Are there really only five Platonic Solids?

New readers:

• At this point you may like to click here. A new window will open where you can find my definition of the 4th dimension. This links to a simple explanation of a 4D cube ... how it relates to a point, a line, a square and an ordinary cube ... and how to draw it ... and how it rolls.

Any number of people have tried in vain to disprove the supposed impossibility of various geometrical items, such as trisecting an angle with ruler and compasses, and have published long screeds on such matters.   Why then are there no crackpots getting busy with their knives and raw potatoes and writing about 3D shapes?   The reason, I think, is that it's fairly easy to see, and convince oneself, that only the five listed in the 8 June 2014 post, the so-called "Platonic Solids", are possible.

Is this true however?   Are there really only five?   Arguably not, I suggest.   What about the sphere?   Just as a circle can be thought of as an infinite number of infinitely short straight lines arranged in circular formation, so a sphere can be thought of as an infinite number of infinitely small faces arranged in spherical formation.   And these faces, of course, if you make them small enough, can be thought of as being bounded by infinitely short straight lines.   All of the same length, since you ask, so that perfect regularity is guaranteed

What you must imagine therefore is a round ball whose surface consists of an infinite number of  infinitely small, straight-edged, regular-shaped faces.   In just the same way that an infinite number of infinitesimal straight lines can, and indeed do, combine to make a circle in two dimensions, so an infinite number of infinitesimally small surfaces can and do combine to make a sphere in three dimensions.   Now, you may complain that the supposed nearly-spherely shape that you're imagining in your mind's eye feels a bit rough and knobbly to you.   If that's your problem, the answer is simple.   You haven't made the faces small enough.   There's an infinite number of them, remember, and they're infinitely small.   If they really were infinite in number, you wouldn't feel the knobbles at all.   The result would be a perfectly smooth sphere.   Indeed, why do I say "would be"?   With an infinite number of the things, the result, like it or not, IS a sphere.

Let's now go into four dimensions.   When thinking of possible 3D solids, a useful approach is to think of a point in 3D space and work out how many of a given 2D shape could be fitted in round that one point.   Consider 2D squares for instance.   Three of them can readily converge at a point, as they do at the corner of an ordinary cube.   There isn't room for four squares to converge however.   If four squares do converge, 4 x 90 degrees = 360 degrees so they fill up the whole plane, leaving nowhere for the volume (of any possible 3D solid) to go.  So if you're building a 3D shape, you can have three squares meeting at a point, but not four or more.

So much for squares.   What about triangles?   There's plenty of room for three to meet at a point.   Think of the corner of a tetrahedron.   There's room for four to meet also, as at the corner of an octahedron.   What about five triangles meeting?   They're equilateral triangles and 5 x 60 degrees = 300 degrees, i.e. less than 360 degrees.   Just OK therefore.   And five triangles do indeed meet at the corner of an icosahedron.   6 x 60 degrees however = 360 degrees and there isn't room for any such corner to exist.   So when you're building a 3D shape out of equilateral triangles, you can't have six or more of the triangles meeting at a point.

So we've done squares and triangles, what about pentagons?   It turns out that there's room for three of them to meet at a point, but there isn't room for four.   Hexagons?   Put three hexagons together at a point, their internal angles are 120 degrees, 3 x 120 = 360, so there isn't room even for as few as three hexagons to form part of a solid shape.   Heptagons, octagons and so on?   Again, no room for them to meet at a point and form part of a 3D shape.

These arguments certainly do not prove that the five Platonic solids exist.   They do show clearly, however, that there can be no more than the five (not counting the sphere, which is a bit of a digression).  And in practice, as it happens, all five do exist.   The one we've not mentioned yet is the dodecahedron, which has three regular pentagons meeting at each corner and turns out to have twelve of these pentagonal faces in all.

All this goes some way to explaining why the people with their knives and their raw potatoes have not spent much time cutting.   They can go through the above thought processes and can see there's no point.

What about the equivalent in four dimensions?