Triangle — Square (4 pieces)
The discovery of this dissection is normally attributed to Henry Ernest Dudeney but may have been first discovered by C. W. McElroy.
Square — Pentagon (6 pieces)
This is not a very elegant solution because of the rather small piece, but it is another example of a TT22 dissection. There are a number of different six piece solutions possible and this raises the question of whether or not a five piece solution exists. There is a range of rectangle shapes that will dissect to a pentagon in just five pieces, but I think it unlikely that anyone will find a five piece solution for the square.
Square — Hexagon (5 pieces)
This new dissection is unusual in that there are aligned edges of the square and the hexagon. I found this dissection after finding the following more complex dissection for the heptagon. The hexagon strip can be formed in a variety of ways. The trick is to form it the correct way so that when the two strips are overlaid, a hexagon edge coincides with a square edge, hence saving a piece.
Square — Heptagon (7 pieces)
This was one of the first dissection improvements I found, and I am particularly proud of finding it. The previous record that I knew of was a 9 piece dissection found by Lindgren. I managed to improve this in 8 pieces in a number of ways, and this made me sure that there had to be a 7 piece solution. The problem is the plain square strip cannot be overlaid over the usual heptagon strip since the square strip is too wide. So I looked for a narrower heptagon strip. The technique I use allows me to produce a range of heptagon strips, but I chose the one that ensures that an edge of the heptagon coincides with an edge of the square. This saves a piece giving a 7 piece record. I don’t believe that a further improvement exists.
Square — Octagon (5 pieces)
This dissection first appeared in the circa 1300AD anonymous Persian manuscript "Interlocks of Similar or Complementary Figures".
Square — Enneagon (9 pieces)
The first of these two dissections was my first solution of this dissection. It suffers from having several short straight lines that don't show up clearly in diagrams of this size. Click on the diagrams to see an enlargement. The second solution is much more elegant.
Square — Decagon (7 pieces)
I like this dissection, although it has some odd shaped pieces.
Square — Hendecagon (10 pieces)
Greg Frederickson suggested that I tried dissecting the hendecagon to a square. This is my best solution after many attempts.
Square — Dodecagon (6 pieces)
Discovered by Harry Lindgren (1951).
Square — Tridecagon (11 pieces)
Square — Tetradecagon (9 pieces)
This dissection includes a family of even sided dissections that continues up to the {28}. Then a similar set of dissections with a one piece overhead continues up to the {40}. After that other methods are required.
Square — Pentadecagon (11 pieces)
Square — Hexadecagon (10 pieces)
Square — Heptadecagon (12 pieces)
Square — Octadecagon (10 pieces)
Square — Enneadecagon (13 pieces)
Compare this dissection with that for the hendecagon, tridecagon, pentadecagon and heptadecagon. Each of these dissections uses basically the same technique.
Square — Icosagon (11 pieces)
Square — 21-gon (14 pieces)
Square — 22-gon (11 pieces)
This is the smallest n other than 12 for which a dissection of {4} to {n} has been found in less than or equal to n/2 pieces.
Square — 23-gon (15 pieces)
Square — 24-gon (12 pieces)
Square — 25-gon (16 pieces)
There is a one piece overhead to this dissection. It may well be possible to solve this in 15 pieces, but I've not been able to find anything better than this 16 piece dissection.
Square — 26-gon (12 pieces)
This is the smallest n for which a dissection of {4} to {n} has been found in less than n/2 pieces.
Square — 27-gon (17 pieces)
There is a one piece overhead to this dissection. It may well be possible to solve this in 16 pieces, but I've not been able to find anything better than this 17 piece dissection.
Square — 28-gon (13 pieces)
Square — 29-gon (17 pieces)
There is a one piece overhead to this dissection. It may well be possible to solve this in 16 pieces, but I’ve not been able to find anything better than this 17 piece dissection.
Square — 30-gon (14 pieces)
Square — 32-gon (15 pieces)
Square — 34-gon (15 pieces)
Square — 36-gon (16 pieces)
Square — 38-gon (16 pieces)
Square — 40-gon (17 pieces)
Square — 42-gon (19 pieces)
This dissection is based around the dissection of the tetradecagon.
Square — 44-gon (20 pieces)
This dissection is based around the dissection of the hexadecagon.
Square — 46-gon (20 pieces)
This dissection is based around the dissection of the hexadecagon.
Square — 48-gon (21 pieces)
This dissection is based around the dissection of the hexadecagon.
Square — 50-gon (21 pieces)
Square — 52-gon (22 pieces)
This dissection is based around the dissection of the octadecagon.
Square — 54-gon (22 pieces)
This dissection is based around the dissection of the octadecagon.
Square — 56-gon (23 pieces)
This dissection is based around the dissection of the octadecagon.
Square — 58-gon (23 pieces)
This dissection is based around the dissection of the octadecagon.
Square — 60-gon (24 pieces)
This dissection is based around the dissection of the octadecagon.