How to determine the shape of the world given that there are no astronomical cycles?

Inspired by this question, how might one determine the shape of the world given a cosmology with no cycles? The moon, sun, stars, and planets are all fixed. The Earth does not rotate. You can't simply walk around it because it is essentially tidally locked; one side is a barren, hellish desert, the other is a frozen wasteland.

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How to determine the shape of the world given that there are no astronomical cycles?

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