Locked Square 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 = Square (7 pieces with 3 turned over)

This is based on the usual four piece dissection of a triangle to a square. In Greg Frederickson’s second book he describes how to convert hinged dissections to twist hinge dissections. By replacing each twist hinge by a lug and a lock (also called tab and blank), twist hinge dissections can be converted to locked dissections. This is an example.

Unfortunately, the conversion of a hinge to a twist hinge adds a piece, and since we have replaced three hinges this dissection requires three extra pieces compared to the unlocked version.

Square = Pentagon (7 pieces with 1 turned over)

Locked dissections would be fairly boring if they could be found simply by converting twist hinged dissections into locked dissections. This dissection is an example of one that was not discovered in this way. The best known twist hinged version of this dissection requires 11 pieces so this is a four piece improvement. It shows the possibility that many locked dissections can be improved.

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.

Square = Heptagon (10 pieces with 4 turned over)

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

Square = Octagon (8 pieces with 3 turned over)

The unlocked version of this dissection is here. The twist hinged version of this dissection requires an extra piece. Here, the central square is held in place by the surrounding pieces.

Square = Decagon (8 pieces with 2 turned over)

This is a rather nice dissection except that most of the lugs are much too small.

Square = Dodecagon (9 pieces with 2 turned over)

Greg Frederickson’s second book shows a twist hinge dissection of a square to a dodecagon. It is derived from this dissection. A simple conversion (and a minor change) gives this locked dissection.

Square = Pentagram (11 pieces with 4 turned over)

This dissection is from the normal 7 piece dissection. There is one rather small irregular lug piece.

Square = Hexagram (8 pieces with 2 turned over)

Square = Octagram {8/2} (8 pieces)

This dissection is another that is not derived from a twist hinged dissection but instead directly from the normal 7 piece dissection. It is an unusual dissection on this page as it does not require any pieces to be turned over. There is a small lug piece that I consider an imperfection.

Square = Octagram {8/3} (11 pieces with 5 turned over)

This dissection is derived directly from the second of these 8 piece dissections. Some of the lugs are smaller than I would like.

Square = Dodecagram {12/3} (10 pieces with 2 turned over)

This dissection is derived directly from the normal 8 piece dissection.