It depends on the context of the conversation. If you're talking to your boss, eye contact is necessary to indicate you are paying attention. With a stranger, it's not really that important. If I am uncomfortable maintaining eye contact, I usually make sure I establish eye contact early and then look away. If the conversation lingers, I will make brief eye contact occasionally to show that I am still invested.
Even if the open windows of science at first make us shiver after the cozy indoor warmth of traditional humanizing myths, in the end the fresh air brings vigor, and the great spaces have a splendor of their own - Bertrand Russell


