Pentagon Dissections

Author : Gavin Theobald

Triangle — Pentagon (6 pieces)

Discovered by Harry Lindgren. Michael Goldberg found the first six piece solution.

This is a rare use of the TT1 method.

Triangle — Pentagon (6 pieces)

This is perhaps not the best example of a dissection of triangle and pentagon, but it is new and it does demonstrate the TT22 method.

Square — Pentagon (6 pieces)

First dissected in 6 pieces by Robert Brodie.

This is not a very elegant solution because of the rather small piece, but it is another example of a TT22 dissection.

There are a number of different six piece solutions possible and this raises the question of whether or not a five piece solution exists. There is a range of rectangle shapes that will dissect to a pentagon in just five pieces, but I think it unlikely that anyone will find a five piece solution for the square.

Pentagon — Hexagon (7 pieces)

Discovered by Ernest Irving Freese.

This is a PP dissection.

Pentagon — Heptagon (9 pieces)

Previously, Harry Lindgren found an 11 piece solution.

This is a PP dissection.

Pentagon — Octagon (9 pieces)

Previously, Harry Lindgren found a different solution.

This is a PP dissection.

Pentagon — Octagon (8 pieces with 1 turned over)

This uses an octagon tessellation formed from strips.

Pentagon — Enneagon (10 pieces)

This dissection uses the method of variable tessellation to give a very nice and very efficient dissection.

Pentagon — Decagon (9 pieces)

Previously, Harry Lindgren found first an 11 piece and then a 10 piece solution.

I very nearly missed finding this PT dissection. The *red* piece is so nearly cut into two, adding another two pieces to this dissection, that I did not think that it was possible. Fortunately I checked and hence obtained another record.

Pentagon — Hendecagon (12 pieces)

This dissection uses the TT2 method twice.

The overlay diagrams on the right show the basic dissection before I modify it to save some pieces.

Pentagon — Dodecagon (10 pieces)

There is room for improvement in this PT dissection. Is a 9 piece solution possible?

Pentagon — Tridecagon (14 pieces)

This dissection uses the TT2 method twice.

The top overlay diagram on the right shows the dissection before I modify it to save a piece.

Pentagon — Tetradecagon (12 pieces)

The long triangle at the bottom left of the pentagon is in fact a quadrilateral with a short fourth side at the bottom.

Pentagon — Pentadecagon (14 pieces)

This dissection uses the TT2 method twice.

Pentagon — Hexadecagon (13 pieces)

Pentagon — Octadecagon (13 pieces)

Pentagon — Icosagon (14 pieces)

Pentagon — Pentagram (9 pieces)

This dissection uses the TT2 method.

Pentagon — Hexagram (8 pieces)

Previously Harry Lindgren discovered a different 8 piece solution.

This is another example of a TT22 dissection.

Pentagon — Octagram {8/2} (11 pieces)

This is a PP dissection.

Pentagon — Octagram {8/3} (9 pieces)

This is a PT dissection.

The thin spike makes this a rather inelegant dissection, but I have not been able to find another 9 piece solution.

Pentagon — Decagram {10/2} (7 pieces)

Previously, Harry Lindgren found an 8 piece solution. This solution was discovered by Greg Frederickson.

Pentagon — Dodecagram {12/2} (12 pieces)

This is a PP dissection.

Pentagon — Dodecagram {12/2} (11 pieces with 3 turned over)

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

Compared to the other dodecagram dissections this one is particularly inefficient. This is due to the dimensions of the pentagon making it impractical to form a pentagon tessellation that can overlay a dodecagram tessellation.

By turning over pieces we can improve the dissection, but a entirely different method is required if the dissection is to be improved further.

Pentagon — Dodecagram {12/3} (12 pieces)

This is a PT dissection.

Pentagon — Silver Rectangle (5 pieces)

This is a TT2 dissection.

Pentagon — Golden Rectangle (6 pieces)

Previously, Harry Lindgren found a different solution.

This is a TT22 dissection.

Pentagon — Domino (6 pieces)

This is a TT2 dissection.

Pentagon — Optimised Rectangle (4 pieces)

Pentagon — Greek Cross (7 pieces)

Discovered by Harry Lindgren.

This is a TT2 dissection.

Pentagon — Latin Cross (8 pieces)

Previously, Harry Lindgren found a different solution.

This is a TT2 dissection.