Triangle - Square (4 pieces)
The discovery of this dissection is normally attributed to Herny 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 coindices 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)
Discovered by Geoffrey Bennett (1926).
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 (10 pieces)
There are several similar tessellations of the tetradecagon. From these can
be obtained a rather narrow strip that can then be dissected to a square using
the PP2 method.
Square - Pentadecagon (11 pieces)
Square - Hexadecagon (11 pieces)
The hexadecagon can be dissected into 5 pieces that form a tessellation. From
this can be obtained a strip that can then be dissected to a square.
Square - Heptadecagon (12 pieces)
Compare this dissection with that for the hendecagon, tridecagon and pentadecagon.
Each of these dissections uses basically the same technique.
Square - Octadecagon (12 pieces)
Note that there is a small twelfth piece at the top of the square. There are other
12 piece dissections of the octadecagon but many of these have an even smaller
piece. It would be nice to find a better dissection.
Square - Enneadecagon (16 pieces / 15 pieces with 1 turned over)
The arrangement of the four large pieces of the enneadecagon to form a strip
is very similar to the strip for a heptagon.
Square - Icosagon (14 pieces)
Square - 21-gon (15 pieces / 14 pieces with 1 turned over)
The arrangement of the four large pieces of the enneadecagon to form a strip
is basically the same as for a heptagon.
Square - 22-gon (14 pieces)
Square - 24-gon (14 pieces)
Square - 26-gon (15 pieces)
Square - 28-gon (16 pieces)
Compare this dissection with that for the icosagon, {22}, {24} and {26}.
These all use more or less the same method. The large pieces are arranged
to form a tessellation that is then overlaid by a tessellation of the square.
Note that a similar dissection is also possible for the {30}.
Square - 30-gon (16 pieces)
Square - 36-gon (17 pieces / 16 pieces with 1 turned over)
This is the smallest n for which a dissection
of {4} to {n} has been found in less than n/2 pieces.
Square - 54-gon (24 pieces)
This dissection is based around the dissection of the octadecagon.
