February 2, 2008

Virtually Real or Really Virtual

Posted by David Corfield

If NASA can have a presence in Second Life with their CoLab project, maybe we at the Café should be thinking about the next step. Posted at February 2, 2008 12:34 PM UTC

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Re: Virtually Real or Really Virtual

we at the Café should be thinking about the next step.

2-presence?
I suspect it’ll be easier to start with strict 2-presence, though obviously we’ll need to understand weak 2-presence (or at least, braided 2-presence) to think about 3-presence.

Posted by: Allen Knutson on February 2, 2008 3:51 PM | Permalink | Reply to this

Re: Virtually Real or Really Virtual

Didn’t have a chance to watch the whole vid, but I want to know why are all these virtual worlds only 3-d. Shouldn’t we be flying through 4-d space? The question is more serious than it sounds: the n-catsters are trying to develop higher dim’l algebra. We have a handle on dim 1, 2 and to some degree 3 because we have geometric intuition in the apparently 3-d world. We don’t have intuition in 4d. Computers can do this for us. So like Joe Cube in Rucker’s Spaceland (sorry no time to look up the URLs), we could be flying about in 4d video games and VR spaces. [Wait till you see my new 4-d Tuxedo!] After we get over the motion sickness (cubes necker reversing can be really disorienting), we might begin to train our right brain, which then can tell our left brain how to do theses things.

Posted by: Scott Carter on February 2, 2008 4:27 PM | Permalink | Reply to this

4-D games and crystals; Re: Virtually Real or Really Virtual

Since I do a lot of math and Physics in 4-D and 5-D, I think the world is ready for a true 4-D computer game. I have discussed this with several professional computer ganme designers. Kids playing it will learn Relativity while very very young, and build stardrives and time machines and Mimzies. Or something. I presume this audience’s familiarity with 4-D graphics (such as the nice interactive polytopes on MathWorld), and am frustrated by the reluctance to my pitches for such a game to companies in a hit-driven industry. I do have professional TV, fil, and CD-Rom writing experience, and book/story/play publishing, but don’t want to go into a side-bar on the skewed long-tail distributions implied by “hit-based industry”).

On a related matter, Torbjörn Larsson ( April 5, 2007) wrote:

“When the basic course in solid state physics covers phonon spectra, it typically starts out with an 1D
lattice model, as in Kittel’s “Introduction to Solid State Physics”. The lowest longitudinal mode (the
uniform mode) would be equivalent to moving the whole crystal, and is typically neglected in a Fourier
analysis with fixed or infinite boundaries. At the course only a few of us noted that, after all it is a simple model and it was outside its purpose.
Of course some excitation modes (hitting) may indeed move the crystal. And others (temperature) will
benefit from a more realistic analysis from averaging et cetera in real, finite crystals. But the interesting reaction was the teacher’s. He simply said
something equivalent to “Don’t bother” and dropped the matter, which wasn’t very helpful at the time. A common test for models is, or should be, to push it
‘til it breaks. It is done to judge if it is inherently wrong or where its domain of applicability is or what would improve it. Apparently that educator wasn’t used to think in that way.”

“The 1D lattice model, as in Kittel’s “Introduction to
Solid State Physics” is not a bad place to start. But there is no such thing (so far as I know) as 1-D magnetism. Magnetism starts in 2-D.”

“Ferromagnetism was first explained at a deep mathematical level by Ising’s model, which also explains weird phenomena such as fractional quantum Hall state accompanied by hysteresis in accord with 2D Ising ferromagnetism. and domain formation.”

“The stuff that PHysics students and EE students learn about ‘left-hand rule’ and Maxwell’s Equations (actually most students now learn Heaviside’s equations) are uniquely dependent on 3-D space plus
1-D time. Only in 3-D and 7-D is there a good cross product.”

“Which is part of why I’m playing with 4-D space + 1-D time crystals.”

“Note that actual substances that form ‘quasicrystals’ in 3-D are best explained as projections of 6-D crystals! Penrose tiles are a 2-D side-effect of 6-D geometry and crystallography.”

“I’ve been reading… a paper by John Horton Conway (yes, that Conway: Game of Life, Surreal Numbers) and Neil J. A. Sloane about multidimensional lattices. Very very weird things happen at 11-D and 24-D. Physics intuition seems to get in the way of seeing things in these spaces.”

“Very simple questions such as ‘what is a crystal?’ have shown me the depths of my ignorance in blindingly clear ways.”

Since then, I’ve twice discussed with Math/Physics professor Barry Simon (Caltech) the unfinished work by Richard Feynman on 4-dimensional ionic hypercube-symmetry crystals, and their perturbative stability, and have emailed back and forth with his son Carl Feynman on certain open questions in 4-D lattice polytope symmetry. And I’ve discussed the matter with professor Tim Poston (whose PhD was from Penrose).

See also the recent issue of Science, on 4-D phase transition in incommeasurate 3-D solid.

Posted by: Jonathan Vos Post on February 2, 2008 8:30 PM | Permalink | Reply to this

Re: 4-D games and crystals; Re: Virtually Real or Really Virtual

There is a plenty of 2n-D games. All the games there the player control party of n characters on the 2D board.

Posted by: serg271 on February 3, 2008 7:08 AM | Permalink | Reply to this

Re: Virtually Real or Really Virtual

One of the things we found with higher dimensional algebra in the 1970s, and which always amazed me, was 2-dimensional rewriting’. This is the power behind the 2-dimensional van Kampen theorem.

Group theory workers on identities among relations use deformations of pictures’ of homotopy elements. A problem is that to go further one needs to record deformations of pictures, which seems to need a 3-dimensional language. Work with Al-Agl and Steiner uses a 3-dimensional rewriting procedure to prove a key braid relation.

It would be good if computers could go further. But for n-dimensional rewriting (general n) one seems to need symbols on a page to model what is going on! Trying to find the context to do this is a key part of the problem. This is in the background of my previous comments on the contrasting properties of crossed complexes, and of globular and cubical omega-groupoids. This variety of representations is important.

Posted by: Ronnie Brown on February 3, 2008 10:44 AM | Permalink | Reply to this

Re: Virtually Real or Really Virtual

for deformations of pictures, is the work of e.g. Scott Carter relevant?

Posted by: jim stasheff on February 3, 2008 1:32 PM | Permalink | Reply to this

Re: Virtually Real or Really Virtual

Jim, thanks for the plug! I think that Alissa Crans, Mohamed Elhamdadi, and Masahico Saito have quite a lot to do with this work. The e.g. makes this implicit, but not explicit.

What we have seen on the higher dimensional algebra side is not as explicit as Ronnie Brown is mentioning but we have a lot of glimpses into this: Take a piece of classical algebra, represent it as diagrams, interpret the diagrams as a categorical internalization, for example, now watch the diagram move to express the axiom. The motion of the diagram can be projected (decategorified), and some extra-structure can be reimposed on the projection of the motion. Then the identities among relations (or similar axioms) can be visualized by watching the singular points of the projection undergo their singularities. The movie move theorem and its smarter brother (C., Rieger, and Saito) are example exploitations of this idea.

The recent stuff enhances this with some cocycles attached at the singular points. I am pretty sure (and I think Masahico agrees) that the whole picture goes through nicely in the Frobenius algebra case because all the identities among relations, and their higher order analogues are controlled by the Stasheff polytopes.

Of course, as a hobby, I try to draw things in as an explicit fashion as possible. And this relates back to Jonathan Van Post’s remark. It takes time and hard work to visualize higher dimensional stuff. I had a conversation with J. H. Conway the other day, and I was surprised at some of the questions that even he could not answer. He mentioned a geometric (4d) proof of Heron’s formula, for example. If higher dimensional problems are hard for him, do the rest of us have a chance? So 4d gaming, 4d fly throughs, and all of these things can help build intuition for higher-dimensional algebra.

Here is a homework problem. Take any suitably short book on Hopf algebras (eg Majid’s big pink) and write all the proofs in diagrammatic form. After finding the diagrammatic proof, then take the kinematics of the proof and elaborate them as “spin foam” pictures. Now are there key axioms, key proofs, etc that manifest themselves in a 3-d branched solid?

In regards to JB’s comment about sending primates into space, I think the original space program did a good thing: It took a fundamentally military competition and turned it into a humanitarian endeavor. Some spin-off technology includes the relays from my keyboard to this forum and back. Maybe not directly moon related, but some organization had to put those tin cans in space. The current proposal to send people to Mars does seem far fetched in light of the overwhelming social problems caused by humanity’s taxing this environment so. Some thought (Asimov?) that our purpose was to get the hell out of Dodge: Space exploration was what we were meant to do. As a species, we tend to foul our homes to the point that they are uninhabitable, and then move on.

NASA has no mandate to address the problems of this planet even if they have the resources to make inroads. We, as scientists, can lobby for that mandate to change.

Posted by: Scott Carter on February 3, 2008 9:18 PM | Permalink | Reply to this

Re: Virtually Real or Really Virtual

Scott Carter wrote:

In regards to JB’s comment about sending primates into space, I think the original space program did a good thing: It took a fundamentally military competition and turned it into a humanitarian endeavor.

Just so there’s no confusion: I completely agree!

I also think it was very important for us to see this:

The Apollo mission made good use of technologies available at the time, before computers really took off. My objections are to a manned Mars mission, which at this point would be a huge step backwards. We don’t need to shoot people into space anymore to see these wonderful things, and study them in marvelous detail:

NASA has no mandate to address the problems of this planet even if they have the resources to make inroads. We, as scientists, can lobby for that mandate to change.

For starters, we should lobby to undo some of the damage
already done by the Bush administration’s wrongheaded focus on the Mars mission, combined with their ostrich-like posture when it comes to human-caused global heating:

NASA is canceling or delaying a number of satellites designed to give scientists critical information on the earth’s changing climate and environment.

The space agency has shelved a $200 million satellite mission headed by a Massachusetts Institute of Technology professor that was designed to measure soil moisture – a key factor in helping scientists understand the impact of global warming and predict droughts and floods. The Deep Space Climate Observatory, intended to observe climate factors such as solar radiation, ozone, clouds, and water vapor more comprehensively than existing satellites, also has been canceled. And in its 2007 budget, NASA proposes significant delays in a global precipitation measuring mission to help with weather predictions, as well as the launch of a satellite designed to increase the timeliness and accuracy of severe weather forecasts and improve climate models. The changes come as NASA prioritizes its budget to pay for completion of the International Space Station and the return of astronauts to the moon by 2020 – a goal set by President Bush that promises a more distant and arguably less practical scientific payoff. Ultimately, scientists say, the delays and cancellations could make hurricane predictions less accurate, create gaps in long-term monitoring of weather, and result in less clarity about the earth’s hydrological systems, which play an integral part in climate change. Posted by: John Baez on February 5, 2008 3:06 AM | Permalink | Reply to this Re: Virtually Real or Really Virtual Which of the current candidates has a clue about this? Posted by: jim stasheff on February 5, 2008 12:57 PM | Permalink | Reply to this Re: Virtually Real or Really Virtual Here is a remark on space travel, for what it’s worth. NASA can have a presence At some point in the video the off-voice says something about “not sending us into space but getting space here to us”. I have always thought that interpreting “space exploration” as “figuring out how to make primates survive a wee bit beyond their biosphere” is mislead, and that one should be content about, and in fact grateful for, having various robots sent out there sending their data back here. Hope I don’t have to turn that statement into an $n$-categorical analogy now, somehow. Posted by: Urs Schreiber on February 2, 2008 5:42 PM | Permalink | Reply to this Re: Virtually Real or Really Virtual Urs wrote: I have always thought that interpreting “space exploration” as “figuring out how to make primates survive a wee bit beyond their biosphere” is misleading, and that one should be content about, and in fact grateful for, having various robots sent out there sending their data back here. Indeed, one of the many backwards steps taken by the current US administration is their crazy plan to send canned primates to Mars. It’ll be incredibly expensive if it ever happens: NASA guesses it’ll cost$104 billion just to get back to the Moon by 2020 — and by 2024, the Government Accounting Office guesses the total price of the manned Mars mission will hit \$230 billion.

With luck, we’ll never carry out this stupid stunt. But, even the planning to do it is already crushing many projects that are far more interesting.

For a more detailed rant on this subject, see my comments here.

Posted by: John Baez on February 3, 2008 2:17 AM | Permalink | Reply to this

Re: Virtually Real or Really Virtual

Bloomberg TV just said that NASA has hired a prominent ad agency but the story hasn’t appeared yet on bloomberg.com

Does anyone know if NASA engineers could potentially work on serious alternative energy projects or would doing so be outside of their abilities or interests?

Posted by: Charlie Stromeyer Jr on February 3, 2008 2:12 PM | Permalink | Reply to this

Re: Virtually Real or Really Virtual

If you were living in Europe around 1500 would you guys have written the following?

“Indeed, one of the many backwards steps taken by the current crowned heads of Europe is their crazy plan to send boxed primates to America. It’ll be incredibly expensive if it ever happens. Spain guesses it’ll cost thousands of pounds of gold just to maintain a presence in the Carribean.

With luck, we’ll never carry out this stupid stunt. But, even the planning to do it is already crushing many projects that are far more interesting.”

Posted by: Jeffery Winkler on February 4, 2008 6:54 PM | Permalink | Reply to this

Re: Virtually Real or Really Virtual

Your analogy is flawed. The monarchs of Europe sent explorers around the globe to take over territory and make money. The practicality of the endeavor was obvious to them, though it certainly involved risks.

And the reason is: they knew Europeans could easily survive on other parts of the Earth. They also soon learned they could subdue the locals with superior weaponry.

Here is a quote from Columbus on his journey: “Many of the men I have seen have scars on their bodies, and when I made signs to them to find out how this happened, they indicated that people from other nearby islands come to San Salvador to capture them; they defend themselves the best they can. I believe that people from the mainland come here to take them as slaves. They ought to make good and skilled servants, for they repeat very quickly whatever we say to them… I could conquer the whole of them with 50 men, and govern them as I pleased.”

Will we see any folks on the Mars mission making similar remarks? No: they’ll be struggling to survive on a chilly airless ball of rock, for no particular good reason.

Until something changes — like genetically engineering ‘men’ that’ll enjoy life on other planets — manned space missions will remain a tired old 1950’s-science-fiction-style vision of the future. There are much cooler things to do now.

Posted by: John Baez on February 5, 2008 12:40 AM | Permalink | Reply to this

Re: Virtually Real or Really Virtual

Perhaps one should regard space programs not as part of science, but as belonging to psychology, because people project their phantasies into it like into a big Rorschach-Test. E.g. here a book on russian SF, in which the author describes the supposed role of gnostic and eschatological thinking on the russian science community and its interest in space. I wonder if escape phantasies, e.g. for the case that consequences of global warming become desastrous, play a similar role in the way americans think about spacetravel?

Posted by: Thomas Riepe on February 6, 2008 2:02 PM | Permalink | Reply to this

Re: Virtually Real or Really Virtual

I’ve often thought this, and that the distance between escape fantasies and eschatology isn’t necessarily all that large…

Posted by: Tim Silverman on February 6, 2008 11:56 PM | Permalink | Reply to this

Re: Virtually Real or Really Virtual

I guess the only difference between attitudes towards human space travel is that one community thinks of it in terms of salvation, the other in terms of comfortable escape opportunities. But „salvation“ needs only to be achieved by few, representing all others, whereas „escape“ would be requested by all. So, a few cosmonauts suffice for the one community, but americans phantasy about space colonies . I wonder if such phantasies of a possible comfortable escape made the leading people in economy and politics accept irresponsible politics the past? Why should leading people in economy not imagine that they could, if necessary, transfer their way to live into space, e.g. like they can transfer their favorite sites for vacation , if the real thing is in a too unpleaseant surrounding?

Posted by: Thomas Riepe on February 7, 2008 10:08 AM | Permalink | Reply to this

Re: Virtually Real or Really Virtual

“the next step”? Get scanned and virtual by the Harvard brain peeler.

Posted by: Thomas Riepe on February 3, 2008 6:00 PM | Permalink | Reply to this

Re: Virtually Real or Really Virtual

BTW, does anyone know if there exists a japanese-for-mathematicians book? I forgot nearly all I ever learned of it.

Posted by: Thomas Riepe on February 7, 2008 3:11 PM | Permalink | Reply to this

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