File talk:Integrality of root systems.svg

Suppose $\beta =\alpha$ ; then $(\alpha ,\beta )=(\beta ,\alpha )$ , but according to the diagram this means $3/2=1$ . --JWB (talk) 13:18, 27 November 2009 (UTC)[reply]

I think the red captions should be reversed so that the outermost red circles are 1/2 and the innermost 2. --JWB (talk) 14:09, 27 November 2009 (UTC)[reply]

Well spotted. I think it would be nice to move the labels slightly: moving the labels pi/4 and 3pi/4 on the vertical lines 1/2 and -1/2 to the other sides of those vertical lines, so we can put the red labels -1 and < α,β > = 1 higher up and closer to the central vertical line so that the bigger label does not overlap a blue line. It would also be nice to add a label in front of the angle pi/2, I guess. I think it's important that we keep everything clear and as symmetric as possible, and with labels clearly visible and not overlapping. -Xedi^Talk 08:00, 29 November 2009 (UTC)[reply]

I made a bunch of improvements:

Indicate angles with labeled green rays instead of labels on points (getting rid of point angle labels you were concerned about)
Indicate Y coordinates of points
Add red circles for >2 to illustrate increase towards origin
Indicate end root used in series A-G with same color as alpha, similar arrowhead, and length √3, √2, or 1
Make dots on X axis hollow to indicate they do not define new roots independent of alpha
Trim whitespace at edges to minimum

I've followed the originator by hand-editing the SVG code, and have improved it in these ways:

Group similar objects to avoid repetition of parameters
Give explanatory names (ids) to the groups
Implement reflected and rotated features by calling groups with transforms rather than repeating definition
Indent by grouping level

Hand-editing with a text editor would be best for any further changes - it seems like Inkscape etc. rewrite the file changing formatting and introducing non-human-readable stuff. --JWB (talk) 23:08, 29 November 2009 (UTC)[reply]

I've made the Y axis both blue and red to indicate both products are zero on that line, and labeled the pi/2 angle, in view of your comments. I've also elaborated the caption under the diagram in Root system. --JWB (talk) 23:33, 29 November 2009 (UTC)[reply]

Ok well, the image is a bit crowded now, but it's nice to see how much information you can fit on there. I don't think it's very readable though. What I meant to say though is that if β is on the vertical line <β,α> the root system isn't necessarily reducible (B2, G2 for example), so it would be nice if that line could also be in black. But I'm not sure how to colour it seeing as it should be red, blue and black technically. It's also a bit annoying that the labels corresponding to the root systems are not entirely correct as you mention in the caption - it is perhaps better to not include too much information as it is hard to put it all on the same diagram. --Xedi^Talk 01:07, 30 November 2009 (UTC)[reply]

I think B₂, G₂ would use the βs marked BF, G and not a β on the Y-axis, and that a β on the Y-axis would be not linked with α by definition. Maybe we are misunderstanding each other here.

I almost captioned that the β marked with a letter would serve as the first root for the root system known by that letter, with α as the second root, with any subsequent roots the same length as α and successive angles being pi/3. However this fails in the single case of F₄ where the α and β would be the middle two of the four roots. I'm still searching for terminology to describe it succinctly. --JWB (talk) 01:29, 30 November 2009 (UTC)[reply]

B₂, G₂ have roots on the Y-axis but those do not form a set of two simple roots with α. --JWB (talk) 02:00, 30 November 2009 (UTC)[reply]

First thanks for correcting my mistake with the red circles (still kicking myself about that). I agree with Xedi that the image is getting a bit crowded now. I don't know why you have added the additional red circles though as I don't think they add to it. I also think the horizontal purple lines are unnecessary since the vertical distance is not really important. As for the letters for types or root system, I think anything technically correct is going to be too complicated for the diagram, and are probably best left for the article to explain. smithers888 (talk) 12:27, 1 December 2009 (UTC)[reply]

I think the additional red circles show that the value is increasing towards the origin, instead of decreasing as one would intuitively expect, and that the additional circles take up only space that was not being used.

It is true the image now goes beyond merely illustrating integrality and further into the definition and classification of root systems. I think my diagram is a good one and that the letters are now technically correct with the revised caption in Root system, but if you think it is too far from your original intent, I will be glad to put it under another filename. --JWB (talk) 14:31, 1 December 2009 (UTC)[reply]

I think it's great to add lots of information to this picture when it can be succinctly explained. I don't really see the purpose of the purple lines, and I definitely think the additional red circles should be removed as it is obvious that they have no influence on the picture (it just adds irrelevant information). If I can just say what I think: I like the idea of having the labels ABCDEFG on the picture, but I'd want them to be entirely correct (and not at the cost of some smudge factor given in the description; the picture should stand for itself). I can't think of any particularly nice way of putting the ABCDEFGH information on the diagram, but I think something should be done to make it clearer on itself. Otherwise, good work on the picture! I think the essential is that the picture contains no redundant information and that it is entirely correct on its own, sadly I cannot provide many good ways of achieving this. --Xedi^Talk 07:01, 2 December 2009 (UTC)[reply]

Thanks for the feedback. I've replaced the purple lines for Y coordinate values with (x,y) near each point, but haven't finished and uploaded the next version yet. I don't think the coordinates are irrelevant, as you use actual values when you do explicit calculations. I believe the labels are entirely correct, however some verbal explanation is still needed - if you take "stand by itself" to its conclusion, you would have to get rid of all the article text! --JWB (talk) 07:28, 2 December 2009 (UTC)[reply]