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alext87

Platonic Solids

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Can anybody describe to be "in basic" terms what is a Platonic solid? And why are there only five?

 

Also what does it mean to say that the faces, angles and edges are all congruent?

 

Thanks

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Well, for example, with a cube, every face is an identical square, any edge or vertex you look at has the same angles as every other one.

 

There are only five because, by definition, it has to be made of regular polygons, and there is a limited number of ways they can be tiled. This is because, in order to make a 3D shape, you need the sum of the angles at each vertex to be less than 360 degrees.

 

For example, taking squares, if you arrange them so that three of their vertexes meet at a point, you get the corner of a cube, which is 3D because they have to "bend back" in order to meet each other without any space in between. If you use four squares, however, you just end up with a flat surface, not a 3D shape, because the sum of the four angles is 360. And you simply can't have more than four without overlapping.

 

So why are there only five? Well, each vertex needs at least 3 faces in order to be 3D (think about 2 - you just end up with a flat surface with zero volume), and their total angles can't be 360 or greater. So, you can have 3, 4, or 5 triangles (corner are 60 degrees, so totals are 180, 240, and 300), or 3 squares (90x3=270), or 3 pentagons (108x3=324). Any greater numbers will result in total angles greater than or equal to 360, and the same for any shapes with more sides. And the angles only get larger as you add sides, so you never have the possibility of a 3d vertex again. The angles of a regular hexagon, for example, are each 120 degrees, so three of them together is already 360.

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Another way to put it, there are only five 3-dimensional forms where each face is exactly the same size and shape (and doesn't have any indents in it).

 

http://en.wikipedia.org/wiki/Platonic_solids

 

Although in 4-dimensional space, there are those five, and one more platonic solid imaginatively called the 24-cell.

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