Question

There is an infinite number of regular polygons but only five regular (Platonic) polyhedra. Of the Platonic polyhedra, 3 have surfaces consisting of triangles, 1 consisting of squares, and 1 consisting of pentagons.

Why do you think is it impossible to create more than five regular polyhedra?

Please express your opinion.

Answer #1

Solution:

This is because of the angle at vertex. Angles at vertex should add up to less than 360 degree. If the total angle is 360 degree, the shape flattenns and it no longer remain a closed solid.

A regular triangle has internal angles of 60°, so we can have:

- 3 triangles (3×60°=180°) meet
- 4 triangles (4×60°=240°) meet
- or 5 triangles (5×60°=300°) meet.

A square has internal angles of 90°, so there is only:

- 3 squares (3×90°=270°) meet.

A regular pentagon has internal angles of 108°, so there is only:

- 3 pentagons (3×108°=324°) meet.

A regular hexagon has internal angles of 120°, but 3×120°=360° which won't work because at 360° the shape flattens out.

So a regular pentagon is as far as we can go.

2. When the inscribed and circumscribed regular polygons of any
common number of sides to a circle have more than four sides, the
difference of their areas is less than the square of the side of
the inscribed polygon.

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