#### DOCUMENTA MATHEMATICA,
Vol. Extra Volume: Optimization Stories (2012), 87-93

** Friedrich Eisenbrand **
Pope Gregory, the Calendar,
and Continued Fractions

The success of many activities of modern civilization crucially depends
on careful planning. Some activities should be carried out during a certain
period of the year. For example: When is the right time of the year to
sow, when is the right time to plow? It is thus no surprise that calendars
are found in literally every ancient civilization.
The earth revolves around the sun in about 365.2422 days. An accurate
calendar can thus not provision the same number of days every year if the
calendar should be synchronous with the seasons. This article is about
the problem of approximating a given number by a rational number with small
denominator, continued fractions and their relationship to the Gregorian
calendar with its leap-year rule that is still in use today and keeps the
calendar synchronized for a very long time.

2010 Mathematics Subject Classification: 11J70, 11Y16, 11A55

Keywords and Phrases: Calendar, Diophantine approximation, continued fractions

Full text: dvi.gz 17 k,
dvi 51 k,
ps.gz 181 k,
pdf 93 k.

Home Page of
DOCUMENTA MATHEMATICA