This is a Y-DNA project for all men named Bürgi, or with a surname along their paternal line thought to be derived from the surname Bürgi or Bürki.

Typically a Swiss surname, Bürgi and Bürki combined are carried by about 0.1% of the population of Switzerland. In 1911, academic Emanuel Friedli wrote that Bürgi/Bürki was a nickname (presumably an Alemannic-style diminutive) of the common given name Burghart, which first appeared in the 14th century. "Burghart," in turn, means "strong/hardy castle/fort."

Perhaps the most famous carrier of the surname was clockmaker and mathematician Jost Bürgi (1552-1632), inventor of cross-beat escapement and the remontoire, and co-inventor of antilogarithms.

Bürgi has long been a common name among Swiss-origin Mennonites, see this article in GAMEO. Many variants have been adopted, including Anglicized variants such as as Berkey, Bergey, Barkey, and Parkey.

Here are maps of the distributions of the two most common variants in Switzerland:
http://www.verwandt.ch/karten/absolut/b%25C3%25BCrgi.html
http://www.verwandt.ch/karten/absolut/b%25C3%25BCrki.html
The most common Anglicized variant is probably Berkey, which is most frequent in Pennsylvania:
http://www.ancestry.com/name-origin?surname=berkey

Many different wappen, or coats of arms, have been taken by different Bürgi/Bürki families, but many share common themes, like fortresses beneath celestial objects. Older examples include:
http://www.query.sta.be.ch/detail.aspx?ID=443681
http://www.query.sta.be.ch/detail.aspx?id=443670
http://www.query.sta.be.ch/detail.aspx?id=443675