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
{R×} Optimised Rectangle
{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 11 9 9 8 13 9 8 11 8 10


Triangle - Hexagon Triangle - Hexagon

Triangle — Hexagon (5 pieces)

Discovered by Harry Lindgren.

This uses the TT2 method. I know of no other 5 piece solution to this dissection.


Square - Hexagon Square - Hexagon

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.


Pentagon - Hexagon Pentagon - Hexagon

Pentagon — Hexagon (7 pieces)

This is a PP dissection.

Discovered by Ernest Irving Freese.


Hexagon - Heptagon Hexagon - Heptagon

Hexagon — Heptagon (8 pieces)


Hexagon - Octagon Hexagon - Octagon
Hexagon - Octagon Hexagon - Octagon

Hexagon — Octagon (8 pieces)

Hexagon - Octagon Hexagon - Octagon

Hexagon — Octagon (7 pieces with 2 turned over)


Hexagon - Enneagon Hexagon - Enneagon

Hexagon — Enneagon (11 pieces)

Hexagon - Enneagon Hexagon - Enneagon

Hexagon — Enneagon (10 pieces with 1 turned over)


Hexagon - Decagon Hexagon - Decagon

Hexagon — Decagon (9 pieces)

Hexagon - Decagon Hexagon - Decagon

Hexagon — Decagon (8 pieces with 3 turned over)


Hexagon - Hendecagon
Hexagon - Hendecagon Hexagon - Hendecagon

Hexagon — Hendecagon (12 pieces)

Hexagon - Hendecagon
Hexagon - Hendecagon Hexagon - Hendecagon

Hexagon — Hendecagon (11 pieces with 1 turned over)


Hexagon - Dodecagon Hexagon - Dodecagon

Hexagon — Dodecagon (6 pieces)

Discovered by Ernest Irving Freese.

This dissection and the following one use the method of completed tessellations. This dissects a hexagon with two small triangles to a dodecagon with the same two small triangles.

Hexagon - Dodecagon Hexagon - Dodecagon

Hexagon — Dodecagon (6 pieces)

This is different to other published solutions in the use of curved pieces. Without the curved pieces it would be necessary to turn over two of the pieces.


Hexagon - Tetradecagon Hexagon - Tetradecagon
Hexagon - Tetradecagon

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


Hexagon - Hexadecagon Hexagon - Hexadecagon

Hexagon — Hexadecagon (12 pieces)


Hexagon - Octadecagon Hexagon - Octadecagon

Hexagon — Octadecagon (12 pieces)

This dissection uses the method of completed tessellations. This dissects a hexagon with two small triangles to a dissected octadecagon with the same two small triangles.


Hexagon - Icosagon Hexagon - Icosagon

Hexagon — Icosagon (13 pieces)


Hexagon - Pentagram Hexagon - Pentagram

Hexagon — Pentagram (9 pieces)


Hexagon - Hexagram Hexagon - Hexagram

Hexagon — Hexagram (6 pieces)

This dissection uses the method of completed tessellations. This dissects a hexagon with two small triangles to a hexagram with the same two small triangles.

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 - Heptagram Hexagon - Heptagram

Hexagon — Heptagram {7/2} (11 pieces)

Hexagon - Heptagram Hexagon - Heptagram

Hexagon — Heptagram {7/2} (11 pieces with 2 turned over)


Hexagon - Octagram Hexagon - Octagram

Hexagon — Octagram {8/2} (9 pieces)


Hexagon - Octagram Hexagon - Octagram

Hexagon — Octagram {8/3} (9 pieces)

Hexagon - Octagram Hexagon - Octagram

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
Hexagon - Enneagram Hexagon - Enneagram

Hexagon — Enneagram {9/2} (13 pieces)


Hexagon - Enneagram Hexagon - Enneagram

Hexagon — Enneagram {9/3} (9 pieces)

Hexagon - Enneagram Hexagon - Enneagram

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


Hexagon - Decagram Hexagon - Decagram

Hexagon — Decagram {10/2} (11 pieces)

Hexagon - Decagram Hexagon - Decagram

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


Hexagon - Dodecagram Hexagon - Dodecagram

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

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


Hexagon - Dodecagram Hexagon - Dodecagram

Hexagon — Dodecagram {12/3} (10 pieces)


Hexagon - Silver Rectangle Hexagon - Silver Rectangle

Hexagon — Silver Rectangle (5 pieces)

The dissection is translational.


Hexagon - Golden Rectangle Hexagon - Golden Rectangle

Hexagon — Golden Rectangle (5 pieces)

The dissection is translational.


Hexagon - Domino Hexagon - Domino

Hexagon — Domino (5 pieces)

The dissection is translational.


Hexagon - Optimised Rectangle

Hexagon — Optimised Rectangle (3 pieces)

The dissection is translational. Each piece has the same area.


Hexagon - Greek Cross Hexagon - Greek Cross

Hexagon — Greek Cross (7 pieces)


Hexagon - Latin Cross Hexagon - Latin Cross

Hexagon — Latin Cross (6 pieces)

Discovered by Harry Lindgren.