(In monocotyledon plants there is usually 3-fold symmetry; in dicotyledons 4- or 5-fold. Animals like starfish often have 5-fold symmetry; higher animals usually only 2-fold symmetry. There are fossils with 7- and 9-fold symmetry.

But what kinds of shapes can folding produce? The pictures below show what happens when the local curvature—which is essentially the local rate of folding—is taken to vary according to several simple rules as one goes along a curve. … In addition to folding, there are other kinds of processes that are made possible by the lack of rigidity in a developing animal.

Probably the most important of these is folding. For folding is not only involved in producing shapes such as teeth surfaces and human ear lobes, but is also critical in allowing flat sheets of tissue to form the kinds of pockets and tubes that are so common inside animals. Folding seems to occur for a variety of reasons.

Protein folding When the molecular structure of proteins was first studied in the 1950s it was assumed that given their amino acid sequences pure minimization of energy would determine their often elaborate overall shapes. But by the 1990s it was fairly clear that in fact many details of the actual processes by which proteins are assembled can greatly affect their specific pattern of folding.

Discrete folding [in biological growth] See page 892 .

Implementation of digit sequences A whole number n can be converted to a sequence of digits in base k using IntegerDigits[n,k] or (see also page 1094 ) Reverse[Mod[NestWhileList[Floor[#/k] &, n, # ≥ k &], k]] and from a sequence of digits using FromDigits[list,k] or Fold[k #1 + #2 &, 0, list] For a number x between 0 and 1, the first m digits in its digit sequence in base k are given by RealDigits[x, k, m] or Floor[k NestList[Mod[k #, 1]&, x, m - 1]] and from these digits one can reconstruct an approximation to the number using FromDigits[{list, 0}, k] or Fold[#1/k + #2 &, 0, Reverse[list]]/k

f[n_] := n f[n - 1]; f[1] = 1 f[n_] := Product[i, {i, n}] f[n_] := Module[{t = 1}, Do[t = t i, {i, n}]; t] f[n_] := Module[{t = 1, i}, For[i = 1, i ≤ n, i++, t ⋆ = i]; t] f[n_] := Apply[Times, Range[n]] f[n_] := Fold[Times, 1, Range[n]] f[n_] := If[n  1, 1, n f[n - 1]] f[n_] := Fold[#2[#1] &, 1, Array[Function[t, # t] &, n]] f = If[#1  1, 1, #1 #0[#1 - 1]] &

Nested radicals Given a list of integers acting like digits one can consider representing numbers in the form Fold[Sqrt[#1 + #2]&, 0, Reverse[list]] . … Repeats of a digit block b give numbers that solve Fold[(#1 2 - #2) &, x, b]  x .

Nevertheless, if one looks at overall shapes into which these proteins fold, there is some evidence that the same patterns of behavior are often seen. But probably such patterns would also occur in purely random proteins—at least if their folding happened in the same cellular apparatus.

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] .