Today is Pi day, and indeed a typical way to impress people with the power of Mathematica is to display Pi to an obscene number of decimal places.
It is the purpose of this note, however, to exhibit a more refined display of power in which Mathematica creates elaborate structures purely from that venerable vessel of vacuity, the empty set, which we specify by
empty = { };
Mathematica is so powerful it can do geat things even when running on empty, so to speak. Consider the list
nonempty = NestList[Subsets, empty, 4]
This is better appreciated in the form
Grid[Transpose@{nonempty}, Frame -> All]
This can, of course be continued, and the quantity
nonempty = NestList[Subsets, empty, 5];
becomes quite large.
Short of an explicit display, nonempty does have some amusing properties:
Length /@ nonempty
{0, 1, 2, 4, 16, 65536}
Length[ToCharacterCode[ToString[#]]] & /@ nonempty
{2, 4, 10, 30, 274, 9109506}`
empty == Flatten@nonempty
True`
It should finally be noted that, as a gesture of mercy to a finite universe, Mathematica refuses to show the result for
NestList[Subsets, empty, 6];
