Is there a sixth platonic solid?

Question

my teacher says there is if you know please tell me what it is called and tell me where i can see it or atleast show me a net of it... thank you to whoever can answer this you are a lifesaver!!!

: D

: P

thanx a lot

Answer 1

Your teacher is wrong there is no sixth platonic solid, however there are analogous 4 dimensional solids, each of these are related to the platonic solids.  I believe the sixth platonic solid your teacher refers to is actually called the octaplex.  The octaplex is actually not a platonic solid it is infact a convex polytope, or something along those lines, google it ;);)

If your teacher argues with you tell him I said so, and if he still says its wrong refer him to this site

http://mathworld.wolfram.com/PlatonicSol...

After all, sometimes its fun to correct the teacher.

Answer 2

The answer is that there are only five Platonic solids in three dimensions.  If you want more, there are various possibilities:

1.  In four dimensions, there are six Platonic solids.  (There are only three in five dimensions, or any number of dimensions higher than five.)

2.  Relaxing various restrictions in the definition of Platonic solid will give you more possibilities.

(a) If you remove the restriction that the solid be a convex polyhedron and allow it to have zero volume, you have the possibility of the regular dihedron, which consists of two regular polygons glued together, one in front and one on the back.

(b) If you remove the restriction that the solid be a convex polyhedron and allow faces and edges to intersect each other, you have the possibility of the Kepler-Poinsot solids.   There are four of these; each has all vertex neighborhoods congruent to each other, all edges congruent to each other, and all faces congruent to each other.

(c) You can represent regular polyhedra by projecting them onto their circumscribed spheres, turning them into regular tessellations of the sphere.  This raises the possibility of the hosohedron.  This consists of the sphere divided by a number of equally spaced meridians, each running from the North pole to the South pole.  (The dihedron can also be viewed as a spherical tessellation.)

(d) Relaxing the polyhedral requirement again in a different way, the usual infinite tiling of the plane made up out of equilateral triangles is regular and made up of regular polygons.  The same is true for the usual tiling of the plane by squares and the usual tiling of the plane by regular hexagons.

(e) If you relax the polyhedral requirement as in (d) and work in a hyperbolic (negatively curved) plane, you will be able to construct a regular tiling of the plane which has 7 equilateral triangles meeting at each vertex.  In fact, you can also make such tilings with m regular n-gons meeting at each vertex, for any positive m and n with 1/m + 1/n < 1/2.

(f) If you relax the requirement that all faces be congruent, keeping the requirement that all faces be regular polygons and all vertex neighborhoods congruent, you get many additional possibilities, including regular triangular, pentagonal, hexagonal, etc. prisms and antiprisms, the square antiprism, and the so-called Archimedean solids.

(g) The duals of the polyhedra in (f) give polyhedra where all faces are congruent, but not necessarily regular polygons.  These include the duals of the Archimedean solids.

Some of the relaxations in (a) through (g) can be combined to get still more possibilities.

3.  Owing to its use in computer graphics, the Utah teapot has been called the sixth Platonic solid as a joke.