Imagine that the radius of these granules is equal to 1. Then the area of each granule is _, i.e. about
3, and the area of the part of the squares that is not covered by starch is (2x2) – 3 = 1. It means that
dough can be obtained with a proportion of starch: butter of 3:1. This is an order of magnitude only, as
a closer packing can be obtained with disks placed in a hexagonal packing or with disks of different
Of course, dough is not two-dimensional, but the same question in three dimension leads to
comparing the volume of spheres (4 _/3_4) and the volume of cubes outside these spheres (2x2x2 –
4 = 4). Here again, the correction for close packing would not change much the result.
One important assumption, in all these calculations, was that disks or spheres have all the same
radius. Two millennia ago, the Greek mathematician Apollonius of Perga asked if it was possible to
cover completely a plane with disks of any size, and in 1934, the American mathematicians M.
Kaushik and M. Warren demonstrated that it was indeed possible.
Figure 3. The model of figure 2 can be improved by considering that starch granules have all
diameters, instead of being disks of the same size. Using these new shapes, it has been
demonstrated in the 30’s that a plane can be fully covered. The implication of dough would be
that any proportion between 0 and 100 % starch can be used in order to prepare “sablee”
pastry. Of course, the consistency would change!
I - 4 (of 17)
This means that sablee pastry could even be obtained by using a proportion of flour: butter equal to
1:0. Of course, this is only mathematics, not cooking, but experiments shows that dough can even be
achieved with a 30:1 proportion; of course, its consistency is not the same as with 1:1 proportion, and
the final product looks more like chapati bread than like traditional sablee pastry. Anyway, the
conclusion of this survey is that recipes are useless, from the technical point of view, as reasoning can
explain them, and even explain how much liberty the cook has.