familiarity / intimacy
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intimacy
A state of close personal connection and mutual understanding between people, based on shared time and experience.
Zwischen den langjährigen Freunden herrschte eine selbstverständliche Vertrautheit.
There was a natural intimacy between the longtime friends.
'Vertrautheit' (familiarity/intimacy) is the noun form of the adjective 'vertraut' (familiar). The ending '-heit' is often used to turn an adjective into an abstract noun that describes a state or quality.
Er ist mit diesem Thema sehr vertraut.
He is very familiar with this topic.
The preposition helps you recognize the meaning. 'Vertrautheit zwischen' (familiarity/intimacy between) people describes a close personal relationship. 'Vertrautheit mit' (familiarity with) something describes knowing a topic or place well.
Die Vertrautheit zwischen den Geschwistern war für jeden sichtbar.
The intimacy between the siblings was visible to everyone.
When it is about personal relationships, people often say 'eine Atmosphäre der Vertrautheit schaffen' (to create an atmosphere of intimacy/familiarity), 'ein Gefühl von Vertrautheit spüren' (to feel a sense of familiarity), or 'eine tiefe Vertrautheit' (a deep intimacy).
In der kleinen Gruppe entstand schnell eine angenehme Vertrautheit.
A pleasant intimacy quickly developed in the small group.
In the meaning of 'etwas gut kennen' (to know something well), 'Vertrautheit' (familiarity) is almost always used with the preposition 'mit' and the dative. This structure describes what the knowledge refers to.
Für diesen Job ist Vertrautheit mit den gängigen Computerprogrammen erforderlich.
For this job, familiarity with common computer programs is required.
closeness / proximity
'die Nähe' and 'die Vertrautheit' can both describe a close relationship. 'die Nähe' emphasizes that people are emotionally close to each other. 'die Vertrautheit' emphasizes that people know each other well and feel safe with each other. It often develops through shared time and experience. 'Vertrautheit' can also mean that you know a place, a topic, or a task well. So 'Nähe' is more suitable for the feeling of connection, while 'Vertrautheit' is more suitable for knowing someone or something well and feeling secure in dealing with them or it.
Nach einigen Monaten im Büro ist eine angenehme Vertrautheit im Team entstanden.
After a few months in the office, a pleasant familiarity developed within the team.
Zwischen den Geschwistern ist trotz der vielen Kilometer eine große Nähe geblieben.
Despite the many kilometers between them, a great closeness has remained between the siblings.