Mixing

Mixing thoughts on the occasion of the 364th batch of bread.

My understanding is that if you mix up fine dry powders in a bowl with your hands, you very quickly achieve a very uniform mix. The composition of one cubic centimetre will be pretty much the same as another. And I imagine it would not be that hard to come up with some statistics about how good the mix is likely to be, perhaps in terms of sizes and proportions of the various powders. Do the varying densities of the powders have a discernible effect? Or the varying sizes of particles?

But if we now wet the powders and make them into a dough, what happens then? What about subsequent mixing? Intuitively, one now thinks that if two small bits in the dough start close together, they are going to stay close together as the dough is subjected to a series of more or less continuous deformations - where I see pulling the dough out into a disc and then folding it up as such a deformation.

Let us think of a small ball in the dough, which we perhaps manage to dye blue in some way which does not affect what it is we are trying to look at, but which does mean that we can track its progress.

Suppose we start with a brick of dough and pull it out into a flatter brick, with perhaps five times the area and one fifth of the thickness of the brick we started with. Our small ball will now be quite disc like. Then we fold the brick back up to the shape it was before - and we assume that our small flat ball is not affected by this folding.

Do the process again and our small ball is now twenty fives times as wide as it was and one twenty fifth of the thickness that it was. Say I do this twenty times and my small ball will have been deformed by some very large number, with my sum giving me 150 billion or so for 5 to the power of 16.

While it remains true that for any number ε > 0, however small, there is a number δ > 0 so that if two bits of dough start less than δ apart they will still be no more than ε apart after mixing - the only catch being that δ is a great deal smaller than ε.

The other catch is that I don't suppose that there were anything like 150 billion particles of powder in the first place. So what has gone wrong?

And what colour would the dough be? Could one see the alternating blue and white layers under a microscope, in the way that I think you can with a damascened sword. Or kitchen knife, as above.