Pentagon Dissections

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 technique.

Square — Pentagon (6 pieces)

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 Harry Lindgren (1964).

Pentagon — Heptagon (9 pieces)

Pentagon — Octagon (9 pieces)

Pentagon — Octagon (8 pieces with 1 turned over)

Pentagon — Enneagon (10 pieces)

Pentagon — Decagon (9 pieces)

I very nearly missed finding this dissection. The 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)

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 dissection. Is a 9 piece solution possible?

Pentagon — Tridecagon (14 pieces)

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)

Pentagon — Hexadecagon (13 pieces)

Pentagon — Icosagon (14 pieces)

Pentagon — Pentagram (9 pieces)

Pentagon — Hexagram (8 pieces)

This is another example of a TT22 dissection.

Pentagon — Octagram {8/2} (11 pieces)

Pentagon — Octagram {8/3} (9 pieces)

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)

Discovered by Greg Frederickson (1974).

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

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

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 technique is required if the dissection is to be improved further.

Pentagon — Silver Rectangle (5 pieces)

Pentagon — Golden Rectangle (6 pieces)

Pentagon — Domino (6 pieces)

Pentagon — Optimised Rectangle (4 pieces)

Pentagon — Greek Cross (7 pieces)

Discovered by Harry Lindgren (1961).

Pentagon — Latin Cross (8 pieces)