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The design task is to assemble shapes, each with dimensions that are Fibonacci numbers, that together form a simple whole, a 13x13 square for example. We start by identifying what collection would tile the space. In other words, one must pick a collection such that the sum of the areas of the shapes is equal to the area of the whole. In addition, the x and y dimensions should be either adjacent in the series (1x2, 2x3, 3x5 ...) or skip-sequential: two-off (1x3, 2x5, 3x8, ...), three-off (1x5, 2x8, 3x13), four-off (1x8, 2x13), and five-off (1x13). This criterion eliminates all but one square from consideration; the exception is 1x1 because it is a special case in which a number is adjacent to itself in the Fibonacci series.
A 13x13 square has an area of 169 but the sum of the areas of the shapes fitting this aesthetic criterion is 377.
To find a collection with the greatest number of shapes, one should eliminate the shapes with the largest areas until the sum of their areas is equal to the area of the whole. Eliminating the 8x13 shape with an area of 104, the 5x13 with an area of 65, and the 3x13 with an area of 39 will give us the desired total area of the collection — viz 377 - 104 - 65 - 39 = 169.
Now that one has identified a collection of shapes that would tile the space, one must find a configuration that fits together appropriately. The simplest method for tackling this step is to make shapes with the proper dimensions and shuffle them around until they fit. Although there are many possible configurations that would fit together to make up the desired overall shape, there is a much larger number of configurations that don't.
The photo to the left is an example of a possible configuration.
Unfortunately, many of the components line up with one another and too many are oriented with the long axis running left-to-right as opposed to up-and-down: 8 vs 4.
After some more shuffling, one gets the configuration to the right.
Fewer components line up with each other and there are as many with the long axis running up-and-down as left-to-right: 6 and 6.
The final step is to pick the stone. Ideally, each component should be of a different kind of stone than every other one.
Two of the myriad sets of choices are shown here.
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