Locked Dissections

Author : Gavin Theobald

Dissection models are normally made up of loose pieces. When knocked, the pieces become disarrayed. One solution to this problem is to have frames for the pieces to fit into. An alternative solution is for the pieces to be interlocked as for a jigsaw puzzle. This is the purpose of locked dissections.

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?

No simple method seems to give best results for these dissections. Twist hinge dissections can be a good starting point, but can frequently be improved upon. This makes these dissections an interesting challenge to find.

With normal dissections I can often be confident that no further improvements can be found, but that is not the case here. So there are lots of opportunities to find improvements or to find new dissections to fill in the gaps in the table. Generally I am looking for dissections that require 12 pieces or less or that require no more than four pieces more than normal dissections.

On this page I do not follow my normal convention of showing when pieces are turned over as these pieces are very common in these dissections and can be rather small.

Triangle — Square (7 pieces)

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. Twist hinges can, by adding a lug and a lock (also called tab and blank), be converted to locked dissections. This is an example.

Triangle — Pentagon (8 pieces)

This dissection is based around the first one here. Compare to see how this one was found.

Triangle — Hexagon (7 pieces)

Greg Frederickson’s second book shows a twist hinge dissection of a triangle to a hexagon. A simple conversion gives this locked dissection.

Triangle — Heptagon (13 pieces)

Greg Frederickson’s third book mentions a twist hinge dissection of a triangle to a heptagon in 15 pieces. I found a simple improvement that reduces this to 13 pieces. Converting this gives this locked dissection. 13 pieces seems a lot as an ordinary dissection only requires 8 pieces. Can you find something better?

Triangle — Octagon (10 pieces)

This dissection is based around the first one here.

Triangle — Hexagram (6 pieces)

I am rather pleased with this dissection. It was an unexpected find.

Triangle — Octagram (12 pieces)

This dissection is based around this dissection. Note that there is a very short edge on the left edge of the triangle just above the spike.

Square — Pentagon (7 pieces)

This page would be fairly boring if all these dissections 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 of the dissections on this page could be improved.

Square — Hexagon (6 pieces)

The twist hinged version of this dissection was discovered by Greg Frederickson.

Square — Heptagon (10 pieces)

The twist hinged version of this dissection was discovered by Greg Frederickson.

Square — Octagon (8 pieces)

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

Square — Dodecagon (9 pieces)

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

Square — Hexagram (7 pieces)

Unlike most of these dissections, this dissection is not derived from a twist hinged dissection but instead directly from the normal 5 piece dissection. It is a shame that there are two such small pieces.

Square — Octagram (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 a shame that the extra piece is so small.

Square — Dodecagram (10 pieces)

This dissection is another that is not derived from a twist hinged dissection but instead directly from the normal 8 piece dissection.

Pentagon — Hexagon (10 pieces)

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. It would be nice to get rid of the small lug piece.

Pentagon — Decagon (11 pieces)

Greg Frederickson’s third book shows a twist hinge dissection of a pentagon to a decagon. A simple conversion gives this locked dissection. Unfortunately half the lugs are rather too small.

Square — Pentagram (11 pieces)

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

Pentagon — Pentagram (11 pieces)

Greg Frederickson’s third book shows a twist hinge dissection of a pentagon to a pentagram. A simple conversion gives this locked dissection.

Hexagon — Dodecagon (7 pieces)

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)

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 cental triangle.

Heptagon — Heptagram (15 pieces)

Greg Frederickson’s third book shows a twist hinge dissection of a heptagon to a heptagram. A simple conversion gives this locked dissection.