What building of particular normal polygons permits them to be faces of the Platonic Solids?

It shows up to me that just Triangles, Squares, and also Pentagons have the ability to "tessellate" (is that the correct word in this context?) to come to be normal 3D convex polytopes.

What building of those normal polygons themselves permit them to faces of normal convex polyhedron? Is it something in their angles? Their variety of sides?

Additionally, why exist even more Triangle-based Platonic Solids (3) than Square- and also Pentagon- based ones? (one each)

In a similar way, is this the very same building that permits particular Platonic Solids to be made use of as "faces" of normal polychoron (4D polytopes)?

0
2019-05-04 18:52:47
Source Share
Answers: 1

The normal polygons that create the Platonic solids are those for which the action of the indoor angles, claim α for ease, is such that $3\alpha<2\pi$ (360 ° )to make sure that 3 (or even more) of the polygons can be constructed around a vertex of the strong.

Normal (equilateral) triangulars have indoor angles of action $\frac{\pi}{3}$ (60 °), so they can be constructed 3, 4, or 5 at a vertex ($3\cdot\frac{\pi}{3}<2\pi$, $4\cdot\frac{\pi}{3}<2\pi$, $5\cdot\frac{\pi}{3}<2\pi$), yet not 6 ($6\cdot\frac{\pi}{3}=2\pi$ - - they tesselate the aircraft).

Normal quadrangles (squares) have indoor angles of action $\frac{\pi}{2}$ (90 °), so they can be constructed 3 at a vertex ($3\cdot\frac{\pi}{2}<2\pi$), yet not 4 ($4\cdot\frac{\pi}{2}=2\pi$ - - they tesselate the aircraft).

Normal governments have indoor angles of action $\frac{3\pi}{5}$ (108 °), so they can be constructed 3 at a vertex ($3\cdot\frac{3\pi}{5}<2\pi$), yet not 4 ($4\cdot\frac{3\pi}{5}>2\pi$).

Normal hexagons have indoor angles of action $\frac{2\pi}{3}$ (120 °), so they can not be constructed 3 at a vertex ($3\cdot\frac{2\pi}{3}=2\pi$ - - they tesselate the aircraft).

Any kind of various other normal polygon will certainly have bigger indoor angles, so can not be constructed right into a normal strong.

0
2019-05-08 06:10:56
Source