Platonic solids are one of the most beautiful and unique solids ever discovered, among other things. In this article, we’ll explore the following topics mentioned in the index and you are encouraged to download and use the pintables and DIY activities related to platonic solids.
Index
Fundamentals of Geometry
History
Mathematical Properties
Take a Quiz on 5 Platonic Solids
Download Coloring Pages
Download Platonic Solids Activity Book
How to Make Platonic Solids using Straws
Fundamentals of Geometry
Lets learn some fundamentals of geometry before we start to explore about the platonic solids.
Polygon
A flat or plane two dimensional closed shape that consists of only straight lines without any curves is called a “polygon.” There are two types of polygons, they are regular and irregular polygons.
In a regular polygon, all the sides and angles of the polygon are equal.
In an irregular polygon, the sides and angles are not equal.
In a polygon, two lines meet at a point with an angle and the point where they meet forms a vertex (or is called a vertex).
When more than two lines meet at a vertex they will form a three dimensional convex shape, which is known as a “polyhedron”.
Poly means many and hedra means face of the solid.
A polyhedron is a three dimensional solid formed by joining polygons together. There are two types of polyhedrons they are regular and irregular polygons.
A regular polyhedron consists of regular polygons.
An irregular polyhedron consists of polygons with different shapes.
There are only five regular polyhedrons, and they are “Platonic solids”
Both polygons and polyhedrons are together called “Polytopes.”
History
Pythogars (560 – 480 BCE) was a Greek philosopher and mathematician who was born in the town of Somas. There is a strong claim that Pythogars is the first person who knew about the cube and tetrahedron solids. Later one of the followers of Pythogars, named Hippasus discovered the dodecahedron. After a few years, Theaetetus (417-369 BCE) another Greek mathematician discovered the other two solids octahedron and icosahedron, and was the first to give the mathematical description of all the five solids. Plato (427- 347 BCE) who lived around the same time as Theaetetus, learnt about the solids. He recognised the beauty and mathematical significance of the solids, he hypothesized all of them, associating each regular with the classic elements of the universe, they are fire, sky, water and earth. It was mentioned in the book Timaeus by Plato. These simple representations made an impact on the people, so the solids are named after him as Platonic solids.
Plato compared the platonics basing on their shapes
1. Tetrahedron is compared with fire as it is sharp like flame
2. Cube is is compared with Earth
3. Octahedron is compared with air as it is soft
4. Icosahedron is compared with water as it flows easily
5. Dodecahedron is compared with the Universe.
Euclid (325- 265 BCE), the father of geometry has given the complete mathematical properties of the platonic solids.
Many centuries later, Johannes Kepler in the 17th century well known for his laws of planetary motion has constructed a structure called Mysterium Cosmographicum. It was a representation of the six planets’ size and the sun which are known during that time, moved through space. Kepler, using the platonic solids, constructed Mysterium cosmographicum, where he nested a sphere inside and outside each platonic solid.
Platonic Solids are five geometrical shapes which are formed by regular polygons, with identical properties. They have the same number of faces meeting the vertex and the angle made at each vertex is also the same.
Mathematical properties of Platonic solids
Tetrahedron consists of 4 equilateral triangles, where 3 triangular faces meet at the same vertex forming a triangular base pyramid shape. It has 4 vertices, 6 edges and 4 faces. The dihedral angle between two faces is 70.53 degrees.
Octahedron consists of 8 equilateral triangular faces, where 4 equilateral triangular faces meet at the same vertex forming a square base. It has 6 vertices, 12 edges and 8 faces. The dihedral angle between two faces is 109.47 degrees.
Icosahedron consists of 20 equilateral triangular faces, where 5 equilateral triangular faces meet at the same vertex forming a pentagonal base. It has 12 vertices, 30 edges and 20 faces. The dihedral angle between two faces is 138.18 degrees.
Cube has 6 square faces, where 3 squares meet at the same vertex. It has 8 vertices, 12 edges and 6 faces. The dihedral angle between the faces is 90 degrees. Cube is also known as Hexahedron.
Dodecahedron consists of 12 pentagonal faces, where 3 pentagonal faces meet at the same vertex. It has 20 vertices, 30 edges and 12 faces. The dihedral angle between two faces is 116.56 degrees.
Why there are only 5 platonic solids not more?
We see that platonic solids are made out of only three regular polygons they are triangle, square and pentagon. To form a vertex of a platonic we need minimum three faces to meet. The sum of angles formed by the vertex must be less than 360 degrees, if it is equal to or greater than 360 degrees it turns out to be a polygon (2D) shape. So considering these points we see that a triangle with 3,4 and 5 sides can form a vertex. A square with 3 faces can form a vertex, and a pentagon with 3 faces will form a vertex. Above the pentagon, next one is a hexagon where the angle between two sides is 120 degrees and joining 3 such sides makes an angle of 360 degree, and as the sides increase the angle between the sides increases which cannot form a vertex. So it is clear that any shape apart from the triangle, square and pentagon cannot form a vertex to be a platonic solid.
| Solid | Polygon Face | Number of faces meeting at vertex | Angle at the vertex |
| Tetrahedron | Equilateral Triangle | 3 | 180 |
| Octahedron | Equilateral Triangle | 4 | 240 |
| Icosahedron | Equilateral Triangle | 5 | 300 |
| Cube | Square | 3 | 270 |
| Dodecahedron | Pentagon | 3 | 324 |
Platonic in nature
Platonic solids can be observed in both macroscale and microscale.
In macroscale, we see them in crystalline structures such as
In microscale, we can see them in microscopic organisms such as the bacteriophage whose head is commonly in the shape of icosahedron (elongated icosahedron).
Most essential thing for all living beings is water and the molecular structure of water (H2O) is tetrahedral. The shape carbon 60 (C60) molecule i.e Fullerene resembles a football used as conductors, lubricants, and in many more vaires fields.
Formulas
Euler’s formula
Euler has given a unified formula, which is valid for many solids.
| Solid | Vertices V | Edges E | Faces F |
| Tetrahedron | 4 | 6 | 4 |
| Cube | 8 | 12 | 6 |
| Octahedron | 6 | 12 | 8 |
| Icosahedron | 12 | 30 | 20 |
| Dodecahedron | 20 | 30 | 12 |
𝛘 (Chi) = V – E + F = 2
Volume of each solid
Inradius
Circumradius
Surface Area
Quiz on 5 Platonic Solids
Take a quiz on the 5 Platonic Solids
Coloring Pages
Download Coloring Pages of 5 Platonic Solids
References
https://ncert.nic.in/textbook/pdf/eemh109.pdf
https://www.math-only-math.com/nets-of-solids.html
Images
By Kukski – Own work, CC BY-SA 4.0, https://commons.wikimedia.org/w/index.php?curid=52162496