Optimised Rectangle Dissections

Author : Gavin Theobald

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The optimised rectangle {R×} is a rectangle of whatever proportions that gives the best dissection for a particular polygon. It is not necessarily the optimal rectangle as a better dissection may one day be found.

{3} Triangle
{5} Pentagon
{6} Hexagon
{7} Heptagon
{8} Octagon
{9} Enneagon
{10} Decagon
{11} Hendecagon
{12} Dodecagon
{13} Tridecagon
{14} Tetradecagon
{15} Pentadecagon
{16} Hexadecagon
{18} Octadecagon
{5/2} Pentagram
{6/2} Hexagram
{7/2} Heptagram
{8/2} Octagram
{8/3} Octagram
{9/3} Enneagram
{10/2} Decagram
{10/3} Decagram
{10/4} Decagram
{12/2}  Dodecagram
{12/5}  Dodecagram
{G} Greek Cross
{L} Latin Cross
3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
3 2 1 4 3 5 4 7 4 9 5 10 7 10 9 9


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
5 4 8 7 6 4 10 10 9 15 8 8 7 6 7 7


Triangle — Optimised Rectangle (3 pieces)
Triangle — Optimised Rectangle (2 pieces with 1 turned over)

The first dissection is hingeable.


Pentagon — Optimised Rectangle (4 pieces)


Hexagon — Optimised Rectangle (3 pieces)

The dissection is translational. Each piece has the same area.


Heptagon — Optimised Rectangle (5 pieces)


Octagon — Optimised Rectangle (4 pieces)


Enneagon — Optimised Rectangle (7 pieces)


Decagon — Optimised Rectangle (4 pieces)


Hendecagon — Optimised Rectangle (9 pieces)


Dodecagon — Optimised Rectangle (5 pieces)


Tridecagon — Optimised Rectangle (10 pieces)


Tetradecagon — Optimised Rectangle (7 pieces)


Pentadecagon — Optimised Rectangle (10 pieces)


Hexadecagon — Optimised Rectangle (9 pieces)


Octadecagon — Optimised Rectangle (9 pieces)


Optimised Rectangle — Pentagram (5 pieces)


Optimised Rectangle — Hexagram (4 pieces)

The dissection is translational.


Optimised Rectangle — Heptagram {7/2} (8 pieces)


Optimised Rectangle — Octagram {8/2} (7 pieces)
Optimised Rectangle — Octagram {8/2} (6 pieces with 2 turned over)


Optimised Rectangle — Octagram {8/3} (4 pieces)


Optimised Rectangle — Enneagram {9/3} (10 pieces)
Optimised Rectangle — Enneagram {9/3} (9 pieces with 2 turned over)


Optimised Rectangle — Decagram {10/2} (8 pieces)


Optimised Rectangle — Decagram {10/3} (8 pieces)
Optimised Rectangle — Decagram {10/3} (7 pieces with 1 turned over)


Optimised Rectangle — Decagram {10/4} (6 pieces)


Optimised Rectangle — Dodecagram {12/2} (7 pieces)


Optimised Rectangle — Dodecagram {12/5} (7 pieces)


Optimised Rectangle — Greek Cross (3 pieces)

The dissection is translational and hingeable.


Optimised Rectangle — Latin Cross (3 pieces)

The dissection is translational.

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