A View of Reykjavík from the Hallgrímskirkja

A light-gray concrete space shuttle pointed to the stars, the Hallgrímskirkja is Reykjavík’s most instantly recognizable landmark. Set atop a hill, the Lutheran church is visible from around the city, and its tower offers one of the best aerial views of Iceland’s capital.

Construction began on the “Church of Hallgrímur” in 1945, just after Iceland won its independence. But the fledgling country had to wait a long time; their grand new church wasn’t finished for another 41 years. The architect,...