Dodecagram {12/2} Dissections

Author : Gavin Theobald

{3} Triangle
{4} Square
{5} Pentagon
{6} Hexagon
{7} Heptagon
{8} Octagon
{9} Enneagon
{10} Decagon
{12} Dodecagon
{5/2} Pentagram
{6/2} Hexagram
{8/2} Octagram
{8/3} Octagram
{10/2}  Decagram
{12/3}  Dodecagram
{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
6 8 1211 8 11 12 14 12 1312


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
14 9 13 1413 17 1 14

Triangle — Dodecagram (6 pieces)

This dissection uses the method of overlaid tessellations.

Discovered by Harry Lindgren.

Square — Dodecagram (8 pieces)

Robert Reid found a 9 piece solution which I improved to 8 pieces, but needed to turn over a piece. This version removes this requirement.

This dissection uses the method of variable tessellations.

Note that there are some very short edges that are not visible in these views. Click on the images to see enlarged versions.

Rectangle {R1.05} — Dodecagram (8 pieces)

These diagrams use the same technique as for the square, but better show the way in which the dissection works. Click on the images to see enlarged versions.

Pentagon — Dodecagram (12 pieces)

Pentagon — Dodecagram (11 pieces with 3 turned over)

Compared to the other dodecagram dissections this one is particularly inefficient. This is due to the dimensions of the pentagon making it impractical to form a pentagon tessellation that can overlay a dodecagram tessellation.

By turning over pieces we can improve the dissection, but a entirely different technique is required if the dissection is to be improved further.

Hexagon — Dodecagram (8 pieces)

This is a modification of a dissection discovered by Greg Frederickson.

Heptagon — Dodecagram (11 pieces)

Octagon — Dodecagram (12 pieces)

Enneagon — Dodecagram (14 pieces)

Decagon — Dodecagram (12 pieces)

Dodecagon — Dodecagram (13 pieces)

Dodecagon — Dodecagram (12 pieces with 1 turned over)

Pentagram — Dodecagram (14 pieces)

Hexagram — Dodecagram (9 pieces)

Discovered by Harry Lindgren. This is a modified version with greater symmetry.

Octagram {8/2} — Dodecagram (13 pieces)

Octagram {8/3} — Dodecagram (14 pieces)

Octagram {8/3} — Dodecagram (13 pieces with 1 turned over)

Decagram {10/2} — Dodecagram (17 pieces)

Dodecagram {12/2} — Dodecagram {12/3} (14 pieces)

Silver Rectangle — Dodecagram (9 pieces)

Golden Rectangle — Dodecagram (9 pieces)

Domino — Dodecagram (9 pieces)

Optimised Rectangle — Dodecagram (7 pieces)

Greek Cross — Dodecagram (10 pieces)

Latin Cross — Dodecagram (11 pieces)

This dissection is a very, very tight fit!