The complexity of a straight line

At Lake of the Woods, Minnesota's border takes an odd jog to the north. The jog includes a few small islands, and part of a peninsula. The U.S. controlled portion of the peninsula is called the Northwest Angle.

This peculiarity is cause by a Ben Franklin mistake while negotiating the original U.S. Canadian border in the Treaty of Paris. The border was supposed to extend west from the boundary waters until it met the Mississippi River. 

However, they were working from a faulty survey and later discovered that the source of the Mississippi doesn't reach that far north. Because of that, the border couldn't be drawn as intended, so in 1818 negotiators solved the problem by drawing the border due south from the westernmost point of the boundary waters to the 49th parallel, and then followed that line west to the Pacific coast.  

This still left a few small problems. Jutting south from Canada into the Lake of the Woods are at least three, and possibly more, small prominences that cross the 49th parallel.  The chart detail below shows three of the enclaves. As you can see, the point of land just west of Elm Point is extremely small.

They're uninhabited so they are little more than a curiosity to cartographers. If you're interested. more information about these peculiar enclaves can be found at Strange Maps. That link also has an interesting link to an article about the short-lived Republic of Indian Stream (1832-1835) which grew out of another border anomaly of the Treaty of Paris.