Platonic Solids and Modular Origami Polyhedra

Material: Square paper (about 10-12 cm in length)

Objective: Students get to know about modular origami, Platonic solids, and stellation.

Topics: Modular origami, Platonic Solids

Presentation

1. Introduce the stellated Platonic solids to the students. These mesmerizing designs draw inspiration from Tomoko Fuse’s renowned work, “The Complete Book of Origami Polyhedra.” Show the students these captivating geometric structures, allowing them to explore and appreciate their intricate beauty.
2. Introduce the concept of Platonic Solids and stellation:

What are Platonic Solids?

Platonic Solids are a unique set of five polyhedra that possess the remarkable property of having all their faces and vertices equal. The five Platonic Solids are the tetrahedron (with 4 equilateral triangle faces), the octahedron (with 8 equilateral triangle faces), the cube (with 6 square faces), the dodecahedron (with 12 regular pentagon faces), and the icosahedron (with 20 equilateral faces).

Origin of Platonic Solids

Platonic Solids, named after the Greek philosopher Plato, are intriguingly connected to the classical elements of the universe: Earth (cube), Fire (tetrahedron), Water (icosahedron), Air (octahedron), and Aether/Ether (linked by Aristotle). Kepler further expanded on this concept in his work Mysterium Cosmographicum by associating these 5 solids and a sphere with the known planets of the heavens (Sun to Saturn). Kepler’s empirical evidence revealed that the distance relationships between the planets somewhat align with the sizes of each solid. Thus, establishing a remarkable mathematical and natural connection between these geometric shapes and the fundamental aspects of the universe.

What is stellation?

Stellation is the process of extending edges in such a way that star-shaped caps are formed from the faces. To calculate the number of caps required, we can use the following formula:

Number of units needed = (number of sides of the polygon cap) x (number of faces of the solid) / 2

For example, in the case of a Dodecahedron model with pentagonal caps, since there are 12 pentagonal faces with 5 sides each, we need 12 x 5 / 2 = 30 units.

3. Instruct students to make a Rose Unit (type A) and when they have finished one, divide them into groups of 3 and instruct them to join the units.

Practice

1. Students determine the number of units needed to construct a model based on the number and shape of each face of the polyhedra.
   1. Students cooperate to make models of their choice.