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The Maxima on-line user's manual

Algebra Calculator

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Phi Calculator

Phi

-- Constant: %phi %phi represents the so-called golden mean, (1 + sqrt(5))/2. The numeric value of %phi is the double-precision floating-point value 1.618033988749895d0.

fibtophi expresses Fibonacci numbers fib(n) in terms of %phi.

f:2*r*sin(%pi/r)/sqrt(((1+r^2)+(1-r^2)*cos(2*%pi/r-2*phi))*((1+r^2)+(1-r^2)*cos(4*%pi/r-2*phi)));
plot3d(at(f,r=1/rI),[rI,1/4,3],[phi,-%pi,%pi],[grid,110,30]);

By default, Maxima does not know the algebraic properties of

     %phi.  After evaluating tellrat(%phi^2 - %phi - 1) and
     algebraic: true, ratsimp can simplify some expressions
     containing %phi.

Examples:

fibtophi expresses Fibonacci numbers fib(n) in terms of %phi.

          (%i1) fibtophi (fib (n));
                                     n             n
                                 %phi  - (1 - %phi)
          (%o1)                  -------------------
                                     2 %phi - 1
          (%i2) fib (n-1) + fib (n) - fib (n+1);
          (%o2)          - fib(n + 1) + fib(n) + fib(n - 1)
          (%i3) fibtophi (%);
                      n + 1             n + 1       n             n
                  %phi      - (1 - %phi)        %phi  - (1 - %phi)
          (%o3) - --------------------------- + -------------------
                          2 %phi - 1                2 %phi - 1
                                                    n - 1             n - 1
                                                %phi      - (1 - %phi)
                                              + ---------------------------
                                                        2 %phi - 1
          (%i4) ratsimp (%);
          (%o4)                           0

By default, Maxima does not know the algebraic properties of

     %phi.  After evaluating tellrat (%phi^2 - %phi - 1) and
     algebraic: true, ratsimp can simplify some expressions
     containing %phi.

          (%i1) e : expand ((%phi^2 - %phi - 1) * (A + 1));
                           2                      2
          (%o1)        %phi  A - %phi A - A + %phi  - %phi - 1
          (%i2) ratsimp (e);
                            2                     2
          (%o2)        (%phi  - %phi - 1) A + %phi  - %phi - 1
          (%i3) tellrat (%phi^2 - %phi - 1);
                                      2
          (%o3)                  [%phi  - %phi - 1]
          (%i4) algebraic : true;
          (%o4)                         true
          (%i5) ratsimp (e);
          (%o5)                           0

There are also some inexact matches for phi. Try ?? phi to see them.

(%o1)                                true
(%i2) 

Phi Example

Related Examples

phi-sin-sqrt

2*sin(phi)*sqrt(1-sin...

Calculate

phi-solve-tan

eq1:V=h*pi/3*(R^2+R*r...

solve(eq1,h);

eq2:h=(R-r)/tan(phi);

Calculate

phi-solve

n:118534081;

phi:111561472;

solve(q^2+q*(phi-n-1)...

Calculate

phi-plot2d-sqrt
plot2d(phi(z),[z,-0.5,0.5]);

L: 1;

phi(z):= sqrt(105/(2*...

phi(-1/2);

Calculate

phi-sin

[r=sin(phi)^2];

Calculate

phi

l:100;

e:2;

nu:0.3;

Calculate

phi-solve-tan

eq1: tan(phi) = a/b*t...

solve(eq1, t);

Calculate

phi-solve

n:118534081;

phi:111561472;

solve(q^2+q*(phi-n-1)...

Calculate

phi-sin-sqrt-trigreduce

2*sin(phi)*sqrt(1-sin...

trigreduce(%);

Calculate

phi-sqrt

phi(z):= sqrt(105/(2*...

phi(-L/2);

phi(L/2);

Calculate