Why dimension of antenna's elements can be 1/2 and 1/4 or wave length and what about 1/3 or 1/8?
Antennas of any length will radiate (or receive) electromagnetic waves. The efficiency of transmission depends upon 3 components: the transmitter, the feed-line and the antenna. To transmit efficiently, the loss in the feed-line should be small. To have small feed-line losses, one wants a power-factor in the feed-line to be near 100% (or the Voltage Standing Wave Ratio to be near 1:1). That is, one wants most of the apparent power to be true power, rather than reactive power. Reactive power increases the loss in the feed-line without adding anything to the radiated power.
Half-wave dipole and quarter-wave &quot;monopole&quot; antennas have purely resistive impedance. This means that the power-factor of the feed-line will theoretically be 100%. Short antennas, such as 1/8 wave-length, will have a strong capacitative reactance. This will cause the power-factor to be far from 100%. One can compensate for this capacitative reactance by adding a inductive &quot;loading coil&quot; at the base of the antenna. In this way, 1/8 wave-length or even 1/8.283947 (i.e. some random number) wave-length antennas are possible.
The following image from the Wikipedia article Dipole Antenna shows the dependence of reactance on normalized dipole length.

If one looks at approximately 0.25 wavelength, (remember this is a dipole), the resistive impedance looks to be less than 50\$\Omega\$ and the capacitative reactance looks to be about 500\$\Omega\$, i.e. about 10 times the resistive impedance.