Locked Decagon 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 = Decagon (10 pieces with 4 turned over)

This dissection is derived from the dissection here.

Square = Decagon (8 pieces with 2 turned over)

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

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.

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

Pentagon = Decagon (11 pieces with 5 turned over)

This version requires the same number of pieces but removes the small lugs. Unlike the previous version, this method does not generalise.

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.

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.

Decagon = Pentagram (10 pieces with 1 turned over)

There is an overhead of 4 pieces compared to the unlocked version suggesting that there is a better dissection waiting to be found.