The Martin Behaim globe shows Cipangu, which is actually the Japanese island of Kyushu as depicted by the Chinese. It also show part of Indonesia. Worth noting is that when Columbus sailed West, he was not looking for "the Indies", but for Cipangu, described by Marco Polo as "the land of gold". 
So it seems that Piri Reis compiled recent south-american data together with older asia-pacific data. The so-called "antartica" part of the Piri reis map appears to match Java as shown on world maps of the same period. Mystery solved, I guess... (check for yourself by comparing the Piri Reis map and the Martin Behaim globe).
This is supported by the fact that the Piri Reis map also seems to show Cipangu off North America, and describes the "antartica" land at the bottom as a warm land.
Anyway, the outlandish claims of the article about the extraordinary accuracy of the map are complete BS. Check it yourself. 
Compare to this 1502 map:
The rest, I cannot make sense of it.
They think that Piri Reis (commander of the navy at that time) obtained maps from ships that passed through the Bosphorus. He was then able to draw a current map of the world from all the diverse information.
They basically explored the world without ever leaving home.
I had an amazing tour guide who, among many other things, explained how everything had two layers of security (at least) built in. For instance, eunuchs in the Harem were always black just in case the castration was botched - that way, you could still tell the child was not the Sultan's. A bit more here: http://jmilton6000.wordpress.com/2006/11/23/the-harem-and-th...
Talk about redundancy.
Didn't know there's actually someone by the name of Piri Reis from Turkey in the real world!
But even better is:
Which links to (comparison to modern accurate maps):