number-pairings

JavaScript number pairing functions

This is a fun project about pairing functions, the coding of two natural numbers into one and vice versa. Now implemented are:

• Pairing for finite fields (both dimensions are limited) (field).
• Pairings for one finite and one infinite dimension (stack_y, stack_x).
• Cantor pairing (Cantor), and a half variant for unordered pairs (half).
• elegant pairing (elegant).
• POTO pairing (poto). The formulas are derived from here.

There is also a composition operator available that can mix arbitrary (finite and infinite) pairings recursively. With this operator it is possible to encode lists of natural numbers into a single number and back.

Installation

Prerequisite: Node.js needs to be installed.

1. Make a folder
2. Open a console in that new folder
3. Install locally with: npm install number-pairings

Usage

const np = require( "number-pairings" )
let pair = np.Cantor()
console.log( pair.split( 100 ) )
// => [ 4, 9 ] ( = [x, y] )
console.log( pair.join( 4, 9 ) )
// => 100 ( = z )

Replace np.Cantor with e.g. np.elegant to try out the elegant pairing.

Build

Build from source: npm run build.

Limitations

• There are no overflow checks. Since z (result of the join functions) is always in the order of a multiplication of x and y (results of the split functions) the library only works up to some numbers. There are no warnings given. Use with caution.
• There are other possible number pairings that could be included. Some of them are not "dense", i.e. more than one pair may encode a number. Here the focus is on dense pairings.

Further links

• Goedel numbering is also related to pairing functions. It also works for more than two input numbers. Coding more numbers can be achieved with this library here by using the composition operator mentioned above.
• Some more background about pairing functions.
• Another Cantor pairing JavaScript code and elegant pairing code on codepen.

License

MIT, see license file in the repository.

Examples

const np = require("number-pairings")
let p = np.Cantor()
p.join( 10, 34 ) // => 1024
p.split( 1024 ) // => [ 10, 34 ]
p = np.elegant()
p.join( 10, 34 ) // => 1200
p.split( 1200 ) // => [ 10, 34 ]
p = np.poto()
p.join( 10, 34 ) // => 70655
p.split( 70655 ) // => [ 10, 34 ]
p = np.half()
p.join( 10, 34 ) // => 605
p.join( 34, 10 ) // => 605
p.split( 605 ) // => [ 10, 34 ]
p = np.field( 2, 3 )
p.join( 1, 2 ) // => 5
p.join( 2, 2 ) // => undefined (out of bound)
p.split( 5 ) // => [ 1, 2 ]
p = np.stack_x( 5 )
p.join( 2, 4 ) // => 14
p.join( 2, 5 ) // => undefined (out of bound)
// note: use stack_y the same way as stack_x
p.split( 14 ) // => [ 2, 4 ]
p = np.composition( [ 2, 3, 4, 5 ] )
p.join( [ 0, 1, 2, 3 ] ) // => 86
p.split( 86 ) // => [ 0, 1, 2, 3 ]
p.join( [ 1, 0, 0, 0 ] ) // => 1
p.join( [ 0, 1, 0, 0 ] ) // => 2
p.join( [ 0, 0, 1, 0 ] ) // => 6
p.join( [ 0, 0, 0, 1 ] ) // => 24