One can coordinatize the plane by choosing three axes at 120 degree angles and representing points by triples $(x,y,z)$ with $x+y+z=0$. Is there an accepted name for this kind of coordinate system? (It corresponds to the standard irreducible representation of $S_3$, but it is not the "standard" way to coordinatize the plane!)
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asked Apr 26, 2022 at 19:59
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5$\begingroup$ I think "barycentric coordinates" is one term that is used for essentially this system, especially in e.g. computer graphics. See for instance en.wikipedia.org/wiki/…. $\endgroup$– Sam HopkinsCommented Apr 26, 2022 at 20:01
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$\begingroup$ Buckminster Fuller spent a long time promoting an approach to geometry based on this idea under the name synergetics. \\ By the way, speaking of names, I think the standard irreducible representation of $S_3$ is more commonly called its reflection representation. $\endgroup$– LSpiceCommented Apr 26, 2022 at 22:34
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1$\begingroup$ @LSpice: "Standard representation" is also a standard name- see groupprops.subwiki.org/wiki/Standard_representation or en.wikipedia.org/wiki/…. $\endgroup$– Sam HopkinsCommented Apr 26, 2022 at 23:09
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1$\begingroup$ @JamesPropp: The only difference is $x+y+z=1$ versus $x+y+z=0$ but that just amounts to subtracting $\frac{1}{3}$ from each coordinate. $\endgroup$– Sam HopkinsCommented Apr 27, 2022 at 0:49
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2$\begingroup$ Alternatively, it really is barycentric coordinates with respect to the equilateral triangle which has vertices $(1,-1,0)$, $(0,1,-1)$, and $(-1,0,1)$. $\endgroup$– Sam HopkinsCommented Apr 27, 2022 at 0:59
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