I was a mathematics teacher at the community college level for decades, while also becoming a textbook author and interactive courseware designer. My vector calculus students loved the look of the 3D surfaces I drew on the whiteboard, and convinced me we could call it art. I have found that art is a good way to promote the beauty of mathematics, and to get people thinking and puzzling before they realize they might be doing (and enjoying) math.
We can fill 3D space with a union of disjoint circles. In this artwork I have organized the circles by spherical shells and slope, and a series of vertical rings. The spherical shells are shown, with each nuclear family of circles avoiding the two intersection points with the vertical red ring. In this way we could define a new coordinate system based on the distance from the origin, the slope of the circle, and the position along the circumference of that circle. (Inspired by Andrzej Szulkin, Amer Math Monthly Nov 1983 pp 640-41. Follow the link for a 3D interactive view!)