The term controversy may be appropriate insofar as "relativists" often complain about not being able to decode the "absolute" way of telling the time, resulting in missed appointments etc.
The same way of giving the time is used also in Slovenia (of course using Slovenian language) perhaps as the remnant of the old Austro-Hungarian times.
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The other variant is relative; this one is also used for multiples of five minutes.
The use of a dot as a separator matches the convention of pronouncing the day and the month as an ordinal number, because ordinal numbers are written in German followed by a dot.
The traditional representation with a dot remains in widespread use, however, and in this format leading zeros are generally omitted from the hours; additionally, the literal string “Uhr” is frequently added – e.g. Just as with the date format, leading zeros appear to be less commonly used in Germany than in Austria and Switzerland although the Austrian Standard ÖNORM recommends the zero for table-form dates only – such as Abfahrt Uhr – and not for running text., especially for formal announcements and exact points in time.
Systematic use of the 24-hour clock by German radio and TV announcers, along with the proliferation of digital clocks, may have been a significant factor in this development.
The controversy between the "absolute" and "relative" ways of giving the time is largely one of regional dialect differences: the "relative" variant (as in "viertel/Viertel vor/auf drei") is the much more common one as it is used in a wide diagonal strip from Hamburg to Switzerland, leaving some of the German south-west and most of eastern Germany as well as the eastern half of Austria with the "absolute" variant (as in "dreiviertel drei" or "drei Viertel drei").
For half-hours, the absolute form as in "halb zwei" is used everywhere.