gap>
gap># Illustration that we can still do the original stuff
gap>
gap> Read("Whiteheadv3.g");
gap> n;#The program is set up for 3-d projective space
4
gap> WA;#GAP understands Whitehead's algebra
<free left module of dimension 
15 over PolynomialRing( Rationals, ["x_1"] ), and ring>
gap> a;#We have a basis for the algebra
CanonicalBasis( <free left module of dimension 
15 over PolynomialRing( Rationals, ["x_1"] ), and ring> )
gap> AsList(a);
[ a1, a2, a3, a4, a1a2, a1a3, a1a4, a2a3, a2a4, a3a4, a1a2a3, a1a2a4, a1a3a4, 
  a2a3a4, a1a2a3a4 ]
gap> a[1];a[2];#Some reference points
a1
a2
gap> p1:=a[1]-4*a[2]+a[3]+2*a[4];#R.N. Tsai's example points
a1+(-4)*a2+a3+(2)*a4
gap> p1 in WA;#Just checking that p1 belongs to Whitehead's algebra
true
gap> p2:=-3*a[1]+2*a[2]+5*a[3]+a[4];
(-3)*a1+(2)*a2+(5)*a3+a4
gap> p3:=-2*a[2]+4*a[3]+3*a[4];
(-2)*a2+(4)*a3+(3)*a4
gap> E1:=p1*p2*p3;#Plane through points p1,p2,p3
(-24)*a1a2a3+(-16)*a1a2a4+(-4)*a1a3a4+(-16)*a2a3a4
gap> p1*E1=Zero(WA);#Check p1 is in E1
true
gap> p2*E1=Zero(WA);#Check p2 is in E1
true
gap> p3*E1=Zero(WA);#Check p3 is in E1
true
gap> L1:=p1*p2;#Line containing p1,p2
(-10)*a1a2+(8)*a1a3+(7)*a1a4+(-22)*a2a3+(-8)*a2a4+(-9)*a3a4
gap> E2:=L1*p3;#Plane containing L1,p3
(-24)*a1a2a3+(-16)*a1a2a4+(-4)*a1a3a4+(-16)*a2a3a4
gap> E1=E2;#Check planes are the same
true
gap> c:=List([1..n],i->Random(Rationals));
[ -1, 1/2, 0, 4 ]
gap> p4:=c[1]*a[1]+c[2]*a[2]+c[3]*a[3]+c[4]*a[4];#Try to make a random point
Error, no method found! For debugging hints type ?Recovery from NoMethodFound
Error, no 1st choice method found for `PROD' on 2 arguments called from
<function>( <arguments> ) called from read-eval-loop
Entering break read-eval-print loop ...
you can 'quit;' to quit to outer loop, or
you can 'return;' to continue
brk> quit;#Oops, GAP wants the coefficients as polynomials
gap> z:=Indeterminate(Rationals);
x_1
gap> c:=List([1..n],i->Random(Rationals)*z^0);
[ -3/2, 1/2, 0, -2/5 ]
gap> IsUnivariatePolynomial(c[1]);#Should be ok this time
true
gap> p4:=c[1]*a[1]+c[2]*a[2]+c[3]*a[3]+c[4]*a[4];#Second try to make random pt
(-3/2)*a1+(1/2)*a2+(-2/5)*a4
gap> p5:=LinearCombination(a{[1..n]},List([1..n],i->Random(F)*z^0));
a1+(-3/2)*a2
gap> p5 in WA;#Another random point, ok
true
gap> L2:=p4*p5;#Line through the new points
(7/4)*a1a2+(2/5)*a1a4+(-3/5)*a2a4
gap> p6:=L2*E1;#Intersection of the line on the first plane
(-83/5*x_1)*a1+(-68/5*x_1)*a2+(-44/5*x_1)*a4
gap> p6*E1=Zero(WA);#Check that the intersection point is in fact on the plane
true
gap> 
gap> #Illustrating the new version of WA as a module over a polynomial ring
gap>
gap> A;#Dual basis made from hyperplanes, x_1 is GAP's output for monomial z
[ (x_1^-1)*a2a3a4, (-x_1^-1)*a1a3a4, (x_1^-1)*a1a2a4, (-x_1^-1)*a1a2a3, 
  (x_1^-1)*a3a4, (-x_1^-1)*a2a4, (x_1^-1)*a2a3, (x_1^-1)*a1a4, 
  (-x_1^-1)*a1a3, (x_1^-1)*a1a2, (x_1^-1)*a4, (-x_1^-1)*a3, (x_1^-1)*a2, 
  (-x_1^-1)*a1, (x_1^-2)*a1a2a3a4 ]
gap> a[1]*A[1];#The hyperplanes really are dual to the points
(x_1^-1)*a1a2a3a4
gap> a[1]*A[2]=Zero(WA);
true
gap> I(a[1])=A[1];#The elliptic polarity I
true
gap> a[1]*I(a[3])=Zero(WA);
true
gap> L:=LinearCombination(A{[1..n]},List([1..n],i->Random(F)*z^0));
(x_1^-1)*a1a2a3+(2*x_1^-1)*a1a2a4+(-x_1^-1)*a1a3a4+(-x_1^-1)*a2a3a4
gap> L in WA;#L is a random hyperplane
false
gap> z*L in WA;#GAP only seems to allow polynomial rings, not Laurent poly's
true
gap> f:=function(x)#Lets make a function to do relections in hyperplane L
> return ((x*I(L))*L+(x*L)*I(L))/(I(L)*L);
> end;
function( x ) ... end
gap> f(p1);#Relect p1 in L
(3/7)*a1+(18/7)*a2+(-27/7)*a3+(-4/7)*a4
gap> f(f(p1))=p1;#The reflection is involutory as expected
true
gap> f(p1*p2)=f(p1)*f(p2);#The reflection is a ring module homomorphism
true
gap> Product(a{[1..n]});#Our pseudonumber z
a1a2a3a4
gap> Product(a{[1..n]},f)=-Product(a{[1..n]});#Showing f has det(f)=-1
true
gap> LogTo();