Alan Peake wrote:
"---what would happen as the antenna became shorter and shorter compared
with the half-wave dipole?"

Terman answers that question on page 871 of his 1955 opus:
"The directive gain of the elementary doublet =1.5." For a resonant wire
of 0.5 lambda, the gain is 1.64.

There`s not much difference in directivity as the doublet shrinks to a
vanishingly small size. The gains shown are power ratios, not dB`s.

Comparison antenna is the isotropic of which Terman says:
"Although an isotropic radiator of coherent waves does not exist because
it cannot satisfy Maxwell`s equations, the properties of such an
imaginary antenna are easily visualized, and the concept of an isotropic
radiator is often found useful in the analysis of antenna systems."
(Page 871 in the 1955 opus.)

Best regards, Richard Harrison, KB5WZI