Calling all Mathematicians - A new idea to Sphere Stacking

Dear Mathematicians,

I’m no mathematician, I’m a Animator. I work in 3D a lot. I have a keen interest in general science and have been studying a variety of disciplines.

I have a Mathematical Problem to an idea about hexagonal close packing.

I have posted this question on the web and even contacted Henry Cohn among other professionals to ask if they could help figure out the math, but not yet had a solution

I posted: The Sphere Stacking Problem states the pyramid is the best way to stack spheres. In hexagonal close-packing there are two simple regular lattices that achieve this highest average density. They are called face-centered cubic (fcc) (also called cubic close packed) and hexagonal close-packed (hcp), based on their symmetry. Both are based upon sheets of spheres arranged at the vertices of a triangular tiling; they differ in how the sheets are stacked upon one another. The fcc lattice is also known to mathematicians as that generated by the A3 root system.

I have come up with a not yet seen solution which may stack spheres in a hexagonal shaped box in a more efficient way, using less surface area for the box (which could save money building the box) Can any mathematicians prove whether this alternative stacking in the two layers in the pictures attached would be a more efficient way than the traditional way? The centre sphere of each layer will always lie on top of each other. but there must be -1 of the centre spheres than the amount of the layers piled up to fit in the box. If the spheres are stacked in a 6 sided box does it beat the amount of spheres to box area in a square or rectangular box?

Any help? Can you share this with any mathematicians you know. If each sphere is 1 diameter what would the percentage of space left over in a hexagonal shaped box beat a square box?

I have already done plenty of math to figure out the size of the box dependant on how many spheres are in the first 2 layers and the positions of each the spheres if the first is placed at 0,0,0 in a 3D space on my computer, but I cant figure out the percentage of space left over, im stumped. There are far too many possibilities and I just don’t have the ability to solve it.

Can you advise how I should proceed? Whether I can give this idea to a student to solve or a teacher to hand out in a lesson.