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 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?
