When renovating the home in the first photo, the architects at Dufner Heighes used the existing geometry of the architecture as an inspiration. The diamond-patterned floor and rug have a gorgeous interplay with the triangle patterns of the fireplace and ceiling.
Unfortunately, we don't all have such striking original architecture to work with, but it's very possible to play around with shapes in other ways, from flooring to furniture. The pattern of a herringbone floor (like the one in the home of Skinny Laminx, fifth photo above) can be echoed in a pillow, or contrasted with a striped rug. A rolling ladder (as in the second photo) can mimic the rectangles in a set of open shelves.
When you're putting together or re-doing a room at home, do you think about the interplay of shapes?
(Images: Martha Stewart (1, 2, and 3); Ngoc Minh Ngo for InStyle Home, via Style Court; SFGirlByBay)
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Yes - I live in a round building, therefore all the rooms are pie shaped with oddly-angled corners and a balcony that is a half-circle...
...so to offset the angles, I've tried to include as many circles in my decorating as reasonably possible: Round living room rug, round end tables, curvy Saarinen Executive Armchairs upholstered in a fabric w/ circles and crescents at the Round dining table, half-round wall-sconces, circular drawer pulls in the kitchen, round mirrors, lots of drum lampshades...
We renovated our bathroom last year and used an interplay of shapes, which ended up looking really neat. We used octagon floor tile, and then the shower curtain really played off of that with circles & diamond shapes throughout. All of fixtures (lighting, faucets, etc..) are all square, as well as the ceramic & glass tile in the tub/shower area and the backsplash.