What is it about arbitrary (shiny) geometric shapes that seem so appealing?

I don’t think they are arbitrary shapes. Apple designers execute everything with intention. The proportions of the original iPod, for example, are based on the golden ratio. That is probably one of the reasons why most people find the iPod aesthetically pleasing.

Right, so there exists particular shapes that are inherently pleasing.
Perhaps the more interesting question is “why do people think the golden ratio is visually appealing?” I understand that it does occur in nature a great deal (which is strange enough as it is) so that somehow may play a role in it. I haven’t got a clue as to why that is though.

Yes, it’s a very interesting subject. Why people find certain proportions appealing, why we find beauty in symmetry. And it goes beyond visuals, too. We find 3— and 5— syllable words and phrases more pleasing to listen to, which is why rhymes and limericks follow that rhythm. A lot of it is based on the Fibonacci sequence. I learned all of this from the book, A Beginner’s Guide to Constructing the Universe by Michael Schneider. It’s a fascinating book, and it even dissects famous paintings to find classical proportions in them.