Strange Spheres in Higher Dimensions - Numberphile 

Parker CIRCLE T-Shirts and Mugs: teespring.com/stores/parker-circle

*"There is a trick that you can use in mathematics called not worrying about it."*

Student: "How do imagine things in the 4th dimension???!!!" Maths Professor: "Easy! You just imagine the nth dimension and let n=4"

"I'm going to call them spheres no matter the dimensions." >Continues calling them circles indefinitely

I absolutely lost it at Parker Circle.

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.

There is a trick that you can use in Mathematics : by not worrying about it.- Matt Parker 2017 #parkertrick

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 😂

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!

Teacher: "Do you call that a circle?" Me: "Yes Sir, that's called a Parker Circle." #ParkerCircle

*Walks into the fruit section, starts comparing the sphericity of oranges*

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.

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.

“There is a trick you can use in mathematics called… not worrying about it.” - Ah! :-D

"That's a Parker's circle" ahahhahahahahahahah

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.

"They've gone a bit beyond kissing" lololol

I wonder how many Parker circles would fit in a Parker square

"there's a trick you can use in mathematics called not worrying about it". Yep.

3:30 who are we to judge circles’ relationships

The Parker shape family is slowly expanding :D

#ParkerCircle

"Imagine your favourite film with a spiky thing in it. It's a bit like that."

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.

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 a simple man, I see Matt Parker and a square and I click

1:07 I thought Matt swore in response to the Parker Circle comment and I almost spat my tea out

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.

Now we can call every badly drawn circle as a Parker circle

ParkerBox containing ParkerCircles ... brilliant

2:50 I don't know why but i kind of expected you to draw a vertical line in the air there

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.

This video only leaves me with one question - what is the audio waveform on Matt's shirt of?

So the Tardis must be working in a 10+ dimensional space in order to be bigger on the inside

Pure class!! I’ll never tire of watching Numb/Compphile vids, or rewatching old ones. Especially those with with Matt in them :)

3:23 I guess you could say those circles are quite into each other :^)

First, a Parker square, then a Parker circle. What's next? A Parker triangle!?

Yay! Now we have Parker squares and circles!

Matt Parker is my fav professor from this channel. Reminds me of my own uncle and high school math teacher.

"In 4d lovely stuff happens..." brilliant add placement. One of my absolute favorite books.

#parkercircle

That pun-tastic seque into the sponsors ad at the end was a work of art.

This video is now 372 days old and it still gets me at 1:16 with "Parker Circle"

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.

8:13 you could have started with 1 dimension. It works too.

Eagerly awaiting the introduction of the #parkertriangle now.

man, between proportional packing, the highest number of regular polytopes aside form 2d, and the ability to make a complete graph utilising two complex number planes, the fourth dimension is the best

"The short moral of the story is that high dimensional spheres are really weird." - Probably the best quote of the year.

Spiky spheres sound like spheres that aren't quite right somehow...Parker Spheres....

3Blue1Brown

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?

The Matt Parker memes are literally the best thing ever, I love this guy.

#ParkerCircle I think Matt will forever be teased with 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

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.

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 was the audio waveform on his tshirt?

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.

So after 9d we start to make our own Tardis? Cool :-)

I called my dog "PI" because he's infinitely constant.

Parker circles are "spikier" than the dimension theyre in and spiky is a Parker description of higher dimensions :)

"The cheapest spheres I could find in a grocery stores" I wonder what are the most expensive hyperspheres out there

2:18 Matt Parker grocery shopping: "Can you direct me to the aisle where I might find your cheapest spheres good sir?"

It's in times like these where I quote Rick Sanchez: "Don't think about it!"

1:33 I spent 5 minutes trying to answer that, he explained it in 30 seconds :')

The 4th dimension has to be so beautiful and symmetrical given the contained sphere being exactly 1

I love videos about higher dimensions. This video ist my most favorite :)

3blue1brown just did this a few videos ago.

Matt, don't listen to the haters- just Parker Square

Really good video. The idea of this paradox is really good brought to me

I have found the pinnacle of entertainment. A grown man taping oranges together.

I died at the Parker circle but only a minute in

1:01 These circles look very good actually, I challenge Brady to draw better ones XD

Probably my favourite numberphile video!

During a practice scholarship paper my class did a question similar to this where we had to work out the area of the circle in the very centre given only values of the outer edges of the square.

Parker circle LOL

I dont understand why parker didnt bring 10 dimensional oranges

First Cliff Stoll and then Matt Parker video this is a great week.

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.

Let's get ahead of him before he makes a video: Parker Platonics (tetrahedron, cube, octahedron...)

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!

Moral of the story is: *_”DON’T_* use padding spheres in 10D and up.”.

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!

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.

@ 8:14 can someone pls make a _graph_ which extends into _complex numbers_ as well??? ((((((:

8:24 My favorite question and answer.

I genuinely lol'd at the sponsor segue at the end

So Matt has also seen that 3Blue1Brown video. :D

i love that math trick: not worry 'bout it

I was waiting for a Parker Circle comment and Brady didn't disappoint.

I want to classify shapes in a synthetic geometrical manner only by their (relative) dimension. For example a rectangle is less two dimensional related to a square which is absolute two dimensional. Or if we reduce proportionally a figure it tends toward zero dimensions. Practically the relative dimension is a result of quantity influence over dimension of figure. Is there any domain or geometrical problems that involve this approach to dimensions?

So in the 0th dimension, the radius is... -1?

Am I right that as number of dimensions approach infinity the n-dimension sphere would have infinitely big radius.

8:20 is set in the background I love finding these Easter eggs in the backgrounds.

SET!!!! it's in the background!! best game ever!!

Thumbs up if you learnt about this from Matt's book before watching this video!

Okay, smart people: First, its the Parker Square Then, the Parker Circle Coming up. Parker Triangle (Illuminati confirmed)

AMAZING!

Well this makes perfect sense! Every dimension you add, you offset the containing spheres. try and take a slice of the 3d box such that it looks like the 2d box and you can't! The original 2d slice is 45 degrees, along the diagonal of the 3d, but the containing circles have retreated from each-other. Do this a few times, and the inner circle can slip out in that 2d slice.

Who are confused and couldn't fully understand here Go to 3Blue1Brown channel and search there about higher dimensional sphere.... He has more illustrative description