Pentagon Dissections

Author : Gavin Theobald

Home Index Pentagon Contact  Colour
{3} Triangle
{4} Square
{6} Hexagon
{7} Heptagon
{8} Octagon
{9} Enneagon
{10} Decagon
{11} Hendecagon
{12} Dodecagon
{13} Tridecagon
{14} Tetradecagon
{15} Pentadecagon
{16} Hexadecagon
{20} Icosagon
{5/2} Pentagram
{6/2} Hexagram
{8/2} Octagram
{8/3} Octagram
{10/2} Decagram
{12/2}  Dodecagram
{R√2} Silver Rectangle
{Rϕ} Golden Rectangle
{R2} Domino
{G} Greek Cross
{L} Latin Cross
3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
6 6 1 7 9 9 8 10 9 12 10 14 12 14 13 14


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 8 11 9 7 1211


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 grey 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 — Greek Cross (7 pieces)

Discovered by Harry Lindgren (1961).


Pentagon — Latin Cross (8 pieces)

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