Platonic Solids
The Five Uniform Polyhedra
What is a Platonic solid?
And what is a Polyhedron?
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A \(polyhedron\) is a 3D shape that consists of a bunch of polygons glued together at the edges to form a volume so there are no gaps, openings, or overlaps.
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A \(regular\ polyhedron\) is a polyhedron where all of the faces are the same shape and size, and it has the same number of polygons meeting at each corner.
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A \(Platonic\ solid\) is one of the 5 regular polyhedron that are "convex". That is, there are no inward dips on the outside.
Remember
Vertices are the corner points where edges meet.
Edges are the straight lines connecting two vertices.
Faces are the flat surfaces enclosed by edges.
Platonic Solids
The 5 Regular Polyhedra
The ancient scholar Plato believed the universe was built from the 5 solids. Four of the solids were used for earth, air, water, and fire, while the remainder was "the fifth element". This is why they are called the Platonic solids.
Tetrahedron
4 vertices, 6 edges, 4 faces
The tetrahedron is the simplest of all Platonic solids. The ancient Greek philosopher Plato associated it with the element fire due to its sharp and piercing nature. The tetrahedron is not a pyramid - a pyramid has a square bottom and triangular sides, but ALL sides of a tetrahedron are triangles.
Cube
8 vertices, 12 edges, 6 faces
The cube, also known as a hexahedron if you're feeling fancy, is the most familiar Platonic solid. Be careful to not call it a box! Boxes can be short, thin, and a wide variety of shapes. A cube has 6 identical square faces.
Octahedron
6 vertices, 12 edges, 8 faces
The prefix "oct" means 8, as in octopus (8 legs), octagon (8 sides), and octave (8 notes). The Octahedron can be described as two pyramids joined at their bases. One place in nature where you will find octahedrons is in crystals! Pyrite, diamonds, and other crystals often form octahedrons.
Icosahedron
12 vertices, 30 edges, 20 faces
The icosahedron is composed of 20 equilateral triangles. The word is a combination of the Greek "eíkosi" which means twenty, and "hedra" meaning face. This solid has the highest number of faces among the 5 Platonic solids. It makes you wonder - if you were to roll all 5 Platonic solids, would the icosahedron roll the longest?
Dodecahedron
20 vertices, 30 edges, 12 faces
The dodecahedron is built using 12 equal sized pentagons. This solid is often associated with the cosmos or universe in Platonic philosophy, symbolizing a sense of mystery and wonder. The name comes from the Greek words for "twelve faces", or "dodeka" + "hedra".
Euler's Formula
A Connection between V, E, and F
The mathematician Leonhard Euler showed that for every convex polyhedron, the number of vertices (V), edges (E), and faces (F) have this relationship:
\[ V - E + F = 2 \]
For example, a dodecahedron has \(V=20\), \(E=30\), and \(F=12\). Plugging these in gives you:
\[ 20 - 30 + 12 = 2 \]
If the polyhedron is not convex or has holes, the relationship may differ, but for simple convex polyhedra like the Platonic solids, Euler's formula holds true.
Why are there only 5 Platonic Solids?
Considering that a platonic solid is a convex polyhedron in which faces are congruent regular polygons and there's the same number of faces meeting at each vertex, the following limit the number of possible Platonic solids:
Regular Polygon Face: The interior angle of a regular polygon must be less than 360∘ because at least three faces must meet at each vertex to form a 3D shape.
Angle Sum Around a Vertex: For a convex polyhedron, the sum of the angles around a vertex must be less than 360∘. This condition limits the number of faces that can meet at a vertex depending on the type of polygon that forms the faces.
Let's go through the possible polygons:
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Equilateral Triangle (interior angle = 60):
- Three triangles: 3×60=180∘ (forms a tetrahedron)
- Four triangles: 4×60∘=240∘ (forms an octahedron)
- Five triangles: 5×60∘=300∘ (forms an icosahedron)
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Square (interior angle = 90):
- Three squares: 3×90∘=270∘ (forms a cube or regular hexahedron)
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Pentagon (interior angle = 108∘):
- Three pentagons: 3×108∘=324∘ (forms a dodecahedron)
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Hexagon (interior angle = 120∘) or polygons with more sides:
- Three hexagons: 3×120∘=360∘ (flat tiling, does not form a 3D shape)
As the number of sides of the polygon increases, the interior angle also increases, and the sum of the angles around a vertex exceeds 360∘, making it impossible to form a 3D shape.
Thus, there are only five combinations of regular polygons that meet the criteria for Platonic solids, resulting in the five unique Platonic solids:
- Tetrahedron: 4 faces, each an equilateral triangle.
- Cube (Hexahedron): 6 faces, each a square.
- Octahedron: 8 faces, each an equilateral triangle.
- Dodecahedron: 12 faces, each a regular pentagon.
- Icosahedron: 20 faces, each an equilateral triangle.
Nets
A net of a three-dimensional object is a two-dimensional representation that, when folded along specific lines or edges, forms the 3D object. It is essentially a flat layout of all the faces of the object, connected in such a way that they can be folded into the shape without any overlaps or gaps. For example, the net of a cube consists of six squares arranged in a cross-like pattern, where each square represents one face of the cube. When folded correctly, the squares come together to form the cube.
Nets are a valuable tool in geometry and spatial reasoning because they help visualize and understand how 3D objects are constructed from their surfaces. They are also used to calculate surface area, as each face of the object is represented in the net, allowing for easy measurement and summation of the areas. Understanding nets can provide deeper insights into the properties of shapes and their dimensions, making them an essential concept in mathematics and engineering.
Here are the nets of the 5 platonic solids.
Review
| Platonic Solid | Number of Faces | Face Shape | Number of Vertices | Number of Edges |
| Tetrahedron | 4 | Equilateral Triangle | 4 | 6 |
| Cube (Hexahedron) | 6 | Square | 8 | 12 |
| Octahedron | 8 | Equilateral Triangle |
6 | 12 |
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Icosahedron |
20 | Equilateral Triangle |
12 | 30 |
| Dodecahedron | 12 | Regular Pentagon | 20 | 30 |
“In the grand blueprint of the cosmos, the gods employed geometry as their divine language. The tetrahedron sparked the flames of fire, the cube laid the earth's foundation, the octahedron whispered to the winds, and the icosahedron flowed through the waters. As for the dodecahedron, the gods used it to roll the dice on the universe's fate. Such is the playful wit of celestial architects.”
Plato's "Lost" Tweets
The Simplex
In mathematics, a simplex is a generalization of a triangle and tetrahedron. The first four simplexes are made from familiar shapes: a point, a line segment, a triangle, and a tetrahedron.
