Square to Even-sided Polygon Dissections

Author : Gavin Theobald

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. For n over 40 and when n is a multiple of 12 or 16, then it is possible to improve slightly on this.

For n from 14 to 40 the formula is: ⌊n/4⌋+6

For n from 42 to 124 an additional two pieces are required: ⌊n/4⌋+8

The general formula is: ⌊n/4⌋+2⌊log₃(n/14)⌋+6.

Square — Hexagon (5 pieces)

The first dissection of a square to a hexagon in 5 pieces was by Paul-Jean Busschop.

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 making this another variable strip. 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 uses the method of completed tessellations. This dissects a square with a small square to an octagon with the same small square.

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

Square — Decagon (7 pieces)

Previously Ernest Irving Freese found an 8 piece solution to this dissection.

This is a PP dissection that uses a variable decagon strip. The two overlays show the strips before and after the modification of the decagon strip that saves a piece.

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

Square — Dodecagon (6 pieces)

This dissection uses the method of overlaid tessellations.

Discovered by Harry Lindgren.

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)

Previously Ernest Irving Freese found a 15 piece solution.

Square — Octadecagon {18} (10 pieces)

The *lime green* 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)

Previously Ernest Irving Freese found a 19 piece solution.

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)

Previously Ernest Irving Freese found a 22 piece solution.

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 — Tetracontaoctagon {48} (19 pieces with 1 turned over)

This is the first of a family of dissections of {n} when n is a multiple of 12.

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 — Pentacontahexagon {56} (21 pieces with 2 turned over)

This is the first of a family of dissections of {n} when n is 8 more than a multiple of 16.

Square — Pentacontaoctagon {58} (22 pieces)

Square — Hexacontagon {60} (23 pieces)

Square — Hexacontagon {60} (22 pieces with 1 turned over)

This dissection was the first that I found that improved on the formula given at the top of this page. It is a shame that it relies on turning over a piece.

Square — Heptacontagon {70} (25 pieces)

Square — Octacontagon {80} (28 pieces)

Square — Octacontatetragon {84} (28 pieces with 1 turned over)

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 require less than n/3 pieces is expected to be the dissection of the {88} in 29 pieces.

Square — Enneacontagon {90} (30 pieces)

If pieces are not allowed to be turned over, then 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)