Hexagon Dissections

Author : Gavin Theobald

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{3} Triangle
{4} Square
{5} Pentagon
{7} Heptagon
{8} Octagon
{9} Enneagon
{10} Decagon
{11} Hendecagon
{12} Dodecagon
{14} Tetradecagon
{16} Hexadecagon
{18} Octadecagon
{20} Icosagon
{5/2} Pentagram
{6/2} Hexagram
{8/2} Octagram
{8/3} Octagram
{9/2} Enneagram
{9/3} Enneagram
{10/2} Decagram
{12/2} Dodecagram
{12/3}  Dodecagram
{R√2} Silver Rectangle
{Rϕ} Golden Rectangle
{R2} Domino
{G} Greek Cross
{L} Latin Cross
3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
5 5 7 1 8 8 7 1110 9 8 1211 6 1110 12 12 13


5/2 6/2 7/2 7/3 8/2 8/3 9/2 9/3 9/4 10/2 10/3 10/4 12/2 12/3 12/4 12/5
9 6 10 9 8 13 9 8 11 8 1312


Triangle — Hexagon (5 pieces)

Discovered by Harry Lindgren (1961).

I know of no other 5 piece solution to this dissection.


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.


Pentagon — Hexagon (7 pieces)

Discovered by Harry Lindgren (1964).


Hexagon — Heptagon (8 pieces)


Hexagon — Octagon (8 pieces)

Hexagon — Octagon (7 pieces with 2 turned over)


Hexagon — Enneagon (11 pieces)

Hexagon — Enneagon (10 pieces with 1 turned over)


Hexagon — Decagon (9 pieces)

Hexagon — Decagon (8 pieces with 3 turned over)


Hexagon — Hendecagon (12 pieces)

Hexagon — Hendecagon (11 pieces with 1 turned over)


Hexagon — Dodecagon (6 pieces)

The first solution is nicely symmetric, but the second is different to other published solutions in the use of curved pieces.


Hexagon — Tetradecagon (11 pieces)
Hexagon — Tetradecagon (10 pieces with 1 turned over)


Hexagon — Hexadecagon (12 pieces)


Hexagon — Octadecagon (12 pieces)


Hexagon — Icosagon (13 pieces)


Hexagon — Pentagram (9 pieces)


Hexagon — Hexagram (6 pieces)

This is my favourite dissection! Greg Frederickson found a similar dissection, but his requires two pieces to be turned over. His solution is given by the overlay on the right. But by extending the two pieces that are turned over using arcs creates two symmetric pieces that no longer need turning over. The same trick can be used for other dissections, but this is the only straight sided dissection known for which curved pieces are essential for an optimum solution.


Hexagon — Octagram {8/2} (10 pieces)


Hexagon — Octagram {8/3} (9 pieces)

Hexagon — Octagram {8/3} (8 pieces with 1 turned over)

The second of these two dissections is a very tight fit! Solving this in just eight pieces was a surprise.


Hexagon — Enneagram {9/2} (13 pieces)


Hexagon — Enneagram {9/3} (9 pieces)

Hexagon — Enneagram {9/3} (8 pieces with 2 turned over)


Hexagon — Decagram {10/2} (11 pieces)

Hexagon — Decagram {10/2} (10 pieces with 2 turned over)


Hexagon — Dodecagram {12/2} (8 pieces)

This is a modification of a dissection discovered by Greg Frederickson.


Hexagon — Dodecagram {12/3} (13 pieces)

Hexagon — Dodecagram {12/3} (12 pieces with 3 turned over)


Hexagon — Silver Rectangle (5 pieces)

The dissection is translational.


Hexagon — Golden Rectangle (5 pieces)

The dissection is translational.


Hexagon — Domino (5 pieces)

The dissection is translational.


Hexagon — Greek Cross (7 pieces)


Hexagon — Latin Cross (6 pieces)

Discovered by Harry Lindgren (1961).

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