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August 27

Barycentric coordinates

In Water pouring puzzle it states "If the number of jugs is three, the filling status after each step can be described in a diagram of barycentric coordinates, because the sum of all three integers stays the same throughout all steps. In consequence the steps can be visualized as some kind of billard moves in the (clipped) coordinate system on a triangular lattice."

Can someone explain what any of that means. -- SGBailey (talk) 15:47, 27 August 2019 (UTC)[reply]

"Barycentric coordinates" would mean around the center of mass, but I don't see how that applies here. SinisterLefty (talk) 16:07, 27 August 2019 (UTC)[reply]

It really makes no sense. The barycentric coordinate system looks like a node graph from graph theory, but it really isn't. I believe the person who wrote that thought of it as a vertex/edge graph and went with something that looks the same. What you can do is create a node/edge graph starting with [0,0,0] (all empty). Send an edge out to [8,0,0], [0,5,0], and [0,0,3]. Then, from each of those vertices, send out edges to every possibility. Eventually, you will have states that already exist in the graph. The number of states is limited, so the graphing process will end at some point. 135.84.167.41 (talk) 16:37, 27 August 2019 (UTC)[reply]

They are referring to a ternary plot, which is a type of barycentric plot, where points in a triangle are located by computing the center of mass resulting from three unequal weights; those weights give the coordinates. If you duplicate the triangle and unfold it along an edge to form mirrored triangles, then any path that bounced off the the common edge of the original triangle has an associated path that continues in a straight line across the common edge into the other triangle. By unfolding again and again, ad inifinitum, you get a triangular lattice of triangles. In this lattice any straight line path can mapped to a bouncing path inside one triangle, a cute and useful mathematical trick. --{{u|Mark viking}} {Talk} 20:39, 27 August 2019 (UTC)[reply]

The key thing I have gathered from all the above is "It really makes no sense". Does anyone feel the article would be worse if I deleted that paragraph? -- SGBailey (talk) 22:03, 27 August 2019 (UTC)[reply]

What I wrote makes sense to me. I added a ref to that section for using barycentric (trilinear) coordinates. --{{u|Mark viking}} {Talk} 22:17, 27 August 2019 (UTC)[reply]

The link in the Mathworld page [1] is more informative than the Mathworld page itself. It's natural to draw the kind of triangular grid shown when you solve the problem, and the coordinate system is a convenient way of describing one, but this type of coordinate system isn't exactly on the standard math curriculum nowadays. I think it comes down what you expect the target audience for the article to be familiar with. --RDBury (talk) 22:49, 27 August 2019 (UTC)[reply]