In the first of a rare trilogy from Shaw he examines the theory of the Dyson sphere, a theoretical construction which is, as might be imagined, a sphere, but one which whose diameter is the same size as the orbit of the Earth. It is believed that such a sphere would be able to contain a star such as our own, and given sufficient orbital rotation, would provide a habitable area on the inner surface equal to several hundred million Earths.
Vance Garamond is a Starflight Captain reporting directly to Elizabeth Lindstrom, President of a starfaring society in which only one other habitable planet, Terranova, has been discovered. Earth is overcrowded and Lindstrom, a psychotic and psychopathic dictator, is parcelling up the new planet and selling strips of it off to the highest bidder.
While awaiting his audience with the President before a routine assignment Vance is asked to entertain her nine-year old son. Distracted by his thoughts, Vance does not see that the boy has climbed up into the arms of a statue and before he can react the boy falls and cracks his head on the pedestal, killing himself instantly.
Realising that his life is now forfeit when the borderline-insane Elizabeth discovers her son's death, Garamond collects his wife and young son and smuggles them aboard his spacecraft. Along with his crew they head out for the stars, knowing that their chances of finding a new habitable world and so being able to escape the President's wrath is minimal.
Garamond has one hope in that he has what amounts to a treasure map; ancient research consisting of a chronological series of alien stellar maps in which a star apparently disappears.
Setting off for this point in space, Garamond discovers that the sun has been encased in a sphere of indestructible material, with an entrance at the equator. Inside, the inner surface has been terraformed and its surface area so large that it would provide the same space as several million Earths.
Radio and radar do not work within the sphere, and it is suggested that its creators meant it as a honeytrap for intelligent life, as the alien races which are discovered living within the sphere have reverted to an idyllic pastoral existence.
It's a gloriously retro novel for its time. Elizabeth's Presidential position has regal overtones quite apart from the symbolic relevance of her name. Other critics have seen the influence of Van Vogt here, and certainly the tone and the scope is redolent of novels such as 'Empire of The Atom' or 'Mission to the Stars' although the characterisation exceeds anything Van Vogt produced in either work.
It is also maybe a response to the 'Big Dumb Object' trend which arguably began with Niven's 'Ringworld' and was followed most famously by Clarke's 'Rendezvous with Rama'. Certainly, it would appear that Shaw's novel was the first major use of a Dyson sphere, the concept of which was later used by other authors such as Stephen Baxter in 'The Time Ships' and on TV in 'Star Trek - The Next Generation'.