In geometry a »Kreis« is a circle. It is either the set of all points on a plane having the same distance (the radius) from a given center point, or the set of all points that have a distance less than the radius.
In geopolitics a »Kreis« is an administrative area. Areas named »Landkreis« or just »Kreis« exist in Germany and obviously also in Zürich.
In Germany a »Kreis« is the geographical area around a »Kreisstadt«. I'm not really sure about administration in Germany (I live in Austria) but I think a Kreis is a level between Gemeinde (municipality) and Bezirk (district):
- Gemeinde (municipality) - Kreis/Landkreis (district) - Regierungsbezirk (district) - Bundesland (state) - Deutschland (Germany)
In Austria it is similar, but we don't have Kreise in Austria. (We only have "Wahlkreise" (election districts) which only are used in election arithmetics but have no administrative meaning.):
- Gemeinde (municipality) - Regierungsbezirk (district) - Bundesland (state) - Österreich (Austria)
I'm also not really sure about Switzerland, but I think this should be correct:
- Gemeinde (municipality) - Kanton (canton/state) - Schweiz (Switzerland)
And in all three countries bigger cities are exceptions of these systems, some of them (Berlin, Vienna, Hamburg, Bremen) are states with their own inner structure.
In France, Switzerland, Belgium, Luxembourg, in the Federation of Bosnia and Herzegovina, in Canada, Syria, Bolivar, Ecuador and El Salvador there also is an administrative layer called »canton« (German: Kanton). In Switzerland this is the same level as states in other federal nations like Germany, Austria, USA, Mexico, India etc. A canton is not always the highest administrative level. For example in France it is the 3rd level (btw: France is not a federal nation, so it doesn't have states.)
Other cities are also subdivided in smaller administrative areas. For example Vienna in Austria is subdivided in 23 districts called »Bezirke« (singular: Bezirk). And they also have numbers. I used to live in »11. Bezirk« for a long time, but also in »10. Bezirk« and in »16. Bezirk«. Those districts have names too (10. Bezirk = Favoriten, 11. Bezirk = Simmering, 16. Bezirk = Ottakring) but in Vienna people are used to say just the number of the district. Also Paris is subdivided in 20 districts, called »arrondissements« which also have numbers, and as far as I know, also in Paris people are used to use just the numbers instead of the names. (The name of the 6th arrondissement is »Luxembourg« and it might be confusing if someone says they live in Luxembourg. It is clearer if they say they live in the 6th arrondissement.)
Kreis (circle) and Kreuz (cross) are not related to each other.
The origin of the word »Kreis« is Old High German »kreiz« (arena, magic circle) which is a cognate of the verb »krizzon« (modern German: kritzeln, English doodle, scribble, scrawl) which originally meant to carve a line. (People carved a line on the ground which was the border of a magic circle or an arena for fights.)
The origin of the German word Kreuz (Old High German kruzi) is latin crux (beam with crossbar, gallows, torture stake) and was used for many centuries only as the name of Jesus' cross (Christian cross). The former German name for this execution device was »Galgen« (gallows), but later the word »Galgen« was used only for an execution device that looks like the greek capital letter Gamma (»Γ«) while the device that looks like a plus sign (»✝«) lost the name »Galgen« and was named »Kreuz« only.