Square to Polygram Dissections

Author : Gavin Theobald

Square — Pentagram (7 pieces)

Philip Tilson was the first to discover a 7 piece solution, but by a different method.

This is a TT2 dissection.

Square — Hexagram (5 pieces)

Discovered by Harry Bradley.

This PP dissection is translational.

Square — Heptagram {7/2} (10 pieces)

This dissection uses the PP method twice.

Square — Heptagram {7/3} (12 pieces)

This dissection uses the TT2 method twice. There is the main heptagram strip and also a narrow strip of triangles. Since both strips repeat at the same rate they can be overlaid side by side.

Square — Octagram {8/2} (7 pieces)

Harry Lindgren found an 8 piece solution, but Greg Frederickson found this improved solution.

Square — Octagram {8/3} (8 pieces)

Harry Lindgren was the first to find an 8 piece dissection. There are many other ways to solve this dissection in 8 pieces. This suggests that a seven piece solution may exist, but if it does, I have been unable to find it.

The second dissection is a PP dissection. The strips’ anchor points are aligned for the sake of greater symmetry, but this is not necessary.

Square — Enneagram {9/2} (11 pieces)
Square — Enneagram {9/2} (10 pieces with 1 turned over)

These are PP dissections.

Square — Enneagram {9/3} (13 pieces)

This is a PP dissection.

Square — Enneagram {9/3} (12 pieces with 3 turned over)

This is a PP dissection.

Square — Enneagram {9/4} (19 pieces with 3 turned over)

This is a PP dissection.

Square — Decagram {10/2} (10 pieces)

This is a PP dissection.

Square — Decagram {10/2} (9 pieces with 1 turned over)

This builds a tessellation from a strip.

Square — Decagram {10/3} (11 pieces)

Square — Decagram {10/3} (10 pieces with 4 turned over)

Both of these are PP dissections.

Square — Decagram {10/4} (10 pieces)

Anton Hanegraaf was the first to discover a 10 piece solution.

The first solution is a PP dissection and the second is a PP2 dissection.

Square — Dodecagram {12/2} (8 pieces)

Robert Reid found a 9 piece solution which I improved to 8 pieces, but needed to turn over a piece. This version removes this requirement.

This dissection uses the method of variable tessellations.

Note that there are some very short edges that are not visible in these views. Click on the images to see enlarged versions.

Rectangle {R_1.05} — Dodecagram {12/2} (8 pieces)

These diagrams use the same technique as for the square, but are better at showing the way in which the dissection works. Click on the images to see enlarged versions.

Square — Dodecagram {12/3} (8 pieces)

Discovered by Greg Frederickson.

This dissection uses the method of overlaid tessellations.

Square — Dodecagram {12/4} (11 pieces)

Discovered by Greg Frederickson.

This dissection is a clever use of the method of completed tessellations.

Square — Dodecagram {12/5} (9 pieces)

Previously Harry Lindgren found a 10 piece solution. This solution was discovered by Greg Frederickson.

Square — Hexadecagram {16/2} (13 pieces)

Square — Hexadecagram {16/2} (12 pieces with 2 turned over)

Square — Icosagram {20/2} (15 pieces)

Square — Icositetragram {24/2} (15 pieces with 1 turned over)