About Sextuplets

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About Sextuplets

 

   Sextuplets are shapes consisting of 25 unit hexes, six copies of which can be tiled using a complete set of mono- through pentahexes. In order to restrict the number of possible shapes, the majority of solutions posted here display some sort of symmetry. All solutions with horizontal mirror symmetry may be connected in at least one way, to form a single 150-unit shape with full symmetry. Unfortunately, the same does not apply to solutions with vertical mirror symmetry, as shown below.

 

   

 

   However, it is possible to form a 24-unit vertically symmetric shape adding the final hex along one of the axes shown in pink. These "offset" shapes can now construct a fully symmetric figure. The same procedure can be applied to a horizontally symmetric

generator, but there are now two possible starting positions on the grid.

 

     

 

 

Related Solutions

 

   Each birotary shape is related to three horizontally and three vertically symmetric shapes as shown below. In this case the central shape on the bottom row also possesses ternary symmetry. The resulting 25 hex shapes cannot necessarily be solved as sextuplets.

Shape Rotate 60 Rotate 120
 
Invert right side Invert right side Invert right side
Flip bottom half Flip bottom half Flip bottom half

 

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