adamnemecek / swiftymath

Pure Math in Pure Swift.

SwiftyMath

The aim of this project is to understand Mathematics by realizing abstract concepts as codes. Mathematical axioms correspond to protocols, and objects satisfying some axioms correspond to structs.

Getting Started

Swift REPL

With Xcode installed, you can run SwiftyMath on the Swift-REPL by:

$ swift build 
$ swift -I .build/debug/ -L .build/debug/ -ldSwiftyMath

Try something like:

:set set print-decls false // suppress print-decl
import SwiftyMath

typealias F5 = IntegerQuotientRing<_5>
F5.printAddTable()
F5.printMulTable()

Creating Your Own Project

1. Initialize a Package

$ mkdir YourProject
$ cd YourProject
$ swift package init --type executable

2. Edit Package.swift

 // swift-tools-version:4.0
 // The swift-tools-version declares the minimum version of Swift required to build this package.
 
 import PackageDescription
 
 let package = Package(
     name: "YourProject",
     dependencies: [
         // Dependencies declare other packages that this package depends on.
-        // .package(url: /* package url */, from: "1.0.0"),
+        .package(url: "https://github.com/taketo1024/SwiftyMath.git", from: "0.1.0"),
     ],
     targets: [
         // Targets are the basic building blocks of a package. A target can define a module or a test suite.
         // Targets can depend on other targets in this package, and on products in packages which this package depends on.
         .target(
             name: "YourProject",
-            dependencies: []),
+            dependencies: ["SwiftyMath", "SwiftyTopology"]),
     ]
 )

3. Edit Sources/YourProject/main.swift

import SwiftyMath

let a = 𝐐(4, 5)  // 4/5
let b = 𝐐(3, 2)  // 3/2

print(a + b)     // 23/10

4. Run

$ swift run

 23/10

Using Mathematical Symbols

We make use of mathematical symbols such as sets 𝐙, 𝐐, 𝐑, 𝐂 and operators ⊕, ⊗ etc. Copy the folder CodeSnippets to ~/Library/Developer/Xcode/UserData/ then you can quickly input these symbols by the completion of Xcode.

Examples

Rational Numbers

let a = 𝐐(4, 5)  // 4/5
let b = 𝐐(3, 2)  // 3/2

a + b  // 23/10
a * b  // 6/5
b / a  // 15/8

Matrices (type safe)

typealias M = Matrix<_2, _2, 𝐙> // Matrix of integers with fixed size 2×2.

let a = M(1, 2, 3, 4)  // [1, 2; 3, 4]
let b = M(2, 1, 1, 2)  // [2, 1; 1, 2]

a + b  // [3, 3; 4, 6]
a * b  // [4, 5; 10, 11]

a + b == b + a  // true: addition is commutative
a * b == b * a  // false: multiplication is noncommutative

Permutation (Symmetric Group)

typealias S_5 = Permutation<_5>

let s = S_5(cyclic: 0, 1, 2) // cyclic notation
let t = S_5([0: 2, 1: 3, 2: 4, 3: 0, 4: 1]) // two-line notation

s[1]  // 2
t[2]  // 4

(s * t)[3]  // 3 -> 0 -> 1
(t * s)[3]  // 3 -> 3 -> 0

Polynomials

typealias P = Polynomial<𝐐>

let f = P(0, 2, -3, 1) // x^3 − 3x^2 + 2x
let g = P(6, -5, 1)    // x^2 − 5x + 6
    
f + g  // x^3 - 2x^2 - 3x + 6
f * g  // x^5 - 8x^4 + 23x^3 - 28x^2 + 12x
f % g  // 6x - 12
    
gcd(f, g) // 6x - 12

Integer Quotients, Finite Fields

typealias Z_4 = IntegerQuotientRing<_4>
Z_4.printAddTable()

+   |   0   1   2   3
----------------------
0   |   0   1   2   3
1   |   1   2   3   0
2   |   2   3   0   1
3   |   3   0   1   2

typealias F_5 = IntegerQuotientField<_5>
F_5.printMulTable()

*   |   0   1   2   3   4
--------------------------
0   |   0   0   0   0   0
1   |   0   1   2   3   4
2   |   0   2   4   1   3
3   |   0   3   1   4   2
4   |   0   4   3   2   1

Algebraic Extension

// Construct an algebraic extension over 𝐐:
// K = 𝐐(√2) = 𝐐[x]/(x^2 - 2).

struct p: _Polynomial {                            // p = x^2 - 2, as a struct
    typealias K = 𝐐
    static let value = Polynomial<𝐐>(-2, 0, 1)
}

typealias I = PolynomialIdeal<p>                   // I = (x^2 - 2)
typealias K = QuotientField<Polynomial<𝐐>, I>      // K = 𝐐[x]/I

let a = Polynomial<𝐐>(0, 1).asQuotient(in: K.self) // a = x mod I
a * a == 2                                         // true!

Homology, Cohomology

import SwiftyMath
import SwiftyTopology

let S2 = SimplicialComplex.sphere(dim: 2)
let H = Homology(S2, 𝐙.self)
print("H(S^2; 𝐙) =", H.detailDescription, "\n")

H(S^2; 𝐙) = {
  0 : 𝐙,    [(v1)],
  1 : 0,    [],
  2 : 𝐙,    [-1(v0, v2, v3) + -1(v0, v1, v2) + (v1, v2, v3) + (v0, v1, v3)]
}

let RP2 = SimplicialComplex.realProjectiveSpace(dim: 2)
let H = Homology(RP2, 𝐙₂.self)
print("H(RP^2; 𝐙₂) =", H.detailDescription, "\n")

H(RP^2; 𝐙₂) = {
  0 : 𝐙₂,    [(v1)],
  1 : 𝐙₂,    [(v0, v1) + (v1, v2) + (v0, v3) + (v2, v3)],
  2 : 𝐙₂,    [(v0, v2, v3) + (v3, v4, v5) + (v2, v3, v5) + (v1, v2, v5) + (v0, v4, v5) + (v1, v3, v4) + (v0, v1, v5) + (v1, v2, v4) + (v0, v2, v4) + (v0, v1, v3)]
}

References

1. Swift で代数学入門
2. Swift で数学のススメ

License

Swifty Math is licensed under CC0 1.0 Universal.