a natural number size kite

consider the type of number 3^(1/4)*2*y = x for natural number y.
When you plug x in for all 3 sides of Heron's formula you get:

and then fill in for x:

So this type of number makes an equilateral triangle have a natural number area. 

Interestingly because of the 3 in 3y^2, that means:

This shape I believe is called a kite, because it is 1/3 of the triangle's area it will have an area of y^2. So it has the same area as a perfect square but a different shape. 

This hexagon has area 6*y^2. 

which is the same as the surface area of this cube:

If you imagine this tiling of hexagons made to look like cubes:

You can see layers of hexagonal tiling would be able to map the surfaces of a volume of cubes in a way that preserves area. Though not shape.