If you do not use any special properties of the dodecahedron, then the solution is not very complex, but rather cumbersome. 

We will look for the vertex coordinates of a dodecahedron inscribed in the unit sphere. We assume that the uppermost and lowermost vertex lies on the axis Oz, the vertex adjacent to the top lies in the plane xOz. First we look for the edge length of the dodecahedron (see figure):

restart;
AB:=2*a*convert(sin(54*Pi/180), radical);
BK:=AB/sqrt(3):
OK:=sqrt(1-BK^2):
EK:=sqrt(a^2-BK^2):
solve(OK+EK=1);
a:=op(select(t->is(t>0), [%]));  # the length of the edge

Then we find the coordinates of the vertex  A  and vertex  B , and  C  using the rotation around the axis Oz:

c:=a/2:  # Cosine of the angle OEA
with(geom3d):
point(E, 0, 0, 1):
point(A,simplify(a*sqrt(1-c^2)), 0, simplify(1-a*c)):
coordinates(A);
rotation(B, A, 2*Pi/3, line(l,[0,0,t],t)):
rotation(C, A, 4*Pi/3, line(l,[0,0,t],t)):
coordinates(B);
coordinates(C);

Next, we find the coordinates of the vertices  F and G, belonging to the face  AFGBE, using a rotation about an axis passing through the origin and center of this face:

EO1:=simplify(a/2/convert(cos(54*Pi/180), radical)):
midpoint(L, A, B):
point(O1, convert(simplify(convert(coordinates(E), Vector)+convert(coordinates(L)-coordinates(E), Vector)*EO1/distance(E,L)), list)):
dsegment(OO1, [point(o, 0,0,0), O1]):
rotation(F, A, 72*Pi/180, line(l1,[o,OO1])):
rationalize(simplify(convert(coordinates(F), radical)));
point(F, %):
rotation(G, A, 144*Pi/180, line(l1,[o,OO1])):
expand(rationalize(simplify(convert(coordinates(G), radical))));
point(G, %):
Pol1:=plottools[polygon]([coordinates(A),coordinates(F),coordinates(G),coordinates(B),coordinates(E)], style=line, thickness=2, color=red):
s1:=plottools[line](coordinates(A),coordinates(B), linestyle=3, color=black):
s2:=plottools[line](coordinates(E),coordinates(O1), linestyle=3, color=black):
s3:=plottools[line](coordinates(o),coordinates(O1), linestyle=3, color=black):
T1:=plots[textplot3d]([[0,0,0,"O"], [op(coordinates(A)),"A"], [op(coordinates(B)),"B"], [op(coordinates(E)),"E"], [op(coordinates(F)),"F"], [op(coordinates(G)),"G"], [op(coordinates(L)),"L"], [op(coordinates(O1)),"O1"]], align=left, color=black, font=[TIMES,ROMAN,16]):
plots[display](Pol1, s1, s2, s3, T1, axes=normal, scaling=constrained, tickmarks=[0, 0, 0], view=[-1.4..1.4,-1.4..1.4,-0.4..1.4]);

Then we find the coordinates of all vertices yet two faces using  rotation of the face  AFGBE  around the axis Oz:

rotation(M, F, 2*Pi/3, line(l,[0,0,t],t)):
rotation(N, G, 2*Pi/3, line(l,[0,0,t],t)):
rotation(P, F, 4*Pi/3, line(l,[0,0,t],t)):
rotation(Q, G, 4*Pi/3, line(l,[0,0,t],t)):
coordinates(M);
expand(rationalize(coordinates(N)));
coordinates(P);
expand(rationalize(coordinates(Q)));
Pol2:=plottools[polygon]([coordinates(B),coordinates(M),coordinates(N),coordinates(C),coordinates(E)], style=line, thickness=2, color=red):
Pol3:=plottools[polygon]([coordinates(C),coordinates(P),coordinates(Q),coordinates(A),coordinates(E)], style=line, thickness=2, color=red):
T2:=plots[textplot3d]([[0,0,0,"O"], [op(coordinates(A)),"A"], [op(coordinates(B)),"B"], [op(coordinates(C)),"C"], [op(coordinates(E)),"E"], [op(coordinates(F)),"F"], [op(coordinates(G)),"G"], [op(coordinates(M)),"M"], [op(coordinates(N)),"N"], [op(coordinates(P)),"P"], [op(coordinates(Q)),"Q"]], align=left, color=black, font=[TIMES,ROMAN,16]):
plots[display](Pol1, Pol2, Pol3, T2, axes=normal, scaling=constrained, tickmarks=[0, 0, 0], view=[-1.4..1.4,-1.4..1.4,-0.4..1.4]);

Finally, we find all the other vertices of the dodecahedron, using the reflection about the plane xOy and rotation around Oz:

X:=plots[display](Pol1, Pol2, Pol3):
Y:=plottools[rotate](plottools[reflect](X, [[0,0,0], [1,0,0], [0,1,0]]), Pi/3, [[0,0,0], [0,0,1]]):
rotation(P1, reflection(P0, P, plane(xOy, z=0, [x,y,z])), Pi/3, l):
rotation(Q1, reflection(Q0, Q, plane(xOy, z=0, [x,y,z])), Pi/3, l):
rotation(F1, reflection(F0, F, plane(xOy, z=0, [x,y,z])), Pi/3, l):
rotation(G1, reflection(G0, G, plane(xOy, z=0, [x,y,z])), Pi/3, l):
rotation(M1, reflection(M0, M, plane(xOy, z=0, [x,y,z])), Pi/3, l):
rotation(N1, reflection(N0, N, plane(xOy, z=0, [x,y,z])), Pi/3, l):
rotation(A1, reflection(A0, A, plane(xOy, z=0, [x,y,z])), Pi/3, l):
rotation(B1, reflection(B0, B, plane(xOy, z=0, [x,y,z])), Pi/3, l):
rotation(C1, reflection(C0, C, plane(xOy, z=0, [x,y,z])), Pi/3, l):
reflection(E1, E, plane(xOy, z=0, [x,y,z])):
T4:=plots[textplot3d]([[op(coordinates(A1)),"A1"], [op(coordinates(B1)),"B1"], [op(coordinates(C1)),"C1"], [op(coordinates(E1)),"E1"], [op(coordinates(F1)),"F1"], [op(coordinates(G1)),"G1"], [op(coordinates(M1)),"M1"], [op(coordinates(N1)),"N1"], [op(coordinates(P1)),"P1"], [op(coordinates(Q1)),"Q1"]], align=left, color=black, font=[TIMES,ROMAN,14]):
P1Q:=plottools[line](coordinates(P1),coordinates(Q), color=red, thickness=2):
Q1F:=plottools[line](coordinates(Q1),coordinates(F), color=red, thickness=2):
F1G:=plottools[line](coordinates(F1),coordinates(G), color=red, thickness=2):
G1M:=plottools[line](coordinates(G1),coordinates(M), color=red, thickness=2):
M1N:=plottools[line](coordinates(M1),coordinates(N), color=red, thickness=2):
N1P:=plottools[line](coordinates(N1),coordinates(P), color=red, thickness=2):
T5:=plots[textplot3d]([[1.7,0,0,"x"], [0,1.7,0,"y"], [0,0,1.7,"z"]], align=above, color=black, font=[TIMES,ROMAN,16]):
plots[display](X, Y, T2, T5, T4, P1Q, Q1F, F1G, G1M, M1N, N1P, axes=normal, scaling=constrained, tickmarks=[0, 0, 0], view=[-1.7..1.7,-1.7..1.7,-1.7..1.7]);

 dodecahedron.mws

aa:=``(delta[1]*n[1])/``(n[f1]*delta[1]);

 If you want to reduce the fraction, then

expand(aa);

solve({30+1144*r^4-832*r^2=0, r>0});

DirectSearch  found  4 solutions in the specified ranges. It is very likely that there is no other positive solutions:

restart;

interface(rtablesize = infinity):

f := r*(8-2*x^2):

g := subs(x = f, f):

h := subs(x = g, g):

eq1 := expand(h-x):

eq2 := expand(diff(h, x)+1):

DirectSearch[SolveEquations]([eq1, eq2], [r = 0 .. 20, x = 0 .. 20], AllSolutions);

You can download  DirectSearch  package from 

http://www.maplesoft.com/applications/view.aspx?SID=101333

Your system is very complicated and I think that it can only be solved numerically, for example, with  fsolve  command. Only this command gives one solution, and the plot shows that there are many solutions. To find other solutions you can use this command with the appropriate option, taking the approximate solutions of the plot:

restart;

f:=r*(8 - 2*x^2);

g:=subs(x=f,f);

h:=subs(x=g,g);

eq1:= expand(h-x);

eq2:= expand(diff(h,x) + 1);

Sol:=fsolve({eq1, eq2});

P:=eval([r,x], Sol):

A:=plottools[disk](P, 0.02, color=green): # Green point is the obtained solution

B:=plots[implicitplot]([eq1, eq2], r=0..1, x=0..2.5, color=[red, blue], numpoints=100000):

plots[display](A, B, scaling=constrained);

Example

plot(sin(3*phi), phi = 0 .. 2*Pi, coords = polar, scaling = constrained, axes = box);

Example:

f:=(x,y)->x^2+x*y+y^3;
D[1](f)(x,y);

 or

f:=x->x^2+x*y+y^3;
D(f)(x);
int(f(x), x);

Maybe instead of  exp(k)  should be  lambda^k? (Poisson formula in probability theory).

In both cases no solution. See the plot:

lambda:=2:
 plot(exp(-lambda)*lambda^k/k!, k=0..10);

Refine formulation of the problem!

without parentheses with Carl's improvement:

P(z) := (1 + z^(-1))^2*(1 + z)^2*(r[1]*z + r[0] + r[1]*z^(-1)):   

add(coeff(P(z),  z, n)*z^`if`(n<=0,-n, convert(-n, symbol)), n= ldegree(P(z),z)..degree(P(z),z));

P(z) := (1 + z^(-1))^2*(1 + z)^2*(r1*z + r0 + r1*z^(-1)); 

add(coeff(P(z),  z, n)*z^(``(-n)), n=-3..3);  

add(coeff(P(z),  z, n)*z^(`if`(n<=0,-n, ``(-n))), n=-3..3);

P(z) := (1 + z^(-1))^2*(1 + z)^2*(r1*z + r0 + r1*z^(-1));
series(P(z), z);

Should be  arctan  instead of  arctg  .

f:=(x,y)->piecewise(x=0 and y=0,139 ,x=1 and y=0,144,x=2 and y=0,149,x=3 and y=0,155,x=4 and y=0,155,x=5 and y=0,155,x=6 and y=0,155,x=7 and y=0,155,x=0 and y=1,144,x=1 and y=1,151,x=2 and y=1,153,x=3 and y=1,156,x=4 and y=1,159,x=5 and y=1,159,x=6 and y=1,159,x=7 and y=1,159,x=0 and y=2,150,x=1 and y=2,155,x=2 and y=2,160,x=3 and y=2,163,x=4 and y=2,158,x=5 and y=2,156,x=6 and y=2,156,x=7 and y=2,156,x=0 and y=3,159,x=1 and y=3,161,x=2 and y=3,162,x=3 and y=3,160,x=4 and y=3,160,x=5 and y=3,159,x=6 and y=3,159,x=7 and y=3,159,x=0 and y=4,159,x=1 and y=4,160,x=2 and y=4,161,x=3 and y=4,162,x=4 and y=4,162,x=5 and y=4,155,x=6 and y=4,155,x=7 and y=4,155,x=0 and y=5,161,x=1 and y=5,161,x=2 and y=5,161,x=3 and y=5,161,x=4 and y=5,160,x=5 and y=5,157,x=6 and y=5,157,x=7 and y=5,157,x=0 and y=6,162,x=1 and y=6,162,x=2 and y=6,161,x=3 and y=6,163,x=4 and y=6,162,x=5 and y=6,157,x=6 and y=6,157,x=7 and y=6,157,x=0 and y=7,162,x=1 and y=7,162,x=2 and y=7,161,x=3 and y=7,161,x=4 and y=7,163,x=5 and y=7,158,x=6 and y=7,158,x=7 and y=7,158):

C:=t->piecewise(t=0, 1/sqrt(2), t>0 or t<0, 1):

F:=(u,v)->1/4*C(u)*C(v)*add(add(f(x,y)*cos((2*x+1)*u*Pi/16)*cos((2*y+1)*v*Pi/16), y=0..7), x=0..7):

F(0,0);

                                                                             1261

Your equation has too many parameters. If you set their values​​, the equation can be solved exactly sometimes, but more often numerically as follows:

restart;

p:=a*sin(omega[P]*t): q:=b*sin(omega[Q]*t+phi[Q]): r:=c*sin(omega[R]*t+phi[R]): s:=d*sin(omega[S]*t+phi[S]):

z:=p+q+r+s:

a,b,c,d,omega[P],omega[Q],omega[R],omega[S],phi[Q],phi[R],phi[S]:=1,2,3,4, Pi/6,Pi/3,Pi/4,2*Pi/3, Pi/2,Pi/4,Pi/4:  # values of parameters

fsolve(z, t); 

               -0.6371104099

f:=a->fsolve({0.5704+0.5211*exp(-0.6569*x)-0.03849*y=a, 0.561+0.5284*exp(-0.6479*x)-0.03929*y=a}):

f(1);

A:=plot(a->rhs(f(a)[1]),2..10, labels=[a, x(a)]):

B:=plot(a->rhs(f(a)[2]),2..10, labels=[a, y(a)]):

plots[display](Array([A, B]));