Platonic Solids
ILLL
Dr Shobha Bagai
University of Delhi

PLATONIC SOLIDS
Trace the following figures on a piece of paper and cut it along the outer edges
marked in red. Fold it along the dotted blue lines so that the red lines join. You get a
solid figure in each case.
Can you name these solids?
Fig. 1
The solid obtained in the first case is a tetrahedron, a cube in the second case and
an octahedron in the last case. The given figures are called the net of the solids.
(a) Tetrahedron
(b) Cube
(c) Octahedron
What's so special about these geometric shapes?
1)
Each formation will have the same shape on every side (equilateral triangle in
case of a tetrahedron and an octahedron and square in case of a cube.
2)
Every line on each of the formations will be exactly the same length.
3)
Every internal angle on each of the formations will also be the same.
4)
Each shape will fit perfectly inside a sphere, all the points touching the edges
of the sphere.
5)
Same number of lines meets at the vertex (three in case of a tetrahedron and
a cube and four in case of an octahedron)

Are these the only solids that exhibit these properties or are there more?
Are these the only five solids exhibiting these properties or are there
more?
Before we answer this question, let us learn more about such solids.
PLATONIC SOLIDS
The solids mentioned above are called the
Platonic solids
or regular solids or
regular polyhedra. They belong to the group of geometric figures called polyhedra. A
polyhedron
is a solid bounded by plane polygons. The polygons are called
faces
.
They intersect in
edges
. The points where three or more edges intersect are called
vertices
.
These solids are named after
Plato
.
Two more solids exhibit these properties (a) Dodecahedron (b) Icosahedron. Dodecahedron is made
up of 12 pentagons and icosahedron of 20 equilateral triangles
Plato
(427 BC – 347 BC), was a Classical Greek philosopher, who,
together with his teacher, Socrates, and his student, Aristotle, helped to lay the
foundations of Western philosophy. Plato was also a mathematician, writer of
philosophical dialogues, and founder of the Academy in Athens, the first institution of
higher learning in the western world
.

Plato wrote about the
Platonic Solids
in his work
Timaeus
c.
360 B.C.
In the first place, then, as is evident to all, fire and earth and water and air are bodies.
And every sort of body possesses solidity, and every solid must necessarily be
contained in planes; and every plane rectilinear figure is composed of
triangles;…………….
And next we have to determine what are the four most beautiful bodies which are
unlike one another, and of which some are capable of resolution into one another; for
having discovered thus much, we shall know the true origin of earth and fire and of
the proportionate and intermediate elements. ………………
Now, the one which we maintain to be the most beautiful of all the many triangles
(and we need not speak of the others) is that of which the double forms a third
triangle which is equilateral; the reason of this would be long to tell; ……………..

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- Fall '09
- Math, Polyhedron, Platonic solid, Johannes Kepler, Regular polyhedron, Icosahedron