This is an introductory article to the Y combinator from `lambda-calculus`. But we won’t mention the `Y combinator` in this article.

This article is the first one of a serie about the Y combinator in `javascript`:

In this article, we are going to show how to write recursive functions in `javascript` (`EcmaScript6`) without giving names to any function.

(If you are curious to see it in other languages, there is a version of the code in `clojure` and `ruby`.)

At first glance it seems impossible: how could we refer to something that we are currently defining without using its name? 1. Live: The code is executed in your browser
2. Interactive: You can modify the code and it is evaluated as you type

It will work only if your browser supports `EcmaScript6` arrow functions.

## Begin with the end in mind

The end result of this article is the recursive implementation of the `factorial` function without using neither names nor loops.

Here is the code:

``````((f => f(f)))
(func => n => (n === 0) ? 1 : (n * func(func)(n - 1)))
(19)
``````

As you can check, no mention of any names.

At first, it feels like magic.

Now, we are going to show the 4 step process that leads to this wonderful piece of code.

(We were inspired by this long but awesome article by Mike Vanier.)

# Step 0: recursive function

Let’s start with the recursive implementation of `factorial`:

``````factorial = n => (n === 0)? 1 : n * factorial(n - 1)
``````
``````factorial(10)
``````

# Step 1: simple generator

Let’s write a function that is a generator of the `factorial` function:

``````factorial_gen = f => (n => ((n === 0) ? 1 : n * f(n - 1)))
``````

One one hand, `factorial-gen` is not recursive.

On the other hand, `factorial-gen` is not the `factorial` function.

But the interesting thing is that if we pass `factorial` to `factorial-gen` it returns the `factorial` function:

``````factorial_gen(factorial)(19)
``````

Before going on reading make sure you understand why it is true that:

`factorial-gen(factorial)` is equivalent to `factorial`

# Step 2: weird generator

Now, we are going to do something very weird: instead of using `func`, we are going to use `(func func)`. Like this:

``````factorial_weird = f => (n => ((n === 0) ? 1 : n * f(f)(n - 1)))
``````

The funny thing now is that if we apply `factorial-weird` to itself we get the `factorial` function:

``````factorial_weird(factorial_weird)(19)
``````

Before going on reading make sure you understand why it is true that:

`factorial-weird(factorial-weird)` is equivalent to `factorial`

# Step 3: Recursion without names

Now, let’s write down the application of `factorial_weird` to itself, using the body of `factorial_weird` instead of its name:

``````factorial_no_names = (f => (n => ((n === 0) ? 1 : n * f(f)(n - 1))))((f => (n => ((n === 0) ? 1 : n * f(f)(n - 1)))))
``````

And we got a recursive implementation of `factorial` without using any names!

We gave it a name just for the convenience of using it.

As you can check, this is a completely valid implementation of `factorial`:

``````[1,2,3,4,5,6,7,8,9,10,11].map(factorial_no_names)
``````

Do you understand why this is equivalent to the code we shown in the beginning of the article?

``````((f => f(f)))
(func => n => (n === 0) ? 1 : (n * func(func)(n - 1)))
(19)
``````

Can you write your own implementation of other recursive functions without names?

In our next article, we are going to show the Y combinator in action in `javascript`.