Square to Even-sided Polygon Dissections

Author : Gavin Theobald

 {6} Hexagon {8} Octagon {10} Decagon {12} Dodecagon {14} Tetradecagon {16} Hexadecagon {18} Octadecagon {20} Icosagon {22} Icosidigon {24} Icositetragon {26} Icosihexagon {28} Icosioctagon {30} Triacontagon {32} Triacontadigon {34} Triacontatetragon {36} Triacontahexagon {38} Triacontaoctagon {40} Tetracontagon {42} Tetracontadigon {44} Tetracontatetragon {46} Tetracontahexagon {48} Tetracontaoctagon {50} Pentacontagon {52} Pentacontadigon {54} Pentacontatetragon {56} Pentacontahexagon {58} Pentacontaoctagon {60} Hexacontagon {70} Heptacontagon {80} Octacontagon {90} Enneacontagon {100} Hectogon
 n 0 1 2 3 4 5 6 7 8 9 n 4 1 6 5 7 5 9 10+n 7 10 6 11 9 11 10 12 10 13 20+n 11 14 11 15 12 16 12 17 13 17 30+n 13 14 14 15 15 40+n 16 18 19 19 20 50+n 20 21 21 22 22 60+n 23 70+n 25 80+n 28 90+n 30 100+n 33

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 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.

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 — Octadecagon {18} (10 pieces)

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

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 — 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 — 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 — Tetracontadigon {42} (18 pieces)

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

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 — 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.