of the real Central Sphere, and thus smaller. The 3D central sphere has room to 'round out' as you move the cross section higher.
Likewise, in 4D, we can take the 3D cross section across the 'bottom layer' of 8 spheres (bottom in terms of the 4th direction), and see that the Central Sphere in the cross-section must just be an off-center cross-section of the 4D Central Sphere - the 4D Central Sphere must 'round out' from it some ways as you move the cross-section upward. Taking this to the logical extreme, then, in each higher dimension, the sphere has more room to round outward from the last off-center cross-section, until in 9D the central sphere fits perfectly within the Box. 9D, essentially, allows for so much 'rounding out' of the Central Sphere that the packing spheres neatly fit within the corners.
It isn't so much that they're 'spiky', rather higher dimensional packing spheres leave a lot more space to 'round out' in - the spheres are no less round, it's just that the packing spheres' cross-sections get smaller much faster as you move the cross sectional plane. They 'get out of the way' faster.
Another way to think of it is to think of the radius of the Central Sphere as a fraction of the Diagonal of the Box. The Diagonal of the Box of dimension D is equal to 2sqrt(D). The ratio is then (sqrt(D) - 1)/2sqrt(D) = 1/2 - 1/(2sqrt(D)). This approaches 1/2 as D gets larger and larger.So the Central Sphere's radius never becomes larger than half the Diagonal of the Box, as we would intuitively expect. It is interesting that it escape the box's sides, but it never escapes the box's corners, ultimately keeping it from expanding unbounded.
Thumbs up if you learnt about this from Matt's book before watching this video!
I can imagine Matt going into the grocery store and being like:
What's the cheapest sphere you guys have?
- Um... we have oranges?
i love that math trick: not worry 'bout it
Added to my quote collection: "There's a trick you can use in mathematics called 'Not worrying about it.'"- Matt Parker
What was the audio waveform on his tshirt?
I don't get what "spikiness" has to do with it. The more dimensions you add, the more room you get corner to corner. In a line of 4 units, the length is 4 = sqrt(16). In the 4-unit square, the length of the diagonal is sqrt(32) . In the cube the length of the distance from most opposite vertices is sqrt(48). You are getting longer distances each time you add a dimension because you have to go another 4 units in a direction orthogonal to what you had before adding the new direction.
Woah dude I totally own that book you wrote! :D I want to read it soon but I have 13 books to read just for coursework this semester so it'll probably have to wait until winter break.
Also this video blew my mind!
Well ya better start reading and stop watching numberphile videos
Similar with me!
did you read it yet
It gets even weirder.
I found the equation for the vulme of a n-sphere and the volume of a n-cube, plugged in this formula for the center sphere, and compared the volume of the sphere vs the volume of the cube. For sufficiently high dimensions, the volume of the center sphere is higher Tha the volume of the containing cube. For instance, at 26 dimensions, the cube has a volume of 4.5035 e15, while the sphere has a volume of 1.21 e17.
Parker circle LOL
It's somewhat intuitive if you compare 4 circles in a square and 8 spheres in a cube. Starting at the cubic condition, to get the 2D condition, you would need to cut a section at 1/4 the height of the cube, but the maximum radius occurs at the center of the cube, or 1/2 the height. Picturing this makes it clear that the radius of a sphere in 3D will be larger than the radius of a circle in 2D, and provides a bit of evidence that the radius will increase as the number of dimensions increase.
So Matt has also seen that 3Blue1Brown video. :D
0:02 capture, contain, investigate
sounds a whole lot like secure contain protect if you ask me
Okay, smart people:
First, its the Parker Square
Then, the Parker Circle
Coming up. Parker Triangle
(Illuminati confirmed)
The parker icosahedron
The parker-klein bottle
That man has a "Set" game box in his boardgame collection. Give him a hug!
@ 8:14 can someone pls make a _graph_ which extends into _complex numbers_ as well??? ((((((:
I love videos about higher dimensions. This video ist my most favorite :)
So in the 0th dimension, the radius is... -1?
and the -1st dimension, the radius is -1 + i
I was waiting for a Parker Circle comment and Brady didn't disappoint.
Am I right that as number of dimensions approach infinity the n-dimension sphere would have infinitely big radius.
Yes, they're getting endlessly "spikier".
Yes, in n dimensions it's sqrt(n)-1, which goes to infinity as n goes to infinity.
As the infinite-box stretches in infinite directions and the sphere is in the middle and actually occupies all of those dimensions, it makes sense that the radius of this sphere can (and must) be infinite to reach through all dimensions, and will in fact never stretch outside the cube.
In infinite dimensions, the concept of measuring distance as simply extending the Pythagorean Theorem does not make sense anymore. Measuring distances that way would be similar to measuring volumes in the 3D space: You can find the area of a square, but its volume would be zero. What we can do is change how we measure things.
If you're interested, one of the fields that study such infinite-dimensional spaces is Functional Analysis, but it's quite advanced if you haven't studied some other areas (especially Topology, Analysis and Linear Algebra)
I do not agree, extending the pythagorean theorem as n approaches infinity makes perfect sense.
How is measuring distance in infinite-dimensions analogous to measuring the volume of a square? Makes no sense to me.
Functional Analysis seems very interesting though, will look it up further!
For me it is sounds like we cannot put this big sphere to that small box, but then we pack the sphere in a bubble wrap, and it goes perfectly.
Would you define it in 1 dimension as the point circles equidistant from the center?
Yes, although in that case the center 1-sphere has a radius of 0.
Abs(x)=the distance formula.
The moment you drew that explanation with square root of 2 was very helpful.
Mathematics in school should have examples like this. This really helped me understand your point.
9:27 does that mean that if you were to take a sphere to the 'infinity'-nth dimension, that it would make another sphere? Thus coming full circle? *punpunpun* haha but still a legit question
No because you are using Parker circular logic.
Weird things happen in infinite dimensions. For example, the box itself would not exist because most of its points would be infinitely far from the origin. The infinite sphere would no longer be compact, and would actually be contractible...
Infinity-nth dimension is probably not a concept, but...
If you consider the radius of the sphere is sqrt(d) - 1, where d is the dimension, you could take the limit of the radius R when d goes to infinity. In that case sqrt(d) - 1 -> Infinity too. That would mean the contained sphere tends to occupy the entire d-dimensional hyperspace as the d -> Infinity.
Wow... The contained sphere would contain the container spheres and the box.
Mathematicians, can we mix high-dimension geometry with analysis like this? I don't think so... It seems so wrong...
First, infinite-dimensional vector spaces are totally a thing. Second, we can't take the formula and extrapolate to infinity naively, because things break in surprising ways. As I said above, the main problem is that the box can't exist because most of its points would be infinitely far from the origin. The proper intuition I think is to imagine the box itself as more spikey as d increases, so that the corners are further and further away while the spheres inside stay a fixed radius. Also other fun things happen, e.g. the sphere becomes contractible but stops being compact.
Third, if you like these thought experiments you would probably enjoy learning functional analysis. It's the study of what still works in vector spaces once we have infinitely many dimensions. Pretty wild.
+Joey Beauvais-Feisthauer The box would exist, but it would not have a meaningful measurable size, because that naively constructed infinite dimensional space would have no "inherited" metric. The corners of the box are defined. They're just all the infinituples (x1, x2, ...) for which each xi = +/-1. I believe a metric can be defined for R x R x R x ..., but it would not be the limit of the metrics on R x R x R x ... x R for finitely many copies of R.
An intuitive way to grasp the "spikiness" of high-dimensional spheres is that the Central Limit Theorem starts to apply. The projected mass of the sphere beings to resemble a normal distribution along each dimension, but shrunk down horizontally so that instead of the tails growing longer and longer, the middle instead gets taller and taller and the arms thinner and thinner. This concept is called "Concentration of Measure" in the literature. Nice video!
"There's a trick you can use in mathematics called 'not worrying about it'."
- The best thing Matt Parker has ever said. Amazingly quotable!