Triangle Dissections

Author : Gavin Theobald

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{4} Square
{5} Pentagon
{6} Hexagon
{7} Heptagon
{8} Octagon
{9} Enneagon
{10} Decagon
{11} Hendecagon
{12} Dodecagon
{13} Tridecagon
{14} Tetradecagon
{16} Hexadecagon
{20} Icosagon
{5/2} Pentagram
{6/2} Hexagram
{7/3} Heptagram
{8/2} Octagram
{8/3} Octagram
{10/2} Decagram
{10/4} 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
1 4 6 5 8 7 8 7 11 8 12 11 12 13


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
7 5 12 9 8 10 10 6


Triangle — Square (4 pieces)

The discovery of this dissection is normally attributed to Henry Ernest Dudeney (1902) but may have been first discovered by C. W. McElroy (1902).

The dissection is hingeable.


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.

Triangle — Pentagon (6 pieces)


Triangle — Hexagon (5 pieces)

Discovered by Harry Lindgren (1961).

I know of no other 5 piece solution to this dissection.


Triangle — Heptagon (8 pieces)


Triangle — Octagon (7 pieces)


Triangle — Enneagon (8 pieces)

Previously Freese found a 9 piece solution to this dissection. In Lindgren's book he states “The conclusion is that if you can beat Freese, you will have found a needle in a haystack”. I found the needle! This was one of the first dissection improvements that I found, and I am particularly pleased with it.


Triangle — Decagon (7 pieces)

I’m very pleased with this dissection. It’s the only dissection in which I use this particular decagon strip.


Triangle — Hendecagon (11 pieces)

Reducing this dissection to 11 pieces was not easy. In the wider hendecagon strip the two large pieces can be varied in size. I vary the size to ensure that a hendecagon edge passes exactly through the intersection of triangle edges at the edge of the triangle strip. This saves a piece. The second overlay diagram shows the initial stage of overlaying the second two strips. I modify the shape of the trapezium of the thinner triangle strip to save a further two pieces.


Triangle — Dodecagon (8 pieces)

It is disappointing that this dissection requires as many as 8 pieces, but I’ve been unable to find anything better.


Triangle — Tridecagon (12 pieces)


Triangle — Tetradecagon (11 pieces)


Triangle — Hexadecagon (12 pieces)


Triangle — Icosagon (13 pieces)


Triangle — Pentagram (7 pieces)


Triangle — Hexagram (5 pieces)

Discovered by Harry Lindgren (1961).

The dissection is hingeable.


Triangle — Heptagram {7/3} (12 pieces)


Triangle — Octagram {8/2} (9 pieces)

This was another dissection that was particularly hard to find. A ten piece dissection was fairly straightforward, but every trick I tried to use to improve the dissection further failed. Finally I found the above dissection. The overlaying of strips shown is not sufficient on its own. A further trick is required to save the last piece.


Triangle — Octagram {8/3} (8 pieces)

This is yet another dissection that requires a trick to save a piece.


Triangle — Decagram {10/2} (10 pieces)


Triangle — Decagram {10/4} (10 pieces)


Triangle — Dodecagram {12/2} (6 pieces)

Discovered by Harry Lindgren (1964).


Triangle — Silver Rectangle (4 pieces)

The dissection is hingeable.


Triangle — Golden Rectangle (4 pieces)

The dissection is hingeable.


Triangle — Domino (4 pieces)

The dissection is hingeable.


Triangle — Greek Cross (5 pieces)

Discovered by Harry Lindgren (1961).


Triangle — Latin Cross (5 pieces)

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