Geometric Dissections

Author : Gavin Theobald

• 2D Dissections
  • Triangle {3}
    • Polygon dissections {3}—{n}
    • Polygram dissections {3}—{m/n}
    • Multiple triangles {3}—n×{3}
    • Triangle series {3}—{m}…{n}
  • Square {4}
    • Even-sided polygon dissections {4}—{2n}
    • Odd-sided polygon dissections {4}—{2n+1}
    • Polygram dissections {4}—{m/n}
    • Other dissections {4}—{x}
    • Two identical polygons {4}—2×{n}
    • Two different polygons {4}—{m}+{n}
    • Square series {4}—{m}…{n}
  • Pentagon {5}
  • Hexagon {6}
    • Polygon dissections {6}—{n}
    • Polygram dissections {6}—{m/n}
    • Multiple triangles {6}—n×{3}
    • Hexagon series {6}—{m}…{n}
  • Heptagon {7}
  • Octagon {8}
  • Enneagon {9}
  • Decagon {10}
  • Hendecagon {11}
  • Dodecagon {12}
  • Pentagram {5/2}
  • Hexagram {6/2}
  • Octagram {8/2}
  • Octagram {8/3}
  • Decagram {10/2}
  • Dodecagram {12/2}
  • Silver Rectangle {R_√2}
  • Golden Rectangle {R_ϕ}
  • Domino {R₂}
  • Optimised Rectangle {R_×}
  • Greek Cross {G}
  • Latin Cross {L}
  • Curved Crosses {G_c} and {L_c}
  • Other
  • Two polygons to one {n}—2×{n}
  • Three way dissections {p}—{q}—{r}
  • Four way dissections {p}—{q}—{r}—{s}
  • Hole dissections
    • Introduction
    • Hole index [p]—[q]
    • Triangular hole [3]
    • Square hole [4]
    • Pentagonal hole [5]
    • Hexagonal hole [6]
    • Octagonal hole [8]
    • Decagonal hole [10]
    • Pentagram hole [5/2]
    • Hexagram hole [6/2]
    • Octagram hole [8/3]
    • Multiple holes [p]—[q]—[r]…
• 3D Dissections
  • Solid dissections
  • Surface dissections
  • Spherical dissections

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.
{R_ϕ} 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.
{G_c} is the curved Greek cross.
{L_c} 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}

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

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

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

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

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

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

Dissections of Square to Two Identical Polygrams: {4}—2×{m/n}

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

Dissections of Two Polygons to One: {n}—2×{n}

Dissections of Two Polygrams to One: {m/n}—2×{m/n}

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

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