Heptagon Dissections

Author : Gavin Theobald

{3} Triangle
{4} Square
{5} Pentagon
{6} Hexagon
{8} Octagon
{9} Enneagon
{10} Decagon
{11} Hendecagon
{12} Dodecagon
{14} Tetradecagon
{5/2} Pentagram
{6/2} Hexagram
{7/2} Heptagram
{7/3} Heptagram
{8/2} Octagram
{8/3} Octagram
{10/2} Decagram
{12/2} Dodecagram
{12/3} Dodecagram
{14/2}  Tetradecagram
{R√2} Silver Rectangle
{Rϕ} Golden Rectangle
{R2} Domino
{R×} Optimised Rectangle
{G} Greek Cross
{L} Latin Cross
3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
8 7 9 8 1 1110 13 11 15 11 15


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
11 9 15 13 12 12 1413 11 13

Triangle — Heptagon (8 pieces)

This is a PP dissection.

Square — Heptagon (7 pieces)

This was one of the first dissection improvements I found, and I am particularly proud of finding it. The previous record that I knew about was a 9 piece dissection found by Lindgren. I managed to improve this in 8 pieces in a number of ways, and this made me sure that there had to be a 7 piece solution. The problem is the plain square strip cannot be overlaid over the usual heptagon strip since the square strip is too wide. So I looked for a narrower heptagon strip. The technique I use allows me to produce a range of heptagon strips, but I chose the one that ensures that an edge of the heptagon coincides with an edge of the square. This saves a piece giving a 7 piece record. I don’t believe that a further improvement exists.

Pentagon — Heptagon (9 pieces)

Hexagon — Heptagon (8 pieces)

Heptagon — Octagon (11 pieces)

Heptagon — Octagon (10 pieces with 1 turned over)

Heptagon — Enneagon (13 pieces)

Heptagon — Decagon (11 pieces)

Heptagon — Hendecagon (15 pieces)

Heptagon — Dodecagon (11 pieces)

Heptagon — Tetradecagon (15 pieces)

Heptagon — Pentagram (11 pieces)

Heptagon — Hexagram (9 pieces)

Heptagon — Heptagram {7/2} (15 pieces)

Heptagon — Heptagram {7/3} (13 pieces)

Discovered by Anton Hanegraaf.

Heptagon — Octagram {8/2} (12 pieces)

Heptagon — Octagram {8/3} (12 pieces)

Heptagon — Decagram {10/2} (14 pieces)

Heptagon — Decagram {10/2} (13 pieces with 1 turned over)

Heptagon — Dodecagram {12/2} (11 pieces)

Heptagon — Dodecagram {12/3} (13 pieces)

Heptagon — Tetradecagram {14/2} (12 pieces)

This is a modification of a dissection found by Greg Frederickson. My variation is more symmetric, but uses the same number of pieces. In Frederickson’s first book he mentions an 11 piece version of this dissection, but he has not been able to find a copy of it. Either he miscounted the number of pieces, or there is a forgotten trick to save a piece. So, can you improve this dissection?

Heptagon — Silver Rectangle (7 pieces)

Heptagon — Golden Rectangle (7 pieces)

Heptagon — Domino (7 pieces)

Heptagon — Optimised Rectangle (5 pieces)

Heptagon — Greek Cross (9 pieces)

Heptagon — Latin Cross (8 pieces)