3 ms·

Let me offer a couple of other ways to view this. The most important point is that higher dimensional space is very roomy. There are many degrees of freedom, m

by hfinney 16y ago

Let me offer a couple of other ways to view this. The most important point is that higher dimensional space is very roomy. There are many degrees of freedom, many directions. Take the first puzzle, the inner sphere that is bigger than the outer ones. One factor is that the 2 by 2 by ... by 2 packing isn't very efficient. It's not even the densest packing in 2 dimensions, less so in 3, and it gets dramatically less so as you go up. With 10 dimensions it is really inefficient so there is a lot of space in the middle. As far as the "cap" not having much volume: this wasn't explained very clearly, what he meant. Picture a circle with radius=1 centered at the origin, and then look at the piece cut off by y > 1/2 (hope that prints ok, I mean y greater than 0.5). That piece has a certain fraction of the total area. Now picture a sphere at the origin, radius 1, and the cap cut off by y > 1/2. That cap will have a smaller fraction of the total volume. Going to higher dimensions, the fraction gets smaller and smaller. But rather than meaning the sphere is "spiky", this is a result of more degrees of freedom. There are many more caps in many more directions on a high dimensional sphere. So each cap has to have less of the volume. Spikiness is really an absurd way to think of it. With practice, I've developed some ability to visualize four dimensional space. Its overwhelming character, as I said, is that it is infinitely and somewhat frighteningly roomy. This would be even more so in higher dimensions.

srean 16y ago
+1 We find similar things intriguing http://news.ycombinator.com/item?id=1848275 http://news.ycombinator.com/item?id=1848275 :) Though this might seem esoteric, this phenomena has practical applications. In the vector space model of information retrieval, documents are modeled as points on a high-d sphere, where d is the size of the vocabulary. So unless one accounts for these effects, there will be fascinating surprises.
RiderOfGiraffes 16y ago
I hear what you say, and what you say is true. Having said that, for me, I don't think your observations really help to develop an intuition that helps when working in a higher dimensional space. My experience is that people know it's roomy, but don't appreciate the consequences. The early comment about there being 10^5400 "places to be" if you have 1000 in each direction for a 1800 dimensional space immediately conveys the idea that space is big, but it's the implications that tend to go missing. People still think of spheres are round, and that "smooth" means "like a landscape in 3D." So while your comments are true, and add usefully to the discussion, to me they don't really convey a helpful visualization.

by hfinney 16y ago

srean 16y ago

+1 We find similar things intriguing http://news.ycombinator.com/item?id=1848275 http://news.ycombinator.com/item?id=1848275 :) Though this might seem esoteric, this phenomena has practical applications. In the vector space model of information retrieval, documents are modeled as points on a high-d sphere, where d is the size of the vocabulary. So unless one accounts for these effects, there will be fascinating surprises.

RiderOfGiraffes 16y ago

I hear what you say, and what you say is true. Having said that, for me, I don't think your observations really help to develop an intuition that helps when working in a higher dimensional space. My experience is that people know it's roomy, but don't appreciate the consequences. The early comment about there being 10^5400 "places to be" if you have 1000 in each direction for a 1800 dimensional space immediately conveys the idea that space is big, but it's the implications that tend to go missing. People still think of spheres are round, and that "smooth" means "like a landscape in 3D." So while your comments are true, and add usefully to the discussion, to me they don't really convey a helpful visualization.