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Is there an elegant way to build a round window with the Radial Cockpit Window? I've tried experimenting with vertical clip plates but it just ends up looking odd.

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I'm gonna suggest a SNOT contraption relying on two 3659 Arch 1x4, two 1x1 round plates in trans clear, and one 2654 2x2 round plate with rounded bottom, also in trans-clear:

contraption CAD drawing

The 2x2 plus 1x1 round plates should just fit within the arches of the arch bricks and provide just enough friction to hold:

Detail of cross-section

If stability is a bigger concern, replace one of the 1x1 round plates with a 2x2 plate in trans-clear. This should keep the window from wiggling:

contraption CAD drawing

There should be enough room inside the 1x4 arch brick to snugly accommodate the 2x2 plate:

Detail of cross-section

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It's also possible to use the (relatively new) part 78666 Arch 1x2 Inverted together with 3659Arch 1x4 to achieve a SNOT-less window frame, for example:

CAD diagram of window assembly

Bear in mind that the curve of the inverted 1x2 arches is not the same as the 1x4 arch, and there'll be a small (about 1/10th of a stud) gap between the inverted arches and the 2x2 dish:

CAD diagram showing gap

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