Geometric Dissections

Author : Gavin Theobald

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Click on any table entry 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 connected 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.

{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] indicates the plane with a {p/q} hole.


Polygon/Polygram Dissections: {m}—{n}

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
11 10 12 1211 15 1615 11
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 15 9 10 6/2
9 7 11 10 13 11 15 13 9 14 11 8/2
8 8 9 9 8 12 6 13 12 12 13 10 1413 8/3
10 10 9 7 1110 1413 1312 17 14 14 15 1312 16 14 10/2
6 8 1211 8 11 12 14 12 1312 14 9 13 1413 17 12/2
4 3 5 5 7 4 9 7 7 7 5 8 5 10 9 9 R√2
4 3 6 5 7 6 9 6 10 7 7 5 8 7 10 9 3 Rϕ
4 3 6 5 7 6 9 7 5 7 5 7 6 7 10 9 3 3 R2
5 4 7 7 9 8 11 10 1312 6 10 8 7 10 13 10 5 5 3 G
5 5 8 6 8 8 10 10 13 7 10 8 1110 11 13 11 5 5 5 7 L

Dissections of Triangle, Square, Pentagon and Hexagon to Polygons: {m}—{n}

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 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 14
6 5 5 7 1 8 8 7 1110 9 8 1211 6 1110 12 12 13

Dissections of Square to Polygons: {4}—{n}

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 20
50+n 20 21 21 22 22
60+n 23
70+n 25
80+n 28
90+n 30
100+n 33

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

Dissections of Rectangles to Polygons/Polygrams: {R}—{n}

3 4 5 6 7 8 9 10 11 12 5/2 6/2 8/2 8/3 10/2 12/2 G L
4 4 1 6 5 7 5 9 7 10 6 7 5 7 8 10 9 8 4 5
R√2 4 3 5 5 7 4 9 7 7 7 5 8 5 10 9 9 5 5
Rϕ 4 3 6 5 7 6 9 6 10 7 7 5 8 7 10 9 5 5
R2 4 3 6 5 7 6 9 7 5 7 5 7 6 7 10 9 3 5

Dissections of Square to Two Identical Polygons: {4}—2×{n}

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×{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
4 12 9 1110 10 16 1514 14

Dissections of Square to Two Different Polygons/Polygrams: {4}—{m}+{n}

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: {n}—2×{n}

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: {m/n}—2×{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
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 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
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