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Copy pathdemo.h
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demo.h
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void demo1 (int const mult) {
octonion::set_default_mult (mult);
cout << "\n\n ========================== ";
cout << "\n\n demo1 using multiplication table " << octonion::get_default_mult();
cout << "\n verify some algebraic laws";
cout << "\n ";
cout << "\n a.abs * b.abs = " << a_abs * b_abs;
cout << "\n (a * b).abs = " << (a * b).abs ();
{
cout << "\n\n The following pairs will be equal (alternate associativity): ";
{
octonion const o1 {(a * a) * b };
octonion const o2 { a * (a * b)};
cout << "\n ";
cout << "\n (a * a) * b = " << o1;
cout << "\n a * (a * b) = " << o2;
} {
octonion const o1 {(a * b) * a };
octonion const o2 { a * (b * a)};
cout << "\n ";
cout << "\n (a * b) * a = " << o1;
cout << "\n a * (b * a) = " << o2 ;
} {
octonion const o1 {(b * a) * a };
octonion const o2 { b * (a * a)};
cout << "\n ";
cout << "\n (b * a) * a = " << o1;
cout << "\n b * (a * a) = " << o2;
}
} {
cout << "\n\n The following pair or quadruple will be equal (conjugate alternate associativity): ";
{
octonion const o1 {(a * b) * ~a };
octonion const o2 { a * (b * ~a)};
cout << "\n ";
cout << "\n ( a * b) * ~a = " << o1;
cout << "\n a * (b * ~a) = " << o2 ;
} {
octonion const o1 {(~a * a) * b };
octonion const o2 { ~a * (a * b)};
octonion const o3 {( b * a) * ~a };
octonion const o4 { b * (a * ~a)};
cout << "\n ";
cout << "\n (~a * a) * b = " << o1;
cout << "\n ~a * (a * b) = " << o2;
cout << "\n ( b * a) * ~a = " << o3;
cout << "\n b * (a * ~a) = " << o4;
}
} {
cout << "\n\n The following pairs will be equal (distributivity): ";
{
octonion const o1 {(a + b) * c};
octonion const o2 {a * c + b * c};
cout << "\n ";
cout << "\n (a + b) * c = " << o1;
cout << "\n a * c + b * c = " << o2;
} {
octonion const o1 {a * (b + c)};
octonion const o2 {a * b + a * c};
cout << "\n ";
cout << "\n a * (b + c) = " << o1;
cout << "\n a * b + a * c = " << o2;
} {
octonion const o1 {(a - b) * c};
octonion const o2 {a * c - b * c};
cout << "\n ";
cout << "\n (a - b) * c = " << o1;
cout << "\n a * c - b * c = " << o2;
} {
octonion const o1 {a * (b - c)};
octonion const o2 {a * b - a * c};
cout << "\n ";
cout << "\n a * (b - c) = " << o1;
cout << "\n a * b - a * c = " << o2;
}
} {
cout << "\n\n The following pairs or triplet will be equal (Moufang identity): ";
{
octonion const o1 { a * (b * (a * c))};
octonion const o2 {((a * b) * a) * c };
cout << "\n ";
cout << "\n a * (b * (a * c)) = " << o1;
cout << "\n ((a * b) * a) * c = " << o2;
} {
octonion const o1 { b * (a * (c * a))};
octonion const o2 {((b * a) * c) * a };
cout << "\n ";
cout << "\n b * (a * (c * a)) = " << o1;
cout << "\n ((b * a) * c) * a = " << o2;
} {
octonion const o1 {(a * (b * c)) * a };
octonion const o2 {(a * b) * (c * a)};
octonion const o3 { a * ((b * c) * a)};
cout << "\n ";
cout << "\n (a * (b * c)) * a = " << o1;
cout << "\n (a * b) * (c * a) = " << o2;
cout << "\n a * ((b * c) * a) = " << o3;
}
} {
cout << "\n\n The following pair will be equal (conjugate Moufang identity): ";
{
octonion const o1 {(~a * (b * c)) * a };
octonion const o2 { ~a * ((b * c) * a)};
cout << "\n ";
cout << "\n (~a * (b * c)) * a = " << o1;
cout << "\n ~a * ((b * c) * a) = " << o2;
}
} {
cout << "\n\n The following pair will in general not be equal (noncommutativity): ";
octonion const o1 {a * b};
octonion const o2 {b * a};
cout << "\n ";
cout << "\n ab = " << o1 << " abs = " << o1.abs();
cout << "\n ba = " << o2 << " abs = " << o2.abs();
} {
cout << "\n\n The following pair will in general not be equal (nonassociativity): ";
octonion const o1 {(a * b) * c };
octonion const o2 { a * (b * c)};
cout << "\n ";
cout << "\n (a * b) * c = " << o1 << " abs = " << o1.abs();
cout << "\n a * (b * c) = " << o2 << " abs = " << o2.abs();
}
}
void demo2 () {
cout << "\n\n ========================== ";
cout << "\n\n demo 2: ";
cout << "\n verify that all 480 products are different";
struct duet {
int n;
octonion o;
duet (int const j, octonion const & p)
: n {j}, o {p}
{ }
bool operator< (duet const & d) const
{ return o < d.o; }
};
std::vector<duet> table;
for (int i {0}; i < 480; ++i) {
octonion const q {a.mult (b, i)};
table.emplace_back (i, q);
}
std::sort (table.begin(), table.end());
for (auto const & i : table)
cout << "\n " << setw (3) << i.n << ": " << i.o;
}
void demo3 () {
cout << "\n\n ========================== ";
cout << "\n\n demo 3: ";
cout << "\n verify invariance";
cout << "\n a * b * a = " << a.mult_aba (b);
for (int i {0}; i < 480; ++i) {
cout << "\n\n mult table = " << setw(3) << i;
octonion const ab {a.mult (b, i)};
octonion const ab_a {ab.mult (a, i)};
cout << "\n (a * b) * a = " << ab_a;
octonion const ba {b.mult (a, i)};
octonion const a_ba {a.mult (ba, i)};
cout << "\n a * (b * a) = " << a_ba;
}
}
void demo4 () {
cout << "\n\n ========================== ";
cout << "\n\n demo 4: ";
cout << "\n verify conjugate invariance";
cout << "\n ~a * a * b = " << a.mult_conj_aab (b);
for (int i {0}; i < 480; ++i) {
cout << "\n\n mult table = " << setw(3) << i;
octonion const c {~a};
octonion const ca {c.mult (a, i)};
octonion const ca_b {ca.mult (b, i)};
cout << "\n (~a * a) * b = " << ca_b;
octonion const ab {a.mult (b, i)};
octonion const c_ab {c.mult (ab, i)};
cout << "\n ~a * (a * b) = " << c_ab;
octonion const ba {b.mult (a, i)};
octonion const ba_c {ba.mult (c, i)};
cout << "\n ( b * a) * ~a = " << ba_c;
octonion const ac {a.mult (c, i)};
octonion const b_ac {b.mult (ac, i)};
cout << "\n b * (a * ~a) = " << b_ac;
}
}
void demo5 () {
cout << "\n\n ========================== ";
cout << "\n\n demo 5: ";
cout << "\n\n examine distributivity for a * b * a and ~a * a * b";
cout << "\n\n The following pairs will be equal (distributivity): ";
{
octonion const o1 {a.mult_aba (b) + a.mult_aba (c)};
octonion const o2 {a.mult_aba (b + c)};
cout << "\n ";
cout << "\n a * b * a + a * c * a = " << o1;
cout << "\n a * (b + c) * a = " << o2;
}
{
octonion const o1 {a.mult_conj_aab (b) + a.mult_conj_aab (c)};
octonion const o2 {a.mult_conj_aab (b + c)};
cout << "\n ";
cout << "\n ~a * a * b + ~a * a * c = " << o1;
cout << "\n ~a * a (b + c) = " << o2;
}
cout << "\n\n The following pairs will in general not be equal (nondistributivity): ";
{
octonion const o1 {a.mult_aba (c) + b.mult_aba (c)};
octonion const o2 {(a + b).mult_aba (c)};
cout << "\n ";
cout << "\n a * c * a + b * c * b = " << o1;
cout << "\n (a + b) * c * (a + b) = " << o2;
} {
octonion const o1 {a.mult_conj_aab (c) + b.mult_conj_aab (c)};
octonion const o2 {(a + b).mult_conj_aab (c)};
cout << "\n ";
cout << "\n ~a * a * c + ~b * b * c = " << o1;
cout << "\n ~(a + b) * (a + b) * c = " << o2;
}
}