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Geometric algebra

Geometric algebra is a draft programming task. It is not yet considered ready to be promoted as a complete task, for reasons that should be found in its talk page.

Geometric algebra is an other name for Clifford algebras and it's basically an algebra containing a vector space ${\displaystyle {\mathcal {V}}}$ and obeying the following axioms:

${\displaystyle {\begin{array}{c}(ab)c=a(bc)\\a(b+c)=ab+ac\\(a+b)c=ac+bc\\\forall \mathbf {x} \in {\mathcal {V}},\,\mathbf {x} ^{2}\in \mathbb {R} \end{array}}}$

The product operation in such algebra is called the geometric product. Elements are called multivectors, while multivectors in ${\displaystyle {\mathcal {V}}}$ are just called vectors.

There are a few simple examples of geometric algebras. A trivial one for instance is simply ${\displaystyle \mathbb {R} }$, where ${\displaystyle {\mathcal {V}}=\mathbb {R} }$. The complex numbers also form a geometric algebra, where the vector space is the one-dimensional space of all purely imaginary numbers. An other example is the space of quaternions, where the vector space is the three-dimensional space of all linear combinations of ${\displaystyle (i,j,k)}$.

The purpose of this task is to implement a geometric algebra with a vector space ${\displaystyle {\mathcal {V}}}$ of dimension n of at least five, but for extra-credit you can implement a version with n arbitrary large. Using a dimension five is useful as it is the dimension required for the so-called conformal model which will be the subject of a derived task.

To ensure the unicity of the solution (that is, up to some isomorphism), we will also restrict ourselves to the so-called euclidean case, where the square of a non-zero vector is positive:

${\displaystyle \forall \mathbf {x} \in {\mathcal {V}},\,\mathbf {x} \neq 0\implies \mathbf {x} ^{2}>0}$.

You can of course, for extra credit, implement the general case. This would require the definition of a parameter for the signature of the metric.

In order to show that your solution uses a vector space of dimension at least five, you will create a function n -> e(n) such that the vectors e(0), e(1), e(2), e(3), e(4) are linearly independent. To do so you will make them orthonormal with the following scalar product:

${\displaystyle \mathbf {x} \cdot \mathbf {y} =(\mathbf {x} \mathbf {y} +\mathbf {y} \mathbf {x} )/2}$

The fact that this so-called inner product defines a scalar product is a consequence of the fourth axiom. To see it one just needs to notice the relation:

${\displaystyle \mathbf {x} \mathbf {y} +\mathbf {y} \mathbf {x} =(\mathbf {x} +\mathbf {y} )^{2}-\mathbf {x} ^{2}-\mathbf {y} ^{2}}$

Once you'll have shown that your vector space is at least of dimension five, you will show that the axioms are satisfied. For this purpose you will pick three random multivectors a, b and c, along with a random vector ${\displaystyle \mathbf {x} }$.

Producing a random vector is easy. Just use a pseudo-random generation function rand and create a vector:

${\displaystyle \mathrm {randomVector} ()=\sum _{i=0}^{4}\mathrm {rand} ()\mathbf {e} _{i}}$

Producing a random multivector is slightly more involved. It is known that when the dimension of ${\displaystyle {\mathcal {V}}}$ is n, then the dimension of the algebra (seen as a vector space with its natural scalar multiplication) is 2n. This means that for n=5 there is a basis of 25 = 32 basis multivectors from which any multivector can be written as a linear combination. Create such a basis ${\displaystyle m_{0},m_{1},\ldots ,m_{31}}$ along with a function producting a random multivector:

${\displaystyle \mathrm {randomMultivector} ()=\sum _{i=0}^{31}\mathrm {rand} ()m_{i}}$

To summarize, to solve this task you will:

• define the inner product of two vectors : ${\displaystyle \mathbf {x} \cdot \mathbf {y} =(\mathbf {xy} +\mathbf {yx} )/2}$.
• define the function e
• verify the orthonormality ${\displaystyle \mathbf {e} _{i}\cdot \mathbf {e} _{j}=\delta _{i,j}}$ for i, j in ${\displaystyle \{0,1,2,3,4\}}$.
• create a function returning a random multivector
• create a function returning a random vector
• verify the axioms for three rarndom multivectors a, b, c and a random vector x.

Optionally, you will repeat the last step a large number of times, in order to increase confidence in the result.

C#

Translation of: D
using System;using System.Text; namespace GeometricAlgebra {    struct Vector {        private readonly double[] dims;         public Vector(double[] da) {            dims = da;        }         public static Vector operator -(Vector v) {            return v * -1.0;        }         public static Vector operator +(Vector lhs, Vector rhs) {            var result = new double[32];            Array.Copy(lhs.dims, 0, result, 0, lhs.Length);            for (int i = 0; i < result.Length; i++) {                result[i] = lhs[i] + rhs[i];            }            return new Vector(result);        }         public static Vector operator *(Vector lhs, Vector rhs) {            var result = new double[32];            for (int i = 0; i < lhs.Length; i++) {                if (lhs[i] != 0.0) {                    for (int j = 0; j < lhs.Length; j++) {                        if (rhs[j] != 0.0) {                            var s = ReorderingSign(i, j) * lhs[i] * rhs[j];                            var k = i ^ j;                            result[k] += s;//there is an index out of bounds here                        }                    }                }            }            return new Vector(result);        }         public static Vector operator *(Vector v, double scale) {            var result = (double[])v.dims.Clone();            for (int i = 0; i < result.Length; i++) {                result[i] *= scale;            }            return new Vector(result);        }         public double this[int key] {            get {                return dims[key];            }             set {                dims[key] = value;            }        }         public int Length {            get {                return dims.Length;            }        }         public Vector Dot(Vector rhs) {            return (this * rhs + rhs * this) * 0.5;        }         private static int BitCount(int i) {            i -= ((i >> 1) & 0x55555555);            i = (i & 0x33333333) + ((i >> 2) & 0x33333333);            i = (i + (i >> 4)) & 0x0F0F0F0F;            i += (i >> 8);            i += (i >> 16);            return i & 0x0000003F;        }         private static double ReorderingSign(int i, int j) {            int k = i >> 1;            int sum = 0;            while (k != 0) {                sum += BitCount(k & j);                k >>= 1;            }            return ((sum & 1) == 0) ? 1.0 : -1.0;        }         public override string ToString() {            var it = dims.GetEnumerator();             StringBuilder sb = new StringBuilder("[");            if (it.MoveNext()) {                sb.Append(it.Current);            }            while (it.MoveNext()) {                sb.Append(", ");                sb.Append(it.Current);            }             sb.Append(']');            return sb.ToString();        }    }     class Program {        static double[] DoubleArray(uint size) {            double[] result = new double[size];            for (int i = 0; i < size; i++) {                result[i] = 0.0;            }            return result;        }         static Vector E(int n) {            if (n > 4) {                throw new ArgumentException("n must be less than 5");            }             var result = new Vector(DoubleArray(32));            result[1 << n] = 1.0;            return result;        }         static readonly Random r = new Random();         static Vector RandomVector() {            var result = new Vector(DoubleArray(32));            for (int i = 0; i < 5; i++) {                var singleton = new double[] { r.NextDouble() };                result += new Vector(singleton) * E(i);            }            return result;        }         static Vector RandomMultiVector() {            var result = new Vector(DoubleArray(32));            for (int i = 0; i < result.Length; i++) {                result[i] = r.NextDouble();            }            return result;        }         static void Main() {            for (int i = 0; i < 5; i++) {                for (int j = 0; j < 5; j++) {                    if (i < j) {                        if (E(i).Dot(E(j))[0] != 0.0) {                            Console.WriteLine("Unexpected non-null sclar product.");                            return;                        }                    } else if (i == j) {                        if ((E(i).Dot(E(j)))[0] == 0.0) {                            Console.WriteLine("Unexpected null sclar product.");                        }                    }                }            }             var a = RandomMultiVector();            var b = RandomMultiVector();            var c = RandomMultiVector();            var x = RandomVector();             // (ab)c == a(bc)            Console.WriteLine((a * b) * c);            Console.WriteLine(a * (b * c));            Console.WriteLine();             // a(b+c) == ab + ac            Console.WriteLine(a * (b + c));            Console.WriteLine(a * b + a * c);            Console.WriteLine();             // (a+b)c == ac + bc            Console.WriteLine((a + b) * c);            Console.WriteLine(a * c + b * c);            Console.WriteLine();             // x^2 is real            Console.WriteLine(x * x);        }    }}
Output:
[-2.7059639813936, -2.65443237395364, -1.03355975191747, 5.431067101183, 9.57183741787636, 7.41390675997241, -8.09043009371666, -7.30180304927878, -1.50642825479215, -4.14594595273162, -1.78857280373918, -0.484382016930444, -7.42401696794793, -4.78491995868705, -8.43252648860165, -4.47485336471182, -2.3458119817208, 3.6184495826099, -7.50802924695147, -2.10106278080463, 7.15782745037037, 7.60049996423655, -10.6945339837475, -6.9874232887485, -4.85603010723139, -9.8225346377117, 3.50534384939214, 3.08239272713895, -9.92019737517891, -10.1065306142574, -8.79795495448311, -4.01442257971653]
[-2.70596398139361, -2.65443237395364, -1.03355975191747, 5.43106710118299, 9.57183741787636, 7.41390675997241, -8.09043009371666, -7.30180304927878, -1.50642825479215, -4.14594595273162, -1.78857280373917, -0.484382016930444, -7.42401696794793, -4.78491995868705, -8.43252648860165, -4.47485336471182, -2.3458119817208, 3.6184495826099, -7.50802924695147, -2.10106278080463, 7.15782745037036, 7.60049996423655, -10.6945339837475, -6.9874232887485, -4.85603010723139, -9.8225346377117, 3.50534384939214, 3.08239272713894, -9.9201973751789, -10.1065306142574, -8.79795495448311, -4.01442257971653]

[-5.35792362178344, -2.83992055857095, -0.00435065203934659, 0.548701864561436, 7.31905167177133, 2.98280880755406, 2.54042778638867, -1.82513292511705, -4.07944497473881, -3.87206774796269, 2.05320674506366, 1.00599045579458, -3.72912260067605, -3.4226426940923, 1.62806109332525, 3.26554642532303, -2.2232872618717, -2.06401927877361, 3.31046340349916, 3.21397515180161, 4.84963949791875, 1.98923326842668, 2.28775873178049, -0.927084258496161, -3.99100687576943, -1.67065148926557, 2.70438629346319, 0.400654313359669, -0.74522009023782, 0.737528754412463, 4.53246775536051, 5.82111882775243]
[-5.35792362178344, -2.83992055857095, -0.0043506520393467, 0.548701864561437, 7.31905167177133, 2.98280880755406, 2.54042778638867, -1.82513292511705, -4.0794449747388, -3.87206774796269, 2.05320674506366, 1.00599045579458, -3.72912260067605, -3.4226426940923, 1.62806109332525, 3.26554642532303, -2.2232872618717, -2.06401927877361, 3.31046340349916, 3.21397515180161, 4.84963949791875, 1.98923326842668, 2.28775873178049, -0.92708425849616, -3.99100687576943, -1.67065148926557, 2.70438629346319, 0.40065431335967, -0.745220090237821, 0.737528754412463, 4.53246775536051, 5.82111882775243]

[-6.85152530876464, -2.6466613868296, -0.814269203555543, -2.63807771948643, 6.89540453623166, 4.70804372948965, 2.78512631505336, 2.03877626337364, -1.77047482836463, 1.52836763170898, 0.71565745873906, 0.886707645932111, -2.25661460785133, -1.08891196061849, 3.44949857952115, 5.8645384965889, -3.09249704978979, -1.41518183096078, 1.87603297737169, 2.45042504250642, 3.66908389117503, 1.85358883025892, 1.23206155683761, -0.216105143607701, -1.88866851071821, -0.131570581491294, 5.7355274883507, 4.22029797577044, -0.212906215521922, -0.323884694190184, 4.41630150883192, 5.94513377054442]
[-6.85152530876464, -2.6466613868296, -0.814269203555542, -2.63807771948643, 6.89540453623166, 4.70804372948965, 2.78512631505335, 2.03877626337364, -1.77047482836463, 1.52836763170898, 0.715657458739061, 0.886707645932111, -2.25661460785133, -1.08891196061849, 3.44949857952115, 5.8645384965889, -3.09249704978979, -1.41518183096078, 1.87603297737169, 2.45042504250642, 3.66908389117503, 1.85358883025892, 1.23206155683761, -0.216105143607701, -1.88866851071821, -0.131570581491294, 5.7355274883507, 4.22029797577044, -0.212906215521922, -0.323884694190183, 4.41630150883192, 5.94513377054442]

[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]

C++

Translation of: D
#include <algorithm>#include <iostream>#include <random>#include <vector> double uniform01() {    static std::default_random_engine generator;    static std::uniform_real_distribution<double> distribution(0.0, 1.0);    return distribution(generator);} int bitCount(int i) {    i -= ((i >> 1) & 0x55555555);    i = (i & 0x33333333) + ((i >> 2) & 0x33333333);    i = (i + (i >> 4)) & 0x0F0F0F0F;    i += (i >> 8);    i += (i >> 16);    return i & 0x0000003F;} double reorderingSign(int i, int j) {    int k = i >> 1;    int sum = 0;    while (k != 0) {        sum += bitCount(k & j);        k = k >> 1;    }    return ((sum & 1) == 0) ? 1.0 : -1.0;} struct MyVector {public:    MyVector(const std::vector<double> &da) : dims(da) {        // empty    }     double &operator[](size_t i) {        return dims[i];    }     const double &operator[](size_t i) const {        return dims[i];    }     MyVector operator+(const MyVector &rhs) const {        std::vector<double> temp(dims);        for (size_t i = 0; i < rhs.dims.size(); ++i) {            temp[i] += rhs[i];        }        return MyVector(temp);    }     MyVector operator*(const MyVector &rhs) const {        std::vector<double> temp(dims.size(), 0.0);        for (size_t i = 0; i < dims.size(); i++) {            if (dims[i] != 0.0) {                for (size_t j = 0; j < dims.size(); j++) {                    if (rhs[j] != 0.0) {                        auto s = reorderingSign(i, j) * dims[i] * rhs[j];                        auto k = i ^ j;                        temp[k] += s;                    }                }            }        }        return MyVector(temp);    }     MyVector operator*(double scale) const {        std::vector<double> temp(dims);        std::for_each(temp.begin(), temp.end(), [scale](double a) { return a * scale; });        return MyVector(temp);    }     MyVector operator-() const {        return *this * -1.0;    }     MyVector dot(const MyVector &rhs) const {        return (*this * rhs + rhs * *this) * 0.5;    }     friend std::ostream &operator<<(std::ostream &, const MyVector &); private:    std::vector<double> dims;}; std::ostream &operator<<(std::ostream &os, const MyVector &v) {    auto it = v.dims.cbegin();    auto end = v.dims.cend();     os << '[';    if (it != end) {        os << *it;        it = std::next(it);    }    while (it != end) {        os << ", " << *it;        it = std::next(it);    }    return os << ']';} MyVector e(int n) {    if (n > 4) {        throw new std::runtime_error("n must be less than 5");    }     auto result = MyVector(std::vector<double>(32, 0.0));    result[1 << n] = 1.0;    return result;} MyVector randomVector() {    auto result = MyVector(std::vector<double>(32, 0.0));    for (int i = 0; i < 5; i++) {        result = result + MyVector(std::vector<double>(1, uniform01())) * e(i);    }    return result;} MyVector randomMultiVector() {    auto result = MyVector(std::vector<double>(32, 0.0));    for (int i = 0; i < 32; i++) {        result[i] = uniform01();    }    return result;} int main() {    for (int i = 0; i < 5; i++) {        for (int j = 0; j < 5; j++) {            if (i < j) {                if (e(i).dot(e(j))[0] != 0.0) {                    std::cout << "Unexpected non-null scalar product.";                    return 1;                } else if (i == j) {                    if (e(i).dot(e(j))[0] == 0.0) {                        std::cout << "Unexpected null scalar product.";                    }                }            }        }    }     auto a = randomMultiVector();    auto b = randomMultiVector();    auto c = randomMultiVector();    auto x = randomVector();     // (ab)c == a(bc)    std::cout << ((a * b) * c) << '\n';    std::cout << (a * (b * c)) << "\n\n";     // a(b+c) == ab + ac    std::cout << (a * (b + c)) << '\n';    std::cout << (a * b + a * c) << "\n\n";     // (a+b)c == ac + bc    std::cout << ((a + b) * c) << '\n';    std::cout << (a * c + b * c) << "\n\n";     // x^2 is real    std::cout << (x * x) << '\n';     return 0;}
Output:
[-5.63542, -6.59107, -10.2043, -5.21095, 8.68946, 0.579114, -4.67295, -6.72461, 1.55005, -2.63952, -1.83855, 2.4967, -4.3396, -9.9157, -4.6942, -3.23625, -2.3767, -4.55607, -14.3135, -14.2001, 9.84839, 3.69933, -3.38306, -7.60063, -0.236772, 0.988399, -0.549176, 6.61959, 4.69712, -5.34606, -12.2294, -12.6537]
[-5.63542, -6.59107, -10.2043, -5.21095, 8.68946, 0.579114, -4.67295, -6.72461, 1.55005, -2.63952, -1.83855, 2.4967, -4.3396, -9.9157, -4.6942, -3.23625, -2.3767, -4.55607, -14.3135, -14.2001, 9.84839, 3.69933, -3.38306, -7.60063, -0.236772, 0.988399, -0.549176, 6.61959, 4.69712, -5.34606, -12.2294, -12.6537]

[-6.27324, -6.7904, 3.10486, -1.14265, 6.38677, 3.57612, -3.90542, -4.17752, -1.36656, -0.0780159, 6.77775, 6.39118, 1.5939, 1.04175, 8.18152, 2.72047, -3.59085, -5.1028, 2.62711, -1.41586, 5.84934, 4.25817, 1.1197, 0.123976, -2.04301, -1.81806, 4.87518, 6.67182, 2.91358, 0.252558, 6.15595, 1.1159]
[-6.27324, -6.7904, 3.10486, -1.14265, 6.38677, 3.57612, -3.90542, -4.17752, -1.36656, -0.0780159, 6.77775, 6.39118, 1.5939, 1.04175, 8.18152, 2.72047, -3.59085, -5.1028, 2.62711, -1.41586, 5.84934, 4.25817, 1.1197, 0.123976, -2.04301, -1.81806, 4.87518, 6.67182, 2.91358, 0.252558, 6.15595, 1.1159]

[-7.29133, -8.2555, 0.985588, -1.48171, 2.4995, 4.5152, -1.1938, -2.29702, -2.34025, -2.16526, 10.2208, 7.0629, 0.552639, -0.437582, 7.18962, 2.63274, -3.25348, -4.07006, -0.883786, -3.09677, 1.1018, 2.91198, -0.0095405, 0.0123323, -2.69156, -1.30815, 3.36179, 3.26852, 3.09518, -0.166247, 6.74016, 3.20827]
[-7.29133, -8.2555, 0.985588, -1.48171, 2.4995, 4.5152, -1.1938, -2.29702, -2.34025, -2.16526, 10.2208, 7.0629, 0.552639, -0.437582, 7.18962, 2.63274, -3.25348, -4.07006, -0.883786, -3.09677, 1.1018, 2.91198, -0.0095405, 0.0123323, -2.69156, -1.30815, 3.36179, 3.26852, 3.09518, -0.166247, 6.74016, 3.20827]

[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]

D

Translation of: Kotlin
import std.exception;import std.random;import std.stdio; auto doubleArray(size_t size) {    double[] result;    result.length = size;    result[] = 0.0;    return result;} int bitCount(int i) {    i -= ((i >> 1) & 0x55555555);    i = (i & 0x33333333) + ((i >> 2) & 0x33333333);    i = (i + (i >> 4)) & 0x0F0F0F0F;    i += (i >> 8);    i += (i >> 16);    return i & 0x0000003F;} double reorderingSign(int i, int j) {    int k = i >> 1;    int sum = 0;    while (k != 0) {        sum += bitCount(k & j);        k = k >> 1;    }    return ((sum & 1) == 0) ? 1.0 : -1.0;} struct Vector {    private double[] dims;     this(double[] da) {        dims = da;    }     Vector dot(Vector rhs) {        return (this * rhs + rhs * this) * 0.5;    }     Vector opUnary(string op : "-")() {        return this * -1.0;    }     Vector opBinary(string op)(Vector rhs) {        import std.algorithm.mutation : copy;        static if (op == "+") {            auto result = doubleArray(32);            copy(dims, result);            foreach (i; 0..rhs.dims.length) {                result[i] += rhs[i];            }            return Vector(result);        } else if (op == "*") {            auto result = doubleArray(32);            foreach (i; 0..dims.length) {                if (dims[i] != 0.0) {                    foreach (j; 0..dims.length) {                        if (rhs[j] != 0.0) {                            auto s = reorderingSign(i, j) * dims[i] * rhs[j];                            auto k = i ^ j;                            result[k] += s;                        }                    }                }            }            return Vector(result);        } else {            assert(false);        }    }     Vector opBinary(string op : "*")(double scale) {        auto result = dims.dup;        foreach (i; 0..5) {            dims[i] = dims[i] * scale;        }        return Vector(result);    }     double opIndex(size_t i) {        return dims[i];    }     void opIndexAssign(double value, size_t i) {        dims[i] = value;    }} Vector e(int n) {    enforce(n <= 4, "n must be less than 5");     auto result = Vector(doubleArray(32));    result[1 << n] = 1.0;    return result;} Vector randomVector() {    auto result = Vector(doubleArray(32));    foreach (i; 0..5) {        result = result + Vector([uniform01()]) * e(i);    }    return result;} Vector randomMultiVector() {    auto result = Vector(doubleArray(32));    foreach (i; 0..32) {        result[i] = uniform01();    }    return result;} void main() {    foreach (i; 0..5) {        foreach (j; 0..5) {            if (i < j) {                if ((e(i).dot(e(j)))[0] != 0.0) {                    writeln("Unexpected non-null scalar product.");                    return;                } else if (i == j) {                    if ((e(i).dot(e(j)))[0] == 0.0) {                        writeln("Unexpected null scalar product.");                    }                }            }        }    }     auto a = randomMultiVector();    auto b = randomMultiVector();    auto c = randomMultiVector();    auto x = randomVector();     // (ab)c == a(bc)    writeln((a * b) * c);    writeln(a * (b * c));    writeln;     // a(b+c) == ab + ac    writeln(a * (b + c));    writeln(a * b + a * c);    writeln;     // (a+b)c == ac + bc    writeln((a + b) * c);    writeln(a * c + b * c);    writeln;     // x^2 is real    writeln(x * x);}
Output:
Vector([-4.60357, -1.66951, 0.230125, -4.36372, 11.0032, 7.21226, -1.5373, -6.44947, -5.07115, -1.63098, 2.90828, 7.1582, -15.5565, -1.31705, 1.3186, -1.07552, -4.04055, -2.16556, -4.41229, 0.323326, 5.03127, -1.36494, -0.915379, -6.86147, -5.87756, -4.31528, 12.4005, 15.6349, -9.54983, -1.08376, 3.60886, 4.17844])
Vector([-4.60357, -1.66951, 0.230125, -4.36372, 11.0032, 7.21226, -1.5373, -6.44947, -5.07115, -1.63098, 2.90828, 7.1582, -15.5565, -1.31705, 1.3186, -1.07552, -4.04055, -2.16556, -4.41229, 0.323326, 5.03127, -1.36494, -0.915379, -6.86147, -5.87756, -4.31528, 12.4005, 15.6349, -9.54983, -1.08376, 3.60886, 4.17844])

Vector([-3.70178, 0.430038, -3.1952, -0.878759, 2.91374, 4.86224, 3.52303, 1.66396, -0.847575, -2.11591, 1.3121, 4.42268, -3.80298, -0.773252, 5.63781, 4.70647, -2.50968, 0.196386, -1.51296, 1.92306, 2.38331, 3.00232, 4.1991, -0.254381, 0.112317, 1.53895, 2.74983, 7.70699, 0.00697711, 0.785638, 7.04352, 3.94291])
Vector([-3.70178, 0.430038, -3.1952, -0.878759, 2.91374, 4.86224, 3.52303, 1.66396, -0.847575, -2.11591, 1.3121, 4.42268, -3.80298, -0.773252, 5.63781, 4.70647, -2.50968, 0.196386, -1.51296, 1.92306, 2.38331, 3.00232, 4.1991, -0.254381, 0.112317, 1.53895, 2.74983, 7.70699, 0.00697711, 0.785638, 7.04352, 3.94291])

Vector([-4.49501, 0.40722, -3.84594, -0.429832, 6.94562, 7.06166, 1.99871, 2.27987, 0.153668, 0.16586, 1.44596, 3.74806, -3.1458, -2.35553, 3.87377, 7.18143, -2.17737, 0.987571, -1.47633, 3.38, 2.24917, 2.68417, 1.81881, 0.886526, 1.28089, 1.63795, 7.06364, 8.61935, 2.78735, -0.226174, 3.83569, 2.1715])
Vector([-4.49501, 0.40722, -3.84594, -0.429832, 6.94562, 7.06166, 1.99871, 2.27987, 0.153668, 0.16586, 1.44596, 3.74806, -3.1458, -2.35553, 3.87377, 7.18143, -2.17737, 0.987571, -1.47633, 3.38, 2.24917, 2.68417, 1.81881, 0.886526, 1.28089, 1.63795, 7.06364, 8.61935, 2.78735, -0.226174, 3.83569, 2.1715])

Vector([0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0])

EchoLisp

We build a CGA based upon a generating quadratic form in R^n. The implementation is general enough, that is ei*ei = +/- 1 , and not restricted to 1. The 5 dimension limit comes from the use of 32 bits numbers to generate all permutations 101... , but this could be improved. The multi-vector multiplication is based on a multiplication table 2^n * 2^n , generated once for all.

 (define e-bits (build-vector 32 (lambda(i) (arithmetic-shift 1 i)))) ;; 1,2,4,..(define (e-index i) ;; index of ei in native vector	(if (zero? i) 0 (arithmetic-shift 1 (1- i)))) (define DIM 0) ;; 2^N(define N 0)(define MultTable null) ;; multiplication table eijk * el.. = exyz..(define SignTable null) ;; sign of products(define Signature null) ;; input quadratic form ;; return "eijk"(define( e-print E  sign )	(string-append		(cond ((= sign 1) " ") ((= sign -1) "- ") (else ""))	  (if (zero? E) "1"	  (for/string ((i  N))	  #:continue (zero? (bitwise-and E (vector-ref e-bits  i)))	  (string-append "e" (1+ i)))))) ;; returns a string a *e1 + b*e2 + .. z*eijk + ..(define (multi-print V (x))	(for/string ((i DIM))	(set! x (vector-ref V i))	#:continue (zero? x)	(string-append " " (if (> x 0) "+" "") x  "*" (e-print i 0))))  ;; generates the multiplication table e_i e__k . * e_j e_l ..==> e_u e_v ...;; E_I and E_J are sets of indices >=1 , increasing order,  represented by a 32 bits number (define (make-mult-table (verbose #f) (result) (swaps) (ej))(when verbose (writeln 'N= N 'DIM= DIM 'Q= Signature))(for* ((E_I (in-range 1 DIM))(E_J (in-range 1 DIM)))		(set! result E_I)		(set! swaps 0)		(for ((j DIM)) ; each bit# in E_J		(set! ej (vector-ref e-bits j))		#:continue (zero? (bitwise-and  ej E_J)) 			(for((s (in-range (1- N) j -1))) ;; count swaps				(when (!zero? (bitwise-and E_I (vector-ref e-bits s)))					  (set! swaps (1+ swaps)))) 		(if (zero? (bitwise-and E_I ej)) ;; e_i * e_j		(set! result (bitwise-ior result ej))		(begin ;; else e_i * e_i		(set! result (bitwise-xor result ej))		(when (= -1 (vector-ref Signature ej)) (set! swaps (1+ swaps)))		))) ;; j loop 		(when verbose (writeln  (e-print E_I 0) '* (e-print E_J 0) 				'= (e-print result  (if (even? swaps) 1 -1)))) 		(matrix-set! MultTable E_I E_J result)		(matrix-set! SignTable E_I E_J (if (even? swaps) 1 -1))		)) ;; multivector operations;; addition is standard vector addition;; multiplication a  b -> c(define (multi-mult  a b)	(define c (make-vector DIM 0))	(for* ((i DIM) (j DIM))		#:continue (zero? (vector-ref a i))		#:continue (zero? (vector-ref b j))		(vector-set! c  			(array-ref MultTable i j) 			(+ 				(* (array-ref  SignTable i j) (vector-ref a i) (vector-ref b j))				(vector-ref c (array-ref MultTable i j)))))		c) ;; pretty print  a • b or a • b • c(define ( • a b (c #f))	(multi-print	(if c  (multi-mult a (multi-mult b c)) (multi-mult a b))))  ;; (Eij i j) ->  return multi-vector eiej 0 <= i <= n(define (Eij i j (coeff 1))	(define Eij (make-vector DIM))	(vector-set! Eij (array-ref MultTable (e-index i) (e-index j)) coeff)	Eij)  ;; Reference : https://en.wikipedia.org/wiki/Clifford_algebra#Real_numbers ;; (make-cga  m p [verbose])  => Algebra A(m p);; Input : a quadratic form Q(x) =  x1*x1 + + xm*xm - xm+1*xm+1 - xm+p*xm+p;; n = m + p = dimension of vector space R^n;; generates an algebra A(m p) of dimension  DIM  = 2^n;; Ex : A(n 0) = use R^n dot product as quadratic form : ei*ei = 1;; Ex : A (0 1) = Complex , e1*e1 = -1 ;  A(0 2) => quaternions ei*ei = -1;;;; Implementation;; limitation n <= 5;; multivectors of A(m p) will be mapped on Vectors  V of dimension 2^n;; V[0] is the scalar part of a multivector.;; Blade of vectors of R^n :  :V[2^(i-1)] = 1 , 0 elsewhere , i in [1 ..n] (define (make-cga m p (verbose #f))(string-delimiter "")	(set! N (+ m p))	(set! DIM (expt 2 N))	(set! MultTable (build-array DIM DIM (lambda(i j) (cond ((zero? i) j)((zero? j) i)(else 0)))))	(set! SignTable (make-array DIM DIM 1))	(set! Signature (make-vector DIM 1)) ;; Q polynomial	(for ((j (in-range m N))) (vector-set! Signature (vector-ref e-bits j) -1)) 	(make-mult-table verbose) DIM )
Output:
 ;; we use indices (1 ... n) in conformity with the Wikipedia reference ;; dimension 2;; (1 + e1e2) • (e1 -2e2) = -e1 -3e2(make-cga 2 0)(define u #(1 0 0 1))(define v #(0 1 -2 0)) (multi-print u) → +1*1 +1*e1e2(multi-print v) → +1*e1 -2*e2(• u v)         → -1*e1 -3*e2 ;; task(make-cga 5 0)(define X #(0 1 -1 0 2 0 0 0 -3 0 0 0 0 0 0 0 -2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ))(multi-print X) → +1*e1 -1*e2 +2*e3 -3*e4 -2*e5(• X X)  →  +19*1 ; with another polynomial(make-cga 0 5)Signature → #( 1 -1 -1 1 -1 1 1 1 -1 1 1 1 1 1 1 1 -1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1)(• X X) → -19*1 ;(make-cga 4 0)(define i (Eij 1 2))(define j (Eij 2 3))(define k (Eij 1 3)) (multi-print i) → +1*e1e2(multi-print j) → +1*e2e3(multi-print k)]→ +1*e1e3(• i i) → -1*1(• j j) → -1*1(• k k) → -1*1(• i j k) → -1*1 (define I (Eij 2 3))😖️ error: define : cannot redefine : I (used in Complex) ;; use II instead (define II (Eij 2 3)) → +1*e2e3(define J (Eij 3 4)) → +1*e3e4(define K (Eij 2 4)) → +1*e2e4 (• II II) → -1*1(• J J)  → -1*1(• K K)  → -1*1(• II J K) → -1*1

Multiplication table for A(3 0)

N= 3 DIM= 8 Q= #( 1 1 1 1 1 1 1 1)
e1 * e1 = 1
e1 * e2 = e1e2
e1 * e1e2 = e2
e1 * e3 = e1e3
e1 * e1e3 = e3
e1 * e2e3 = e1e2e3
e1 * e1e2e3 = e2e3
e2 * e1 = - e1e2
e2 * e2 = 1
e2 * e1e2 = - e1
e2 * e3 = e2e3
e2 * e1e3 = - e1e2e3
e2 * e2e3 = e3
e2 * e1e2e3 = - e1e3
e1e2 * e1 = - e2
e1e2 * e2 = e1
e1e2 * e1e2 = - 1
e1e2 * e3 = e1e2e3
e1e2 * e1e3 = - e2e3
e1e2 * e2e3 = e1e3
e1e2 * e1e2e3 = - e3
e3 * e1 = - e1e3
e3 * e2 = - e2e3
e3 * e1e2 = e1e2e3
e3 * e3 = 1
e3 * e1e3 = - e1
e3 * e2e3 = - e2
e3 * e1e2e3 = e1e2
e1e3 * e1 = - e3
e1e3 * e2 = - e1e2e3
e1e3 * e1e2 = e2e3
e1e3 * e3 = e1
e1e3 * e1e3 = - 1
e1e3 * e2e3 = - e1e2
e1e3 * e1e2e3 = e2
e2e3 * e1 = e1e2e3
e2e3 * e2 = - e3
e2e3 * e1e2 = - e1e3
e2e3 * e3 = e2
e2e3 * e1e3 = e1e2
e2e3 * e2e3 = - 1
e2e3 * e1e2e3 = - e1
e1e2e3 * e1 = e2e3
e1e2e3 * e2 = - e1e3
e1e2e3 * e1e2 = - e3
e1e2e3 * e3 = e1e2
e1e2e3 * e1e3 = e2
e1e2e3 * e2e3 = - e1
e1e2e3 * e1e2e3 = - 1

Go

Translation of: JavaScript
package main import (    "fmt"    "math/rand"    "time") type vector []float64 func e(n uint) vector {    if n > 4 {        panic("n must be less than 5")    }    result := make(vector, 32)    result[1<<n] = 1.0    return result} func cdot(a, b vector) vector {    return mul(vector{0.5}, add(mul(a, b), mul(b, a)))} func neg(x vector) vector {    return mul(vector{-1}, x)} func bitCount(i int) int {    i = i - ((i >> 1) & 0x55555555)    i = (i & 0x33333333) + ((i >> 2) & 0x33333333)    i = (i + (i >> 4)) & 0x0F0F0F0F    i = i + (i >> 8)    i = i + (i >> 16)    return i & 0x0000003F} func reorderingSign(i, j int) float64 {    i >>= 1    sum := 0    for i != 0 {        sum += bitCount(i & j)        i >>= 1    }    cond := (sum & 1) == 0    if cond {        return 1.0    }    return -1.0} func add(a, b vector) vector {    result := make(vector, 32)    copy(result, a)    for i, _ := range b {        result[i] += b[i]    }    return result} func mul(a, b vector) vector {    result := make(vector, 32)    for i, _ := range a {        if a[i] != 0 {            for j, _ := range b {                if b[j] != 0 {                    s := reorderingSign(i, j) * a[i] * b[j]                    k := i ^ j                    result[k] += s                }            }        }    }    return result} func randomVector() vector {    result := make(vector, 32)    for i := uint(0); i < 5; i++ {        result = add(result, mul(vector{rand.Float64()}, e(i)))    }    return result} func randomMultiVector() vector {    result := make(vector, 32)    for i := 0; i < 32; i++ {        result[i] = rand.Float64()    }    return result} func main() {    rand.Seed(time.Now().UnixNano())    for i := uint(0); i < 5; i++ {        for j := uint(0); j < 5; j++ {            if i < j {                if cdot(e(i), e(j))[0] != 0 {                    fmt.Println("Unexpected non-null scalar product.")                    return                }            } else if i == j {                if cdot(e(i), e(j))[0] == 0 {                    fmt.Println("Unexpected null scalar product.")                }            }        }    }     a := randomMultiVector()    b := randomMultiVector()    c := randomMultiVector()    x := randomVector()     // (ab)c == a(bc)    fmt.Println(mul(mul(a, b), c))    fmt.Println(mul(a, mul(b, c)))     // a(b + c) == ab + ac    fmt.Println(mul(a, add(b, c)))    fmt.Println(add(mul(a, b), mul(a, c)))     // (a + b)c == ac + bc    fmt.Println(mul(add(a, b), c))    fmt.Println(add(mul(a, c), mul(b, c)))     // x² is real    fmt.Println(mul(x, x))}
Output:

Sample output:

[1.6282881498413662 0.2490818261523896 -0.8936755921478269 -1.555477163901491 6.47589688756284 8.705633181089887 -1.28416798750558 -3.0984267307080446 -14.384954438859133 -13.511137120485879 6.071767421804147 1.099627765550034 -0.40641746354718655 -3.3593459129408076 -2.8089033176352967 -3.003641914720827 4.223517526463662 5.7807271315990185 -4.921271185053852 -5.698203073886508 -0.5449956221104395 3.2199941835007997 -0.4168598688210261 -1.6164380014352773 -13.447900615475964 -11.892642419707807 5.484071302025009 2.781324432176212 5.237445180167182 0.4643791234551212 -7.986755945938485 -2.9272187129576714]
[1.628288149841367 0.24908182615239083 -0.8936755921478254 -1.5554771639014895 6.475896887562836 8.705633181089889 -1.2841679875055805 -3.0984267307080433 -14.384954438859129 -13.51113712048588 6.071767421804145 1.099627765550032 -0.40641746354718755 -3.359345912940806 -2.8089033176352958 -3.003641914720825 4.223517526463663 5.7807271315990185 -4.921271185053855 -5.698203073886506 -0.5449956221104393 3.2199941835008032 -0.4168598688210253 -1.6164380014352775 -13.44790061547596 -11.89264241970781 5.484071302025003 2.781324432176209 5.237445180167183 0.464379123455121 -7.986755945938484 -2.9272187129576706]
[-4.652496095413123 -6.651741805769786 -0.18044192849719706 -1.118756694503706 -1.4545868044605725 0.10199724090664991 0.5018587820915257 2.3004721822960024 -1.813996268087529 0.38357415506855985 7.882236705126414 4.377167004918281 0.338317137066833 1.0631923204534859 5.08861779773926 4.611434580371178 -5.277764644396049 -7.961720991272197 -1.27063408303169 -1.2002120748969933 -0.3251154212726659 2.005000622005424 1.0505371909084391 1.9822320823801767 -2.271503682913346 -0.2403902213900877 6.6269980604812275 4.006018365857085 -0.15074367863748028 -0.48557428903338595 5.291057793190274 2.7751733146879394]
[-4.652496095413124 -6.651741805769786 -0.18044192849719662 -1.1187566945037064 -1.454586804460573 0.10199724090665008 0.5018587820915265 2.3004721822960037 -1.8139962680875295 0.38357415506856163 7.882236705126415 4.377167004918281 0.338317137066834 1.0631923204534859 5.088617797739261 4.611434580371178 -5.277764644396049 -7.9617209912721965 -1.2706340830316898 -1.200212074896994 -0.3251154212726654 2.0050006220054257 1.050537190908438 1.9822320823801762 -2.2715036829133437 -0.24039022139008648 6.626998060481226 4.006018365857084 -0.15074367863747984 -0.4855742890333855 5.291057793190275 2.7751733146879403]
[-4.682894668443622 -7.686899263290272 -0.2500585680601418 -0.8779897639638435 2.579501108403806 2.901924320921563 -0.4542430696469483 1.0374201754917136 -2.588607371991848 -3.229485794697976 7.444924967244714 4.93089687101153 -2.0124785310408284 0.46939350205418884 6.568153651157447 5.710192967946121 -1.41530169954937 -4.536345225213684 0.5240731948596796 1.5123734256564463 1.767786974613905 2.9960861930917018 0.969306657461082 1.036924529835111 -1.456640437477983 -1.3085896429723467 2.9678738261068895 1.3051596528834055 -3.8197616441989037 -1.3030506523824616 4.7818448875243895 4.122121121578954]
[-4.682894668443622 -7.686899263290271 -0.2500585680601421 -0.877989763963844 2.5795011084038055 2.9019243209215633 -0.4542430696469487 1.0374201754917127 -2.5886073719918494 -3.2294857946979736 7.444924967244715 4.93089687101153 -2.0124785310408275 0.46939350205418884 6.568153651157447 5.710192967946121 -1.415301699549369 -4.536345225213685 0.5240731948596802 1.512373425656447 1.7677869746139057 2.9960861930917013 0.9693066574610807 1.036924529835109 -1.456640437477983 -1.3085896429723465 2.9678738261068895 1.305159652883405 -3.819761644198904 -1.3030506523824616 4.781844887524388 4.122121121578957]
[2.1206022437357284 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0]


J

Sparse arrays give better performance for this task, than dense arrays, and support relatively arbitrary dimensions, but can be a bit quirky because current implementations of J do not support some features for sparse arrays. We add multivectors x and y using x + y. Also, the first element of one of these multivectors represents the "real valued" or "scalar component" of the multivector.

Implementation:

NB. indices are signed machine integersvzero=: 1 $.2^31+IF64*32odim=. 2^.#vzero ndx01=:1 :0: NB. indexed update of numeric rank 1 sparse y NB. creating rank 2 sparse result NB. using scalar values from x and scalar inds from m NB. where x, m are rank 0 or 1 NB. (this works around a spurious error in sparse handling) n=. #x,.m x ((i.n),&.> m)} n#,:y) NB. specify that all axes are sparse, for better displayclean=: (2;[email protected]#@$) $. ] gmul=:4 :0"1 xj=. ,4$.x  yj=. ,4$.y if. 0= xj *&# yj do. vzero return. end. b=. (-##:>./0,xj,yj)&{."1@#: xb=. b xj yb=. b yj rj=. ,#.xb~:"1/yb s=. ,_1^ ~:/"1 yb *"1/ 0,.}:"1 ~:/\"1 xb vzero (~.rj)}~ rj +//. s*,(xj{x)*/yj{y ) gdot=: (gmul + gmul~) % 2: obasis=:1 (2^i.odim)ndx01 vzeroe=: {&obasis Explanation: We work with sparse vectors of length 2147483647 on 32 bit machines and of length 9223372036854775807 on 64 bit machines. These are the largest representable vector lengths in current J implementations. J allows negative indices and J uses an index value corresponding to the length of the list to indicate value not found when searching. Thus, current J implementations use signed machine integers for performance and correctness reasons and these are the largest vector lengths we can use in J. Except, for the purpose of this task, we must pretend that these are "multivectors" instead of vectors - task vectors have a length of log2 the multivector length. So technically speaking, we can only represent 30 element (or less) vectors on 32 bit J and 62 element (or less) vectors on 64 bit J. (Actually, we can represent 31 element vectors on 32 bit J and 63 bit vectors on 64 bit J, but there are hypothetical operations involving the last element which perhaps would be hindered by the fact that the multivector length is not 2147483648 or 9223372036854775808 -- but fortunately, none of this is actually relevant.) Since these multivectors are silly large, and almost all zeros, we use a sparse representation and only concern ourselves with non-zero values, and their indices. For addition on multivectors we use J's + (and "multivectors" includes "task vectors"). For "geometric multiplication" (whose relationship to geometry seems obscure, at best) we define something analogous to an inner product. More specifically, we look at the base 2 representation of the argument indices to find the result index and to find whether we negate the product of those two values. For each pair of values in the left and right argument we use the base 2 representation of the indices to determine what to do with the product of the corresponding two non-zero values: • The result index for that product is the bit-wise exclusive-or of the two indices (the corresponding bit in the result index is 1 when the corresponding bits in the argument indices are different, 0 otherwise). • We negate the product depending on the bits of the left and right argument index. For each 1 bit of the right argument index, we count the number of more significant bits in the left argument index. If this total is odd, we negate the product (but not if it's even). Note that we don't actually have to form a sum to do this - exclusive-or is an adequate replacement for sum if we only need to know whether it's odd or even. Where we have more than one non-zero value pair contributing to the same product index we sum those values. (Of course, this would apply to value pairs whose result is zero, but since typically there's something approaching 85070591730234615847396907784232501249 of those, we don't actually compute all those zeros...) Task examples:  NB. test arbitrary vector being real (and having the specified result) clean gmul~ +/ (e 0 1 2 3 4) gmul 1 _1 2 3 _2 (0 ndx01) vzero 0 │ 19 NB. required orthogonality clean gdot&e&>/~i.40 0 0 │ 11 1 0 │ 12 2 0 │ 13 3 0 │ 1 NB. i j k i=: 0 gmul&e 1 j=: 1 gmul&e 2 k=: 0 gmul&e 2 i gmul i0 │ _1 j gmul j0 │ _1 k gmul k0 │ _1 i gmul j gmul k0 │ _1 NB. I J K I=: 1 gmul&e 2 J=: 2 gmul&e 3 K=: 1 gmul&e 3 I gmul I0 │ _1 J gmul J0 │ _1 K gmul K0 │ _1 I gmul J gmul K0 │ _1 K-J10 │ 112 │ _1 I gmul J+K10 │ 112 │ _1 Note that sparse arrays display as indices | value for elements which are not the default element (0). So, for example in the part where we check for orthogonality, we are forming a 5 by 5 by 9223372036854775807 array. The first two dimensions correspond to arguments to e and the final dimension is the multivector dimension. A 0 in the multivector dimension means that that's the "real" or "scalar" part of the multivector. And, since the pair of dimensions for the e arguments whose "dot" product are 1 are identical, we know we have an identity matrix. Java Translation of: Kotlin import java.util.Arrays;import java.util.Random; public class GeometricAlgebra { private static int bitCount(int i) { i -= ((i >> 1) & 0x55555555); i = (i & 0x33333333) + ((i >> 2) & 0x33333333); i = (i + (i >> 4)) & 0x0F0F0F0F; i += (i >> 8); i += (i >> 16); return i & 0x0000003F; } private static double reorderingSign(int i, int j) { int k = i >> 1; int sum = 0; while (k != 0) { sum += bitCount(k & j); k = k >> 1; } return ((sum & 1) == 0) ? 1.0 : -1.0; } static class Vector { private double[] dims; public Vector(double[] dims) { this.dims = dims; } public Vector dot(Vector rhs) { return times(rhs).plus(rhs.times(this)).times(0.5); } public Vector unaryMinus() { return times(-1.0); } public Vector plus(Vector rhs) { double[] result = Arrays.copyOf(dims, 32); for (int i = 0; i < rhs.dims.length; ++i) { result[i] += rhs.get(i); } return new Vector(result); } public Vector times(Vector rhs) { double[] result = new double[32]; for (int i = 0; i < dims.length; ++i) { if (dims[i] != 0.0) { for (int j = 0; j < rhs.dims.length; ++j) { if (rhs.get(j) != 0.0) { double s = reorderingSign(i, j) * dims[i] * rhs.dims[j]; int k = i ^ j; result[k] += s; } } } } return new Vector(result); } public Vector times(double scale) { double[] result = dims.clone(); for (int i = 0; i < 5; ++i) { dims[i] *= scale; } return new Vector(result); } double get(int index) { return dims[index]; } void set(int index, double value) { dims[index] = value; } @Override public String toString() { StringBuilder sb = new StringBuilder("("); boolean first = true; for (double value : dims) { if (first) { first = false; } else { sb.append(", "); } sb.append(value); } return sb.append(")").toString(); } } private static Vector e(int n) { if (n > 4) { throw new IllegalArgumentException("n must be less than 5"); } Vector result = new Vector(new double[32]); result.set(1 << n, 1.0); return result; } private static final Random rand = new Random(); private static Vector randomVector() { Vector result = new Vector(new double[32]); for (int i = 0; i < 5; ++i) { Vector temp = new Vector(new double[]{rand.nextDouble()}); result = result.plus(temp.times(e(i))); } return result; } private static Vector randomMultiVector() { Vector result = new Vector(new double[32]); for (int i = 0; i < 32; ++i) { result.set(i, rand.nextDouble()); } return result; } public static void main(String[] args) { for (int i = 0; i < 5; ++i) { for (int j = 0; j < 5; ++j) { if (i < j) { if (e(i).dot(e(j)).get(0) != 0.0) { System.out.println("Unexpected non-null scalar product."); return; } } } } Vector a = randomMultiVector(); Vector b = randomMultiVector(); Vector c = randomMultiVector(); Vector x = randomVector(); // (ab)c == a(bc) System.out.println(a.times(b).times(c)); System.out.println(a.times(b.times(c))); System.out.println(); // a(b+c) == ab + ac System.out.println(a.times(b.plus(c))); System.out.println(a.times(b).plus(a.times(c))); System.out.println(); // (a+b)c == ac + bc System.out.println(a.plus(b).times(c)); System.out.println(a.times(c).plus(b.times(c))); System.out.println(); // x^2 is real System.out.println(x.times(x)); }} Output: (-14.826385115483191, -10.223187918212313, -15.02996653452487, -14.46670632198489, 9.367877413249497, 16.476783705852135, -15.946557036515442, -6.398255774639048, 1.5560496352407314, -7.135570742872677, -3.926278194944716, 4.7948511296793646, -8.132546330778185, -6.8139397609050025, -4.203632573891533, -0.5216302370612196, -11.240590807272154, -4.724378457579242, -16.477497153082254, -16.781194046381223, 3.715894934067395, 14.220125782905383, -12.846081825616357, -2.8313637859206557, 3.149468439469149, -5.163082715280577, -7.174869565605504, -3.34527279512389, -11.232806169860876, -9.975821980348016, -3.7795503524017735, -1.0610782061627755) (-14.826385115483196, -10.223187918212307, -15.029966534524872, -14.46670632198489, 9.367877413249495, 16.476783705852135, -15.946557036515443, -6.398255774639046, 1.5560496352407327, -7.135570742872677, -3.92627819494472, 4.794851129679368, -8.132546330778192, -6.813939760904995, -4.203632573891537, -0.5216302370612151, -11.240590807272161, -4.724378457579236, -16.477497153082247, -16.781194046381223, 3.7158949340673937, 14.220125782905386, -12.846081825616348, -2.831363785920656, 3.1494684394691537, -5.163082715280574, -7.174869565605505, -3.3452727951238828, -11.232806169860877, -9.975821980348009, -3.7795503524017757, -1.0610782061627715) (-6.53148126676475, -3.686197874035778, -0.7257324996849555, 4.278415831115293, 6.812402569345597, 1.3919745454991994, -2.3737314163466, -2.534117136854738, -0.02773680252709232, -2.959894343894277, 9.079951306324414, 3.5445539013571175, 4.855317638943012, 2.7949884209613094, 3.4702307051832215, 5.9614310877021195, -6.273887650229915, -3.3318300067614355, 2.5996846757588705, 5.753030636602611, 8.66307309332676, 4.103168227184037, -2.0758275707527396, -0.7429391125916148, -1.7949737037366886, -2.709591610956401, 10.9623759848287, 5.951981557918796, 5.367767313008209, 1.288037481749198, 4.141675073009981, 7.643531800306178) (-6.5314812667647475, -3.686197874035778, -0.7257324996849563, 4.278415831115293, 6.812402569345599, 1.3919745454991996, -2.3737314163465992, -2.5341171368547366, -0.02773680252709232, -2.9598943438942764, 9.079951306324414, 3.54455390135712, 4.8553176389430135, 2.7949884209613103, 3.470230705183222, 5.96143108770212, -6.273887650229915, -3.3318300067614346, 2.5996846757588723, 5.753030636602611, 8.66307309332676, 4.103168227184039, -2.075827570752741, -0.742939112591615, -1.7949737037366884, -2.7095916109564, 10.9623759848287, 5.951981557918795, 5.367767313008202, 1.2880374817491977, 4.141675073009981, 7.64353180030618) (-7.736834353560962, -2.502599400656683, -1.3572044140931654, 0.08823728154205668, 6.035469456565145, -0.12760675585521175, 1.6863493427502443, 0.3572366658217211, -0.5993428822081044, -1.9667973240788488, 9.136744013591118, 5.590676320343252, 5.026876608331699, 2.930873432370117, 3.3588635932299327, 7.0096288327820115, -9.169529600001841, -2.648288994190627, 1.0293696456484516, 1.403189013821808, 6.487492648530549, 2.70613886829337, -0.1101834481621051, 0.9130516717900701, -1.5135624840660704, -0.03875540986063708, 7.7402678101288025, 3.7385632770106736, 2.9147287828638784, -1.2493549934763781, 2.9769191225149667, 7.2051875611073894) (-7.736834353560959, -2.502599400656682, -1.357204414093165, 0.08823728154205956, 6.035469456565146, -0.12760675585521244, 1.686349342750246, 0.35723666582172275, -0.5993428822081048, -1.9667973240788483, 9.136744013591116, 5.590676320343253, 5.026876608331699, 2.9308734323701158, 3.358863593229933, 7.009628832782012, -9.169529600001841, -2.648288994190626, 1.0293696456484527, 1.4031890138218066, 6.487492648530545, 2.7061388682933707, -0.11018344816210468, 0.9130516717900696, -1.5135624840660693, -0.038755409860637524, 7.7402678101288025, 3.738563277010674, 2.914728782863878, -1.249354993476379, 2.976919122514966, 7.205187561107389) (1.9374102817883092, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0) JavaScript var GA = function () { function e(n) { var result = []; result[1 << n] = 1; return result; } function cdot(a, b) { return mul([0.5], add(mul(a, b), mul(b, a))) } function neg(x) { return mul([-1], x) } function bitCount(i) { // Note that unsigned shifting (>>>) is not required. i = i - ((i >> 1) & 0x55555555); i = (i & 0x33333333) + ((i >> 2) & 0x33333333); i = (i + (i >> 4)) & 0x0F0F0F0F; i = i + (i >> 8); i = i + (i >> 16); return i & 0x0000003F; } function reorderingSign(a, b) { a >>= 1; var sum = 0; while (a != 0) { sum += bitCount(a & b); a >>= 1; } return (sum & 1) == 0 ? 1 : -1; } function add(a, b) { var result = a.slice(0); for (var i in b) { if (result[i]) { result[i] += b[i]; } else { result[i] = b[i]; } } return result; } function mul(a, b) { var result = []; for (var i in a) { if (a[i]) { for (var j in b) { if (b[j]) { var s = reorderingSign(i, j) * a[i] * b[j]; // if (i == 1 && j == 1) { s *= -1 } // e0*e0 == -1 var k = i ^ j; if (result[k]) { result[k] += s; } else { result[k] = s; } } } } } return result; } return { e : e, cdot : cdot, neg : neg, add : add, mul : mul };}(); And then, from the console: var e = GA.e, cdot = GA.cdot; for (var i = 0; i < 5; i++) { for (var j = 0; j < 5; j++) { if (i < j) { if (cdot(e(i), e(j))[0]) { console.log("unexpected non-nul scalar product"); } } else if (i === j) { if (!cdot(e(i), e(j))[0]) { console.log("unexpected nul scalar product"); } } }} function randomVector() { var result = []; for (var i = 0; i < 5; i++) { result = GA.add( result, GA.mul([Math.random()], e(i))); } return result;}function randomMultiVector() { var result = []; for (var i = 0; i < 32; i++) { result[i] = Math.random(); } return result;} var a = randomMultiVector(), b = randomMultiVector(), c = randomMultiVector();var x = randomVector(); // (ab)c == a(bc)console.log(GA.mul(GA.mul(a, b), c));console.log(GA.mul(a, GA.mul(b, c))); // a(b + c) == ab + acconsole.log(GA.mul(a, GA.add(b, c)));console.log(GA.add(GA.mul(a,b), GA.mul(a, c))); // (a + b)c == ac + bcconsole.log(GA.mul(GA.add(a, b), c));console.log(GA.add(GA.mul(a,c), GA.mul(b, c))); // x² is realconsole.log(GA.mul(x, x)); Output: [-7.834854130554672, -10.179405417124476, 5.696414143584243, -1.4014556169803851, 12.334288331422336, 11.690738709598888, -0.4279888274147221, 6.226618790084965, -10.904144874917206, -5.46919448234424, -5.647472225071031, -2.9801969751721744, -8.284532508545746, -3.3280413654836494, -2.2182526412098493, 0.4191036292473347, 3.0485450100607103, -0.20619687045226742, 2.1369938048939527, 3.730913391951158, 10.929856967963905, 8.301187183717643, -4.874133827873075, 0.7918650606624789, -8.520661635525103, -7.732342981599732, -6.494750491582618, -2.458749173402162, 3.573788336699224, 2.784339193089742, -1.6479372032388944, -0.35120747879544256] [-7.83485413055467, -10.179405417124475, 5.696414143584248, -1.4014556169803827, 12.334288331422337, 11.690738709598893, -0.4279888274147213, 6.226618790084964, -10.90414487491721, -5.46919448234424, -5.647472225071032, -2.9801969751721726, -8.284532508545746, -3.3280413654836507, -2.218252641209847, 0.41910362924733874, 3.048545010060707, -0.20619687045226748, 2.136993804893955, 3.7309133919511575, 10.929856967963904, 8.301187183717648, -4.8741338278730755, 0.7918650606624811, -8.520661635525107, -7.732342981599734, -6.494750491582625, -2.45874917340216, 3.5737883366992262, 2.7843391930897443, -1.6479372032388935, -0.351207478795442] [-4.5157935996060425, -3.9762419076273514, -2.653425845411889, -1.2899302330562412, 6.161562884801266, 3.664812215240675, -0.4471521091019873, 2.39303455739218, -1.6486347268701103, 1.156714478904937, 4.5859158357958965, 6.879356425817299, 1.3341425863947358, 5.641350122882839, 6.378155334673649, 6.466962714879142, -3.645688408496504, -1.9659188980662032, 1.3062519818876646, 1.7973392350972788, 2.4770203476100843, 1.258017836002405, 1.3794942194985413, 3.993871627961031, -3.3620439843097127, -0.4228490927003264, 0.27245046364398495, 3.813642689561589, 2.6785051915908604, 5.409359105713415, 2.9578168177883555, 4.425426168284635] [-4.515793599606042, -3.976241907627351, -2.653425845411889, -1.2899302330562417, 6.161562884801263, 3.664812215240676, -0.44715210910198766, 2.393034557392179, -1.6486347268701103, 1.156714478904937, 4.585915835795897, 6.8793564258172974, 1.3341425863947352, 5.641350122882839, 6.378155334673649, 6.466962714879143, -3.645688408496502, -1.9659188980662032, 1.3062519818876661, 1.7973392350972783, 2.4770203476100843, 1.258017836002407, 1.379494219498544, 3.99387162796103, -3.3620439843097127, -0.42284909270032545, 0.2724504636439853, 3.8136426895615894, 2.67850519159086, 5.409359105713415, 2.9578168177883555, 4.425426168284636] [-5.8903316026132755, -6.619647679486295, -1.8140191326116537, -2.519531799741982, 6.604158362571294, 6.352401943423508, 0.9412086471616096, 3.719341486246096, -2.209111542028446, 1.9980997124233557, 5.717878641652222, 7.351597777237362, -2.9037939632499974, 1.497897713658653, 6.811544238648882, 5.861907187665564, -3.2638975880372363, -2.2659714695119115, 1.227221599808634, 0.8343365341022846, 2.72461491531054, 2.728833585944902, 2.226404227376565, 3.888097816250177, 0.35867175462798684, 2.3965356477571302, -1.7151608532791172, 1.403673323043394, -2.1441532262277607, 2.5435142440445646, 2.00110597707534, 1.9825972651495558] [-5.8903316026132755, -6.6196476794862935, -1.8140191326116533, -2.5195317997419817, 6.604158362571292, 6.352401943423505, 0.9412086471616091, 3.719341486246094, -2.209111542028446, 1.9980997124233555, 5.71787864165222, 7.351597777237364, -2.9037939632499974, 1.4978977136586529, 6.81154423864888, 5.861907187665565, -3.263897588037235, -2.2659714695119124, 1.2272215998086353, 0.8343365341022843, 2.72461491531054, 2.7288335859449018, 2.226404227376565, 3.8880978162501783, 0.3586717546279864, 2.396535647757131, -1.715160853279117, 1.4036733230433938, -2.1441532262277603, 2.543514244044564, 2.0011059770753405, 1.9825972651495565] [3.193752260485546, 3: 0, 5: 0, 6: 0, 9: 0, 10: 0, 12: 0, 17: 0, 18: 0, 20: 0, 24: 0] Jsish From Javascript. /* Geometric Algebra, in Jsih */ var GA = function () { function e(n) { var result = []; result[1 << n] = 1; return result; } function cdot(a, b) { return mul([0.5], add(mul(a, b), mul(b, a))); } function neg(x) { return mul([-1], x); } function bitCount(i) { // Note that unsigned shifting (>>>) is not required. i = i - ((i >> 1) & 0x55555555); i = (i & 0x33333333) + ((i >> 2) & 0x33333333); i = (i + (i >> 4)) & 0x0F0F0F0F; i = i + (i >> 8); i = i + (i >> 16); return i & 0x0000003F; } function reorderingSign(a, b) { a >>= 1; var sum = 0; while (a != 0) { sum += bitCount(a & b); a >>= 1; } return (sum & 1) == 0 ? 1 : -1; } function add(a, b) { var result = a.slice(0); for (var i in b) { if (result[i]) result[i] += b[i]; else result[i] = b[i]; } return result; } function mul(a, b) { var result = []; for (var i in a) { if (a[i]) { for (var j in b) { if (b[j]) { var s = reorderingSign(i, j) * a[i] * b[j]; // if (i == 1 && j == 1) { s *= -1 } // e0*e0 == -1 var k = i ^ j; if (result[k]) result[k] += s; else result[k] = s; } } } } for (var i = 0; i < result.length; i++) result[i] = (result[i]) ? result[i] : 0; return result; } return { e:e, cdot:cdot, neg:neg, add:add, mul:mul };}(); if (Interp.conf('unitTest')) { var e = GA.e, cdot = GA.cdot; for (var i = 0; i < 5; i++) { for (var j = 0; j < 5; j++) { if (i < j) { if (cdot(e(i), e(j))[0]) { console.log("unexpected non-nul scalar product"); } } else if (i === j) { if (!cdot(e(i), e(j))[0]) { console.log("unexpected nul scalar product"); } } } } function randomVector() { var result = new Array(32).fill(0); for (var i = 0; i < 5; i++) { result = GA.add( result, GA.mul([Math.random()], e(i))); } return result; } function randomMultiVector() { var result = new Array(32).fill(0); for (var i = 0; i < 32; i++) { result[i] = Math.random(); } return result; } Math.srand(0); var a = randomMultiVector(), b = randomMultiVector(), c = randomMultiVector(); var x = randomVector(); ; a;; b;; c; // (ab)c == a(bc); GA.mul(GA.mul(a, b), c);; GA.mul(a, GA.mul(b, c)); // a(b + c) == ab + ac; GA.mul(a, GA.add(b, c));; GA.add(GA.mul(a,b), GA.mul(a, c)); // (a + b)c == ac + bc; GA.mul(GA.add(a, b), c);; GA.add(GA.mul(a,c), GA.mul(b, c)); // x² is real; x;; GA.mul(x, x);} /*=!EXPECTSTART!=a ==> [ 0.1708280361062897, 0.7499019804849638, 0.09637165562356742, 0.8704652270270756, 0.5773035067951078, 0.7857992588396741, 0.6921941534586402, 0.3687662699204211, 0.8739040768618089, 0.7450950984500651, 0.4460459090931117, 0.3537282030933753, 0.7325196320025391, 0.2602220010828802, 0.3942937749238773, 0.7767899512256164, 0.845035137580286, 0.5757882004827763, 0.7155385951686632, 0.08300424607387669, 0.4558251286597574, 0.1099468141806454, 0.5452280238165734, 0.3906865706486755, 0.5685854232144472, 0.9590664494883754, 0.8677190964591368, 0.1631895102523586, 0.2755089268683157, 0.2603610948720672, 0.9240947418691654, 0.435922637102685 ]b ==> [ 0.7894608655520905, 0.1276170168117403, 0.08220568604043521, 0.9406420164478462, 0.02557492625301805, 0.1542109313278246, 0.382182425278156, 0.1547366996666923, 0.5293334181169804, 0.8768484910832512, 0.4306114383383992, 0.2639062263420797, 0.313594499023214, 0.7700916858547231, 0.107390883054105, 0.7710422551956455, 0.7051955588944487, 0.2186396587077581, 0.7617939928559956, 0.4117130455789564, 0.648826822929113, 0.929956254907367, 0.5024185655986706, 0.6874406794288674, 0.4360909481473279, 0.6083009079497401, 0.5765586336353685, 0.6326217107140693, 0.4634256476287426, 0.6322437896100865, 0.1382949326493268, 0.9607614179251911 ]c ==> [ 0.1443750010878411, 0.4466830710677812, 0.3245844598453793, 0.9525840775924692, 0.358183910481177, 0.3982082224922436, 0.1012815422323818, 0.9550857356528795, 0.9846169476710749, 0.5759700814514872, 0.8659137353852593, 0.1498758912327744, 0.9091504302816595, 0.6512525086400416, 0.06386085476492198, 0.9549993865624558, 0.9662627642756583, 0.7855433852139697, 0.8051648178950011, 0.5712536797957526, 0.2825857199542448, 0.9625510519679636, 0.5794770589463063, 0.4368586269705688, 0.3754361745372741, 0.9234076091930206, 0.02869938006311301, 0.7688599777155005, 0.7231067717175854, 0.5909693498129656, 0.4258885385798372, 0.6421430590830184 ]GA.mul(GA.mul(a, b), c) ==> [ 1.541918542712994, 4.115829300591508, 6.756686908265433, 1.237835072374462, 10.35373455425321, 15.27788368572946, -10.02377587769663, -7.963603479812443, -4.551812682204639, -2.063312081493453, 6.501355951540042, 7.264555440540969, -8.486452071298562, -4.822494555299121, -4.759231263111541, -10.77800945200822, -7.362166274344194, -4.396761648257096, -4.546442029824131, -5.48725057173969, 7.12307450722373, 12.97562340826319, -15.0237414847673, -14.51557782429663, -4.883430414017697, -4.901409415740756, 7.906069132188758, 6.734848434665599, -9.783158191139472, -6.519424552890571, -2.139880496917612, -9.858645000418766 ]GA.mul(a, GA.mul(b, c)) ==> [ 1.541918542712995, 4.11582930059151, 6.756686908265435, 1.23783507237446, 10.35373455425321, 15.27788368572945, -10.02377587769663, -7.963603479812441, -4.551812682204639, -2.06331208149345, 6.501355951540043, 7.264555440540969, -8.486452071298562, -4.822494555299119, -4.759231263111545, -10.77800945200822, -7.362166274344201, -4.396761648257097, -4.546442029824129, -5.48725057173969, 7.12307450722373, 12.97562340826319, -15.0237414847673, -14.51557782429663, -4.883430414017695, -4.901409415740756, 7.906069132188759, 6.734848434665598, -9.783158191139472, -6.519424552890572, -2.139880496917614, -9.858645000418766 ]GA.mul(a, GA.add(b, c)) ==> [ -4.618544334910429, -1.861355248134714, -1.344616248842651, -0.2493092523303422, 2.805076632862419, 7.011456497501809, -2.595903465496111, -2.470668530528046, 2.219481357349965, 1.819582156863502, 7.750350045338696, 7.672479893390893, -0.4578513259004232, -0.2416326854384544, 6.428349361134747, 6.656718281504274, -5.564788398161922, -3.02924871963873, -0.2716453262784845, 2.840935952141899, 0.0859121820261296, 4.694618143418901, -5.096817957116206, -0.4737762603304853, -1.385313600482476, -3.455075469132893, 6.134561343411066, 4.384932175234187, -0.6284984442281132, -1.060102630996634, 6.555893195756837, 8.364446730582344 ]GA.add(GA.mul(a,b), GA.mul(a, c)) ==> [ -4.618544334910429, -1.861355248134714, -1.344616248842651, -0.2493092523303427, 2.805076632862418, 7.011456497501809, -2.595903465496112, -2.470668530528046, 2.219481357349964, 1.819582156863503, 7.750350045338696, 7.672479893390893, -0.4578513259004234, -0.2416326854384541, 6.42834936113475, 6.656718281504276, -5.564788398161921, -3.029248719638729, -0.2716453262784859, 2.840935952141899, 0.08591218202612916, 4.6946181434189, -5.096817957116205, -0.4737762603304851, -1.385313600482476, -3.455075469132893, 6.134561343411068, 4.384932175234186, -0.6284984442281131, -1.060102630996634, 6.555893195756836, 8.364446730582346 ]GA.mul(GA.add(a, b), c) ==> [ -4.443021937525184, -3.701220635832722, -0.6818565879668145, 0.239151370235406, 3.458768472629645, 6.988109725549518, -3.684631030647005, -0.8507257226324562, 2.724894692072849, 4.331141304914477, 6.87324809064668, 5.39527557728919, -1.338732261664156, 0.5042890433991951, 3.524762387495515, 6.138868250703226, -5.595434900923999, -5.095415369801449, -2.668271896337068, 0.6569878447501574, -0.03973787332027184, 4.991106401646302, -5.122908058541086, -1.273551488694048, -2.981638879697476, -2.047004560822915, 6.135097056892218, 5.062423121634416, -1.370651481167422, -0.9463086012211376, 5.574441079445841, 8.892885715782024 ]GA.add(GA.mul(a,c), GA.mul(b, c)) ==> [ -4.443021937525184, -3.701220635832721, -0.6818565879668153, 0.2391513702354063, 3.458768472629644, 6.988109725549519, -3.684631030647005, -0.8507257226324556, 2.724894692072848, 4.331141304914478, 6.873248090646679, 5.39527557728919, -1.338732261664156, 0.5042890433991953, 3.524762387495516, 6.138868250703224, -5.595434900923999, -5.095415369801452, -2.668271896337069, 0.6569878447501573, -0.03973787332027234, 4.991106401646301, -5.122908058541086, -1.273551488694048, -2.981638879697477, -2.047004560822915, 6.135097056892217, 5.062423121634419, -1.370651481167422, -0.946308601221137, 5.57444107944584, 8.892885715782022 ]x ==> [ 0, 0.7467627291006274, 0.06901854981410338, 0, 0.05329494862236217, 0, 0, 0, 0.7743136455713469, 0, 0, 0, 0, 0, 0, 0, 0.7035034807394887, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ]GA.mul(x, x) ==> [ 1.659737254471485, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ]=!EXPECTEND!=*/ Output: prompt$ jsish -u geometricAlgebra.jsi
[PASS] geometricAlgebra.jsi

Julia

Translation of: Kotlin
using GeometryTypesimport Base.* CliffordVector = Point{32, Float64} e(n) = (v = zeros(32); v[(1 << n) + 1] = 1.0; CliffordVector(v)) randommultivector() = CliffordVector(rand(32)) randomvector() = sum(i -> rand() * e(i), 0:4) bitcount(n) = (count = 0; while n != 0 n &= n - 1; count += 1 end; count) function reorderingsign(i, j)    ssum, i = 0, i >> 1    while i != 0        ssum += bitcount(i & j)        i >>= 1    end    return iseven(ssum) ? 1.0 : -1.0end function Base.:*(v1::CliffordVector, v2::CliffordVector)    result = zeros(32)    for (i, x1) in enumerate(v1)        if x1 != 0.0            for (j, x2) in enumerate(v2)                if x2 != 0.0                    s = reorderingsign(i - 1, j - 1) * x1 * x2                    k = (i - 1) ⊻ (j - 1)                    result[k + 1] += s                end            end        end    end    return CliffordVector(result)end function testcliffordvector()    allorthonormal = true    for i in 0:4, j in 0:4        i < j && all(iszero, e(i) * e(j)) != 0.0 && (allorthonormal = false)        i == j && !all(iszero, e(i) * e(j)) == 0.0 && (allorthonormal = false)    end    println("e(i) * e(j)  are orthonormal for i, j ϵ [0, 4]: ", allorthonormal)     a, b, c = randommultivector(), randommultivector(), randommultivector()    x = randomvector()     @show (a * b) * c ≈ a * (b * c)    @show a * (b + c) ≈ a * b + a * c    @show (a + b) * c ≈ a * c + b * c     isreal(x) = x[1] isa Real && all(iszero, x[2:end])    @show isreal(x * x)end  testcliffordvector()
Output:
e(i) * e(j)  are orthonormal for i, j ϵ [0, 4]: true
(a * b) * c ≈ a * (b * c) = true
a * (b + c) ≈ a * b + a * c = true
(a + b) * c ≈ a * c + b * c = true
isreal(x * x) = true


Kotlin

Translation of: Go
fun bitCount(i: Int): Int {    var j = i    j -= ((j shr 1) and 0x55555555)    j = (j and 0x33333333) + ((j shr 2) and 0x33333333)    j = (j + (j shr 4)) and 0x0F0F0F0F    j += (j shr 8)    j += (j shr 16)    return j and 0x0000003F} fun reorderingSign(i: Int, j: Int): Double {    var k = i shr 1    var sum = 0    while (k != 0) {        sum += bitCount(k and j)        k = k shr 1    }    return if (sum and 1 == 0) 1.0 else -1.0} class Vector(private val dims: DoubleArray) {     infix fun dot(rhs: Vector): Vector {        return (this * rhs + rhs * this) * 0.5    }     operator fun unaryMinus(): Vector {        return this * -1.0    }     operator fun plus(rhs: Vector): Vector {        val result = DoubleArray(32)        dims.copyInto(result)        for (i in 0 until rhs.dims.size) {            result[i] += rhs[i]        }        return Vector(result)    }     operator fun times(rhs: Vector): Vector {        val result = DoubleArray(32)        for (i in 0 until dims.size) {            if (dims[i] != 0.0) {                for (j in 0 until rhs.dims.size) {                    if (rhs[j] != 0.0) {                        val s = reorderingSign(i, j) * dims[i] * rhs[j]                        val k = i xor j                        result[k] += s                    }                }            }        }        return Vector(result)    }     operator fun times(scale: Double): Vector {        val result = dims.clone()        for (i in 0 until 5) {            dims[i] = dims[i] * scale        }        return Vector(result)    }     operator fun get(index: Int): Double {        return dims[index]    }     operator fun set(index: Int, value: Double) {        dims[index] = value    }     override fun toString(): String {        val sb = StringBuilder("(")        val it = dims.iterator()        if (it.hasNext()) {            sb.append(it.next())        }        while (it.hasNext()) {            sb.append(", ").append(it.next())        }        return sb.append(")").toString()    }} fun e(n: Int): Vector {    if (n > 4) {        throw IllegalArgumentException("n must be less than 5")    }    val result = Vector(DoubleArray(32))    result[1 shl n] = 1.0    return result} val rand = java.util.Random() fun randomVector(): Vector {    var result = Vector(DoubleArray(32))    for (i in 0 until 5) {        result += Vector(doubleArrayOf(rand.nextDouble())) * e(i)    }    return result} fun randomMultiVector(): Vector {    val result = Vector(DoubleArray(32))    for (i in 0 until 32) {        result[i] = rand.nextDouble()    }    return result} fun main() {    for (i in 0..4) {        for (j in 0..4) {            if (i < j) {                if ((e(i) dot e(j))[0] != 0.0) {                    println("Unexpected non-null scalar product.")                    return                } else if (i == j) {                    if ((e(i) dot e(j))[0] == 0.0) {                        println("Unexpected null scalar product.")                    }                }            }        }    }     val a = randomMultiVector()    val b = randomMultiVector()    val c = randomMultiVector()    val x = randomVector()     // (ab)c == a(bc)    println((a * b) * c)    println(a * (b * c))    println()     // a(b+c) == ab + ac    println(a * (b + c))    println(a * b + a * c)    println()     // (a+b)c == ac + bc    println((a + b) * c)    println(a * c + b * c)    println()     // x^2 is real    println(x * x)}
Output:
(-6.38113123172589, -6.025395204580336, 0.5762054454373319, -2.224611121553874, -0.03467815839340305, -0.6665488550665257, -3.012105902847624, 0.7315782457554153, -4.183528079943369, -1.8391037440709876, -0.137654892093293, 0.10852885457965271, -6.021317788342983, -5.486453322362711, -3.524908677069778, -1.030729377561671, -6.858194536947578, -8.724962937014816, 0.4660400096706247, -1.6434599565678671, 4.212637141194194, 2.916899539720754, -1.365566480297562, 1.898991559248674, -2.5943503153384517, -0.7167616808942235, 1.2152416665362584, 2.9936787524618067, -5.394453145898911, -4.180356766796923, -6.622391097517418, -4.249450373116712)
(-6.381131231725885, -6.025395204580336, 0.5762054454373331, -2.2246111215538744, -0.034678158393403866, -0.6665488550665265, -3.012105902847625, 0.7315782457554131, -4.183528079943374, -1.83910374407099, -0.13765489209329423, 0.1085288545796525, -6.021317788342984, -5.486453322362712, -3.524908677069777, -1.0307293775616722, -6.858194536947579, -8.724962937014814, 0.46604000967062464, -1.6434599565678658, 4.212637141194197, 2.916899539720756, -1.3655664802975644, 1.8989915592486726, -2.5943503153384526, -0.716761680894223, 1.2152416665362615, 2.9936787524618076, -5.394453145898909, -4.1803567667969235, -6.622391097517417, -4.24945037311671)

(-5.001996874378833, -5.342863347345118, 0.5930301138931318, 1.8495242515856356, 5.226690714239807, 5.60029170092836, 1.9264816320512175, 2.9647400073622383, -1.231074336132209, -0.03574406666207321, 2.0490593688751666, 2.709392811857615, -0.534934380183112, 0.9566513059906786, 0.581531676517317, 2.6867651675849284, -6.072314695960932, -5.780622914352332, -2.264083859970755, -0.29904973719224437, 3.1693202958893196, 3.710947660874322, -0.3876119148699613, 0.28047758363117253, 0.4891346543855399, 1.1102262337923456, 1.9059090529704015, 2.2175302024987404, 3.6850877929174954, 4.798530127158409, 0.6243236143375706, 2.5910748978446576)
(-5.001996874378833, -5.3428633473451175, 0.5930301138931321, 1.8495242515856363, 5.22669071423981, 5.600291700928359, 1.9264816320512164, 2.9647400073622387, -1.231074336132209, -0.035744066662073595, 2.0490593688751657, 2.7093928118576143, -0.5349343801831126, 0.9566513059906783, 0.5815316765173175, 2.6867651675849284, -6.072314695960929, -5.78062291435233, -2.264083859970754, -0.2990497371922432, 3.169320295889319, 3.7109476608743237, -0.3876119148699616, 0.2804775836311723, 0.4891346543855406, 1.1102262337923452, 1.9059090529704013, 2.21753020249874, 3.6850877929174954, 4.798530127158408, 0.6243236143375701, 2.591074897844657)

(-3.583945607396393, -3.5239300960595137, 0.801943558725803, 2.9686354622847357, 4.064529783603304, 3.6456241887862015, 0.5954304494585552, 1.2819925842824296, 0.6768216109361052, 0.352207714198545, 2.843092732483635, 2.680239849424965, -0.04943045827884884, 0.19653339086911384, 2.875432393192088, 2.8657275256839823, -4.689497942711508, -3.42309859067624, -2.830665114554725, -0.017676578013020444, 3.635425127096899, 3.10264457980534, -1.2355052114199154, 0.02671272974814648, 2.5200281685651413, 2.2030677245192627, 2.727374336905742, 1.7812768180621927, 1.901781821035387, 2.8306551025240863, 3.9572153730448463, 2.9611369534401093)
(-3.5839456073963945, -3.5239300960595115, 0.8019435587258037, 2.968635462284734, 4.064529783603305, 3.6456241887862024, 0.5954304494585542, 1.2819925842824302, 0.6768216109361052, 0.3522077141985451, 2.843092732483634, 2.6802398494249644, -0.04943045827884848, 0.1965333908691129, 2.875432393192088, 2.8657275256839827, -4.689497942711507, -3.4230985906762403, -2.8306651145547246, -0.017676578013019806, 3.6354251270968985, 3.1026445798053395, -1.2355052114199154, 0.026712729748146535, 2.5200281685651413, 2.2030677245192627, 2.7273743369057417, 1.7812768180621923, 1.9017818210353874, 2.830655102524086, 3.957215373044847, 2.9611369534401097)

(2.6501903002573224, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0)

Phix

Translation of: Go
function bitCount(integer i)    return sum(sq_eq(sprintf("%b",i),'1'))  -- (idea cribbed from Python)end function function reorderingSign(integer i, j)    i = floor(i/2)    integer tot := 0    while i!=0 do        tot += bitCount(and_bits(i,j))        i = floor(i/2)    end while    return iff(and_bits(tot,1)==0 ? 1 : -1)end function function add(sequence a, b)    return sq_add(a,b)end function function mul(sequence a, b)    sequence result = repeat(0,32)    for i=1 to length(a) do        if a[i]!=0 then            for j=1 to length(b) do                if b[j]!=0 then                    atom s := reorderingSign(i-1, j-1) * a[i] * b[j]                    integer k = xor_bits(i-1,j-1)+1                    result[k] += s                end if            end for        end if    end for    return resultend function function cdot(sequence a, b)    return mul({0.5}, add(mul(a, b), mul(b, a)))end function function e(integer n)    if n>4 then crash("n must be less than 5") end if    sequence result = repeat(0,32)    result[power(2,n)+1] = 1.0    return resultend function --function neg(sequence x) -- (not actually used here)--  return mul({-1}, x)--end function function randomVector()    sequence result = repeat(0,32)    for i=0 to 4 do        result = add(result, mul({rnd()}, e(i)))    end for    return resultend function function randomMultiVector()    sequence result = repeat(0, 32)    for i=1 to 32 do        result[i] = rnd()    end for    return resultend function for i=0 to 4 do    for j=0 to 4 do        if i < j then            if cdot(e(i), e(j))[1] != 0 then                crash("Unexpected non-null scalar product.")            end if        elsif i == j then            if cdot(e(i), e(j))[1] == 0 then                crash("Unexpected null scalar product.")            end if        end if    end forend for sequence a := randomMultiVector(),         b := randomMultiVector(),         c := randomMultiVector(),         x := randomVector(),         xsq = mul(x, x) procedure test(string txt, sequence a, b)--  bool eq = (a==b)                        -- no!    bool eq = (sprint(a)==sprint(b))        -- ok!    printf(1,"%-20s: %s\n",{txt,iff(eq?"true","false")})end procedure  test("(ab)c == a(bc)",mul(mul(a, b), c),                      mul(a, mul(b, c))) test("a(b + c) == ab + ac",mul(a, add(b, c)),                           add(mul(a, b), mul(a, c))) test("(a + b)c == ac + bc",mul(add(a, b), c),                           add(mul(a, c), mul(b, c))) test("x^2 is real",xsq,xsq[1]&repeat(0,31))
Output:

Note the comparison of string representations of floats, rather than the floats themselves. That effectively ensures they all match to 10 significant digits, and avoids a few tiny (~1e-16) discrepancies.

(ab)c == a(bc)      : true
a(b + c) == ab + ac : true
(a + b)c == ac + bc : true
x^2 is real         : true


Python

Translation of: D
import copy, random def bitcount(n):    return bin(n).count("1") def reoderingSign(i, j):    k = i >> 1    sum = 0    while k != 0:        sum += bitcount(k & j)        k = k >> 1    return 1.0 if ((sum & 1) == 0) else -1.0 class Vector:    def __init__(self, da):        self.dims = da     def dot(self, other):        return (self * other + other * self) * 0.5     def __getitem__(self, i):        return self.dims[i]     def __setitem__(self, i, v):        self.dims[i] = v     def __neg__(self):        return self * -1.0     def __add__(self, other):        result = copy.copy(other.dims)        for i in xrange(0, len(self.dims)):            result[i] += self.dims[i]        return Vector(result)     def __mul__(self, other):        if isinstance(other, Vector):            result = [0.0] * 32            for i in xrange(0, len(self.dims)):                if self.dims[i] != 0.0:                    for j in xrange(0, len(self.dims)):                        if other.dims[j] != 0.0:                            s = reoderingSign(i, j) * self.dims[i] * other.dims[j]                            k = i ^ j                            result[k] += s            return Vector(result)        else:            result = copy.copy(self.dims)            for i in xrange(0, len(self.dims)):                self.dims[i] *= other            return Vector(result)     def __str__(self):        return str(self.dims) def e(n):    assert n <= 4, "n must be less than 5"    result = Vector([0.0] * 32)    result[1 << n] = 1.0    return result def randomVector():    result = Vector([0.0] * 32)    for i in xrange(0, 5):        result += Vector([random.uniform(0, 1)]) * e(i)    return result def randomMultiVector():    result = Vector([0.0] * 32)    for i in xrange(0, 32):        result[i] = random.uniform(0, 1)    return result def main():    for i in xrange(0, 5):        for j in xrange(0, 5):            if i < j:                if e(i).dot(e(j))[0] != 0.0:                    print "Unexpected non-null scalar product"                    return                elif i == j:                    if e(i).dot(e(j))[0] == 0.0:                        print "Unexpected non-null scalar product"     a = randomMultiVector()    b = randomMultiVector()    c = randomMultiVector()    x = randomVector()     # (ab)c == a(bc)    print (a * b) * c    print a * (b * c)    print     # a(b+c) == ab + ac    print a * (b + c)    print a * b + a * c    print     # (a+b)c == ac + bc    print (a + b) * c    print a * c + b * c    print     # x^2 is real    print x * x main()
Output:
[2.646777769717816, -5.480686120935684, -8.183342078006843, -9.178717618666656, -0.21247781959240397, -3.1560121872423172, -14.210376795019405, -7.975576839132462, -2.963314079857538, -8.128489630952732, 8.84291288803876, 6.849688422048398, -3.948403894153645, -6.3295864734054295, 0.858339386946704, 0.04073276768257372, -0.8170168057484614, -5.987310468330181, -5.089567141509365, -6.5916164371098205, 1.066652018944462, -0.7553724661211869, -16.61957782752131, -10.74332838047719, -0.22326945346944393, -5.502857138805277, 11.833089760883906, 11.020055749901102, -3.7471254230186233, -3.5483496341413763, 7.788213699886802, 5.385261642366723]
[2.6467777697178145, -5.480686120935683, -8.183342078006845, -9.178717618666658, -0.2124778195924022, -3.156012187242318, -14.210376795019414, -7.975576839132467, -2.9633140798575406, -8.128489630952735, 8.842912888038763, 6.849688422048397, -3.9484038941536435, -6.329586473405431, 0.8583393869467044, 0.04073276768257594, -0.8170168057484637, -5.987310468330179, -5.089567141509367, -6.591616437109819, 1.0666520189444642, -0.7553724661211856, -16.61957782752132, -10.743328380477191, -0.22326945346944407, -5.502857138805281, 11.83308976088391, 11.0200557499011, -3.7471254230186246, -3.5483496341413807, 7.7882136998868035, 5.385261642366723]

[-4.45416862548425, -4.007728055936393, 2.2599167577088886, -0.5008511526260965, 6.404388731518947, 3.553799027487341, -1.4559344997025763, 1.200629741306491, 0.28084755190469957, 1.6881017142666677, 4.622735951152484, 2.042861306698215, -1.1859431529346458, -0.23268120656473645, 3.3088499800790308, 6.272881551311293, -6.283417063207868, -6.059129990387314, 0.9004412097752342, 0.540839567518711, 2.9772708785233157, 2.2875667777090625, 1.6049404915540564, 4.551625497411361, 3.4613544600410537, 4.601629280006443, 4.934029921827034, 3.0257810667323715, -1.280636387447562, -0.21306231994217983, 6.089428007073901, 8.128821330734313]
[-4.454168625484251, -4.007728055936392, 2.259916757708888, -0.5008511526260957, 6.404388731518948, 3.553799027487342, -1.455934499702577, 1.2006297413064915, 0.2808475519046991, 1.688101714266668, 4.622735951152485, 2.0428613066982155, -1.1859431529346458, -0.23268120656473568, 3.30884998007903, 6.272881551311293, -6.283417063207869, -6.059129990387315, 0.9004412097752316, 0.5408395675187101, 2.9772708785233157, 2.287566777709062, 1.6049404915540577, 4.551625497411363, 3.4613544600410533, 4.601629280006441, 4.934029921827035, 3.025781066732372, -1.2806363874475617, -0.21306231994217995, 6.0894280070738995, 8.128821330734311]

[-4.713407548689167, -5.470701972164317, 0.2834720300902142, -1.8843569665485043, 2.8883659013514302, 3.6276355158044815, -2.7614493177411843, 0.14095340884428206, -1.191066813989091, 1.1017933922740846, 2.995254519836379, 1.5249479602578073, 2.153333527417164, 1.999841187299821, 6.220565393668025, 8.730809912614522, -7.61200478654088, -9.557862449328784, 1.432009511995673, 0.14006605762543944, 1.450154388175902, 2.5115288790301835, -1.5609458922816675, 1.9148273860716452, 3.56683599400551, 3.9854109527025505, 3.6838872880534086, 3.059534508634617, 3.237048050921782, 2.674541802512928, 6.252577743980652, 7.309261452099341]
[-4.713407548689167, -5.470701972164316, 0.28347203009021416, -1.8843569665485043, 2.8883659013514302, 3.6276355158044815, -2.761449317741185, 0.14095340884428198, -1.1910668139890919, 1.101793392274085, 2.9952545198363785, 1.5249479602578067, 2.153333527417164, 1.9998411872998219, 6.220565393668025, 8.730809912614523, -7.612004786540879, -9.557862449328784, 1.4320095119956717, 0.14006605762543944, 1.4501543881759016, 2.511528879030183, -1.5609458922816672, 1.9148273860716456, 3.5668359940055105, 3.985410952702549, 3.683887288053409, 3.0595345086346164, 3.2370480509217803, 2.6745418025129273, 6.252577743980652, 7.30926145209934]

[0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0]

Raku

(formerly Perl 6)

Here we write a simplified version of the Clifford module. It is very general as it is of infinite dimension and also contains an anti-euclidean basis @ē in addition to the euclidean basis @e.

unit class MultiVector;subset UIntHash of MixHash where .keys.all ~~ UInt;has UIntHash $.blades;method narrow {$!blades.keys.any > 0 ?? self !!  ($!blades{0} // 0) } multi method new(Real$x) returns MultiVector { self.new: (0 => $x).MixHash }multi method new(UIntHash$blades) returns MultiVector { self.new: :$blades } multi method new(Str$ where /^^e(\d+)$$/) { self.new: (1 +< (2*0)).MixHash }multi method new(Str  where /^^ē(\d+)$$/) { self.new: (1 +< (2*$0 + 1)).MixHash } our @e is export = map { MultiVector.new: "e$_" }, ^Inf;our @ē is export = map { MultiVector.new: "ē$_" }, ^Inf; my sub order(UInt:D$i is copy, UInt:D $j) { (state %){$i}{$j} //= do { my$n = 0;	repeat {	    $i +>= 1;$n += [+] ($i +&$j).polymod(2 xx *);	} until $i == 0;$n +& 1 ?? -1 !! 1;    }} multi infix:<+>(MultiVector $A, MultiVector$B) returns MultiVector is export {    return MultiVector.new: ($A.blades.pairs, |$B.blades.pairs).MixHash;}multi infix:<+>(Real $s, MultiVector$B) returns MultiVector is export {    return MultiVector.new: (0 => $s, |$B.blades.pairs).MixHash;}multi infix:<+>(MultiVector $A, Real$s) returns MultiVector is export { $s +$A } multi infix:<*>(MultiVector $, 0) is export { 0 }multi infix:<*>(MultiVector$A, 1) returns MultiVector is export { $A }multi infix:<*>(MultiVector$A, Real $s) returns MultiVector is export { MultiVector.new:$A.blades.pairs.map({Pair.new: .key, $s*.value}).MixHash}multi infix:<*>(MultiVector$A, MultiVector $B) returns MultiVector is export { MultiVector.new: do for$A.blades -> $a { |do for$B.blades -> $b { ($a.key +^ $b.key) => [*]$a.value, $b.value, order($a.key, $b.key), |grep +*, ( |(1, -1) xx * Z* ($a.key +& $b.key).polymod(2 xx *) ) } }.MixHash}multi infix:<**>(MultiVector$ , 0) returns MultiVector is export { MultiVector.new }multi infix:<**>(MultiVector $A, 1) returns MultiVector is export {$A }multi infix:<**>(MultiVector $A, 2) returns MultiVector is export {$A * $A }multi infix:<**>(MultiVector$A, UInt $n where$n %% 2) returns MultiVector is export { ($A ** ($n div 2)) ** 2 }multi infix:<**>(MultiVector $A, UInt$n) returns MultiVector is export { $A * ($A ** ($n div 2)) ** 2 } multi infix:<*>(Real$s, MultiVector $A) returns MultiVector is export {$A * $s }multi infix:</>(MultiVector$A, Real $s) returns MultiVector is export {$A * (1/$s) }multi prefix:<->(MultiVector$A) returns MultiVector is export { return -1 * $A }multi infix:<->(MultiVector$A, MultiVector $B) returns MultiVector is export {$A + -$B }multi infix:<->(MultiVector$A, Real $s) returns MultiVector is export {$A + -$s }multi infix:<->(Real$s, MultiVector $A) returns MultiVector is export {$s + -$A } multi infix:<==>(MultiVector$A, MultiVector $B) returns Bool is export {$A - $B == 0 }multi infix:<==>(Real$x, MultiVector $A) returns Bool is export {$A == $x }multi infix:<==>(MultiVector$A, Real $x) returns Bool is export { my$narrowed = $A.narrow;$narrowed ~~ Real and $narrowed ==$x;} ###########################################  Test code to verify the solution:  ########################################### use Test; plan 29; sub infix:<cdot>($x,$y) { ($x*$y + $y*$x)/2 } for ^5 X ^5 -> ($i,$j) {    my $s =$i == $j ?? 1 !! 0; ok @e[$i] cdot @e[$j] ==$s, "e$i cdot e$j = $s";}sub random { [+] map { MultiVector.new: :blades(($_ => rand.round(.01)).MixHash)    }, ^32;} my ($a,$b, $c) = random() xx 3; ok ($a*$b)*$c == $a*($b*$c), 'associativity';ok$a*($b +$c) == $a*$b + $a*$c, 'left distributivity';ok ($a +$b)*$c ==$a*$c +$b*$c, 'right distributivity';my @coeff = (.5 - rand) xx 5;my$v = [+] @coeff Z* @e[^5];ok (\$v**2).narrow ~~ Real, 'contraction';

Visual Basic .NET

Translation of: C#
Option Strict On Imports System.Text Module Module1     Structure Vector        Private ReadOnly dims() As Double         Public Sub New(da() As Double)            dims = da        End Sub         Public Shared Operator -(v As Vector) As Vector            Return v * -1.0        End Operator         Public Shared Operator +(lhs As Vector, rhs As Vector) As Vector            Dim result(31) As Double            Array.Copy(lhs.dims, 0, result, 0, lhs.Length)            For i = 1 To result.Length                Dim i2 = i - 1                result(i2) = lhs(i2) + rhs(i2)            Next            Return New Vector(result)        End Operator         Public Shared Operator *(lhs As Vector, rhs As Vector) As Vector            Dim result(31) As Double            For i = 1 To lhs.Length                Dim i2 = i - 1                If lhs(i2) <> 0.0 Then                    For j = 1 To lhs.Length                        Dim j2 = j - 1                        If rhs(j2) <> 0.0 Then                            Dim s = ReorderingSign(i2, j2) * lhs(i2) * rhs(j2)                            Dim k = i2 Xor j2                            result(k) += s                        End If                    Next                End If            Next            Return New Vector(result)        End Operator         Public Shared Operator *(v As Vector, scale As Double) As Vector            Dim result = CType(v.dims.Clone, Double())            For i = 1 To result.Length                Dim i2 = i - 1                result(i2) *= scale            Next            Return New Vector(result)        End Operator         Default Public Property Index(key As Integer) As Double            Get                Return dims(key)            End Get            Set(value As Double)                dims(key) = value            End Set        End Property         Public ReadOnly Property Length As Integer            Get                Return dims.Length            End Get        End Property         Public Function Dot(rhs As Vector) As Vector            Return (Me * rhs + rhs * Me) * 0.5        End Function         Private Shared Function BitCount(i As Integer) As Integer            i -= ((i >> 1) And &H55555555)            i = (i And &H33333333) + ((i >> 2) And &H33333333)            i = (i + (i >> 4)) And &HF0F0F0F            i += (i >> 8)            i += (i >> 16)            Return i And &H3F        End Function         Private Shared Function ReorderingSign(i As Integer, j As Integer) As Double            Dim k = i >> 1            Dim sum = 0            While k <> 0                sum += BitCount(k And j)                k >>= 1            End While            Return If((sum And 1) = 0, 1.0, -1.0)        End Function         Public Overrides Function ToString() As String            Dim it = dims.GetEnumerator             Dim sb As New StringBuilder("[")            If it.MoveNext() Then                sb.Append(it.Current)            End If            While it.MoveNext                sb.Append(", ")                sb.Append(it.Current)            End While            sb.Append("]")            Return sb.ToString        End Function    End Structure     Function DoubleArray(size As Integer) As Double()        Dim result(size - 1) As Double        For i = 1 To size            Dim i2 = i - 1            result(i2) = 0.0        Next        Return result    End Function     Function E(n As Integer) As Vector        If n > 4 Then            Throw New ArgumentException("n must be less than 5")        End If         Dim result As New Vector(DoubleArray(32))        result(1 << n) = 1.0        Return result    End Function     ReadOnly r As New Random()     Function RandomVector() As Vector        Dim result As New Vector(DoubleArray(32))        For i = 1 To 5            Dim i2 = i - 1            Dim singleton() As Double = {r.NextDouble()}            result += New Vector(singleton) * E(i2)        Next        Return result    End Function     Function RandomMultiVector() As Vector        Dim result As New Vector(DoubleArray(32))        For i = 1 To result.Length            Dim i2 = i - 1            result(i2) = r.NextDouble()        Next        Return result    End Function     Sub Main()        For i = 1 To 5            Dim i2 = i - 1            For j = 1 To 5                Dim j2 = j - 1                If i2 < j2 Then                    If E(i2).Dot(E(j2))(0) <> 0.0 Then                        Console.Error.WriteLine("Unexpected non-null scalar product")                        Return                    End If                ElseIf i2 = j2 Then                    If E(i2).Dot(E(j2))(0) = 0.0 Then                        Console.Error.WriteLine("Unexpected null scalar product")                        Return                    End If                End If            Next        Next         Dim a = RandomMultiVector()        Dim b = RandomMultiVector()        Dim c = RandomMultiVector()        Dim x = RandomVector()         ' (ab)c == a(bc)        Console.WriteLine((a * b) * c)        Console.WriteLine(a * (b * c))        Console.WriteLine()         ' a(b+c) == ab + ac        Console.WriteLine(a * (b + c))        Console.WriteLine(a * b + a * c)        Console.WriteLine()         ' (a+b)c == ac + bc        Console.WriteLine((a + b) * c)        Console.WriteLine(a * c + b * c)        Console.WriteLine()         ' x^2 is real        Console.WriteLine(x * x)    End Sub End Module
Output:
[0.574263754349833, -1.04033308353824, -0.776121961159351, 2.00792496222078, 3.16921091358851, 6.73557610270275, -13.4327886840529, -7.74195172209513, -7.21674283588092, -5.86045645950882, 2.22312083899981, 1.32215440847646, -6.14796858424355, -6.31067579900569, -8.24016821785677, -7.13138966213629, 1.70037903072841, -1.61670055512076, -1.53920826677554, 0.0830404550813257, 3.93618132588852, 7.40029585163889, -6.36594388310886, 2.14605789872596, -12.3353817163416, -9.51720072572794, 4.57164353732716, 3.11601890765627, -3.45821025466289, -4.2333151576991, -5.88080545860622, -6.53549242641427]
[0.574263754349836, -1.04033308353825, -0.776121961159348, 2.00792496222078, 3.16921091358851, 6.73557610270275, -13.4327886840529, -7.74195172209514, -7.21674283588092, -5.86045645950883, 2.2231208389998, 1.32215440847646, -6.14796858424355, -6.31067579900569, -8.24016821785676, -7.13138966213629, 1.70037903072841, -1.61670055512076, -1.53920826677554, 0.0830404550813244, 3.93618132588852, 7.40029585163889, -6.36594388310886, 2.14605789872596, -12.3353817163416, -9.51720072572794, 4.57164353732716, 3.11601890765627, -3.4582102546629, -4.2333151576991, -5.88080545860622, -6.53549242641427]

[-4.47297646716049, -5.51675159908628, -2.0422292306007, -2.10280367601528, 4.45365145957714, 3.36803601730516, -1.14571129439412, -0.425208328771765, 1.55145295736126, 1.59542278752789, 5.96209903766593, 3.43000773717995, -1.93701906000176, -0.282031417434974, 2.27435916727966, 5.57909608271093, -4.5313823118833, -5.23492105300004, -1.61573387363407, -2.77225954890565, 2.7749883436186, 2.47317184825279, -1.66854919690689, -1.04001004327011, 0.314038463700274, 0.354414497531512, 6.65882395905864, 4.84274526210668, -1.26994943017024, 0.867830239054973, 5.4316876690133, 8.85070336472853]
[-4.47297646716048, -5.51675159908628, -2.0422292306007, -2.10280367601528, 4.45365145957714, 3.36803601730516, -1.14571129439412, -0.425208328771764, 1.55145295736126, 1.59542278752789, 5.96209903766592, 3.43000773717995, -1.93701906000176, -0.282031417434974, 2.27435916727966, 5.57909608271093, -4.5313823118833, -5.23492105300004, -1.61573387363407, -2.77225954890565, 2.7749883436186, 2.47317184825279, -1.66854919690689, -1.04001004327011, 0.314038463700274, 0.354414497531512, 6.65882395905864, 4.84274526210668, -1.26994943017024, 0.867830239054973, 5.4316876690133, 8.85070336472853]

[-5.22293180574195, -5.83486719133951, -1.69911127655253, -0.239172378976677, 2.88101068267363, 3.97970294192276, -3.56156896393757, -2.61790431554284, -0.736020241689734, -0.776473740587332, 3.0171422268862, 2.15992705303214, -3.20043960767084, -0.644050812810021, 0.714338263909879, 1.74591314489993, -4.38202839888319, -3.26051280533051, -4.05531615492948, -2.93703039040279, 3.56583825504966, 4.17636250632614, -1.11524349520128, -3.21090579244331, 1.27680857237575, 2.37418947249269, 4.48848909104827, 3.7920821695994, -3.71929857788004, 0.286175312382222, 6.0485010014393, 8.27326094456212]
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