Assuming it is not completely irrelevant, when does one write the numbers with words?
Sie isst drei Äpfel
Sie isst 3 Äpfel.
Assuming that there is a rule, is it just an aesthetics issue? Or is it absolutely mandatory to follow it?
The answer strongly depends of the intended accuracy, the nature of the numbers and the type of text. My private rule of thumb is the following:
For scientific texts, manuals, etc., I would write out 1 to 12, if they are natural numbers (in contrast to real numbers), and nothing else.
However, I would not write out cardinal numbers that refer to an enumeration system:
Finally, I would not write out any natural numbers that are followed by an abbreviated unit:
In stories, poems, etc. I would additionally write out 1, 1.5 and 2 to 12 as well as 20, 30, …, 100, 200, …, 1000, …, if they are vague information. As a rule of thumb for everything beyond 12: If x could be replaced by x+1 (ignoring the correctness of the information), do not write x out.
In both cases, the style should not switch in the middle of a list or similar:
Richtig ist, dass kleine Zahlen bis 12 ausgeschrieben werden, aber es gibt Ausnahmen, nämlich wenn sie in einem technischen/mathematischen Kontext vorkommen und wenn sie mit einer Einheit vesehen sind:
Bei Namensbestandteilen wird die Zahl römisch geschrieben:
Im Datum werden auch kleine Zahlen durch Ziffern ausgedrückt:
I would write mostly digits (also in other languages), except when you have tow consecutive numbers in a sentence. In that case I write one (mostly the first one) in words, so that the reader knows they're two separate numbers:
There is a rule: Natural numbers from one to twelve are written as words. It goes back to the former common usage of the duodecimal system. (Compare: dozen; feet and inch)
However there are some special cases as noted in other answers to your question. If you use words in general you have to write these numbers as words, if not you write them as digits (drei Kilogramm, 3 kg).