Platonic Solids Nader Abbasi 18/09/2018

Introduction Since the earliest times, human mind has been fascinated by the five objects known as “Platonic Solids”. These are convex solids which have congruent faces and are called regular polyhedra. Due to symmetry and aesthetic beauty, these objects have been subject of study of the best human minds for thousands of years. 18/09/2018

Platonic Solids Platonic solids is another name for regular polyhedra. Regular polyhedra have regular polygon faces and identical vertices. The five Platonic solids are the only convex regular polyhedra. 18/09/2018

Euclid’s proof At least three polygons are needed to make a solid angle. The smallest regualr polygon is an equilateral triangle. Such an angle can be constructed with 3,4,and 5 equilateral triangles. With 6 equilateral triangle the result lies flat. 18/09/2018

Square Solids The next regular polygon is square 3 squares around a point forms a solid angle, but with 4 squares the result is a flat surface. 3 squares around a point is the only possible case for making a solid angle with squares. 18/09/2018

Pentagonal Solid The next polygon is the pentagon, 3 of which around a point make a solid angle. There is no room for 4 pentagons, even to lie flat. Like the square, 3 is the only combination to form a solid angle. 18/09/2018

Polyhedra Limit! The number of regular polyhedra is limited to 5. The next regular polygon is the hexagon. 3 hexagon lie flat and do not form a solid angle and there is no other regular polygon that 3 of them can meet at a point and form a solid angle. 18/09/2018