But in 1964 Robert Berger demonstrated that this was not the case, and constructed a set of about 20,000 tiles that could cover the plane only in a nested fashion. … Penrose's tiles can cover the plane only in a nested pattern that can be constructed from a substitution system that successively subdivides each tile, as shown on page 932 . … In addition, in no case has a simple set of tiles been found which force a pattern more complicated than a nested one.

Nested patterns may occur in flocks of birds such as geese. Fairly regular nested space-filling curves are sometimes seen in the eating paths of caterpillars.

In the first two examples shown α is a quadratic irrational, so that the continued fraction is repetitive, and the pattern obtained is purely nested.

(m) is a nested pattern seen on page 583 .

There is an accumulation of limit cycles at a ≃ 3.569946 where the system has a special nested structure.

Previous approaches [to complexity] Before the discoveries in this book, nested and sometimes even repetitive behavior were quite often considered complex, and it was assumed that elaborate theories were necessary to explain them.

Paperfolding sequences The sequence of up and down creases in a strip of paper that is successively folded in half is given by a substitution system; after t steps the sequence turns out to be NestList[Join[#, {0}, Reverse[1 - #]] &, {0}, t] .

Notations [for logical primitives] Among those in current use are (highlighted ones are supported directly in Mathematica): The grouping of terms is normally inferred from precedence of operators (typically ordered  , ¬ , ⊼ , ∧ , ⊻ , ⊽ , ∨ ,  ), or explicitly indicated by parentheses, function brackets, or sometimes nested underbars or dots.

It is straightforward to construct a nested digit sequence using for example the substitution systems on page 83 , but the point is that such a digit sequence never corresponds to a number that can be obtained by the mathematical operation of taking roots.

And indeed, if one takes the patterns from successive steps and stacks them on top of each other to form a three-dimensional object, as in the picture below, then this object has a very regular nested structure.