For the dissections of the regular polygon {n} for even n greater than or equal
to 14 then there is a formula for the number of pieces.

n |
Pieces |

14 to 40 |
⌊n/4⌋+6 |

42 to 124 |
⌊n/4⌋+8 |

126 to 376 |
⌊n/4⌋+10 |

378 to 1132 |
⌊n/4⌋+12 |

1134 to 3400 |
⌊n/4⌋+14 |

### 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 more complex dissection
of the square and 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.

The dissection is translational.

### Square — Octagon (5 pieces)

This dissection first appeared in the circa 1300AD anonymous Persian manuscript
“Interlocks of Similar or Complementary Figures”.

### Square — Decagon (7 pieces)

I like this dissection, although it has some odd shaped pieces.

### Square — Dodecagon (6 pieces)

Discovered by Harry Lindgren (1951).

### Square — Tetradecagon {14} (9 pieces)

This dissection is within a family of even sided dissections that continues up to
the {40}.

### Square — Hexadecagon {16} (10 pieces)

### Square — Octadecagon {18} (10 pieces)

The
piece has a leftward pointing spike near its top.
It would nice to find an alternative dissection without this blemish.

### Square — Icosagon {20} (11 pieces)

### Square — Icosidigon {22} (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 — Icositetragon {24} (12 pieces)

### Square — Icosihexagon {26} (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 — Icosioctagon {28} (13 pieces)

### Square — Triacontagon {30} (13 pieces)

The piece at the top left of the square has a very thin downward spike along
the left edge of the square. The angle at the tip of this spike is less than
a tenth of a degree. (This means that the left edge of the shaded piece is
not quite vertical). Also there is a very short edge at the far right end of
the same piece. Neither of these features can easily be seen even at high
magnification.

### Square — Triacontadigon {32} (14 pieces)

### Square — Triacontatetragon {34} (14 pieces)

### Square — Triacontahexagon {36} (15 pieces)

### Square — Triacontaoctagon {38} (15 pieces)

### Square — Tetracontagon {40} (16 pieces)

### Square — Tetracontadigon {42} (18 pieces)

This dissection is the first within a family of even sided
dissections that continues up to about {124}.

### Square — Tetracontatetragon {44} (19 pieces)

### Square — Tetracontahexagon {46} (19 pieces)

### Square — Tetracontaoctagon {48} (20 pieces)

### Square — Pentacontagon {50} (20 pieces)

### Square — Pentacontadigon {52} (21 pieces)

### Square — Pentacontatetragon {54} (21 pieces)

Note that the two lines that appear to cross at left of centre of the
54-gon do not in fact do so.

### Square — Pentacontahexagon {56} (22 pieces)

### Square — Pentacontaoctagon {58} (22 pieces)

### Square — Hexacontagon {60} (23 pieces)

### Square — Heptacontagon {70} (25 pieces)

### Square — Octacontagon {80} (28 pieces)

### Square — Enneacontagon {90} (30 pieces)

This is the smallest n for which a dissection
of {4} to {n} has been found in less than or equal to n/3 pieces.
The first dissection to be in less than n/3 pieces is expected to be the
dissection of the {94} in 31 pieces.

Note that there is a thin spike at the top left of the piece at the
bottom left corner of the square.

### Square — Hectogon {100} (33 pieces)