#hermann weyl

[noosphe-re]

M. Basovník and O. Semerák, Institute of Theoretical Physics, Faculty of Mathematics and Physics, Charles University in Prague, Prague, Czech Republic; published 4 August 2016

[funeral]

[T]hough the idea of an existing completely transcendental world is the basic principle of all formalism, each mathematical formalism has at every step the characteristics of being incomplete [which means that every mathematical theory is consistent in itself but is incomplete, at the borders are questions which are not self-evident, are not clear, and are incomplete] in so far as there are always problems, even of a simple arithmetical nature, which can be formulated in the frame of a formalism, but which cannot be decided by deduction within the formalism itself.

We are therefore not surprised that in an isolated phenomenal existence a piece of nature surprises us by its irrationality and that one cannot analyse it completely. As we have seen, physics therefore projects everything which exists onto the background of possibility or probability.

Hermann Weyl, Philosophy of Mathematics and Natural Science, qt. in Marie-Louise Von Franz, On Divination and Synchronicity: The Psychology of Meaningful Chance

[isomorphismes]

What is a Weyl group?

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).

[atsoukalidis]

“Beauty is bound up with symmetry”

Hermann Weyl

[truthtm]

The introduction of numbers as coordinates is an act of violence.

Hermann Weyl

[oldquotes]

'Mathematics has the inhuman quality of starlight, brilliant and sharp, but cold.' -Hermann Weyl | Visit Jar of Quotes for more image quotes.

[sibyllinemagician]

...Science would perish without a supporting transcendental faith in truth and reality, and without the continuous interplay between its facts and constructions on the one hand and its imagery of ideas on the other.

Hermann Weyl, Philosophy of Mathematics and Natural Science

[noosphe-re]

M. Basovník and O. Semerák, Institute of Theoretical Physics, Faculty of Mathematics and Physics, Charles University in Prague, Prague, Czech Republic; published 4 August 2016