Hexagon Dissections

Author : Gavin Theobald

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 (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 (7 pieces)

Discovered by Ernest Irving Freese.

This is a PP dissection.

Hexagon — Heptagon (8 pieces)

Previously Harry Lindgren discovered an 11 piece solution.

This is a PP dissection.

Hexagon — Octagon (8 pieces)

Previously Harry Lindgren discovered a 9 piece solution. He would be annoyed with himself for missing the first of these two solutions.

The first solution is a PP dissection using a hexagon strip from a tessellation. The second solution is a TT2 dissection.

Hexagon — Octagon (7 pieces with 2 turned over)

This dissection uses the method of overlaid tessellations with both tessellations formed from strips.

Hexagon — Enneagon (11 pieces)

Previously Harry Lindgren discovered a 14 piece solution. David Paterson found a 12 piece solution and Anton Hanegraaf found a different 11 piece solution.

This is a PP dissection.

Hexagon — Enneagon (10 pieces with 1 turned over)

This dissection uses the method of overlaid tessellations with the hexagon tessellation formed from a strip and with a variable enneagon tessellation. These methods combine to give a particularly efficient dissection.

Hexagon — Decagon (9 pieces)

Previously Harry Lindgren discovered a different 9 piece solution.

This is a PP dissection.

Hexagon — Decagon (8 pieces with 3 turned over)

This dissection overlays a decagon strip over a hexagon tessellation formed from strips.

Hexagon — Hendecagon (12 pieces)

This dissection uses the TT2 method twice.

Hexagon — Hendecagon (11 pieces with 1 turned over)

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

Hexagon — Hexadecagon (12 pieces)

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 (13 pieces)

Hexagon — Pentagram (9 pieces)

Previously Harry Lindgren discovered a 10 piece solution.

This is a TT2 dissection.

Hexagon — Hexagram (6 pieces)

Previously Harry Lindgren discovered a 7 piece solution.

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 {7/2} (11 pieces)

This is a PP dissection. The pieces are a very tight fit in two different places.

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

This replaces the hexagon strip with a tessellation. This gives a little more room to shift pieces around so this solution is not quite such a tight fit. The down side is that two pieces must now be turned over.

Hexagon — Octagram {8/2} (9 pieces)

Hexagon — Octagram {8/3} (9 pieces)

This is a PP dissection.

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

This dissection replaces the hexagon strip with a tessellation. This page describes this dissection.

This dissection is a very tight fit at four different points! Solving this in just eight pieces was a pleasant surprise.

Hexagon — Enneagram {9/2} (13 pieces)

This is a TT2 dissection. A PP dissection is used to insert the two rhombuses which have been merged into a single parallelogram.

Hexagon — Enneagram {9/3} (9 pieces)

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

Previously Greg Frederickson found a variation of these dissections.

These dissections uses the method of overlaid tessellations.

Hexagon — Decagram {10/2} (11 pieces)

This is a TT2 dissection.

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

This dissection forms two tessellations from strips and then overlays them.

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

Firstly, Harry Lindgren found a 10 piece solution. This dissection is a minor modification of a dissection discovered by Greg Frederickson.

These dissections uses the method of overlaid tessellations.

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

Hexagon — Silver Rectangle (5 pieces)

This is a PP2 dissection.

The dissection is translational.

Hexagon — Golden Rectangle (5 pieces)

This is a PP2 dissection.

The dissection is translational.

Hexagon — Domino (5 pieces)

This is a PP2 dissection.

The dissection is translational.

Hexagon — Optimised Rectangle (3 pieces)

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

Hexagon — Greek Cross (7 pieces)

Harry Lindgren was the first to find solutions for this dissection. My dissection is yet another way to solve this.

This is a PT2 dissection.

Hexagon — Latin Cross (6 pieces)

Discovered by Harry Lindgren.

This is a PP dissection.