gives the last element of FoldList[f,x,list].


is equivalent to Fold[f,First[list],Rest[list]].


represents an operator form of Fold that can be applied to expressions.


  • You can use Throw to exit from Fold before it is finished.
  • Fold[f][list] is equivalent to Fold[f,list].
  • Fold[f][x,list] is equivalent to Fold[f,x,list].


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Basic Examples  (4)

Successively apply a function f to a seed x and the elements of a list:

Use Fold with List to create nested ordered pairs:

Multiply elements one element at a time:

Start from the first element of the list:

Scope  (8)

Use a pure function with Fold:

Use the operator form of Fold on one argument:

Use the operator form of Fold on two arguments:

Created nested powers with left-associativity:

Created nested powers with right-associativity:

Perform two subsequent permutations:

Perform a chain of cross products:

The head of the third argument need not be List:

Use Throw to exit a Fold early:

Applications  (9)

Create a nested polynomial (Horner form):

HornerForm directly produces this output:

Form a continued fraction:

Form a number from digits:

Form an alternating sum:

Form a binary tree:

Form a left-branching binary tree:

Form a function composition:

Apply an indexed sequence of functions:

Successively partition a list:

Properties & Relations  (9)

Folding with an empty list does not apply the function at all:

Equivalently, this does not apply the function at all:

Fold returns the last element of FoldList:

Fold requires a function f that takes the seed in its first argument:

To use the seed in the second argument, use ReverseApplied:

Fold takes list elements from left to right:

Use Reverse to take elements from right to left:

Combine Reverse and ReverseApplied:

Folding an additional element is equivalent to applying the function to the additional element:

Fold[f,x,{a,b,}] is equivalent to Fold[f,f[x,a],{b,}]:

Functions that ignore their second argument give the same result as in Nest:

FoldList can be computed with NestWhileList:

Fold[Append,{},list] is equivalent to list:

Possible Issues  (1)

An empty list cannot be folded without a seed:

However, the action of FoldList is well defined:

Neat Examples  (3)

An explicit form of the primitive recursive function r[z,r[s,r[s,r[s,p[2]]]]] [more info]:

Generate all subsets of a set:

Find all possible sums of any of the elements of a list of numbers:

The fourth SwinnertonDyer polynomial [more info]:

Wolfram Research (1991), Fold, Wolfram Language function, (updated 2016).


Wolfram Research (1991), Fold, Wolfram Language function, (updated 2016).


Wolfram Language. 1991. "Fold." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2016.


Wolfram Language. (1991). Fold. Wolfram Language & System Documentation Center. Retrieved from


@misc{reference.wolfram_2024_fold, author="Wolfram Research", title="{Fold}", year="2016", howpublished="\url{}", note=[Accessed: 19-July-2024 ]}


@online{reference.wolfram_2024_fold, organization={Wolfram Research}, title={Fold}, year={2016}, url={}, note=[Accessed: 19-July-2024 ]}