Where is the coil on the linear monopole? Seems to be a flawed
example to me. The helical example is loaded, but the linear is not?
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Regardless of the loss in a loading coil for the linear monopole (and
actually the helix would need one also), the salient point here is that even
though the conductor in the helix is 3.14 X longer than in the monopole, the
radiation resistance and pattern of the two forms for a given frequency is
the same for the same end-end length -- as shown in the comparison.

It is the relation of the radiation resistance to the other resistances
(mainly the matching network loss and the r-f ground terminal loss) that
determines the radiation efficiency of the complete antenna system.

Some might think that the radiation resistance of the helix in this model is
determined by the length of the conductor from which it is wound, but that
isn't the case.