Say you've spent several days exploring the city and you're desperate for some greenery, then you'll never need to go far within Edinburgh. Arthur's Seat and Calton Hill are very central and a brief diversion will get you to the Botanic Gardens near Inverleith Park. However, if you're looking for beauty, variation and space, you need to head a little south, to Blackford Hill.
Chances are you'll arrive via Observatory Road, where you can't help but notice the triumphal arch at the entrance to the street, dedicated to George Harrison - the local politician, not the Beatle. It was during Harrison's tenure as Lord Provost that the council acquired Blackford Hill as a public park. When he died a few years later, they chose to commemorate him here. Once through the arch, prepare for a steep incline of sometimes as much as 15%, before discovering the Victorian observatory which is the street's namesake.
This is the Royal Observatory, Edinburgh (ROE), built in 1896. It replaced the previous observatory on Calton Hill, which was made that ever less useful by the light pollution of the steadily growing city. In addition to charting the skies, places like this played a daily role in determining the correct time. Noon was defined in relation to the sun's position, which was then relayed to e.g. the ships in the harbour. To this day, Edinburgh has two such remaining signals: the time ball on the Nelson Monument and the One O'Clock Gun at the castle. Both of these were originally prompted by the observatory.
Now proceed towards the top of the hill. With 164 metres, it may not be a dizzying height, but it's certainly higher than the castle, of which you have an unimpeded view. Upon close inspection, you'll find remnants of a prehistoric hill fort. There are no trees in this area, so your views of the city are second to none. This is one of Edinburgh's "seven hills" and there's an annual race that visits each of them in order. Fun fact: as long as you stick to the order, you can take any (legal) route you please. And even if you don't encounter runners, there will be plenty of people out walking their dogs.
Walking roughly north from the summit, the steep sides of the hill will lead you down to Blackford Pond, a Victorian feature that's very popular among families with children. Had you gone south instead, then you'd soon enter the adjacent Hermitage of the Braid, as this woodland area is called. At its lowest point, you'll find the Braid Burn, a charming stream that can surprise you with the occasional strength of its current. It drains into the sea at Portobello, but it disappears under ground once you get to the nearby residential areas. Yet if you walk just a little downstream, you'll pass a disused quarry and Agassiz's Rock. Both are popular among climbers, but Agassiz's Rock is also of scientific interest. This is where in 1840, the Swiss geologist Lousi Agassiz discovered evidence of ancient land ice. Based on this and other evidence, he concluded that big parts of the northern hemisphere must once have been covered with ice.
Alternatively you can turn the other way and head upstream, towards the equally charming Hermitage of Braid. Despite what its name might suggest, this isn't the home of a recluse. Rather, it was built in 1785 as a country retreat for a wealthy lawyer and is a Category A listed building. It can be visited, as it nowadays houses a visitor's centre. Nearby you'll also find an old ice house and a 'doocot' which could nest nearly 2,000 pigeons, which would have been kept for their meat. And should you wish to visit all of this in style, then there's even a post to hitch your horse!