Spherical Dissections

Author : Gavin Theobald

Home Index ≪ Spherical Contact ☰

The dissections on this page are dissections on the surface of a sphere. These are more tricky than planar dissections since, for example, the angles of a triangle do not up to 180°.

Below “Triangle 90°” refers to a triangle with all angles of 90°. Similiarly for other shapes. A lune is a two sided polygon (or digon). All lines are “great circles” that are the shortest distance between two points.

There are two views of each dissection. The first is an orthographic view that is close to what you would see with an actual sphere with the dissection on the surface. The second is a stereographic view that preserves angles. The view is less distorted and can make the dissection easier to understand.

Click on any diagram to see a scalable pdf version.

Triangle 72 - Lune 18 Triangle 72 - Lune 18

Triangle 72° — Lune 18° (4 pieces)

Twenty triangles cover a sphere without dissection.

Square 105 - Lune 30 Square 105 - Lune 30

Square 105° — Lune 30° (4 pieces)

Pentagon 120 - Lune 30 Pentagon 120 - Lune 30

Pentagon 120° — Lune 30° (4 pieces)

Twelve pentagons cover a sphere without dissection.

Triangle 90 - Lune 45 Triangle 90 - Lune 45

Triangle 90° — Lune 45° (2 pieces)

This dissection is hingeable.

Eight triangles cover a sphere without dissection.

Square 112.5 - Lune 45 Square 112.5 - Lune 45

Square 112½° — Lune 45° (4 pieces)

This dissection is cyclically hingeable.

Triangle 90 - Square 112.5 Triangle 90 - Square 112.5

Triangle 90° — Square 112½° (5 pieces)

Triangle 100 - Lune 60 Triangle 100 - Lune 60

Triangle 100° — Lune 60° (4 pieces)

This dissection is cyclically hingeable.

Square 120 - Lune 60 Square 120 - Lune 60

Square 120° — Lune 60° (2 pieces)

Six squares cover a sphere without dissection.

Triangle 100 - Square 120 Triangle 100 - Square 120

Triangle 100° — Square 120° (6 pieces)

Square 120 - 2 x Pentagon 120 Square 120 - 2 x Pentagon 120

Square 120° — 2 × Pentagon 120° (4 pieces)

This dissection is hingeable.

Triangle 120 - Lune 90 Triangle 120 - Lune 90

Triangle 120° — Lune 90° (3 pieces)

This dissection is hingeable.

Four triangles cover a sphere without dissection.

Square 135 - Lune 90 Square 135 - Lune 90

Square 135° — Lune 90° (4 pieces)

This dissection is cyclically hingeable.

Pentagon 144 - Lune 90 Pentagon 144 - Lune 90

Pentagon 144° — Lune 90° (3 pieces)

It is possible to dissect four pentagons to cover a sphere in 6 pieces.

Hexagon 150 - Lune 90 Hexagon 150 - Lune 90

Hexagon 150° — Lune 90° (6 pieces)

It is possible to dissect four hexagons to cover a sphere in 14 pieces.

Heptagon 154.286 - Lune 90 Heptagon 154.286 - Lune 90

Heptagon 154.286° — Lune 90° (7 pieces)

This dissection is hingeable.

It is possible to dissect four heptagons to cover a sphere in 16 pieces.

Octagon 157.5 - Lune 90 Octagon 157.5 - Lune 90

Octagon 157½° — Lune 90° (7 pieces)

It is possible to dissect four octagons to cover a sphere in 18 pieces.

Enneagon 160 - Lune 90 Enneagon 160 - Lune 90

Enneagon 160° — Lune 90° (8 pieces)

This dissection is hingeable.

It is possible to dissect four enneagons to cover a sphere in 20 pieces.

Decagon 162 - Lune 90 Decagon 162 - Lune 90

Decagon 162° — Lune 90° (8 pieces)

It is possible to dissect four decagons to cover a sphere in 22 pieces.

Triangle 120 - Square 135 Triangle 120 - Square 135

Triangle 120° — Square 135° (6 pieces)

This dissection is hingeable.

Triangle 120 - Pentagon 144 Triangle 120 - Pentagon 144

Triangle 120° — Pentagon 144° (5 pieces)

Square 135 - Pentagon 144 Square 135 - Pentagon 144

Square 135° — Pentagon 144° (7 pieces)

Triangle 120 - 2 x Triangle 90 Triangle 120 - 2 x Triangle 90

Triangle 120° — 2 × Triangle 90° (4 pieces)

This dissection is hingeable.

Triangle 120 - 3 x Pentagon 120 Triangle 120 - 3 x Pentagon 120

Triangle 120° — 3 × Pentagon 120° (6 pieces)

This dissection is hingeable.

Pentagon 144 - 3 x Pentagon 120 Pentagon 144 - 3 x Pentagon 120

Pentagon 144° — 3 × Pentagon 120° (7 pieces)

This dissection is hingeable.

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