High-Dimensional Spaces Are Counterintuitive, Part Onehttp://blogs.msdn.com/b/ericlippert/archive/2005/05/11/high-dimensional-spaces-are-counterintuitive-part-one.aspxA friend of mine over in Microsoft Research pointed out to me the other day that high-dimensional spaces are really counterintuitive. He'd just attended a lecture by the research guys who wrote this excellent paper and we were geeking out at a party abouten-USTelligent Evolution Platform Developer Build (Build: 5.6.50428.7875)re: High-Dimensional Spaces Are Counterintuitive, Part Onehttp://blogs.msdn.com/b/ericlippert/archive/2005/05/11/high-dimensional-spaces-are-counterintuitive-part-one.aspx#9620537Sat, 16 May 2009 04:17:12 GMT91d46819-8472-40ad-a661-2c78acb4018c:9620537Carlos G.<p>Well it CAN be defined both ways although I normally use the topologist definition. But you are right, it depends on how you view it. That is what makes the hypersphere group so special (you have probably heard of the circle as "the perfect shape") from the rest. The reason why I use the topologist definition is the following: </p>
<p>Picture two-dimensional beings living in a two-dimensional universe. Now according to geometer's definition this would mean in a circle. According to the geometer's definition, this would mean they live in a circle. But how can two-dimensional beings live in a one-dimensional bent line? So let's take the topologists definition. If we use it then they would be living in a two-dimensional surface of a sphere (picture it as a beachball with ants on it). </p>
<p>This matches as an analogy with out universe, which is said to be bent on itself in a four dimensional way. Even if the ants go on and on around the beachball thinking they are going in a "straight" line, they are actually going AROUND the curved surface of the beach ball. So when we move in a "straight" line through our 3-dimensional space, we are actually "hypercurving" on the fourth dimension because our universe IS curved. </p>
<p>Now if the "hypercurved" universe theory is correct, Wormholes can be easily explained. Perhaps you think, how can we possibly go to another distant region of the universe through a shorter distance? The answer becomes more obvious using a lower-dimensional analogy. Let's go back to the ants on the beach ball. But now imagine the beach ball has a tube which opens holes on two opposite ends of the ball. </p>
<p>Now if the ant wants to go from Point A (near the first hole) to Point B (near the second hole) it would have to go around the entire half of the beach ball to get there. But in the presence of the Wormhole (which would be best described as a hollow cylinder in that two-dimensional universe), it could just go through its inside and get to the other side covering less distance. </p>
<p>Other phenomena can be explained with this analogy too. Picture Blackholes as this cylinder Wormhole but instead of continueing to another region of that "universe" it ends in a pinched end (called like in our universe a theoretical "singularity" as it ends in a point with no dimensions). Now if the ant where to inevitably get caught into that black hole, it would go down and down until it was squashed by the deep endpoint. </p>
<p>So when we travel through these fluctuations on the surface of the two-dimensional sphere universe we move through a higher dimension. The subatomic quantum fluctuations themselves are four-dimensional in our universe, warping and stretching (which is why it is called quantum "foam"). Another amazing property of the sphere group is this: </p>
<p>Take that two-dimensional sphere and put squash it onto a flat surface (or like taking a picture of it). Now it will have spatially colapsed from a two-dimensional sphere into a one-dimensional one (also called a circle). Now do it again this time, but this time push this "disk" by the side in order to form a line. Now we all of course know that a line has one dimensions, but it then we notice it has decreased dimensionally one more level, so it really is a zero-dimensional line. </p>
<p>This means that a zero-dimensional sphere is what would be a one-dimensional cube. Another way to define this strange zero-dimensional "line" is like this: A two dimensional-sphere is the collection of points in a three-dimensional SPACE at the same distance of a chosen center. So (after skipping the one-dimensional sphere) we go to the zero-dimensional sphere, which is the collection of points in a one-dimensional space. </p>
<p>So this means that the sphere is in a linear universe, and so spreads out at the same distance from a center, in other words a line. Topology, although sometimes a complex subject, is I believe the next step of geometry. It defines laws that govern the manifolds of higher-dimensional orders, helping us comprehend more profoundly about our own universe. </p>
<p>Circles/spheres are so much different from the polygons, since they define a point with a certain line with one endpoint that "sweeps" out at all the available dimensions. I recommend reading "THE ELEGANT UNIVERSE" by BRIAN GREENE explaining about supersymmetrical strings, hidden dimensions inside our universe, and the search for the so called "Theory of Everything". It's great when you learn the stuff it will tell you. </p>
<div style="clear:both;"></div><img src="http://blogs.msdn.com/aggbug.aspx?PostID=9620537" width="1" height="1">re: High-Dimensional Spaces Are Counterintuitive, Part Onehttp://blogs.msdn.com/b/ericlippert/archive/2005/05/11/high-dimensional-spaces-are-counterintuitive-part-one.aspx#436306Thu, 07 Jul 2005 04:46:05 GMT91d46819-8472-40ad-a661-2c78acb4018c:436306Eric LippertI suspect that your avered hatred of nitpicking is merely a rhetorical device.
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<br>If you're going to nitpick though, you really ought to make sure that you are correct before you indulge.
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<br>As you are well aware, topologists define "n-sphere" WRT the dimensionality of the surface. What you are apparently unaware of is the unfortunate fact that geometers define "n-sphere" WRT the dimensionality of the space in which the sphere is embedded.
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<br>As this is a series of articles about the geometry, not the topology, of spheres -- no self-respecting topologist would talk about the volume of a sphere! :) -- I deliberately chose to use the geometers' definition.
<br><div style="clear:both;"></div><img src="http://blogs.msdn.com/aggbug.aspx?PostID=436306" width="1" height="1">re: High-Dimensional Spaces Are Counterintuitive, Part Onehttp://blogs.msdn.com/b/ericlippert/archive/2005/05/11/high-dimensional-spaces-are-counterintuitive-part-one.aspx#436164Wed, 06 Jul 2005 21:03:10 GMT91d46819-8472-40ad-a661-2c78acb4018c:436164Adam GI hate to nitpick, but a circle is a 1-sphere (it's just a bent 1-dimensional line), a "normal" sphere is a 2-sphere (it's just a bent plane), and so on...<div style="clear:both;"></div><img src="http://blogs.msdn.com/aggbug.aspx?PostID=436164" width="1" height="1">How does a four-dimensional world look like?http://blogs.msdn.com/b/ericlippert/archive/2005/05/11/high-dimensional-spaces-are-counterintuitive-part-one.aspx#417713Mon, 16 May 2005 07:04:34 GMT91d46819-8472-40ad-a661-2c78acb4018c:417713AntimailEric Lippert is running an eye-catcher series on High Dimensional spaces (see here part one and two)....<div style="clear:both;"></div><img src="http://blogs.msdn.com/aggbug.aspx?PostID=417713" width="1" height="1">re: High-Dimensional Spaces Are Counterintuitive, Part Onehttp://blogs.msdn.com/b/ericlippert/archive/2005/05/11/high-dimensional-spaces-are-counterintuitive-part-one.aspx#417034Fri, 13 May 2005 03:13:50 GMT91d46819-8472-40ad-a661-2c78acb4018c:417034Eric LippertAh, that's a good one. There are many classic math vs physics jokes. That's a good blog topic...<div style="clear:both;"></div><img src="http://blogs.msdn.com/aggbug.aspx?PostID=417034" width="1" height="1">re: High-Dimensional Spaces Are Counterintuitive, Part Onehttp://blogs.msdn.com/b/ericlippert/archive/2005/05/11/high-dimensional-spaces-are-counterintuitive-part-one.aspx#417021Fri, 13 May 2005 02:31:39 GMT91d46819-8472-40ad-a661-2c78acb4018c:417021Bryce KerleyI always end up thinking about "â€”And He Built a Crooked House" by Heinlein: <a rel="nofollow" target="_new" href="http://www.scifi.com/scifiction/classics/classics_archive/heinlein/heinlein1.html">http://www.scifi.com/scifiction/classics/classics_archive/heinlein/heinlein1.html</a><div style="clear:both;"></div><img src="http://blogs.msdn.com/aggbug.aspx?PostID=417021" width="1" height="1">re: High-Dimensional Spaces Are Counterintuitive, Part Onehttp://blogs.msdn.com/b/ericlippert/archive/2005/05/11/high-dimensional-spaces-are-counterintuitive-part-one.aspx#416931Thu, 12 May 2005 20:57:40 GMT91d46819-8472-40ad-a661-2c78acb4018c:416931BrianThere's a joke about this:
<br>
<br>Two physicists who have just attended a lecture on superstrings are talking about imagining high-dimension systems. The first one says, "Four dimensions doesn't give me any problem, but I'm shaky with five, and I can't really visualize six-dimensional space at all."
<br>The second one admits that he can't go past 6 either.
<br>They stop a passing mathematician, and ask him if he can visualize 9-dimensional space (the subject of the lecture). "Sure," he says, "no problem."
<br>The physiscists are astonished. "How!"
<br>"I just imagine n-dimensional space and set n=9."<div style="clear:both;"></div><img src="http://blogs.msdn.com/aggbug.aspx?PostID=416931" width="1" height="1">re: High-Dimensional Spaces Are Counterintuitive, Part Onehttp://blogs.msdn.com/b/ericlippert/archive/2005/05/11/high-dimensional-spaces-are-counterintuitive-part-one.aspx#416919Thu, 12 May 2005 19:54:13 GMT91d46819-8472-40ad-a661-2c78acb4018c:416919Eric LippertIndeed, I have a small collection of homages to Flatland -- Burger's "Sphereland", Stewart's "Flatterland" (which I did not enjoy nearly as much as his other books), and my favourite, Dewdney's "The Planiverse".
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<br>Flatland is a classic, and at a buck-fifty for the Dover Thrift edition, it should be in every math geek's personal library.<div style="clear:both;"></div><img src="http://blogs.msdn.com/aggbug.aspx?PostID=416919" width="1" height="1">re: High-Dimensional Spaces Are Counterintuitive, Part Onehttp://blogs.msdn.com/b/ericlippert/archive/2005/05/11/high-dimensional-spaces-are-counterintuitive-part-one.aspx#416772Thu, 12 May 2005 09:15:28 GMT91d46819-8472-40ad-a661-2c78acb4018c:416772Stuart DootsonHave you read Edwin Abbot's 'Flatland'? (see <a rel="nofollow" target="_new" href="http://www.alcyone.com/max/lit/flatland/">http://www.alcyone.com/max/lit/flatland/</a>) - that is based around the same problem, except from the viewpoint of 2-d 'beings'.
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<br>Also interesting is 'Tangents', a short story by Greg Bear (I've got it in a collection called 'The Mathenauts' - all the mathematical science-fiction shorts that Rudy Rucker could find in 1987 - no, I'm not a nerd really :-)<div style="clear:both;"></div><img src="http://blogs.msdn.com/aggbug.aspx?PostID=416772" width="1" height="1">re: High-Dimensional Spaces Are Counterintuitive, Part Onehttp://blogs.msdn.com/b/ericlippert/archive/2005/05/11/high-dimensional-spaces-are-counterintuitive-part-one.aspx#416599Wed, 11 May 2005 22:54:04 GMT91d46819-8472-40ad-a661-2c78acb4018c:416599mikebThis reminds me of a funny little bit in an old Simpsons episode:
<br>
<br> Lisa: Well, where's my Dad?
<br> Frink: Well, it should be obvious to even the most dim-witted
<br> individual who holds an advanced degree in hyperbolic
<br> topology, n'gee, that Homer Simpson has stumbled into...[the
<br> lights go off] **the third dimension**.
<br> Lisa: [turning the lights back on] Sorry.
<br> Frink: [drawing on a blackboard] Here is an ordinary square --
<br> Wiggum: Whoa, whoa -- slow down, egghead!
<br> Frink: -- but suppose we exte-end the square beyond the two
<br> dimensions of our universe (along the hypothetical Z axis,
<br> there).
<br>Everyone: [gasps]
<br> Frink: This forms a three-dimensional object known as a "cube", or a
<br> "Frinkahedron" in honor of its discoverer, n'hey, n'hey.
<br>
<br>
<br>(the above was lifted from <a rel="nofollow" target="_new" href="http://www.snpp.com">http://www.snpp.com</a>)<div style="clear:both;"></div><img src="http://blogs.msdn.com/aggbug.aspx?PostID=416599" width="1" height="1">re: High-Dimensional Spaces Are Counterintuitive, Part Onehttp://blogs.msdn.com/b/ericlippert/archive/2005/05/11/high-dimensional-spaces-are-counterintuitive-part-one.aspx#416561Wed, 11 May 2005 21:37:40 GMT91d46819-8472-40ad-a661-2c78acb4018c:416561Jason AbbottAnother way to analogize to higher geometric dimensions is to recognize that each dimension bisects the higher dimension and is perpendicular to the axis added by the higher dimension (or maybe that's too obvious to be helpful).
<br>
<br>An infinite line will bisect an infinite plane. The axis added for a plane is perpendicular to the line. An infinite plane will bisect an infinite solid, which added another perpendicular axis. An infinite solid will bisect an infinite hyper-solid, which has an additional axis perpendicular to the three we are familiar with.
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<br>Or in your sphere terms, the n-sphere can always bisect the equal radius n+1 sphere.<div style="clear:both;"></div><img src="http://blogs.msdn.com/aggbug.aspx?PostID=416561" width="1" height="1">