Square to Even-sided Polygon Dissections

Author : Gavin Theobald

Home Index Square — {2n} Contact  Colour
{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 14 12 15 12 16 13 16
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 — Hexadecagon {16} (10 pieces)


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 — 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 - 42-gon Square - 42-gon

Square — Tetracontadigon {42} (18 pieces)

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


Square - 44-gon Square - 44-gon

Square — Tetracontatetragon {44} (19 pieces)


Square - 46-gon Square - 46-gon

Square — Tetracontahexagon {46} (19 pieces)


Square - 48-gon Square - 48-gon

Square — Tetracontaoctagon {48} (20 pieces)


Square - 50-gon Square - 50-gon

Square — Pentacontagon {50} (20 pieces)


Square - 52-gon Square - 52-gon

Square — Pentacontadigon {52} (21 pieces)


Square - 54-gon Square - 54-gon

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 - 56-gon Square - 56-gon

Square — Pentacontahexagon {56} (22 pieces)


Square - 58-gon Square - 58-gon

Square — Pentacontaoctagon {58} (22 pieces)


Square - 60-gon Square - 60-gon

Square — Hexacontagon {60} (23 pieces)


Square - 70-gon Square - 70-gon

Square — Heptacontagon {70} (25 pieces)


Square - 80-gon Square - 80-gon

Square — Octacontagon {80} (28 pieces)


Square - 90-gon Square - 90-gon

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.


Square - 100-gon Square - 100-gon

Square — Hectogon {100} (33 pieces)

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