@asa12 

If you insist in not using polar coordinates, try:

L:=NULL;
kmax:=0; # increase it to see more of the spiral
for ii from 0 by 5 to 365 do
print('ii'=ii);
for k from 0 to kmax do
eq1 := sqrt(x^2 + y^2) - arctan(y,x) = 2*k*Pi;
eq2 := y = evalf((-Pi/4 - ii*Pi/180)/Pi)*(-1.0)*x;
xy := fsolve({eq1, eq2});
#print(xy):
L:=L,eval([x,y],xy);
od:od:
plot([L],style=point);

@acer 

I do not think that it is very useful to use Maple as a funky typewriter. It is very nice that Maple offers the possibility to use true mathematical notations (as in the Physics or DifferentialGeometry packages). But here the mathematics involved is close to 0.

There are so many things to be done mathematically in Maple! For presentations, we have LaTeX, and a better export to it would be nice. The vector 3d graphics should also have a higher priority.

@Markiyan Hirnyk 

It is easy to modify convexhull to work properly with irrationals.
I did it, but I don't post here because in the past such code was deleted motivated by copyright.

You forgot with(geometry).

@Markiyan Hirnyk 

Solid people try showstat(convexhull).

@Markiyan Hirnyk 

In convexhull the coordinates are converted to floats, so the irrationals are not treated properly.

My favorite  "Next Number" Puzzle:

1  2  4   8  16  ?

"Correct" answer: 31

Proof:

with(CurveFitting):
PolynomialInterpolation([[0,1],[1,2],[2,4],[3,8],[4,16]], x):
eval(%,x=5);
      31

@taro 

In an ideal world, simplify (and a fortiori simplify(...,size)  which is now the default) should be enough.

@acer 

If f is a multivalued mapping, in many situations Maple returns z0 as a solution for f(z)=0, if for some branch f(z0)=0.

I don't think that this behavior is going to change soon. Of course, sqrt is not multivalued (being defined in the help page as principal branch) but it is treated as such.
Note that your equation, for x <>0 reduces to csgn(x)=1 but
solve(csgn(x)=1, x) ;

gives

Warning, solutions may have been lost
                      RootOf(csgn(_Z) - 1)

The correct solution of solve should be

   0,  RootOf(csgn(_Z) - 1)

but I am skeptical about such an answer.

Edit. The Maple answer for your second example is correct if the above "some branch" approach is adopted 

@Markiyan Hirnyk 

In computing the Groebner basis the monomial order is essential in the elimination process.

@Markiyan Hirnyk 

You have the wrong order. Why did you change my correct one?

BTW, Kitonum's solution is equivalent to my "A more direct ...".

@Markiyan Hirnyk 

Groebner:-Basis([x -(u-v)^2, y -  u^2+3*v^2, z - (1/2)*v*(u-2*v)],plex(u,v,x,y,z));
     [x-y+4*z, v^4-v^2*y+8*v^2*z+4*z^2, v^3+2*u*z-v*y+4*v*z, u*v-2*v^2-2*z, u^2-3*v^2-y]

@Markiyan Hirnyk 

It is the standard Brioschi formula for the curvature: https://en.wikipedia.org/wiki/Gaussian_curvature
(it is not really "code" needing comments).

@tomleslie 

Yes, Maxima is an open source branch of the old Macsyma.

You have joined the 5 matrices into a large 2x10 matrix and computed its singular values.
For me it's a nonsense.
@Adam Ledger