Find the equivalent resistance of the triangular bipyramid between the points 

A) A and C 

B) D and E 

Assume the resistance of each brach to be R

Resistor is connected in between A and C but not connected in between D and E.

(a) When a cell is connected in between A and C, the resistors betwen D, B and B, E will become ineffective [This is by symmetry or wheatstone’s beidge principle] becaus the points D, B and E will be at same potential. Now the equivalent circuit will be

It can be proved that ${\mathrm{R}}_{\mathrm{AC}}=\frac{2\mathrm{R}}{5}$

(b) When a cell is connected between D, E the points A, C will be at same potential. [This is also by symmetry or principle of balanced wheatstone’s

bridge circuit i.e. $\left.\frac{{\mathrm{R}}_{\mathrm{EA}}}{{\mathrm{R}}_{\mathrm{AD}}}=\frac{{\mathrm{R}}_{\mathrm{EC}}}{{\mathrm{R}}_{\mathrm{CD}}}\right]$ and (B, C) will become ineffective.

Now the equivalent circuit is clearly ${\mathrm{R}}_{\mathrm{DE}}=\frac{2\mathrm{R}}{3}.$