I was wanting to pack superellipses tightly, so I worked out how to get two of them to be tangent to each other, and this allows me to create superellipses that are as large as possible and so fill space well. For this image, I turned off the requirement that the superellipses not be inside one another, and I think it has a nice look to it.
These are all 4-exponent superellipses (i.e., their shapes are the same as that of x4+y4=1).