No. The only rigid polygon is a triangle. Any other needs diagonals - which break up the polygon into triangles.
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A pair of regular hexagons.
Soccer balls have different patterns, but if you have both regular pentagons and regular hexagons it must have 12 pentagons and 20 hexagons.
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Yes
Yes you can. Envision a beehive - it is made of regular hexagons.