Platonic Solid

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Platonic Solids
Fields: Polyhedra and Geometry
Image Created By: Abram
Website: Sewing Up Math

Platonic Solids

The platonic solids are five regular polyhedra that have faces constructed of congruent convex regular polygons.

Basic Description

Alternatively called regular solids or regular polyhedra, platonic solids are "convex polyhedra with equivalent faces composed of congruent convex regular polygons". The five existing platonic solids include the cube, dodecahedron, icosahedron, octahedron, and tetrahedron. Collectively with the Kepler-Poinsot solids, these shapes are more commonly referred to as the "cosmic figures" [1].

A More Mathematical Explanation

Note: understanding of this explanation requires: *Geometry

Definition of a Platonic Solid

The attributes of symmetry that a shape must have in order to [...]

Definition of a Platonic Solid

The attributes of symmetry that a shape must have in order to define it as a platonic solid are:

  • The same number of faces meet at each vertex.
  • The shape must be convex. Thus, the angle that is created by the shapes at the vertex must be below 360 degrees.

The Five Platonic Solids

Schläfli symbols: A convenient way to describe Platonic solids

With the Schläfli symbol system, platonic solids are described with the use of the notation {P,Q}. P represents the number of edges on each face and Q represents the number of faces that meet at each vertex. For example: A cube is constructed of squares with four sides each, with three squares meeting at each vertex, so the Schläfli symbol for a cube would be {4,3}.

This notation is useful because a given Schläfli symbol can only describe one Platonic solid, although some Schläfli symbols do not correspond to any actual Platonic Solid. Thus, the {4,3} Schläfli symbol for a cube cannot refer to any other platonic solid and symbols like {3,6} refer to shapes that are not classified as platonic solids.

A Summary of the Five Platonic Solids

Solid Base Shapes Schläfli symbol Vertices Edges Faces Angle
Tetrahedron 3 Triangles {3,3} 4 6 4 180°
Octahedron 4 Triangles {3,4} 6 12 8 240°
Icosahedron 5 Triangles {3,5} 12 30 20 300°
Cube 3 Squares {4,3} 8 12 6 270°
Dodecahedron 3 Pentagons {5,3} 20 30 12 324°

This table shows the properties of each platonic solid. By comparing the Base Shape and the Schläfli symbol, it is possible to see how the P and Q values of the symbol were derived. The 'Angle' column signifies the total angle calculated through the interior angle of the shape that meets at the vertex and the number of such shapes present at the vertex. Through observing the angles, it is apparent that the total angle must be below 360° in order for the shape to be classified as a platonic solid.

Another detail to note is that all Platonic Solids, as polyhedra that do not intersect themselves, also follow Euler's formula in the relationship between their vertices, faces and edges. With the aid of this formula, one can determine any of the attributes with the formula provided that the other two are known:

Vertices + Faces = Edges + 2

The images below are clearer visual representations of the platonic solids present in the page image.

Images From Tom Getty's page on Platonic Solids[2]

Duals of Platonic Solids

Duals are created by placing a dot at the center of each face of a platonic solid and then connecting adjacent dots with line segments to form an outline. When this process is performed on a Platonic solid, a platonic solid is created. If this process is then repeated on the created dual, then the third Platonic solid created takes the same form as the original. The Duals of Platonic Solids are all types of Dual Polyhedron.

Platonic Solids Duals
tetrahedron tetrahedron
cube octahedron
octahedron cube
dodecahedron icosahedron
dodecahedron icosahedron

Calculating the Platonic Solid's Attributes

Determining Different Radii

Below is a table that presents the values for the inradius (rd), midradius (ρ) and circumradius (R) based on the value of the edge length (a) for the specific platonic solid. The differing radii are determined by the insphere, which touches the faces of the dual solid that it is inscribed within; the midsphere, which touches the edges of both the original polyhedron and its duals; and the circumsphere, which touches the vertices of the original solid respectively[3]

solid inradius - rd midradius - ρ circumradius - R
tetrahedron \frac{1}{12}a\sqrt{6} \frac{1}{4}a\sqrt{2} \frac{1}{4}a\sqrt{6}
octahedron \frac{1}{6}a\sqrt{6} \frac{1}{2}a \frac{1}{2}a\sqrt{2}
icosahedron \frac{1}{12}a(3\sqrt{3}+\sqrt{15}) \frac{1}{4}a(1+\sqrt{5}) \frac{1}{4}a\sqrt{12 + 2 \sqrt{5}}
cube \frac{1}{2}a \frac{1}{2}a\sqrt{2} \frac{1}{2}a\sqrt{3}
dodecahedron \frac{1}{20}a\sqrt{250 + 110 \sqrt{5} } \frac{1}{4}a(3+\sqrt{5}) \frac{1}{4}a(\sqrt{15} + \sqrt{3})

These values are derived by noting the relation that the circumsphere and insphere obey:

Rr_d = \rho^2

The formulas to determine the values are as follows:


Surface Area and Volume of the Solid

Below is a table of equations used to find the area of a face, the surface area and the volume of a platonic solid based on the length of of an edge (a) [4].

solid Area of One Face Surface Area Volume
tetrahedron \frac{1}{4}a^{2}\sqrt{3} a^{2}\sqrt{3} \frac{1}{12}a^{3}\sqrt{2}
octahedron \frac{1}{4}a^{2}\sqrt{3} 2a^{2}\sqrt{3} \frac{1}{3}a^{3}\sqrt{2}
icosahedron \frac{1}{4}a^{2}\sqrt{3} 5a^{2}\sqrt{3} \frac{5}{12}a^{3}(3+\sqrt{5})
cube a^{2} 6a^{2} a^{3}
dodecahedron \frac{1}{4}a^{2}\sqrt{25+10 \sqrt{5} } 3a^{2}\sqrt{25+10 \sqrt{5} } \frac{1}{4}a^{3}(15+7\sqrt{5})

Why It's Interesting

The ancient Greeks believed that all matter physical was created of the five platonic solids as well as such matter having mystical properties represented by their connections to air, fire, earth, water and ether. Approximately 2500 years ago, the Pythagoreans studied three of these solids: the tetrahedron, the hexahedron (cube) and the dodecahedron. Meanwhile they also worked on developing a proof that only 5 such solids existed while trying to continue to develop mathematical postulations on the the octahedron and dodecahedron. The latter work was later brought to light by Theaetetus. Furthermore, before Plato, these polyhedra were called "Pythagorean solids" -- however, he detailed the properties of the solids heavily in “Timaios" while also assigning the elements to the solids [5].

Image source [6]

Although the Greeks are generally credited with the discovery of the platonic solids, there is evidence, like stone carvings, that resemble the platonic solids dated back to at least 1000 B.C. in Scotland [7]. It is also known that in the Vedic times (3000 to 1000 B.C.) [8], Indians had already classified the same elements of fire (Agni), water (Apa), earth (Prithvi), air (Maya) and ether (Akasha) as the components of the material world.

Plato was very interested in the polyhedra that would later be called the Platonic Solids because they are "the only perfectly symmetrical arrangements of a set of (non-planar) points in space". For four of the Platonic Solids, though, Plato concieved their corresponding elements based on observations of packed atoms and molecules. For example, pyrite is found in the form of tetrahedron crystals, thus connecting the solid to the element of fire [9].

The platonic solids are also the standard polyhedrons used for dice.

Platonic Solid Element State Properties[10]
Tetrahedron Fire Plasma Hot and Dry
Octahedron Air Gas Hot and Humid
Icosahedron Water Liquid Cold and Humid
Cube Earth Solid Cold and Dry
Dodecahedron Universe Vacuum

Chart information except for Properties [11]

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