PBS Space Time A question visualising Einstein's way of thinking about gravity, that is, 3D object with mass will bend the space-time curvature around it, isn't that space-time curvature created in 4D space? Isn't bending the 4th dimension a reason for gravity? #ImANoob
PBS is doing amazing work with their digital content. I'm blown away with the quality of channels like PBS SpaceTime. This is definitely the future of educational content and an invaluable public service.
Why does it have to be a public service? Why can't it be funded with private money? If you enjoy the series, why don't you just give them money? It's completely unnecessary to steal tax money from other people to fund your favorite project, when it can be funded like every other RU-vid channel in existence. You want education? Buy education. You want security? Buy security. You want public television programming? Buy television programming.
starrychloe yo im a libertarian too but no need to shove your liberty down peoples throats. PBS does good work okay? I mean taxation is theft but chill. This is a good cause. XD like you are waaay too upset about an education channel man
4 physical dimensions is not hard to imagine at all. These people do an awful job. basically you choose a 3 dimensional point then add another axis v in the center of that xyz point and travel towards v. size is all an illusion. there are infinite points inside any given point.
In a way, it's not exceptionally "diffucult" to stack them. The question that'll really bake your noodle from now-on is "how do you *slice* 100D oranges?"
@@DoKtaTre: With a knife. As long as an object's number of dimensions is >=1, a knife can slice it, because 1 is the number of dimensions describing the (hypothetical) cutting edge.
7:08 this sounds super exotic but after thinking about it, I realized that the math is actually trivial with Pythagoras' theorem. The 2^n balls all have radius 1/4, but the distance from their center to the center of the cube is sqrt(n)/4. So the center ball has radius sqrt(n)/4-1/4. When this becomes > 1/2, the center ball grows out of the cube. And with n=9, this is exactly equal. It might have been nice to show this in the episode, or at least hint at the fact that anyone can do this for themselves with high school math.
David de Kloet No, haha. I was just watching this video for the first time and scrolling through the comments afterwards when I saw yours. Random coincidence I guess.
After I realized this, I came to the comments to see if anyone else had described it. Thanks for going over it! If anyone's wondering how Pythagoras' theorem gets you there, the distance from corner to corner of the square in 2D is dist_2 = sqrt(1^2 + 1^2) = sqrt(2), and to add another dimension, you can just form a right triangle with the previous diagonal and an edge in the new dimension's direction, so dist_n = sqrt((dist_(n-1))^2 + 1^2) = sqrt((sqrt(n-1))^2 + 1^2) = sqrt(n-1 + 1) = sqrt(n). The distance between the centres of the opposite-corner spheres is half that, and from the centre of the box to the centre of one of the spheres is half that, so sqrt(n)/4.
"The distance between the centres of the opposite-corner spheres is half that, and from the centre of the box to the centre of one of the spheres is half that, so sqrt(n)/4." I accept your sqrt(n). However, you need to prove this.
I have always sucked at math. Even though I really love trying to learn about all of this stuff, its usually confusing. But I feel like I actually understand this now. Great video!
I don't think Mrs. Houston-Something knows anything about this topic and is reading a script, much like I forgot the second part of her name because I wasn't listening. Although, when she started talkin about curves I got interested again.
@@Dr.FeelsGood firstly, it's Dr. Kelsey Houston-Edwards. She's done her PhD from Cornell University and is an Asst. Prof at Olin College. I don't comprehend why you have to assume that she doesn't know what she's talking about. It'll be helpful not to have a sexist bias next time
@@pursuitsoflife.6119 My apologies. If that is the case, then I regret my hasty judgement in this scenario. That's amazing that she is so talented as a speaker, content creator, doctor, and beautiful to boot. What a powerful combination!
I. Tsasecret so glad to learned how to stack them too, they kept rolling around my kitchen in directions I couldn't comprehend. I tripped on one and nearly broke my leg in 8 dimensions, I'm pretty sure my health insurance doesn't cover that...
You guys are so inspiring. I honestly say these shows are changing the world bit by bit. I wish you success and I hope I can learn a lot more from you in the future! Best wishes from Hungary! :)
This is one of my most favorite math paradox! I remember working this problem in Calculus!! :-D It got me to thinking about a hyper-dimensional version of the Gabriela's horn paradox. Where you take the curve y = 1/x from 1 to infinity and revolve that curve about the x axis. If you tried to fill the resulting shape with paint you can do it. But if you tried to paint the outside you would never have enough paint because the surface area diverges. So what happens when you take the 3d horn shape and revolve it about the Y axis? To make a 4-dimensional hyper-horn? Or take the 4-dimensional hyper-horn and revolve it about the Z axis? What happens to the divergent-convergant ratio of surface area to volume at these higher dimensions? Does it remain a paradox or are we left with varying levels of infinities?
I like it! She doesn't speak too fast or too slow, she speaks clearly and is easy to understand. I don't know what the show is called "infinite series" but you've earned my subscription
I applaud you at PBS for presenting programming that is not for idiots. Unfortunately, I am too stupid for this. I'll stick to mostly grasping astrophysics over at Space Time.
Key word: "mostly". Astrophysics is largely graspable without too much math, but it's kinda like prowling around your living room by only starlight. Having a decent footing in math would be being able to turn the lights on, by that analogy.
I don't consider myself stupid, I am a signals analyst by trade and I enjoy SpaceTime but it hurt my brain trying to visualize what she was talking about. I really look forward to trying again on the next episode!
Wait, so you guys started a math channel over a month ago, and have an actual mathematician hosting it, and I'm only just hearing about it NOW??? This is unacceptable!! Time to go binge watch them all. :) PS: Great job so far Kelsey! Very much looking forward to more episodes.
She could have mentioned why the diagonal distance increases as the dimension of the hyper cube increases.. Pythagorean theorum ! Then how do we calculate the vacant hyper volume of a hyper cube after it is packed with hyper spheres? I hyper wish to see that !
That's just the difference between the hypercube volume (base^n) and the hypersphere volume (algebraic extension of circle area, sphere volume,... in function of radius = base/2)
@timwins31 we can represent it in 3 dimensions, and even tho it might be accurate, we just dont have the frame of reference to understand it, like we would a cube on a piece of paper
We can't draw a 3D object on a 2D paper, that's just the illusion of your mind calling it a 3D object. As 3-spatial dimensional beings, we cannot draw a 4D object on a 3D model and understand it. That's like saying a 2D being would be able to see 3 dimensions by having it drawn.
You can’t do that. We only are able to visualize a 3D object on 2D paper because we draw shadows and other light effects. But how will you draw light effects that come from the 4th dimension?!
Perhaps with the use of mirrors or holograms a 3-d object can be enhanced to offer a glimpse of its 4-d counterpart. Incomplete, like a "3-d" drawing, but could be useful or at least a cool toy.
I heard about it on the end of a Physics Girl video. :D I'm not sure if there was a computer science option there, but I'm definitely glad Infinite Series now exists. And if you haven't already, you should really check out *****!
Several years ago my cousin Patricia was studying math that involved higher dimensions, and geodesic topology. Or something…. A professor told her it was a waste of time, and that her work would only be of interest to Gods and Aliens. Yet Patricia is still working on it despite the criticism. I applaud her efforts and I applaud this show for talking about similar topics.
The hyper sphere bursting through the 10th dimension and going all sea urchin looking sort of mimics my understanding and mental picture of how stars morph from one state to another, and the Big Bang for that matter. If it’s a helix or perfect wave graph of demolition and creation over the LONG course of time these dimensional rules make sense in a perfect order kind of way. The same way quantum stuff is intuitive on the surface you know
Understanding the 4th dimension is basically looking at yourself. You look in the mirror and you see your physical self, but you also have thoughts(which exist electromagneticly) and you have a electro magnetic field that surrounds your body, produced by your heart so we are 4th dimensional beings. Our body's being a physical shape but only another shape which is within more bigger layers of the same shape which are just energy. BUT IS VERY MUCH THERE JUST AS YOUR BODY IS. and learning to see these energetic aspects of yourself being your "aura" and beginning to identify back with your aura as we're supposed to, is entering 4d than 5d than higher dimensions. Basically looking at a person and seeing them within another energy them within another energy them within another energetic them. And this is where ghost come from. The energetic aspects of a human just no physical body and that explains us. We just have a body right now
Kelsey said "We only know the best arrangements [to pack spheres] in dimensions 2, 3, 8, and 24." Well, though it be trivial, we also know the best way to pack spheres in dimensions 0 and 1. A 0D sphere is a point and a 1D sphere marks a line segment.
Just discovered this show. Love it. Stage tip: wondering what to do with your hands? Best thing to do is....nothing. Let those hands drop to your sides.
Cool!! The way a hyper sphere as described here kinda makes me feel like our universe is one and we are stuck watching the 3D while the dimensional order of the universe goes up. Need to learn more! thank you for making all these videos!
I am going to echo the sentiment in the comments below, as I was already going to before reading them. My first thought was, "How did I miss this until now?" Now I see that this is new and right up there with Spacetime and Physics Girl. Space Time covers A lot of Astrophysics and science misc. Physics Girl covers the experimentalist side of thing. And now we have this. All of this work is immensely appreciated. These three and Numberphile are my favorite channels. If I count Numberphile as a special elective, can I get a "PBS in Math and Physics"? Hmm... that was a joke while I was writing it.
physics girl has some good content sometimes but her channel name and her way of presentation make me cringe.. i feel like it's a fluffy flowery channel that's more suitable for girls
I've never visualized hyper spheres, but I have tried to do the same for hyper cubes. I have made a 3^n model of a hyper cube where each dimension adds another •3 to the coordinate to find its hypervolume(?). I can only model it however as the closest I can think to see it is the perspective model and one other I don't remember the name of. All I can do is conceptualize lines within the polytope.
It's relatively easy to understand that higher dimensions are degrees of freedom. Thinking of dimensions as right angles could be a limitations. A higher dimension just encapsulates lower dimension, like sphere encapsulates circle. I am not sure why we focus on spheres, hexagonal shape has all neighbours clearly defined without gaps. Higher dimensions and their shapes could be defined well at 0 to 1 fraction level. At fraction level, behaviour of multiplication and division changes. Using slope calculation to find points on a line also may not work correctly and line may start to bend as we explore deeper in fraction level. Thus at fraction level higher dimensions could be more visible.
I like everything about this channel! From the brilliant content to the warm nature of her presentation. Annoyed at myself for not having it found earlier.
Imagine a baseball being thrown at your face, the ball gets bigger and bigger as time passes, you could think of the baseball as a 4 dimensional object where time is the 4th dimension. Think about this until it hits you :)
tl;dr: It's great to talk "about" the math, but it would be *far* better to actually *do* some math! While the details of higher-dimensional optimal sphere packing are understood by few without a math PhD (and, thus, 99.999% of RU-vid viewers, including me), it would be great if an accessible proof (or at least demonstration) for the 2D case could be presented, along with mentioning the math tools and techniques needed to work in higher dimensions. One missed opportunity may have been to directly calculate the space-filling percentage for a given (simple) packing (or, better yet, comparing a pair of simple n-D packings/arrangements). This should be simpler than finding or proving an optimal packing, yet still give a hands-on feel for working in higher dimensions. Just calculating the volume of an n-D sphere would probably have been worth doing (en.wikipedia.org/wiki/Volume_of_an_n-ball). If you connect the centers of a minimal (tile-able) example of an optimal sphere packing, what kind of geometrical object results? In 2D it is a triangle (2D -> 3 faces), in 3D a tetrahedron (3D -> 4 faces). What about in 8D and 24D? The "packing" *IS* the geometrical object described by the arrangement of the hypersphere centers, right? Why not make this explicit? Don't fear the math! While many of us may have forgotten most of the math we learned since turning 12, it would be great to have motivations (and pathways) to reclaim lost math skills and develop new ones. I suppose this really involves identifying your target audience. But don't aim low! I'd recommend a multi-level approach that "informs without alienating" so that the math-less and the math-ful can both enjoy the video. And all viewers can be inspired to become more math-ful. At the very least, we all should learn some new math vocabulary, even if we don't learn the math itself! Becoming "buzz-word compliant" is important: Even if I know nothing about, say, "SU(3)", if I see that text used in two different contexts, I should be able to infer that those contexts may be related (though perhaps only by using common math/geometry). That's useful! Vocabulary matters. Take a look at PBS Digital's Space Time for examples of how to handle "real" math at various levels of abstraction. Matt really has a knack for showing the "whole shebang", then taking it apart and showing how some parts have useful (and more accessible) approximations. But then, he gets to leave the math domain (for physics), and you don't! Numberphile is simply awesome in this area. Perhaps definitive. How can/should Infinite Series compare to and/or be different from Numberphile? Is Infinite Series a "math news" channel, or should it be more? Should it be limited to describing the exterior appearance of recent math developments, or should it try to bite into the sweet, juicy math interior? I'm not suggesting that Infinite Series have anything like formal, pedagogical math tutorials, but instead provide more along the lines of mathematical sketches, recipes and approximations that could be explored by those with sufficient curiosity and determination. Once an Infinite Series video reaches its intended run time, an ideal addition would be to include FMI links to additional learning resources. There's no need to reinvent any wheels. But there is a need to curate online math resources applicable to Infinite Series topics. PLEASE be sure to *always* include direct links to the original and fundamental papers involved (or full citations when the paper isn't freely available online). Some of us like to admire the pretty LaTeX math squiggles while reading the abstract, introduction and conclusions. Regarding the specific content of this video, there are MANY videos that provide very accessible visual explorations of n-dimensional geometry. Was there a need to repeat this well-worn path? Why talk about slicing/viewing higher-dimensional solids through lower dimensions? This felt like a fruitless detour to wind up at "hyperspheres are weird". I'd recommend focusing on issues directly related to getting a handle on hypersphere packing. I mean, if you want to do "An Intro to Hypergeometry" video, please do so! But that wasn't your stated topic for this video. There may have been an actual fail in this video: I believe it was a mistake to say that a higher-dimensional n-sphere "bursts through" the enclosing n-cube. It may have been much more useful and far more illustrative to say that the "completely contained" sphere gets smaller in higher dimensions! This would *directly* lead to a great illustration of *why* packings in ever higher dimensions fill a lower percentage of the available volume. While it was correct to say that optimal packings are known for only 2, 3, 8 and 24D, it is also important to generalize beyond this to say that very efficient packings have been identified in hundreds (?) of dimensions, but that proofs of optimality have been developed for only these 4 specific ones. The best-known packings for other dimensions can often state how closely they approach the theoretical optimum when the optimal packing and proof are not yet known. Infinite Series has a great mission statement: "Infinite Series [is] a show that tackles the mysteries and the joy of mathematics. From Logic to Calculus, from Probability to Projective Geometry, Infinite Series both entertains and challenges its viewers to take their math game to the next level." Please provide concrete steps along the path to that "next level". Looking forward to the next episode!
Just had a way-too-cute idea: Short videos narrowly focused on describing/explaining the fundamentals of a particular math domain could be called the "Terms of the Infinite Series"! Get it? Cool, amiright? Each "Terms" video could start with the applicable vocabulary ("terms" or terminology), then go on to illustrate what it means (the math) and how it is used (including both low- and high-level worked examples). "Terms" videos would remove some of the explanatory burden from the main Infinite Series videos, permitting them to reference the appropriate "Terms" videos without having to recapitulate them. However, "Terms" videos would only be needed when specific high-quality domain-specific videos don't already exist on RU-vid. Always reference those when available. Stand on the shoulders of (RU-vid) giants. Don't reinvent the (domain video) wheel. Create new "Terms" videos only as a last, necessary resort.
Lot of good points there BobC. ... I imagine it's not as hard to show the minimal void shape that results from packing tangent circles with their centers on a triangle, (or spheres centered on tetrahedron vertices), since the sphere radius defines the minimum concave curvature of the void space. However, can the resulting polygon fully tile space? Equilateral triangles *can* fully tile 2D space. However, tetrahedrons *can't* fully tile 3D space. So it would seem the solution and proof for 2D is much more obvious than for 3D. I suppose this is why the proof for higher-dimensions is a more challenging problem than it would at first appear.
Nice! Thanks for all the thoughtful comments and feedback. I totally hear what you're saying about "talking about" versus "doing" math, and, to the extent possible, we're going to *do* tons of cool math! Sphere packing is tough business, even in two and three dimensions. Here's the links to Viazovska's recent proofs: arxiv.org/abs/1603.04246 arxiv.org/abs/1603.06518 There's also two articles linked to in the description that provide great, readable descriptions of her phenomenal work. I also wanted to point you to a reference about the central sphere that "bursts through" the box. I had heard about it recently from a friend, but I just found this great column that also references it: www.americanscientist.org/libraries/documents/201110101628308738-2011-11CompSciHayes.pdf. The column mentions a ton of those cool facts about hyperspheres that I thought you might enjoy. I'll be curious to read your comments next week!
one can actual proof quite easy the volume goes to zero for the ball, when n approaches infinity, thats why the n=30 case has this weird example. probably need to try to read the original paper :)
Yeah, I wrote my reply quickly, relying more on intuition then actuals. I thought things didn't blow up until after 3D, but it turns out 3D is also tough. I've only encountered the general nSphere or nCube packing issue when looking at efficient and robust nD data encodings for transmission protocols. I really don't know much at all about the underlying geometry, but would love to learn.
PBS is rockin the joint lately wow. I can't stop watching all their science videos. Incredibly stimulating. On a side note, she has incredible skin! I think she's an alien pretending to be human, "haha I'm gonna go star in one of their primitive educational videos, it's gonna be hillllllllarious guys"
@@bodhisattva9762 I was joking. However, a proper dose of DMT happens in a mental environment with more than 3 degrees of freedom. A high enough dose of Salvia happens in an environment with less than 3 degrees of freedom. It is literally impossible, regardless of how much one might have taken of any other hallucinogen, to imagine such an experience. Had you, you'd get the joke.
@@bodhisattva9762 hallucinations or not, what you see is geometric configurations of increasing complexity by dose, in both ordered and chaotic configurations. this video explains it well ru-vid.com/video/%D0%B2%D0%B8%D0%B4%D0%B5%D0%BE-loCBvaj4eSg.html I have experience with LSD personally, and with it I experienced at the least the impression of 4 dimensional visual perception of my surroundings as well as.. an extremely altered perception of time. When I say 4 dimensional visual perception this is what I mean: the way you typically see with a 3d eye is by painting a three dimensional world across a two dimensional canvas (the retina), so it follows that a 4d eye would operate by painting a 4d world across a 3d canvas, yielding the effect of a three dimensional visual field, where you would view not flat pictures but more like a hologram of all visual information within your line of sight oriented relative to the focus of your gaze. You could see more of any 3d object you focus on than the face that you look at - think of how when you look at the face of an object you are able to see its entirety of its 2 dimensional front & depending on the orientation you may or may not be able to see its depth too, so too would a 4d eye see a whole 3d object (if the entirety is within line of sight) and also depending on orientation its 4 dimensionality. As for the perception of time, well you develop a real emotional understanding of "Eternity". Past, present, future, possibility, certainty, eventuality, all IS.
as a person who is enthused by body language, i must ask you to put your hands down! Your hand gestures are the equivalent of saying "yeah, um, like so... yeah". That being said, this is great work. I subscribed ;)
I visualize a hypersphere like I visualize the afterlife: I know it's probably there, I have no idea what it actually looks like, and it is absolutely terrifying to comprehend.
You can easily talk about, for example, 7-dimensional space by saying each point can be in 3-dimensional space, with a symbol, size, color, and rotation. It's an easy way to disentangle mathematical dimensions from physical dimensions. So, you could describe a 7-dimensional point as (10, 4, -3, "A", 20mm, red, 30°). And you can visually put it on a chart along with other symbols to reinforce the idea. You don't have to get abstract so soon; it's a tangible way to get someone's head around higher dimensions before launching into the harder math.
Disentanglement is the wrong way. The quantummathematical principia behind it is conform with our reality. It just need 4 dimensions to solve the problem. You need a proof? Have fun: oi68.tinypic.com/28lfac4.jpg *This picture is copyrighted and Iam the respective owner! The picture can just be used for educative purposes, but before you have to contact me, the owner!* Quaternions are the mathematical tool which you need to describe this.
@@G4mm4G0bl1n This is nonsense. That picture was copied by every machine between the original site and my phone, including every internet router between them. I'm only the last in a long chain of copies. You have no copyright over that.
@@Arcsecant But you arent the owner and the picture is digital marked to my ownage! Copyright has different models of licensing. You have the permission to copy the file to your hard disc to see it by yourself and you can also use it for educational purposes, but you have no permition to change or reuploading the file in any way! I also got the mathematical definition for the vectors, which just a few people can calculate. Dont try to be an smartass, because you arent one.
@@G4mm4G0bl1n You can't "own" a picture. You were the guy who made it in the first place, but my copying it doesn't deprive you of anything. It's like taking a photograph of a painting. It's my photograph! This is what it means to live in a free country. This isn't Russia.
@@Arcsecant I have "drawed" and "developed" this picture and this makes me the copyright holder! No, its a drawed picture like the mona lisa and I allow you to make a photo. So you just have a copy in another format of the original, because the original is not a JPG file. So I got the canvas handpainted version and you got just a crap photography. The file you have downloaded has also a digital mark and I can exactly find my images with my signature tough the www. :) This is the internet and the posted imagefile is *my intellectuell property!* What I can see is that people becomes jealous about my work, because their are too dumb to understand how the nature of determinism works! Im also not a russian, Im a german like the originators which discovered the basics of Quantumphysics, but the real pioneer in Quantumphysics was Nikola Tesla. Read more about his math and how he invented his patents with the numbers he found. ;) www.intmath.com/blog/wp-content/images/2016/06/tesla-map-to-multiplication.jpg *P.S* - Dont waste so much time with Overwatch and Fortnite. This will make your brain slushy! Try to solve this calcul: *cosd(6×10^995)* (cosd = Cosine Degree) If you arent able to solve it with your own knowledge, try to use your computer or calculator. :) If you cant get the solution then be sure that my posted picture is copyrighted!
For three dimensional unit sphere. It is the set of all points for which x^2+y^2+z^2=1 holds. Similarly 4d hypersphere is the set of all points for which x^2+y^2+z^2+w^2=1 holds. We can visualise the unit hypersphere by plotting x^2+y^2+z^2=1-w^2. Basically it will be a 3d animation showing the plot for x,y and z as we vary the values of w. Turns out that it's just the set of all spheres centred around the original with radii between [0,1] for values of w in range of [1,0].
Incorrect. We will define a perfectly symmetric shape with minimal space. Call it SHOCKING In 1 dimensional space, it is 1 point, called S(1) In 2D space, it is a circle, called S(2) In 3D space, it is a sphere, called S(3) In 4-dimensional space, it is 1 %@#$#, called S(4) In one-dimensional space, we replace a point into a segment D S(2,D) = Pi.D^2/4 (Circle) S(3,D) = Pi.D^3/6 (sphere) S(4,D) = Pi.D^4/8 = Pi.D^3.D.6/(6.8) = (3/4.D).Pi.D^3/6 S(n,D) = Pi.D^n/(2.n) = (Pi.D^3/6).D^n-3.6/(2.n) = (3/n).D^n-3.Pi.D^3/6 So: %@#$# contains (3/4)xD x sphere with a diameter of 1 D Good luck!😁
I saw this somewhere, but basically you can draw any dimension of a cube onto a 2D sheet of paper. You won't know how it behaves or moves, but you will begin to understand at least 1 of its possible shapes. It works like this: *1D:* Draw a straight line *(Line)* *2D:* Draw another parallel straight line and connect the ends *(Square)* *3D:* Draw another square but offset towards one corner, then connect the corners of the 2 squares *(Cube)* *4D:* Draw another cube offset to one corner of the original and connect the corners of the two cubes *(Hypercube)* *5D:* Draw another hypercube and connect the corners of the two hypercubes I think you get it by now. Basically, you just draw the same thing and connect the corners for as long as you want to get the next dimension. As said before, you wont know how the shape moves or behaves, but you will know at least one of its possible shapes.
Basically. Useless information that does not relate towards reality. Just a means to compare 2D to 3D plus mix them around. At least that's what I gathered since too boring to watch whole vid of meaningless spheres.
@ Metal Gear, shame you did not have enough patience to watch for a single minute. Because, in less than that, it was explained that it does have real world applications for computers, cellphones and the internet. Even a tuna has a longer attention span than that!
Awesome video! I used circle and sphere packing with variable radii and Poisson jittering to try and optimally sample the surface of a spherical planet when distributing objects like trees, mountains, rocks, etc. Since I started with the larger objects and proceeded to fill the gaps with the smaller objects, I managed an average 72% efficiency. The Poisson distribution resulted in a nice, 'natural' distribution, free of the grid-like tendencies of our previous system.
You can't take 3 apples from 2, but with the insertion of a concept, math makes it real. Such, it is. What is time beyond thought? Math is an amazing tool that has been turned into a belief system in science, as if math is the reason, when it merely perfectly describes. It doesn't even have to directly address causality to describe and predict perfectly.
I'm a little surprised that you didn't use other properties to represent higher dimensions, such as color or temperature. so, a 4d sphere would be like a solid 3d sphere, but when you cut into it, the inside would be more and more red (for instance)
Romaji hello, i think this way is really promising and i have been thinking on it for several years now when time was here to dwindle on this mathematical theme of hypergeometry. And i find it quite efficient to generalize the common spheres to 3d and 4d, with a sort of algorithm or procedure of construction. Actually, 1d, 2d spheres are references to test that procedure and then apply it to 3d spheres and on! basically a 3d sphere would equate to a continuous sequence of 2d spheres that intersect all on a 1d sphere (circle) like a 2d sphere can be considered as a collection of 1d spheres that intersect on a 0d sphere (2 points at diameter distance from each other), which is the common "longitudinal view" of a 2d sphere 😃. This may look completely weird as what one imagines in one's mind cannot easily be captured by another imagining mind 😅😅😅
i think it is mathematically named Hopf fibration, see en.m.wikipedia.org/wiki/Hopf_fibration 😆! the continuous sequence could or would be parameterized with complex numbers to denote the analog of an (hyper)angle with which the 2d spheres are placed to form the 3d sphere
I use this method for some visualization. But it is important to note that this only allows visualizing a set of 4d points that is a function - you only get one color per 3d point. However a sphere is not a function, it has a top and bottom, so two values/colors for most coordinates. The other problem is occlusion: our eyes only see the outside of 3d objects - a 2d projection. This can be somewhat overcome with translucency - but that makes the image harder to understand.
Well it should be possible for a being in 'n' dimension to draw a 'n-1' dimensional image. A 4th dimensional creature can draw the shadow of a 4D object in 3D. We have done that to but that shadow doesn't help us visualize what the actual 4D object is.
The worms that live in line-land can only see 0-dimensional points as flat-landers pass through their realm. Flat-landers on a plane can only see points, and lines as solids pass through their domain. So our hyper-buddies are like X-ray machines when staring down at us. I like explaining higher dimensions through drawing perpendicular axes. We can barely see the 3rd z-axis in our minds on a 2-d chalkboard because the z-axis is written as a perpendicular slant to the x and y axes. Plotting the points of higher objects is tough or impossible because of our limited 2-D vision. Some imagine time as the 4th dimension. Like, a series of 3-D spaces moving through a fourth degree of freedom. maybe, I don't know but it is cool. Then 5-D are sections of 4-D spaces moving in another degree of freedom. Some would say alternate 4-D timelines? Some neat stuff!
correction the y axis is perpendicular to x axis for 2 dimensions the z axis is perpendicular on both x and y for 3 dimensions and for higher dimensions the same rule applies
That is not necessary. The axis don't need to be perpendicular, just linearly independent. If the axis aren't linearly independent the generated space will have less dimensions than the amount of axis.
I'm sorry, but, wot? Perhaps there exist "hyper-dimensional" beings that are laughing their ass off, because we are going "WTF" with something that they perceive as natural as we perceive three-dimensional objects as natural. Fascinating stuff.
But perhaps these hyper dimensional beings are rather completely unaware of our existence, no even our possibility as they would have no frame of reference to how 3 dimensions would be perceived or could even have life, just as we view the impossibility of the 2nd dimension due to our 3D viewpoint.
Humanoids are primitive that's why there are so many wars. We fight over everything. I don't know why they don't to just to that planet with ocean of natural gas and just live there.. lol
*Probably easiest way to imagine 4D object:* think that there is a slider, and ball next to it. As you move the slider, the ball morphs slowly into a box and back. The slider just moved object in 4D space!
Dat Epic Fish I always found this to be the easiest way as well. Particularly useful for visualizing higher-dimensional solids like the simplex (4D tetrahedron) and understanding why their properties still apply!
This is dope. But would be nice if she would have showed us sphere stacking problems and solved one, if only vaguely. What does the math look like. Show some numbers!
As a physicist, the way I envision higher dimensions is by first considering the 3 dimensional case. Once I understand that, I assume I understand higher dimensional cases and continue on with equations.
How to confuse and embarrass Geometry Professors THE NATURAL CIRCLE AND ITS SQUARE ABSOLUTE INCONTROVERTIBLE SIMPLE ARITHMETIC Given a "Diametric Distance" of 120-centimetres. 1. Multiply the 120-centimetre Diametric Distance by 3. 2. The length of distance to the Circle's Edge is 360-centimetres. 3. The length of distance to the Circle's Edge is 360 degrees. 4. Each degree of distance to the Circle's Edge is 1-centimetre in length. Squaring the Circle 5. Multiply the 120-centimetre “Diametric Distance” by 4, the perimeter length of the Circles Square is 480-centimetres. 6. The Circle is 360-centimetres and 360 Degrees in length, which is three-quarters of the length to its 480-centimetre perimeter Square. Simply Three times the length of…A Line…is the length of the lines Circle. Four times the length of… A Line…is the length of the lines Square. Questions 1. When we look at the shape of a bright yellow full moon as it is being silhouetted against the dark background of the night sky, does the full moon have a circumference - circumferential outline? Answer No, it does not; the full moon is a yellow coloured round circular area of shape; which is being contrasted against the greater surrounding area, of the darkness of the night sky. to produce a round silhouetted circular shape that does not possess an outline. 2. If we take a black marker pen and draw a black circle at the centre of a sheet of yellow A4 paper, does the yellow round circular shape in the middle of the paper have a circumference - outline? Answer No, it does not; the yellow round circular shape of area in the middle of the paper is being contrasted against the surrounding area of blackness belonging to the circumferential thickness of another circumventing black circular shape. And the circumferential thickness of the area to the black circular shape is its turn is being contrasted against the lighter background of the rest of the yellow A4 paper. Question 3. When we look at a tree in the brightness of daylight, does the shape of the tree possess an outline? Answer No, it does not; the darker area belonging to the shape of the tree is being contrasted against the greater surrounding area, of the brightness of daylight and the blueness of the sky. Simply Shapes are not geometric; they are the visual forms of things that exist in nature, which are made visually manifest by the presence of a surrounding and contrasting background. And the surrounding and contrasting backgrounds are made visually manifest according to six aspects of visibility; shades of darkness, shades of brightness, shades over distance, shades of perspective, shades of colour, shades of texture. In nature as opposed to Euclidean applied geometry and mathematics in physics, there is no such thing as a circumference outline or a line. Sidebar All things in nature are comprised of primal electromagnetic particles, larger particles, and larger groups of sub-atomic particles which form atoms and all of these particles *invisibly coexist at the quantum level*, in a perpetual state of interactive motion. In order for the particles and the atoms at the quantum level to be able to manifest at the molecular level of visible structures, there has to be a vastly larger gravitational body present, which first draws them into its gravitational field; and then gravitationally compresses and **aligns the atoms together interactively** to form solid molecular structures. At the level of our 20-20 vision molecular structures (e.g. elemental crystals and solid bodies) do appear to possess straight linear aspects to their structures, however as any electron microscope will confirm appearances, are deceptive. However, we do not need an electron microscope to confirm that this is the nature of all things, all we need to do is look out into the night sky toward the constellations of the stars. And there, although we see what appear to be the stars formed into linear shapes and patterns, there are no actual lines between them, for it is we are who are responsible for aligning them in the imagination of our minds- eye. Concerning two-millennia of disingenuous Grecian-Roman Euclidean education (brainwashing). Quote: Stuart Close For those who believe no proof is necessary, for those who do not believe no proof is possible (You can take a horse to water, but you cannot make it drink). Reality Versus Fiction The genius of stupidity is that the stupid are too stupid to realise, that they are too stupid to be the geniuses; they stupidly assume themselves to be. The genius of intelligence is when the intelligent reach a point whereby they are so humbled in the face of the awe-inspiring intelligence of our Cosmic Mother Nature, as to realise. There is no such thing as to any one of us being a genius, for a tendency toward genius, lies-only within the realms of the ingenuity and the genius of our Cosmic Mother Nature. www.fromthecircletothesphere.net www.geometry-mass-space-time-.com
CURVED GEOMETRY SIMPLIFIED FOR EVERYONE PARENTS - TEACHERS - ARTS - CRAFTS - TRADES - CONSTRUCTION - TECHNICAL - PROFESSIONS - SCIENCES THE NATURAL CIRCLE AND ITS SQUARE Given a "Diameter Distance" of 120-centimetres. 1. Multiply the 120-centimetre Diameter Distance by 3. 2. The length of Distance to the length of the Circle's Circuit is 360-centimetres. 3. The length of Distance to the length of the Circle's Circuit is 360-degrees. 4. Each degree of Distance to the length of the Circle's Circuit is 1-centimetre in length. Squaring the Circle 5. Multiply the 120-centimetre Diametric Distance by 4, the Perimeter Length of the Circles Square is 480-centimetres. 6. The Circle is both 360-centimetres & 360 Degrees in length, which is three-quarters of the length to the Circles 480-centimetres perimeter square. Simply Three times the length of…A Line…is the length of the lines Circle. Four times the length of… A Line…is the length of the lines Square. THREE TIMES THE RADIUS SQUARED Using a 120-centimeter diameter (Diameter Distance) multiply the diameter by 120, this will yield the sum of 14, 400 square centimeters to the square of the diameter. Using the radius (Radius Distance) of the diameter of 60-centimeters multiply the radius by 60, this will yield the sum of 3,600 square centimeters to the square of the radius. Multiply the 3,600 square centimeters square of the radius by 3, this will yield the sum of 10,800 square centimeters to the area of the circle, which is three-quarters of the 14,800 square centimeters of the square of the circle’s diameter. Reader Self Evidence? Given that you have taken your time, and have read and understood all of the above, could I or anyone now convince you that a circle is not three times its diameter length? If not, and you are convinced by the proof I have provided, that the length of a circle is three times its diameter length. Then you may ask yourself and others, why it is? Despite this knowledge having been made available internationally on my website www.geometry-mass-space-time.com for the last eight years. And my having provided an open challenge to the international world of academia as a whole, or for anyone or any group to disprove the simple arithmetic I have provided. And additionally, despite the website having received more than 400,000 hits from worldwide, and this knowledge having also been broadly posted in TED Talks, on RU-vid, to Magazines and otherwise communicated to many hundreds of persons of academia via the www, internet, and snail mail: Not one the millions of so-called academics belonging to the disingenuous Grecian-Roman founded Old Boy - Big Brother - Ruling Classes University Establishments. Has stepped forward to take up the challenge, or has had the guts to admit as to the disingenuous nature of their Euclidean, Theological and Theoretical based false teachings, or provide any transparency as to their ultimate intentions in doing so. www.fromthecircletothesphere.net INTRODUCTION On reading the website content list on the drop-down menu, you might think "whoa this stuff is way over my head" I assure you it is not. If you (and obviously as you are reading this page, you do) have the four basic skills of being able to add, subtract, multiply, and to divide numbers, then none of the elementary arithmetic that follows, will be beyond your mathematical abilities or comprehension, and as such will prove to be self evidently true. To begin, and for the sake of credibility regarding the subject matter, we should first refer to some recent discoveries made in the Fields of Archaeology. Reference: The Guardian Aug 24, 2017: Sumerian Trigonometry Tablet Discovery Mathematical secrets of ancient tablet unlocked after nearly a century ...www.theguardian.com › Science › Mathematics Reference: Ancient Babylonians Used Geometry To Track Jupiter Thousands Of ...www.iflscience.com/space/babylonian-astronomers-used-geometry-study-sky/ It is historical fact, as this and other discoveries made in the fields of archaeology continue to confirm, that over a period extending back in time to more than four thousand years ago, that it was the ancient civilization of Sumeria who was the progenitors of The first alphabet, of writing, of mathematics, of geometry, of differential geometry, of architecture, of engineering, of astrophysics, of clocks, and 3,600 seconds per hour, to the 360-degree twenty-four hour day. And far toward the opposite extreme as to the amoral narcissistic and hedonistic Greeks having been the progenitors of civilization, democracy and just about everything else (that could not be nailed down) as well. It was the intellectually challenged Greek armies of the barbarian Alexander of Macedonia, and their later allies the Romans, who were responsible for the destruction and the loss of thousands of years of knowledge, and of delivering the greatest blow to civilization and human progress ever known. To the point, as the newly discovered clay tablets of Sumeria serve to prove, and despite all of the disingenuous tripe and propaganda, which the Western Grecian-Roman (Capitalistic - Fascistic) Universities have continued to dish up over the last two millennia. Euclid was not the father of geometry, and Archimedes was not the eureka genius he has been made out to be. And the exemplary proof of this is, that despite my not being a Grecian-Roman University (Old-Boy Approved) Euclidean taught geometer or mathematician, and as such a victim of their disingenuous teachings. Unlike them and all of their students for more than two thousand years, who have barely been able to approximate the length to a simple circle. I have here on this homepage, by use of no more than self-evident - irrefutable - simple arithmetic which all of you can understand, I have published e.g. A. Four methods for finding the exact area of a circle. B. The method for finding the exact areas of rings. C. The twelve-step method for calculating the exact surface areas and volumes of spheres and ovoids. D. The Number of two-dimensional degrees to the surface area of a sphere. E. The number of three-dimensional degrees to the surface area of a sphere. And as such it should also be noted: That given that the simple arithmetic I have used is self-evidently true and irrefutable, so it logically and rationally follows that all of this work is solely my intellectual property, and subject to my copyright, and any dispensations I may choose to make regarding that copyright. Reference: How many humans have lived in the past 2013 years? - Quorawww.quora.com/How-many-humans-have-lived-in-the-past-2013-years And thanks to Euclid and Archimedes, to this very day, not one of any of the millions of Geometer's worldwide is able to carry out the two simple sums that would have been known to any Sumerian child four-thousand years ago, which are that 3 x a straight line is a circle, 4 x a straight line is a square. DRAWN LINES A drawn line is a visually apparent length of distance, that has been artificially made apparent, by applying an overlaying and contrasting "linear - area" of shade, color, or texture, to the distance between any two given points on the drawing surface. Or to "out - line", and so "enclose an amount of surface area", in the form of a shape, by applying an overlaying and contrasting "linear - area" of shade, colour, or texture, to sub-divide the surface area, into "three distinct parts of the drawing surface. Questions 1. When we look at the shape of a bright yellow full moon as it is being silhouetted against the dark background of the night sky, does the full moon have a circumference - circumferential outline? Answer No, it does not; the full moon is a yellow colored round circular area of shape; which is being contrasted against the greater surrounding area, of the darkness of the night sky. to produce a round silhouetted circular shape that does not possess an outline. 1. If we take a black marker pen and draw a black circle at the center of a sheet of yellow A4 paper, does the yellow round circular shape in the middle of the paper have a circumference - outline? Answer No, it does not; the yellow round circular area of shape in the middle of the paper is being contrasted against the surrounding area of blackness belonging to the circumferential thickness of another circumventing black circular shape. And the circumferential thickness of the area to the black circular shape is in its turn is being contrasted against the lighter background of the rest of the yellow A4 paper. Question 1. When we look at a tree in the brightness of daylight, does the shape of the tree possess an outline? Answer No, it does not; the darker area belonging to the shape of the tree is being contrasted against the greater surrounding area, of the brightness of daylight and the blueness of the sky. Simply Shapes are not geometric; they are the visual forms of things that exist in nature, which are made visually manifest by the presence of a surrounding and contrasting background. And contrasting backgrounds are made visually manifest according to six aspects of lighting and visibility; shades of darkness, shades of brightness, shades over distance, shades of perspective, shades of color, shades of texture. www.fromthecircletothesphere.net www.geometry-mass-space-time.com
We can represent this in the spherical coordinate system. Sphere is 2 dimentional, azimut and inclementation, while circle one dimensional - angle. 3rd dimension of sphere can be radius/xoffset/yOffset/zOffset out of euclidian center of 3D coordinates. The analogy of 4D in this video is the radius way. This is the most easiest way to think in 6 dimentions.
I visualise a hypersphere as a planet with an atmosphere, slowly cycling from the inside out, and back in with geological, hydrological, carbon and other cycles. So we seem to live in a relative Flatland (at least until space exploration) on the surface of a slow-cycling hypersphere that is relatively smoother than the surface of a billiard ball.
I was trying to laterally connect practical application for all of this and wasn't having much luck. I could just be too ignorant at this current juncture but I think this is just having fun with math. The universe doesn't have a need for math or does it live in Dimensions. Everything exists all at once and everything is in constant flux and motion. It's all happening in one dimension from a perspective of the universe. Everything is the same thing. Of course the universe isn't receiving much of anything and the universe is not out there with a whiteboard inventing languages to describe itself.... Unless you consider us the universe doing those tasks but of course we are not likely to be the most complex forms of consciousness.. as if the majority of our forms are even fully conscious in the first place?
A Photon has no valence shells and no electrons inside. Quantumentanglement & Quantuminformationtechnics works with Bosons. en.wikipedia.org/wiki/Boson
