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I've been experimenting with vertex colors on the default cube. I've painted three vertices, as is visible in the picture:

vertex painting behaves differently on different faces

My question is, why does the upper right face visible in the picture form a stripe where opposite vertices are colored, while the the other two visible faces are colored differently?

It seems that in any given quad, coloring the top left and bottom right vertices will form a stripe in the middle, and the other two vertices will color in the other parts. My assumption is that this is because each quad is really two triangles, and the coloring is more greatly "stretched" along the longer side, creating the stripe in the middle. Is this correct?

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    $\begingroup$ you can check this is due to the inner triangles of the quad faces : enter edit mode, select all and press ctrl+T $\endgroup$ – lemon Aug 5 '16 at 5:46
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Indeed, this is due to the triangles which make up the quads. The 45° angles, two of which make up two of the corners of the quad, receive vertex coloring differently than the 90° angle. Basically, the corners made of two 45° angles get twice the amount of coloring.

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Subdivide each face in Edit Mode.

This will solve your problem, since it adds another vertice between the two corner vertices

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