Three Way Dissections

Author : Gavin Theobald

{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 (10 pieces)

Triangle — Square — Hexagon (9 pieces)

Discovered by Harry Lindgren.

Triangle — Square — Octagon (11 pieces)

Triangle — Square — Dodecagon (11 pieces)

Triangle — Square — Pentagram (12 pieces)

Triangle — Square — Hexagram (8 pieces)

The dissection is hingeable.

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

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

Triangle — Pentagon — Hexagon (12 pieces)

Triangle — Pentagon — Hexagram (11 pieces)

Triangle — Hexagon — Octagon (12 pieces)

Triangle — Hexagon — Dodecagon (12 pieces)

Triangle — Hexagon — Hexagram (11 pieces)

Triangle — Heptagon — Hexagram (12 pieces)

There is a small twelfth piece that can just be seen at the centre of the hexagram. I have not yet found a way to remove this blemish.

Triangle — Octagon — Hexagram (11 pieces)

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

Square — Pentagon — Hexagon (12 pieces)

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

Square — Pentagon — Octagon (12 pieces)

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

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 (12 pieces)

Square — Hexagon — Heptagon (12 pieces)

Square — Hexagon — Octagon (11 pieces)

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

Square — Hexagon — Dodecagon (11 pieces)

Square — Hexagon — Hexagram (10 pieces)

The dissection is translational.

Square — Octagon — Dodecagon (11 pieces)

Square — Octagon — Hexagram (11 pieces)

Square — Dodecagon — Hexagram (12 pieces)

Hexagon — Dodecagon — Hexagram (12 pieces)