Hopefully you'll hit quite a few of the myriad green spaces in London while we're there. There's no shortage of parks -- including Green, St. James, Hyde, Regents, and Kensington Gardens. One of my personal favorites is Holland Park.
But for absolute vastness of greenery, you can't beat Hampstead and Richmond. The park (at that time considered a manor) dates back to the 13th century. At that point it was well outside the city of London. Today, boroughs surround the park on all sides. Nonetheless, get to the center of the park and you'll feel like you're in the woods somewhere in Ithaca.
The park is famous most for the wildlife that lives there. It also affords a phenomenal view of London from Henry VIII's mound. It's a great respite for the jumble of city life.
As for Hampstead Heath -- it's a bit more city-bound but also gives you a touch of the country. Again, the views from the Heath (specifically from Parliament Hill) are fantastic. There are also some pretty cool pubs in the neighborhood to whet your whistle.
Both of these places are great ways to stay in the city, but get out of it all at the same time.
Directions to Hampstead Heath: Take the Piccadilly Line towards Cockfosters. Change to the Northern Line at Leicester Square. Get off at the Hampstead tube stop.
Directions to Richmond Park: Take the District Line towards Richmond. Get off at the Richmond tube station. Then take the 371 or 65 bus to the pedestrian gate at Petersham.
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