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 only display dissections of the polygons and polygrams that can be dissected in twelve or less pieces. 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 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. 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 with 1 turned over)

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

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.

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

Triangle — Octagon (10 pieces with 3 turned over)

This dissection is based around the first one here.

Triangle — Decagon (10 pieces with 4 turned over)

This dissection is derived from the dissection here.

Triangle — Pentagram (11 pieces with 4 turned over)

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

Triangle — Hexagram (6 pieces)

I am rather pleased with this dissection. It was an unexpected find. It is an unusual dissection on this page as it does not require any pieces to be turned over.

Triangle — Octagram (12 pieces with 4 turned over)

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 with 1 turned over)

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 with 2 turned over)

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

Square — Heptagon (10 pieces with 4 turned over)

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

Square — Octagon (8 pieces with 3 turned over)

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 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. 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 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 (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 (10 pieces with 2 turned over)

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

Pentagon — Octagon (11 pieces with 2 turned over)

An imperfection of this dissection is the small lug piece.

Pentagon — Decagon (11 pieces with 5 turned over)

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, although there are tricks for enlarging them.

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

Pentagon — Pentagram (11 pieces with 5 turned over)

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

The same method will work for dissecting a {p} to a {p/q} in 2p+1 pieces, but only for lower values of q.

Hexagon — Heptagon (11 pieces with 4 turned over)

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

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 long thin spike, a very short spike and a thin projection. Nevertheless, I am pleased to be able to do this dissection is 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 cental triangle.

Octagon — Decagon (11 pieces with 4 turned over)

Many of the lugs of this dissection are too small. There are 6 small lugs visible, but actually there are another 4 that are the tenth of the size of these ones. Only at maximum zoom do they become visible.

Octagon — Decagon (12 pieces with 5 turned over)

This dissection removes the minute lugs of the previous dissection, but at the cost of an extra piece. Many of the lugs of this dissection are still too small, but I have not yet found a way to remove this imperfection.

Triangle — Hexagon — Dodecagon (13 pieces with 6 turned over)

This is a rather nice 3 way dissection. It only requires one extra piece compared to an unlocked dissection.

Triangle — Hexagon — Hexagram (13 pieces with 6 turned over)

Another 3 way dissection, although not quite as clean as the last. It only requires two extra pieces compared to an unlocked dissection.