Geometric Dissections Index

Author : Gavin Theobald

Click on any table entry below to see that dissection.

Where there are superscript entries in the tables, these are for dissections requiring pieces to be turned over.

{n} is an n-sided polgon.
{p/q} is an p-sided polygram (star) formed by connecting each vertex with those q distant.

{R√2} is the silver rectangle: a rectangle of proportions 1:√2 or approximately 1:1.4142.
{Rϕ} is the golden rectangle: a rectangle of proportions 1:(1+√5)/2 or approximately 1:1.6180.
{R2} is the domino: a rectangle of proportions 1:2.
{R×} is the optimised rectangle: a rectangle of whatever proportions that produces the best dissection.

{G} is the Greek cross.
{L} is the Latin cross.
{Gc} is the curved Greek cross.
{Lc} is the curved Latin cross.

[n] is the plane with a {n} hole.
[p/q] is the plane with a {p/q} hole.

{p}–{q} is the dissection of {p} to {q}.
{p}={q} is the locked dissection of {p} to {q}.
[p]–[q] is the hole dissection of [p] to [q].

Polygon/Polygram Dissections: {p}–{q}

3
4 4
6 6 5
5 5 7 6
8 7 9 8 7
7 5 9 8 8 7 1110 8
8 9 10 1110 13 12 9
7 7 9 9 8 11 10 13 10
8 6 10 6 11 10 1413 1211 12
7 7 9 9 11 10 14 6 12 5/2
5 5 8 6 9 8 11 9 9 10 6/2
9 7 11 9 12 11 15 12 9 13 10 8/2
8 8 9 9 8 12 6 13 12 12 13 10 13 8/3
10 10 9 7 1110 1413 1312 17 14 14 15 1312 1614 14 10/2
6 8 1211 8 11 12 14 12 1312 14 9 13 1413 17 12/2
10 8 12 10 13 12 14 12 15 11 13 14 14 12/3
5 4 7 7 9 8 11 10 6 10 8 7 10 13 10 10 G
5 5 8 6 8 8 10 10 7 10 8 10 11 13 11 7 L

Dissections of Triangle, Square, Pentagon and Hexagon to Polygons: {p}–{q}

3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
3 1 4 6 5 8 7 8 7 11 8 12 11 12 1312 13
4 4 1 6 5 7 5 9 7 10 6 11 9 11 10 12 10 13 11
5 6 6 1 7 9 9 8 10 9 12 10 14 12 14 13 13 14
6 5 5 7 1 8 8 7 1110 9 8 1211 6 1110 12 12 13

Dissections of Square to Polygons: {4}–{p}

4 n
0 1 2 3 4 5 6 7 8 9
n 4 1 6 5 7 5 9
10+n 7 10 6 11 9 11 10 12 10 13
20+n 11 14 11 14 12 15 12 16 13 16
30+n 13 14 14 15 15
40+n 16 18 19 19 2019
50+n 20 21 21 2221 22
60+n 2322 23 24 24 25
70+n 25 2625 26 27 27
80+n 28 28 2928 29 30
90+n 30 31 31 3231 32
100+n 33 33 34 34 3534

Dissections of Polygons to Polygrams: {p}–{m/n}

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 16/2 20/2 24/2
3 7 5 12 9 8 10 10 6 10
4 7 5 10 12 7 8 1110 1312 19 10 9 1110 10 8 8 11 9 1312 15  15
5 9 8 11 9 7 1211 12
6 9 6 11 9 9 8 13 9 8 11 8 10
7 11 9 15 13 12 12 1413 11 13
8 10 8 11 6 1312 12 12  16
9 14 11 15 13 19  19 17 14
10 6 9 12 12 14 21 12 14
12 12 9 9 12 14 1312 12  25 10

Dissections of Rectangles to Polygons/Polygrams: {R}–{p} and {R}–{p/q}

3 4 5 6 7 8 9 10 12 5/2 6/2 8/2 8/3 10/2 12/2 12/3 G L
4 4 1 6 5 7 5 9 7 6 7 5 7 8 10 9 8 8 4 5
R√2 4 3 5 5 7 4 9 7 7 7 5 8 5 10 9 9 9 5 5
Rϕ 4 3 6 5 7 6 9 6 7 7 5 8 7 10 9 9 5 5
R2 4 3 6 5 7 6 9 7 5 7 5 7 6 7 10 9 8 3 5
R× 3 2 1 4 3 5 4 7 4 5 5 4 7 6 4 8 7 8 3 3

Dissections of Polygons to Multiple Polygons: {p}–n×{q}

n
p q 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
3 3 5 6 4 9 11 11 13 9 16 18 18 19 16
6 6 9 16 24
4 4 4 6 4 9 12 12 9 16 20 16
12 8 14 16 25
5 5 9 6 12 18 14 26
6 3 4 9 10 13 6 16 12  15 2221 22
6 98 6 6 1413 1615 12 1918 12 24 18 19 19
7 7 1110 21 21
8 8 8 10 8 17 20 24 17
9 9 1916 14 10 28 28
10 10 16 23 10 17
12 4 7 9 14 18 23
12 10 14 12 21 24 28 47

Dissections of Square to Two Identical Polygons: {4}–2×{p}

3 4 5 6 7 8 9 10 11 12
4 5 4 8 7 11 10 13 10 14 8

Dissections of Square to Two Identical Polygrams: {4}–2×{p/q}

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
4 12 9 1110 10 16 1514 14

Dissections of Square to Two Different Polygons/Polygrams: {4}–{p}+{q}

3
6 4
8 7 5
8 7 9 6
10 9 12 11 7
9 8 11 10 13 8
10 9 11 11 13 12 10
8 7 9 9 11 10 11 12
10 10 12 11 14 12 14 11 5/2
9 8 10 10 12 11 12 10 12 6/2
9 8 11 10 13 12 13 10 13 11 8/3

Dissections of Two Polygons to One: {p}–2×{p}

3 4 5 6 7 8 9 10 11 12 13 14 15 16
5 4 9 9 8 1110 8 1916 16 23 10 2521 2622 2824 16

Dissections of Two Polygrams to One: {p/q}–2×{p/q}

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 11 12 13 12 18

Dissections of Three Polygons/Polygrams: {p}–{q}–{r}

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

Polygon/Polygram Locked Dissections: {p}={q}

3
7 4
8 7 5
7 6 10 6
13 10 14 11 7
10 8 11 10 8
10 8 11 10 11 10
9 7 12
11 11 11 15 10 5/2
6 8 7 6/2
8 11 17 8/2
12 11 14 17 8/3
10 21 10/2
11 15 13 25 12/2
10 11 25 12/3

Polygon/Polygram Hole Dissections: [p]–[q]

3
3 4
5 4 5
4 3 5 6
6 6 6 7
5 3 6 5 8
6 9
5 4 6 5 6 10
6 4 4 12
5 5 5 5 5/2
3 4 6 5 6 6 6 6/2
7 6 5 7 6 8/2
5 4 6 5 5 6 6 6 8/3
5 6 6 12/2