
greifen
to grab
To take, grasp, or seize something with the hand.
Er griff nach dem Buch, das auf dem obersten Regal lag.
He reached for the book that was on the top shelf.
to grab / take effect
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to grab
To take, grasp, or seize something with the hand.
Er griff nach dem Buch, das auf dem obersten Regal lag.
He reached for the book that was on the top shelf.

to resort to
To take a certain measure or use a certain means, often in response to a difficult situation.
Wenn die Kopfschmerzen nicht besser werden, muss ich wohl zu einer Tablette greifen.
If the headache doesn't get better, I'll probably have to resort to a tablet.

to take effect
To become effective or come into application (referring to a law, rule, or measure).
Die neue Verkehrsregel greift ab dem kommenden Montag.
The new traffic rule takes effect starting next Monday.

to catch
To catch or arrest someone.
Die Polizei konnte den Einbrecher noch am Tatort greifen.
The police were able to catch the burglar at the scene of the crime.

to spread
To spread quickly and uncontrollably (e.g., a disease, a fire, an idea).
Die Grippewelle begann in der Schule schnell um sich zu greifen.
The wave of flu began to spread quickly through the school.
'Greifen' (to grab) is the base word for many important verbs such as 'begreifen' (to understand), 'angreifen' (to attack), or 'eingreifen' (to intervene). The basic idea of 'grasping' or 'comprehending' is often recognizable.
Es ist schwer zu begreifen, was passiert ist.
It is hard to understand what happened.
To emphasize movement toward an object, the construction 'greifen nach' + dative is often used. If you take the object directly, an accusative object can also follow.
Das Kind griff nach der Hand seiner Mutter.
The child reached for his mother's hand.
The expression 'zu etwas greifen' (to resort to something) means to choose a certain means or measure, often out of necessity. The noun is in the dative.
Bei starken Schmerzen sollte man zu einem Medikament greifen.
For severe pain, one should resort to medication.
In this meaning, 'greifen' (to take effect) is used impersonally, and the subject is always an abstract concept such as a law, rule, or reform. It means 'wirksam werden' (to become effective) or 'gelten' (to apply).
Die neue Regelung greift ab dem ersten Januar.
The new regulation takes effect on the first of January.
The expression 'um sich greifen' (to spread) is a fixed expression and means that something spreads quickly and often uncontrollably. It is mostly used for negative things such as fire, diseases, or panic.
Die Angst vor dem Virus griff schnell um sich.
Fear of the virus spread quickly.
to apply / be valid
Gelten describes a rule as valid or applicable to a case. Greifen emphasizes that under certain conditions it actually becomes effective or is put into practice. Gelten fits general validity, while greifen fits taking effect in a specific case.

Bei einer verspäteten Lieferung ins Ausland greift die Regel sofort.
If a shipment abroad is delayed, the rule takes effect immediately.
Die Regel gilt für alle Lieferungen ins Ausland.
The rule applies to all shipments abroad.