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