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For studying these solids in detail, geometers realized that they had to abstract them into certain ideal solids. Some of these were Sphere, Cylinder, Cone, Cuboid, Cube etc idealized from the common objects around us. There were no real objects which actually matched these above solids. For instance, though the Sun appears like a perfect sphere or circle, or a tree appears like a cylinder, in reality they are not perfect spheres or cylinders.
By studying these idealized 3 D objects visually they came up with a set of 3 elements which all of them had– faces, edges & corners.
Face in geometry means a "flat surface." Surfaces which are "not flat" are called "curved surfaces."
Hence a sphere has "no faces" but "one curved surface." A cone has a face on which it can rest and a curved surface. A cylinder has 2 faces and one curved surface.
Edges can be straight or curved.
These elements themselves were related - Edge was formed when 2 faces or a curved surface & a face meet. When 2 flat faces met the result was a straight edge. When one face meets a curved surface, the result is a curved edge. Corner was formed when 2 edges met.
Some of the Faces & Edges in a solid may not even meet. Examples are the opposite flat faces of a cylinder and opposite edges of a cube.
This term “met” evolved later as a geometrical term “Intersect”. All solids which could be described in terms of Faces, Edges & Corners were called Regular Solids.
These ideal Regular solids could be described in terms of the number of Faces, Edges & Corners which they have. We present them as a table.
Out of these Regular Solids, Greeks also developed the idea of “Platonic Solids” which were 5 in number.
As the study of geometry progressed, a number of Regular Solids were invented which were more complicated that the 4 solids described above. Many of them were also given standard names like prism, pyramid, octahedron etc.
Leonard Euler (18th century) found out a simple formula connecting faces, edges & vertices (corners) for a subclass of solids which were known as convex polyhedra. Polyhedra are Regular solids which do not have curved surfaces or edges. It was
F + V − E = 2
(Number of Faces + Number of Vertices is equal to 2 more than the number of Edges.)
Interestingly the last theorem in Elements is a proof that only 5 Platonic solids are possible! Euler’s Formula can also be used to prove that there can only be 5 Platonic Solids! We can even think of Euler’s Formula as an indication of the limitation imposed by the 3-dimensional space which we live in, on the properties of certain 3 dimensional objects in it.
Greeks also studied shapes which could be drawn on a flat surface. These were 2-dimensional figures. This study came to be known as Plane Geometry. It was Plane Geometry which brought Greeks everlasting fame and immortalized Euclid’s name.
Geometry & Emotions
For some reasons which need to be researched, geometrical shapes seem to be related to our emotions.
Most jovial and nice people in movies are portrayed with soft & round features whereas most villains seem to have sharp triangular features!
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