A Swift playground for Abstract Algebra. This project is intended to understand Algebra by implementing abstract concepts and playing with concrete objects such as Numbers, Matrices, Polynomials, etc.
Open SwiftyAlgebra.xcworkspace and press
We make use of mathematical symbols such as sets ๐, ๐, ๐, ๐ and operators โ, โ etc. Copy the folder CodeSnippets to ~/Library/Xcode/UserData/ then you can quickly input these symbols by the completion of Xcode.
let a = ๐(4, 5) // 4/5
let b = ๐(3, 2) // 3/2
a + b // 23/10
a * b // 6/5
b / a // 15/8typealias M = Matrix<_2, _2, ๐> // Matrix of integers with fixed size 2ร2.
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 noncommutativetypealias 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 -> 0typealias P = Polynominal<๐>
let f = P(0, 2, -3, 1) // x^3 โ 3x^2 + 2x
let g = P(6, -5, 1) // x^2 โ 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 - 12typealias 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
// Construct an algebraic extension over ๐:
// K = ๐(โ2) = ๐[x]/(x^2 - 2).
struct p: _Polynomial { // p = x^2 - 2, as a struct
typealias K = ๐
static let value = Polynomial<๐>(-2, 0, 1)
}
typealias I = PolynomialIdeal<p> // I = (x^2 - 2)
typealias K = QuotientField<Polynomial<๐>, I> // K = ๐[x]/I
let a = Polynomial<๐>(0, 1).asQuotient(in: K.self) // a = x mod I
a * a == 2 // true!let S2 = SimplicialComplex.sphere(dim: 2)
let H = Homology(S2, ๐.self)
print("H(S^2; ๐) =", H.detailDescription, "\n")H(S^2; ๐) = {
0 : ๐, [(v1)],
1 : 0, [],
2 : ๐, [-1(v0, v2, v3) + -1(v0, v1, v2) + (v1, v2, v3) + (v0, v1, v3)]
}
let RP2 = SimplicialComplex.realProjectiveSpace(dim: 2)
let H = Homology(RP2, ๐โ.self)
print("H(RP^2; ๐โ) =", H.detailDescription, "\n")H(RP^2; ๐โ) = {
0 : ๐โ, [(v1)],
1 : ๐โ, [(v0, v1) + (v1, v2) + (v0, v3) + (v2, v3)],
2 : ๐โ, [(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)]
}
Swifty Algebra is licensed under CC0 1.0 Universal.