File:Integrality of root systems.svg

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Summary

DescriptionIntegrality of root systems.svg	English: Conditions on the placement of roots in a root system. Given a root $\alpha$ in a root system, this image shows the possible positions for another root $\beta \neq \pm \alpha$ within the 2-dimensional subspace that they span. The blue lines show the possible half-integer values of ${\frac {(\alpha ,\beta )}{(\alpha ,\alpha )}}$ and the red circles show the possible half-integer values of ${\frac {(\alpha ,\beta )}{(\beta ,\beta )}}$ . The black line shows $(\alpha ,\beta )=0$ which are all (except 0) possible values for $\beta$ . The black points show possible values for $\beta$ and are labeled with the angle ${\widehat {\alpha ,\beta }}$ .
Source	Own work
Author	Smithers888

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File history

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	Date/Time	Dimensions	User	Comment
current	19:32, 8 December 2009	750 × 400 (10 KB)	JWB	More tweaking
	04:26, 4 December 2009	750 × 450 (9 KB)	JWB	Scrap purple lines in favor of (x,y) coordinate labels at points
	23:31, 29 November 2009	750 × 450 (7 KB)	JWB	Make Y axis both blue and red to indicate both products zero on that line Label pi/2 angle
	22:54, 29 November 2009	750 × 450 (7 KB)	JWB	Indicate angles with labeled green rays instead of labels on points 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
	07:47, 29 November 2009	800 × 550 (15 KB)	Sam Derbyshire	Corrected height of vertical lines and changed the colour of the vertical line through 0 to reflect that any point on it is a valid choice of β.
	21:40, 27 November 2009	800 × 550 (7 KB)	JWB	Correct labeling of red circles. Add couple of additional labels. Trim uninteresting outer portions.
	12:15, 27 November 2009	900 × 800 (7 KB)	Smithers888	{{Information \|Description={{en\|1=Conditions on the placement of roots in a root system. Given a root <math>\alpha</math> in a root system, this image shows the possible positions for another root <math>\beta \neq \pm \alpha</math> within the 2-dimensiona