Can an art piece have both symmetrical and asymmetrical balance?

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Yes. If a piece isn’t PERFECT in its approach to balance then there is probably some element of asymmetry. One little bit could be off–slightly out of balance visually– and while we might call the piece symmetrical, it would, in fact, not be perfect, so it would be asymmetrical. Take our faces for example, and maybe I mentioned this already. It’s one of the points about this that I love. It’s well known. If you take a photo of half your face and align it with a mirror image, you will not look like you because the two sides may look symmetrical, but in fact, they aren’t. So that’s a good example of working with symmetry but recognizing and embracing asymmetry. Once again, a rule were learn and then were can adjust, break or (with a wink and a nod), use it to make a piece more interesting!

Q: In the attached photo, I see the piece as symmetrical because of the division of the plane, and I also see it as asymmetrical because of the color balance of light against dark.

You are right about that.