“I’ve got a nice place here,” he said, his eyes flashing about restlessly.
Tom Buchanan’s bragging on his own nice place instead of someone else complementing it. It suggests that he’s generally insecure. More specifically, because he lives reasonably close to the “new rich” across the bay, he feels like he has to show off everything even though he comes from “old money.” This might explain some of his public stunts, such as bringing a bunch of polo ponies from Lake Forest just because. Secretly Tom isn’t very sure of himself, his image, or his status within this changing society.
Not only is Tom unsure of the image he is meant to portray to society, but by complimenting himself on his own house, he is giving himself assurance on his masculinity. As Nick mentions elsewhere in the chapter, he “felt that Tom would drift on forever seeking, a little wistfully, for the dramatic turbulence of some irrecoverable football game.” Nick perceives Tom as a man who hit his prime early, and is therefore trying to recreate that thrill through his lavish wealth. So when Fitz mentions that Tom’s “eyes [are] flashing about restlessly” this could mean that Tom is constantly looking for something new: for something bigger and better.
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