President Trump announced on Tuesday that he would postpone a planned trip to Denmark because Prime Minister Mette Frederiksen refuses to consider selling Greenland to the United States. Ms. Frederiksen called the idea “absurd.” It might be, but it was also revealing. In seeking to purchase Greenland, Mr. Trump did more than rattle an ally. He demonstrated how little he understands the shape of American power.

Mr. Trump is right that Greenland is valuable. It has vast stores of zinc, copper, iron ore and uranium — all of which are becoming more accessible with global warming. It lies conveniently between North America and Eurasia. But his notion that the way to access this value is to buy it from another country is a throwback to the 19th century. Then, the United States bought or conquered a great deal of land, from the Louisiana Purchase in 1803 to the Philippine annexation in 1899. That pattern of forthright acquisition ended in the middle of the 20th century, though, as colonized people worldwide rebelled against empire and the United States found ways to achieve its ends without large land grabs.

The United States today gets key commodities like zinc, copper and iron ore on international markets. It does so with confidence, because many of its trading partners are connected to it via an intricate system of trade pacts and military alliances. Those alliances are backstopped by hundreds of American military bases around the world. The Pentagon lists 514 overseas bases, though there are surely more.

The United States’ network of power is hard to see. Maps show states, not the defense pacts and trade agreements that connect other countries to the United States, or the hundreds of foreign dots, some of them secret, over which Washington claims jurisdiction. Even if they were on the map, they would be hard to make out. Mash all the known base sites together and they total an area not much larger than Houston.