#lie groups

laser etched glass | 80x80mm

V⁷ is filled by geodesics, each an eighth of a great circle on the 8-sphere, which run between points of M⁵ and M⁴ with no overlap along their interiors. A circle’s worth of these geodesics originate from each point of M⁵, leaving it orthogonally, and a 2-torus’s worth of these geodesics arrive at each point of M⁴, also orthogonally.

Besides John Baez's  explanation   , I like the one in Coxeter's Regular Polytopes chapter 5. He calls the phenomenon we are describing The Dihedral Kaleidoscope. Take an image in the plane Joan Miró, Women & Birds at Sunrise and reflect it across any of the (half-open) semicircle's worth of options, of lines-thru-the-origin, that you could reflect it across. Call the action of doing this (however you chose the angle) A. Doing A twice is the same as leaving the figure alone, whatever A you chose. But what if you choose two different lines-thru-the-origin to reflect across? Let's call them A≠B. Now these two reflections will interact in some way.† In most cases, A and B will be pointed askew so that they "miss" each other meaning the infinite sequence ABABABAB... never terminates. But there is just one arrangement of two mirrors A & B that will "line up" in the sense that ABAB brings you back to the start. † (Mathematicians dub this interaction a "reflection group" because a   sequence of reflections forms a "generalised multiplication table",   meaning (1) the way I parenthesise sequential reflections doesn't   matter, and (2) reflections are reversible. [any reflection--however you  rotate the "mirror"--is its own opposite, so that's an easy property to  verify.] You can look up the other two "group axioms" on Wikipedia;  making those work is basically a technicality, unlike the deep facts that make special reflection angles special.) If you're doodling the answer or the group-structure to yourself on paper I recommend marking four corners of a square with a,b,c,d. Then use a different colour for each A and B arrow →. (That will make the group structure clear, I think.) Figuring out which mirror angles work is probably easier to think about than to try to draw. But I thought for this answer would look cooler if I pulled the group structure back onto a Miró; hope you like it this way. (And I'll leave it to you to doodle out B then A then B, as well as the other alternatives.) As you add more & more mirrors ABCDE, the angles they should be at to not miss each other follow a predictable pattern. Every mirror you add in this way adds one o to the o―o―o―o―…―o pattern (as drawn in Baez week230). This pattern is called [math]A_n[/math] (n being how many mirrors you put up). (So you can also doodle the reflections of a pentagon, hexagon, .... see What is a group in group-theory? and isomorphismes for more pictures.) What if I were to do something analogous, instead of with a plane figure, with a statue? ↑ The "Lion-Man". Artist unknown, but s/he lived circa 42,000 years ago (=21 Jesuses ago) in the Swabian alps. The figure is famous because it is the most ancient physical proof of human imagination: whoever carved this statue, envisaged something that does not exist in the physical world. (Hint, hint: Dynkin diagrams also do not exist in the physical world.) Well, all of the plane rotations would still group together in the same fashion. So we could still draw Dynkin diagrams like o―o―o―…―o but could also add in more types of reflections, like a "flip-upside-down in the vertical direction" move.    (Let's now rename the old planar reflections A₁, A₂, … to make room for new letters coming from the new dimension. How about calling the upside-down / vertical one U or V?) Besides adding the "upside-down man" reflection, there are other ways to add mirrors that stay in synch / not askew with all of the totality of other mirrors that are already present. There are also some higher-dimensional analogues as well.  (This is one of the harder things to think about in >3D. And also quite hard to think about in 3D, in my opinion. I wrote a blurb about how to visualise higher dimensions and the reflection-group / Buildings view is still on the to-do list. So normally I would say "many dimensions are easier than you think!" but not in this case. For example if you drew a bunch of sticks |||||| -- let's say twenty-seven (http://www.math.harvard.edu/~lurie/papers/thesis.pdf)   --- and marked the + and ‒ ends of each, what reflections would be easy to do by swapping the ± to ∓, and which could you not do that way?) The 120-cell, Schlaefli symbol 5,3,3, physical model by, I believe, P. S. Donchian. That was like a pre-summary of what Coxeter says. Here are a few screenshots from the google preview of Regular Polytopes which explain it better. (You can read the whole chapter on google preview.) Note that this is different to the reflections (which don't go thru the origin) in Thurston's Geometry and Topology of 3-Manifolds: ↑ isomorphismes has more views of this image and a link to GT3M (on msri website). † Maryam Mirzakhani and Alexander Eskin's recent work (   I believe is the relevant IAS link) discusses "billiard-ball dynamics" (they say this is a sort of familiar, but naïve, instantiation of what they do) with a frictionless billiard ball's path. (Strangely after a century of work this is still unsolved.) But again these are not the reflections-thru-the-origin of the so-called "reflection groups" (Coxeter's dihedral kaleidoscope).

Liam O'Raifeartaigh, group structure of gauge theories

Bill Fulton and Joe Harris show inclusions and equivalences among various geometric ideas in continuous linear algebra