Three Way Dissections

Author : Gavin Theobald

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{3}-{4}-{5} Triangle — Square — Pentagon
{3}-{4}-{6} Triangle — Square — Hexagon
{3}-{4}-{8} Triangle — Square — Octagon
{3}-{4}-{12} Triangle — Square — Dodecagon
{3}-{4}-{5/2} Triangle — Square — Pentagram
{3}-{4}-{6/2} Triangle — Square — Hexagram
{3}-{4}-{8/2} Triangle — Square — Octagram
{3}-{4}-{12/2} Triangle — Square — Dodecagram
{3}-{5}-{6} Triangle — Pentagon — Hexagon
{3}-{5}-{6/2} Triangle — Pentagon — Hexagram
{3}-{6}-{8} Triangle — Hexagon — Octagon
{3}-{6}-{12} Triangle — Hexagon — Dodecagon
{3}-{6}-{6/2} Triangle — Hexagon — Hexagram
{3}-{7}-{6/2} Triangle — Heptagon — Hexagram
{3}-{8}-{6/2} Triangle — Octagon — Hexagram
{3}-{6/2}-{12/2}  Triangle — Hexagram — Dodecagram
 
{4}-{5}-{6} Square — Pentagon — Hexagon
{4}-{5}-{8} Square — Pentagon — Octagon
{4}-{5}-{12} Square — Pentagon — Dodecagon
{4}-{5}-{6/2} Square — Pentagon — Hexagram
{4}-{6}-{7} Square — Hexagon — Heptagon
{4}-{6}-{8} Square — Hexagon — Octagon
{4}-{6}-{12} Square — Hexagon — Dodecagon
{4}-{6}-{6/2} Square — Hexagon — Hexagram
{4}-{8}-{12} Square — Octagon — Dodecagon
{4}-{8}-{6/2} Square — Octagon — Hexagram
{4}-{12}-{6/2} Square — Dodecagon — Hexagram
 
{6}-{12}-{6/2} Hexagon — Dodecagon — Hexagram
4
10 5
9 12 6 {3}
7
11 12 8
11 12 12
12 5/2
8 11 11 12 11 6/2
12 8/2
11 11 12/2
5
1211 6
12 7 {4}
1211 1110 8
12 11 11 12
5/2
12 10 11 12 6/2
8/2
12/2

There is a very large number of possible three way dissections that can be performed. To limit this number I have restricted myself to looking for three way dissections of the regular polygons and polygrams that can be solved in twelve or fewer pieces.


Triangle - Square - Pentagon

Triangle — Square — Pentagon (10 pieces)


Triangle - Square - Hexagon

Triangle — Square — Hexagon (9 pieces)

Discovered by Harry Lindgren (1964).
Triangle - Square - Octagon

Triangle — Square — Octagon (11 pieces)


Triangle - Square - Dodecagon

Triangle — Square — Dodecagon (11 pieces)


Triangle - Square - Pentagram

Triangle — Square — Pentagram (12 pieces)


Triangle - Square - Hexagram

Triangle — Square — Hexagram (8 pieces)

The dissection is hingeable.
Triangle - Square - Octagram

Triangle — Square — Octagram {8/2} (12 pieces)


Triangle - Square - Dodecagram

Triangle — Square — Dodecagram {12/2} (11 pieces)


Triangle - Pentagon - Hexagon

Triangle — Pentagon — Hexagon (12 pieces)


Triangle - Pentagon - Hexagram

Triangle — Pentagon — Hexagram (11 pieces)


Triangle - Hexagon - Octagon

Triangle — Hexagon — Octagon (12 pieces)


Triangle - Hexagon - Dodecagon

Triangle — Hexagon — Dodecagon (11 pieces)


Triangle - Hexagon - Hexagram

Triangle — Hexagon — Hexagram (11 pieces)


Triangle - Heptagon - Hexagram

Triangle — Heptagon — Hexagram (12 pieces)

There is a small twelfth piece that can just be seen at the centre of the hexagram. I’ve not yet found a way to remove this blemish.
Triangle - Octagon - Hexagram

Triangle — Octagon — Hexagram (11 pieces)


Triangle - Hexagram - Dodecagram

Triangle — Hexagram — Dodecagram {12/2} (11 pieces)


Square - Pentagon - Hexagon

Square — Pentagon — Hexagon (12 pieces)

Square - Pentagon - Hexagon

Square — Pentagon — Hexagon (11 pieces with 2 turned over)


Square - Pentagon - Octagon

Square — Pentagon — Octagon (12 pieces)

Square - Pentagon - Octagon

Square — Pentagon — Octagon (11 pieces with 1 turned over)


Square - Pentagon - Dodecagon

Square — Pentagon — Dodecagon (12 pieces)

There is a small thin twelfth piece that can just be seen at the top left of the dodecagon.
Square - Pentagon - Hexagram

Square — Pentagon — Hexagram (12 pieces)


Square - Hexagon - Heptagon

Square — Hexagon — Heptagon (12 pieces)


Square - Hexagon - Octagon

Square — Hexagon — Octagon (11 pieces)

Square - Hexagon - Octagon

Square — Hexagon — Octagon (10 pieces with 2 turned over)


Square - Hexagon - Dodecagon

Square — Hexagon — Dodecagon (11 pieces)


Square - Hexagon - Hexagram

Square — Hexagon — Hexagram (10 pieces)

The dissection is translational.
Square - Octagon - Dodecagon

Square — Octagon — Dodecagon (11 pieces)


Square - Octagon - Hexagram

Square — Octagon — Hexagram (11 pieces)


Square - Dodecagon - Hexagram

Square — Dodecagon — Hexagram (12 pieces)


Hexagon - Dodecagon - Hexagram

Hexagon — Dodecagon — Hexagram (12 pieces)

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