Platonic solids. Platonic polyhedra.

Platonic solids
This group of five geometric solids was given the name of the ancient Greek scientist Plato. Mathematicians call these five polyhedra “regular,” but in ordinary speech we most often call them the Platonic solids.
 
 

First, let us identify these five geometric objects — or, in the terminology of mathematicians, geometric solids.

tetrahedrontetrahedron
octahedronoctahedron
cubecube(hexahedron)
dodecahedrondodecahedron
icosahedronicosahedron
  
 

First of all, why are these five geometric solids called regular polyhedra?
This is quite easy to remember. The sides of regular polyhedra are regular polygons, and in turn, regular polygons are those in which all sides are equal (e.g., triangle, square) and have equal angles between adjacent sides. This is why they are called "regular".

 

What is the connection with Plato?
Most likely, the ancient Greek scientist Plato was not related to the discovery of these remarkable polyhedral, but Plato had another talent. In the modern world, Plato could be called the popularizer of regular polyhedra. The greatest contribution Plato made is that he told people about the existence of regular polyhedra.

If this is all he had done, however, the majority would have quickly forgotten about them. But Plato endowed these seemingly simple objects with incredible strength and mystical meaning and brought them to the forefront of his teaching.

In an attempt to explain the nature of everything that exists, Plato considered five regular polyhedra to be fundamental principles for the structure of each of the elements:

- fire correlated with the tetrahedron;
- air correlated with the octahedron;
- earth correlated with the hexahedron;
- water correlated with icosahedron;
- and the dodecahedron corresponded to the Universe.

The chroniclers of the ancient world wrote down everything in detail and, as a result, developed a whole scientific treatise, both for contemporaries of Plato and for all subsequent generations.

It was the power of Plato's philosophy and mystical postulates that were entrenched in the minds of ordinary people, which inextricably linked by these polyhedra with the ideas of Plato. And, at some point, people began to talk about five regular polyhedra as Platonic solids.

 

What is the basis for the name?

There is a universal basis for each name of these regular polyhedra.

Title

Base word

tetrahedron  

tetra (latin) - four
octahedron   octa (latin) - eight
hexahedron or cube   hexa (latin) - six
dodecahedron   dodeca (latin) - twelve
icosahedron   icosi (latin) - twenty

What kind of geometric shapes can a polyhedron be made of?

All of Plato's polyhedra can be represented as a combination of regular polygons.

1. Tetrahedron: 4 triangles
Tetrahedron
= 4 triangle side of a tetrahedron
 
2. Octahedron: 8 triangles
Octahedron
= 8 triangle side of a tetrahedron
 
 
3. Hexahedron (common name cube): 6 regular quadrangles
cube
= 6  square cube side
 
4. Dodecahedron: 12 pentagons
Dodecahedron
= 12pentagon side dodecahedron
 
5. Icosahedron: 20 triangles
Icosahedron
= 20triangle side of a icosahedron
 

Properties of Platonic Solids

 dihedral angle tetrahedrondihedral angle tetrahedron

The dihedral angle of a tetrahedron.

Two adjacent faces of the tetrahedron are joined to each other at an angle of 70.53°.

top of tetrahedronThree triangular faces converge at the same vertex of the tetrahedron. The three-dimensional angle between the three faces (solid angle of a tetrahedron at the vertex) is Ω = 0,55.

dihedral angle octahedrondihedral angle octahedron

The dihedral angle of an octahedron.

Two adjacent faces of the octahedron are joined to each other at an angle 109,47°.

top of octahedronFour triangular faces converge at the same vertex of the octahedron. The three-dimensional angle between the four faces (solid angle of a octahedron at the vertex) is Ω = 1,36.

 dihedral angle cubedihedral angle cube

Dihedral angle of a cube.

Two adjacent faces of the cube are joined to each other at an angle 90°.

top of cubeThree quadrangles faces converge at the same vertex of the cube. The three-dimensional angle between the three faces (solid angle of a cube at the vertex) is Ω = 1,57.

dihedral angle icosahedrondihedral angle icosahedron

Dihedral angle of an icosahedron.

Two adjacent faces of the icosahedron are joined to each other at an angle 138,19°.

top of icosahedronFive triangular faces converge at the same vertex of the icosahedron. The three-dimensional angle between the five faces (solid angle of a icosahedron at the vertex) is Ω = 2,63.

dihedral angle dodecahedrondihedral angle dodecahedron

The dihedral angle of the dodecahedron.

Two adjacent faces of the dodecahedron are joined to each other at an angle 116,57°.

top of dodecahedronThree pentagonal faces converge at the same vertex of the dodecahedron. The three-dimensional angle between the three faces (solid angle of a dodecahedron at the vertex) is Ω = 2,96.

The surface area, volume, radius of the inscribed and described sphere can be found here.

Sizes of polyhedra

To create a collection of polyhedra, we will need to adhere to certain conditions, so the sizes will be comparable and the models can be easily compared with each other.

One of the possible options is to create models that fit into the sphere of a given size.

platonic solids in sphere

faces of platonic solids

 

The sides of the polygons should have the following proportions:

This is how all 5 regular polyhedra will look like in this case.Platonic solids

Here you can download shape patterns to create all five Platonic solids with dimensions that allow you to place each geometric solid inside a 100 mm diameter sphere:

- download the tetrahedron net;

- download octahedron net;

- download the dodecahedron net;

- download icosahedron net;

- download cube net (hexahedron net).

 

 

face Platonic solids

Another option is to set a single side length for all the polygons from which the model will be assembled. Here are the proportions of polygons with a single side length:

- triangle;

- square;

- pentagon

And here is how the collection of polyhedra — Platonic solids, made of polygons with a single side length — will look:

Platonic solids

Here you can download shape patterns to create all five Platonic solids with dimensions that allow you to build each geometric solid with a side length of 50 mm:

- download octahedron net;

- download the dodecahedron net (two identical sheets are required!);

- download the net of the cube (hexahedron net);

- download icosahedron net;

- download tetrahedron net.

 

Ready set for assembly

You can make all five models of Platonic solids using the parts from the "Magic Edges" set.

regular polyhedra

For ease of assembly, all models have a rib construction, which makes it possible to assemble them even with beginner-level mathematics.

The dimensions are chosen so that any of the polyhedra can be inscribed in a sphere with a diameter of 110 mm.

 

Rotation of all regular polyhedra

Assembly of polyhedra of this set

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