Question

Given a Platonic solid \(P\) you can construct a new polyhedron whose vertices are the centers of the faces of \(P .\) This new polyhedron is called the dual of \(P\) and it turns out that it is also a Platonic solid. For each of the five types of Platonic solids, identify the dual.

Short answer

1. Tetrahedron – self-dual, 2. Cube – Octahedron, 3. Octahedron – Cube, 4. Dodecahedron – Icosahedron, 5. Icosahedron – Dodecahedron.

Step-by-step solution

Understanding Platonic Solids

Platonic solids are highly symmetrical, convex polyhedra with faces made up of congruent regular polygons. There are only five such solids: the tetrahedron, cube, octahedron, dodecahedron, and icosahedron.

Identifying Dual Pairs

For each Platonic solid, its dual is another Platonic solid whose vertices correspond to the centers of the faces of the original solid. The goal is to identify these dual relationships.

Finding the Dual of the Tetrahedron

The tetrahedron is a unique Platonic solid because it is self-dual. This means that if you connect the centers of a tetrahedron’s faces, you get another tetrahedron.

Finding the Dual of the Cube

The dual of the cube is the octahedron. By connecting the centers of the faces of a cube, you form an octahedron.

Finding the Dual of the Octahedron

Conversely, the dual of the octahedron is the cube. Therefore, these two solids form a dual pair.

Finding the Dual of the Dodecahedron

The dual of the dodecahedron is the icosahedron. Connecting the centers of a dodecahedron’s faces results in an icosahedron.

Finding the Dual of the Icosahedron

The dual of the icosahedron is the dodecahedron, thus completing the dual pair, just like the cube and octahedron.

Key concepts

Dual Polyhedron

A dual polyhedron is a fascinating concept where one polyhedron is directly related to another through duality. Imagine starting with a polyhedron whose faces are regular polygons. By focusing on the center of each face and connecting these centers, you derive a new shape. The resulting shape is called the dual polyhedron. This relationship is truly unique as each Platonic solid has exactly one dual, and interestingly enough, the dual of a Platonic solid is also a Platonic solid.

To better understand, consider any of the five Platonic solids — tetrahedron, cube, octahedron, dodecahedron, and icosahedron. For each, you can find a dual shape by visualizing or physically constructing the new vertices at the face centers. The dual shapes possess unique properties and share symmetrical beauty. This concept offers a deep insight into the inherent harmony and symmetry in Platonic solids.

Tetrahedron

The tetrahedron is an intriguing Platonic solid due to its unique property of being self-dual. Here's why this is significant: when you connect the centers of its four triangular faces, you again derive a tetrahedron. This means the dual of a tetrahedron is a tetrahedron itself!

A tetrahedron features:

• Four faces
• Four vertices
• Six edges

This symmetry and simplicity make the tetrahedron remarkable. The self-duality highlights the perfect balance between its dimensions and symmetry. It's a perfect introduction to the beauty of dual polyhedra and their balanced properties.

Cube and Octahedron Duality

The relationship between the cube and the octahedron is an excellent example of duality. The cube, known for its six square faces, forms its dual by connecting the face centers, resulting in the octahedron. Conversely, performing the same operation on an octahedron leads you back to a cube.

Here's a breakdown:

• The cube has six faces, eight vertices, and twelve edges.
• The octahedron has eight faces, six vertices, and twelve edges.

Their geometrical relationship demonstrates a balance of vertices and faces, with each transition maintaining 12 edges. This pair clearly exemplifies the elegance of duality, transforming simple geometrical operations into symmetrical beauty.

Dodecahedron and Icosahedron Duality

The dodecahedron and icosahedron offer a captivating duality dynamic. Starting with a dodecahedron, which consists of twelve pentagonal faces, connecting the centers of these faces gives rise to the icosahedron. The icosahedron, in turn, transforms back into the dodecahedron when you connect the centers of its twenty triangular faces.

Key characteristics include:

• The dodecahedron has twelve faces, twenty vertices, and thirty edges.
• The icosahedron has twenty faces, twelve vertices, and thirty edges.

This duality shows how two complex shapes can be deeply interconnected while maintaining classical geometry principles. By observing such relationships, we uncover the inherent order in the symmetrical structures of the Platonic solids.