In particular, four of the first five Presidents (and the tenth) hailed from Virginia, and the exception was the only one-term President in that group.
The District was originally composed of two counties: Alexandria (land from Virginia) and Washington (land given from Maryland).
In the early years, the original thirteen colonies were a loose coalition whose only uniting factor was the desire to be independent from Great Britain. Apart from a Welcome sign on each side and possibly a toll booth (on toll roads), there are no practical artificial delineations between the states, though as in other countries, rivers sometimes act as natural borders for states; it's awfully hard to miss the Mississippi and Missouri Rivers.
As the central government — which bounced between cities for 27 years before the founding of the District of Columbia in 1801 — grew in power, a war was fought over whether the federal government or the individual states should have the bulk of power. On highways, there is often a visitors' bureau, one on each side, with about 100 pamphlets on tourist activities, and usually a volunteer to give directions to these state-revenue-enhancing locales.
There used to be several towns within the counties until Washington was consolidated as one city and Alexandria was given back to Virginia due to slavery and other disagreements, as well as the theories that the seat of government wouldn't get big enough to need it and that the District would not develop much of a local population.
Both ended up happening, of course, and several major buildings such as The Pentagon are across the Potomac River in what is now known as Arlington, Virginia.