I hate eye contact. I just don't get what it's about, I can't read anything from it. I actually have difficulty focusing my eyes when I look into someone's face, and I can't seem to gather my thoughts. But over the years I've learned that it's critically important to most people, so I've learned to make some strong eye contact at the beginning of an encounter, then when I talk I let my eyes wander off. When something significant occurs, like the other person starts talking, I make contact again briefly and then let my eyes go away. A psychologist once described my eye contact as 'normal but intermittent'.
As far as the difficulty of having a serious conversation while thinking about making eye contact, I usually have three or four different 'threads' of thought going on in my head while I'm conversing. I think this is at least part of why I get overwhelmed when in a group and just clam up.
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Thought is just part of a world that's shaped out of feeling