You may well know of the Reuleaux triangle, which is made by forming an arc centred on each corner of an equilateral triangle. You can see how it is a "curve of constant width" - in other words, however you turn it it always has the same width:

So, if it always has the same width, it will rotate happily within a square:

However this is certainly too square for us here. So here's the Reuleaux pentagon rotating even more happily within a hexagon:


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