tag:blogger.com,1999:blog-11295132.post3004944029623856646..comments2014-11-13T21:22:20.572-08:00Comments on A Neighborhood of Infinity: What's all this E8 stuff about then? Part 1.Dan Piponihttps://plus.google.com/107913314994758123748noreply@blogger.comBlogger33125tag:blogger.com,1999:blog-11295132.post-64523311131399177332008-07-04T00:17:00.000-07:002008-07-04T00:17:00.000-07:00Jeremy,2 angles certainly won't do for 3D rotation...Jeremy,<BR/><BR/>2 angles certainly won't do for 3D rotations even if it might seem obvious to you. You can read about it (with animations) <A HREF="http://en.wikipedia.org/wiki/Flight_dynamics" REL="nofollow">here</A>.<BR/><BR/>In n dimensions you need n(n-1)/2 angles.sigfpehttp://www.blogger.com/profile/08096190433222340957noreply@blogger.comtag:blogger.com,1999:blog-11295132.post-56882037231808401512008-07-03T14:38:00.001-07:002008-07-03T14:38:00.001-07:00Dear Sigfpe,As a non-mathematician it's pretty obv...Dear Sigfpe,<BR/>As a non-mathematician it's pretty obvious to me that if you can use an angle to describe a 2D rotation - then you can use two angles to describe a 3D rotation - so I'm not sure your explanation leading to 6 numbers needed to describe 4 dimensions is very convincing.<BR/>Otherwise many thanks for the article.jeremyclarkukhttp://www.blogger.com/profile/10284558857355269311noreply@blogger.comtag:blogger.com,1999:blog-11295132.post-41788448140408502182008-07-03T14:38:00.000-07:002008-07-03T14:38:00.000-07:00Dear Sigfpe,As a non-mathematician it's pretty obv...Dear Sigfpe,<BR/>As a non-mathematician it's pretty obvious to me that if you can use an angle to describe a 2D rotation - then you can use two angles to describe a 3D rotation - so I'm not sure your explanation leading to 6 numbers needed to describe 4 dimensions is very convincing.<BR/>Otherwise many thanks for the article.jeremyclarkukhttp://www.blogger.com/profile/10284558857355269311noreply@blogger.comtag:blogger.com,1999:blog-11295132.post-18382599981023687722008-07-03T14:37:00.000-07:002008-07-03T14:37:00.000-07:00Dear Sigfpe,Many thanks for a great article. Howev...Dear Sigfpe,<BR/>Many thanks for a great article. However to a non-mathematician it's pretty obvious that if you can define a 2D rotation using an angle, then you can define a 3D rotation using two angles. So your explanation leading to 6 numbers for 4 dimensional space is not very convincing. Have I missed something ?jeremyclarkukhttp://www.blogger.com/profile/10284558857355269311noreply@blogger.comtag:blogger.com,1999:blog-11295132.post-46709560715067597732008-07-03T14:34:00.000-07:002008-07-03T14:34:00.000-07:00Dear Sigfpe,As a non-mathematician it's pretty obv...Dear Sigfpe,<BR/>As a non-mathematician it's pretty obvious to me that if you can use an angle to describe a 2D rotation - then you can use two angles to describe a 3D rotation - so I'm not sure your explanation leading to 6 numbers needed to describe 4 dimensions is very convincing.<BR/>Otherwise many thanks for the article.jeremyclarkukhttp://www.blogger.com/profile/10284558857355269311noreply@blogger.comtag:blogger.com,1999:blog-11295132.post-18360047263370494012008-04-21T14:38:00.000-07:002008-04-21T14:38:00.000-07:00I went back through some of your archives, so forg...I went back through some of your archives, so forgive the untimeliness of this post. Hopefully that won't be too hard, since most of the mathematics was timely a century ago ;-)<BR/><BR/>First, this article is great. It explains in comparably lay terms about groups and symmetry. (I always wondered what a Lie group was before this post, and now I at least have an idea of what flavor they are).<BR/><BR/>My only confusion came about at the very end, where you began to talk about differentiability. (Calculus was always my weakness!) Correct me if I'm wrong, but when we talk about a group G being lie, we're really saying that a function f:R->G is differentiable (in some manner or another)?<BR/><BR/>Also, took me a second to grok the idea at the beginning of when two transformations are different. In some hypothetical future revision of this blog post, it might make it a little easier to explain that the equivalence between transformations is the same as equivalence of functions: that f = g iff for all points p, p transformed under f is the same as p transformed under g.Tac-Ticshttp://www.blogger.com/profile/14070765709653635739noreply@blogger.comtag:blogger.com,1999:blog-11295132.post-56730080988705026802007-11-23T05:44:00.000-08:002007-11-23T05:44:00.000-08:00Great job: my niece is going to use it, here in It...Great job: my niece is going to use it, here in Italy, for a small work at school.<BR/>Looking forward for the second part.<BR/>LucaAnonymousnoreply@blogger.comtag:blogger.com,1999:blog-11295132.post-27838722926722715852007-11-20T22:35:00.000-08:002007-11-20T22:35:00.000-08:00Great post. Thoroughly understandable and fun. S...Great post. Thoroughly understandable and fun. Seems this is a typo: "Its <B>length</B> is the direction of rotation and its <B>length</B> is the rate of change of angle around that axis"Kenn Knowleshttp://www.blogger.com/profile/04205298776211790050noreply@blogger.comtag:blogger.com,1999:blog-11295132.post-68217785954831836632007-11-20T17:51:00.000-08:002007-11-20T17:51:00.000-08:00Garrett,The next part comes out when I manage to c...Garrett,<BR/><BR/>The next part comes out when I manage to come up with some kind of informal layman's description of what the weights of a representation are. I have some ideas. Probably Friday as I'll have the day off.sigfpehttp://www.blogger.com/profile/08096190433222340957noreply@blogger.comtag:blogger.com,1999:blog-11295132.post-86187903705129906122007-11-20T17:37:00.000-08:002007-11-20T17:37:00.000-08:00Oops, sorry Dan. I thought you were someone else. ...Oops, sorry Dan. I thought you were someone else. I'll blame it on my inbox trying to kill me.<BR/><BR/>When's part 2 coming out?Garretthttp://deferentialgeometry.orgnoreply@blogger.comtag:blogger.com,1999:blog-11295132.post-15966653592409176012007-11-20T16:04:00.000-08:002007-11-20T16:04:00.000-08:00Hi sigfpe,This is an impressive post.Is there any ...Hi sigfpe,<BR/><BR/>This is an impressive post.<BR/><BR/>Is there any way to do this relatively easily in 3D rather than in 2D?<BR/><BR/>For example, some engineers appear to use 3D helical functions rather than 2D elliptical functions.Doughttp://www.blogger.com/profile/07643919214761722345noreply@blogger.comtag:blogger.com,1999:blog-11295132.post-77794810040155402992007-11-20T08:37:00.000-08:002007-11-20T08:37:00.000-08:00Since this post is still a lot about groups, I thi...Since this post is still a lot about groups, I think I can advise to your readers this cool freeware : <A HREF="http://groupexplorer.sourceforge.net/" REL="nofollow">GroupExplorer</A><BR/><BR/>Unfortunately, I don't know any good software to help you draw your ADE diagrams for your next post :-)alpheccarhttp://www.blogger.com/profile/14645433315403867431noreply@blogger.comtag:blogger.com,1999:blog-11295132.post-32616479939849516892007-11-20T06:26:00.000-08:002007-11-20T06:26:00.000-08:00So the cube is actually a subset of SO(3).A bit un...<I>So the cube is actually a subset of SO(3).</I><BR/><BR/>A bit unclear: you should probably say "the symmetry group of the cube" or something similar. The cube itself isn't a subset of SO(3)!<BR/><BR/>But generally a really nice article. Look forward to Part 2.Davidhttp://www.blogger.com/profile/03396164169951516460noreply@blogger.comtag:blogger.com,1999:blog-11295132.post-66605189123289376972007-11-19T23:24:00.000-08:002007-11-19T23:24:00.000-08:00We used B to mean a 180 degree rotation around the...<I>We used B to mean a 180 degree rotation around the x-axis. If we apply that rotation to any point in space (x,y,z) it gets mapped to (x,-y,-z). Similary A maps the point (x,y,z) to (x,z,-y).</I><BR/><BR/>I don't understand. Wouldn't 180 degrees (B) be (x,y,-z) or (x,y,+z), and wouldn't 90 degrees (A) be (x,-y,-z) or (x,+y,+z) or (x,-y,+z) or (x,+y,-z)?Anonymousnoreply@blogger.comtag:blogger.com,1999:blog-11295132.post-73735407863426009272007-11-19T22:28:00.000-08:002007-11-19T22:28:00.000-08:00William? Who's William?William? Who's William?sigfpehttp://www.blogger.com/profile/08096190433222340957noreply@blogger.comtag:blogger.com,1999:blog-11295132.post-39436871312309644862007-11-19T22:25:00.000-08:002007-11-19T22:25:00.000-08:00William, this is great! Nice job.William, this is great! Nice job.Garretthttp://deferentialgeometry.orgnoreply@blogger.comtag:blogger.com,1999:blog-11295132.post-48580232517890012502007-11-19T09:20:00.000-08:002007-11-19T09:20:00.000-08:00So far I've only said roughly what a Lie group and...So far I've only said roughly what a Lie group and Lie algebra are, and that E8 is an example of these things. But there are infinitely many Lie groups and Lie algebras. In part II, besides talking more about Lie groups, I hope to zoom in a bit and single out E8 from all of the others and say something about what makes it special. In part III I'll then try to show why group theory is so important to physics and how properties of Lie groups can be (or might be) interpreted as properties of the real world.sigfpehttp://www.blogger.com/profile/08096190433222340957noreply@blogger.comtag:blogger.com,1999:blog-11295132.post-13718253073324388972007-11-19T09:14:00.000-08:002007-11-19T09:14:00.000-08:00Mark,Is your browser acting up? I'm pretty sure I ...Mark,<BR/><BR/>Is your browser acting up? I'm pretty sure I removed all occurrences of the word 'dihedral' as soon as you pointed out my error. I decided not to give this group any name (other than cube group) because I don't really want to spend much time on finite groups. But thanks for pointing the issue.sigfpehttp://www.blogger.com/profile/08096190433222340957noreply@blogger.comtag:blogger.com,1999:blog-11295132.post-84636640536047665972007-11-19T08:15:00.000-08:002007-11-19T08:15:00.000-08:00Thank you for putting the time in to write this ex...Thank you for putting the time in to write this explanation! I have read a little bit of group theory and played around with a book/software package called "Exploring Abstract Algebra with Mathematica," but did not get that far. Anyway, I am referring my friends to your blog for a basic introduction to E8. Most of them will probably get through the first few paragraphs and kind of fade out, but that's OK... it's better than no understanding!Paul R. Pottshttp://www.blogger.com/profile/04401509483200614806noreply@blogger.comtag:blogger.com,1999:blog-11295132.post-89208053053662536622007-11-19T08:08:00.000-08:002007-11-19T08:08:00.000-08:00You didn't correct the thing about the cube group ...You didn't correct the thing about the cube group being the dihedral group of order 24. It isn't. It's the symmetric group of order 24.<BR/><BR/>To see this, consider the action of the group on the four diagonals of the cube. Any of the 24 possible permutations of the diagonals can be effected by a single rotation of the cube. Thus the group is S4.Mark Dominushttp://www.blogger.com/profile/17698641253266210249noreply@blogger.comtag:blogger.com,1999:blog-11295132.post-91526322176140263322007-11-19T07:46:00.000-08:002007-11-19T07:46:00.000-08:00Hi!This has been posted at http://www.physicsforum...Hi!<BR/>This has been posted at http://www.physicsforums.com/showthread.php?p=1510223#post1510223 Layman's explanation wanted.<BR/><BR/>I can see that you are doing a much better job than what I was attempting.<BR/><BR/>Time for the pros to take over.<BR/>JalAnonymousnoreply@blogger.comtag:blogger.com,1999:blog-11295132.post-12331535073171429552007-11-18T22:14:00.000-08:002007-11-18T22:14:00.000-08:00Thanks for a beautifully clear article. I'm lookin...Thanks for a beautifully clear article. I'm looking forward to the next parts.Gregory Brownhttp://www.blogger.com/profile/15726193040616078049noreply@blogger.comtag:blogger.com,1999:blog-11295132.post-22183013825308295932007-11-18T21:36:00.000-08:002007-11-18T21:36:00.000-08:00Thanks for writing this I am thoroughly enjoying i...Thanks for writing this I am thoroughly enjoying it.<BR/><BR/>"If we apply that rotation to any point in space (x,y,z) it gets mapped to (x,-y,-z). Similary A maps the point (x,y,z) to (x,z,-y)." <BR/><BR/>Should it be (x,-y,z) at the end?Anonymousnoreply@blogger.comtag:blogger.com,1999:blog-11295132.post-12819285843274441152007-11-18T09:17:00.000-08:002007-11-18T09:17:00.000-08:00Thanks for the corrections, most of which I've app...Thanks for the corrections, most of which I've applied.<BR/><BR/>Pseudonym, the bivector thing is one of my favourite nitpicks, but today I'm using the word 'vector' to mean 'an element of a vector space', so it doesn't apply. The word vector is a little overloaded isn't it.sigfpehttp://www.blogger.com/profile/08096190433222340957noreply@blogger.comtag:blogger.com,1999:blog-11295132.post-16899883747925837502007-11-18T08:03:00.000-08:002007-11-18T08:03:00.000-08:00I'm really curious where all this is leading to, a...I'm really curious where all this is leading to, as I haven't heard about this E8 buzz before.<BR/><BR/>The cube rotations are not a dihedral group.<BR/>The third paragraph of the section "Rates of change and Lie algebras" ends with a sentence with a typo: "Its length is ... and its length is ..."ChristianSnoreply@blogger.com