Locked Hexagon Dissections

Author : Gavin Theobald

The small figures in the table above indicate the overhead compared to an ordinary dissection. They give an indication of where improvement may be possible. Is it possible to reduce this overhead to zero?

Triangle = Hexagon (7 pieces with 3 turned over)

Greg Frederickson’s second book shows a twist hinge dissection of a triangle to a hexagon. A simple conversion gives this locked dissection. Replacing twist hinges by lugs and locks does not add extra pieces so this is an efficient dissection.

The same method will work for dissecting any {n} to a {2n} in 2n+1 pieces.

Square = Hexagon (6 pieces with 2 turned over)

The twist hinged version of this dissection was discovered by Greg Frederickson. It is derived from this dissection.

Pentagon = Hexagon (10 pieces with 3 turned over)

Greg Frederickson’s third book mentions a twist hinge dissection of a pentagon to a hexagon in 13 pieces that can be directly converted to a 13 piece locked dissection. But using a totally different method I arrive at the above 10 piece solution.

Hexagon = Heptagon (11 pieces with 4 turned over)

This derives from an alternative dissection of a hexagon to heptagon.

Hexagon = Octagon (10 pieces with 5 turned over)

I am pleased with this dissection. It was not very easy to find.

Note that the diagram show the dissection with six pieces turned over, but by simply replacing one of the polygons with its mirror image reduces this to four pieces.

Hexagon = Decagon (10 pieces with 4 turned over)

This derives from the ordinary dissection of a hexagon to decagon. There are a few imperfections with this dissection: a narrow waist, a very short spike and a thin projection. Nevertheless, I am pleased to be able to do this dissection with so few pieces.

Hexagon = Dodecagon (7 pieces with 3 turned over)

Greg Frederickson’s second book shows a twist hinge dissection of a hexagon to a dodecagon. A simple conversion gives this locked dissection, but pieces are saved as we don't need to hinge the cental triangle.

Hexagon = Hexagram (7 pieces with 3 turned over)

Greg Frederickson’s second book shows a twist hinge dissection of a hexagon to a hexagram. A simple conversion gives this locked dissection, but pieces are saved as we don't need to hinge the central triangle.

Hexagon = Octagram {8/2} (11 pieces with 4 turned over)

This dissection is derived from the normal 9 piece dissection. There are some small lugs but there are also some even smaller ones that are barely visible even at high magnification.

Hexagon = Octagram {8/2} (12 pieces with 6 turned over)

This dissection removes the imperfections of the previous one, but at the cost of an extra piece.

Hexagon = Dodecagram {12/3} (11 pieces with 5 turned over)

This dissection is derived from a slightly modified 10 piece dissection.