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?

Regular shapes in various numbers of dimensions

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.

Here is a series of numbers, which are in this order for a reason.   (Some of them, I suppose, are arguable, in that you could claim they should be different, but these ones will do for me.)   Here they are. 

1;  1;  infinity;  5;  6;  3;  3;  3;  3;  and guess the next number.

Any ideas?   What they are supposed to be, and indeed what they are at least in my mind, is the numbers of convex regular shapes, bounded by straight lines, that are possible in various numbers of dimensions.   So the numbers of dimensions corresponding to the above are: 

0;  1;  2;  3;  4;  5;  6;  7;  8;  9;  ?

In zero dimensions the only shape possible is a point.   Hence the answer in zero dimensions is 1.   Now, this is what I mean by arguable.   Is a point, a simple dot on a piece of paper, truly a shape?   And  even if it is a shape, is it convex and is it bounded by straight lines?   Arguably not.   In which case the first number in my series should be 0, zero.

What about one dimension?   Here the only shape possible is a straight line.   Is that a convex shape bounded by straight lines?   I think arguably it is.   So for my part I'd leave the second number in the series as 1.   But you may feel differently.

In two dimensions we are relatively untrammelled.   We can draw an equilateral triangle, a square, a pentagon, a hexagon and so on.   Whatever number of edges we want, however big, it is theoretically possible to draw a regular polygon (that's what they're called) which is convex and has that number of edges, or straight line sides.   So the third number in the series is infinity.

Now, there is a special feature of infinity.   If we were in the business of drawing these dreaded regular polygons, we could draw them, or at any rate imagine them, as big or small as we like.   We could have a regular thousand-sided figure, or a million- or trillion-sided figure if we prefer, with each of its regular straight line sides a mile long, a millimetre long, or whatever we choose.   But what if we draw a circle?

It is perfectly permissible to think of a circle, say a circle one inch in diameter, as consisting of an infinite number of infinitely short straight lines all joined together in such a way as to make a circle. So, on that way of looking at it, a circle is in fact an example of a regular convex polygon with an infinite number of sides.   This may be relevant when we look at higher dimensions.

It is surprising and extraordinary in a way, is it not,  after the infinite pastures of two dimensions, to find that the regular convex shapes in three dimensions are, as Lewis Carroll put it, "provokingly few in number".   They consist of the tetrahedron, which has four triangular faces, the cube, with its six square faces, the octahedron, with its eight triangular faces, the dodecahedron, with its twelve pentagonal faces, and finally the icosahedron which has twenty triangular faces.    Surely someone could create some extra regular shapes if they put their mind to it and had a sharp knife and a raw potato?

What on earth is the 4th dimension?

New readers may start here.

WELL, WHAT ON EARTH IS THE 4th DIMENSION?

Good question.   I define it as follows.   We have 3 dimensions we all know about – call them North-South, East-West and up-down.   They’re all at right angles to each other.   The fourth dimension is a direction at right angles to all three of them.

That is a direction in which, as the late and much-missed math writer and puzzle expert Martin Gardner put it, we humans “cannot even point”, however hard we try.   So it’s not easy to imagine.   The best way to start to picture it, I suggest, is to imagine that you’re a pondskater.   You live on the pond surface and you can skate as much as you want N-S, or E-W, or anywhere between those two.   But you’re stuck on the surface of your pond and you have no concept at all of “up” (or indeed of down, come to that).   You've no reason to think that 'up' or 'down' even exists.

You’re an intelligent pondskater however and you take math lessons.   Your teacher (from another world above or below the pond we assume) teaches you about Pythagoras.   Pythagoras, of course, works perfectly well on the pond surface.   3 squared + 4 squared = 5 squared just as much there as anywhere else, the teacher explains.   Then he or she says something like this.“You, being a pondskater, can’t imagine ‘up’, which is a direction at right angles to N-S and E-W.   But just suppose, in your imagination, that there was such a direction.   Pythagoras would work just as well in 3 dimensions as he does in 2.   Say you started at point X and you found you could get to a mysterious point Y by skating 2 metres North from X, then 3 metres East, then mysteriously jumping 6 metres ‘up’.   Now you could find how far away you were at your new point Y, from X where you started, by adding 2 squared + 3 squared + 6 squared (= 4 + 9 + 36) and taking the square root.   The square root of 49 is 7 so you’d know that at Y you’d be exactly 7 metres away from X.”

(In parenthesis:  we all know about Pythagorean triples such as 3, 4 and 5 where 3 squared + 4 squared = 5 squared.   What about Pythagorean quadruples?   I’ve just found one by accident, namely 2, 3, 6 and 7 where all the numbers are integers – how many more are there?)

We humans are lucky enough to live in 3 dimensions so we can see that the pondskater is mistaken in thinking that “up” doesn’t exist.   He/she can’t see it, or imagine it, or even point in that direction.   It’s there nevertheless.   And what I’m saying is that we ought to be able to think of a 4th dimension which is analogous to the pondskater’s third.   It’s at right angles to N-S and E-W and up-down.   We can’t point to it, but we can surely imagine it.   If the pondskater can imagine “up”, surely we ought to be able to imagine “fourwaurds” (or whatever we’re going to call it)?

Or suppose you were an animal warden, in charge of cats and dogs.   You could count them and keep a 2D chart of their numbers, cat numbers along the horizontal axis and dog numbers along the vertical.   Maybe your job would be to make sure by careful breeding that the plot of the total number of animals (cats plus dogs) fell on some arbitrary line on such a chart.   Cats plus dogs = 100 for instance.   Add elks and you could keep a 3D chart of the numbers.   Add foxes and, in theory at least, a 4D chart of the numbers must exist.   Come to that you could add goats and get a 5D chart, horses and get a 6D version.   And so on.

To suggest that the 4th dimension doesn’t exist is equivalent to saying that foxes don’t exist.   And clearly they do.   Ask any chicken farmer.

New readers:

• At this point you may like to click here.   A new window will open where you can browse for 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 ... all in the context of trying to understand the curious rotating graphic at the top of this blog.

(Another parenthesis.   What about Pythagorean quintuples?   Are there any?   Suppose you’re living in 4 dimensions and you travel A metres North, then B metres East, then C metres up, then D metres “fourwaurds” such that you find yourself, let us say, E metres from where you started.   To work out what E is, you add the squares of A, B, C, and D and then take the square root. Question:  are there any values of A to E such that all five quantities are integers?   And if the answer is yes, is there an infinite number of these Pythagorean quintuples (as there is with Pythagorean triples)?   And what about, in 5 dimensions, Pythagorean sextuples.   Are there any?   I leave the thought with you.)

In hyperspace, can we speak of “3D-type rotation”?

See http://eusebeia.dyndns.org/4d/vis/09-rot-1.html. It bothers me a bit to find this splendid and excellent website (which I shall call eusebeia for short) speaking of a 4-cube (or indeed anything which exists in 4D) executing “3D-like rotation”.

Why? Well, maybe I’ve misunderstood something but this is how I see it. Consider life, and rotation, in 2D. All you need is a single point – the centre (or center if you insist) of rotation – and (given the speed, the sense and the radius) the rotation is 100% defined.

Now what of 3D? You might think that rotation in 3D, e.g. of a weight whirled round on a piece of string, could also be about a fixed, pivotal point. But no! Take a given weight, string length and speed of whirling and assume that all you know is a pivotal point. In that case you find that you’ve no idea which plane the whirling rotation is taking place in. It could be in a vertical North-South plane, a vertical East-West plane, or a horizontal plane. Or indeed in any one of an infinity of planes in between. You find that you must know the axis of rotation in order to define it.

Now go into 4D. Any simple, non-composite (i.e. non-Clifford) rotation is now about a plane. Hence the name, planar rotation. So if all you know is the axis of rotation, say the North-South axis, you don’t know enough. Any rotation which seems to be about an N-S axis could in fact, in 4D, be about the NSEW plane, or the North-South-up-down plane, or about the North-South-fourwaurds-backwaurds plane. Or indeed about any one of an infinity of planes which include the N-S axis and lie in between the planes just mentioned. (I use the words “fourwaurds” and “backwaurds” – or F and B for short – to mean plus and minus in the direction of the fourth dimension. See the original post.)

In the light of all this, therefore, I suggest that it is imprecise – in fact inappropriate – to speak of “3D-like” rotation in a 4D hyperspace world. Now, I admit that there are some varieties of rotation which do truly take place in a 4D world and which might truly, at first sight and to our human eyes, appear to be 3D-rotation-like. But this appearance, I suggest, is misleading.

If we limit ourselves to orthogonals, there are 6 types of rotation in 4D. Let us use NSEW to denote compass points, UD for up and down, and F and B for fourwaurds and backwaurds as explained above. So the 6 types of 4D rotation are about the following 6 planes: NSEW; NSUD; NSFB; EWUD; EWFB; and UDFB.

Consider an ordinary 3D 3-cube existing in our familiar 3D space, with its edges running NS, EW and UD. Now rotate it, say, about the NSFB plane. This will appear (to a 3D person living in 3D who can’t see into the 4th dimension) to be a “3D-like” rotation. Please note however that this rotation clearly cannot be 100% 3D-like (at least as far as the all-seeing mathematician is concerned – and (s)he of course is the one who set up this rotation initially and caused us all these headaches). Why not? Because the plane about which the rotation takes place includes the mysterious FB – fourwaurds and backwaurds – axis.

Exactly the same applies if we take that same 3D 3-cube and rotate it about the EWFB plane. Ditto for rotation of it about the UDFB plane. The rotations may appear to us humans to be 3D-like. But they’re not really. At least I think not anyway.

Six ways to planar-rotate a 4-cube

In an earlier post I implied that I could show the “skeleton” of a rotary 4-cube with just four still pictures of it with its corners lettered A to R (leaving out I and O). This is wrong of course, or incomplete anyway, because  there are six different ways in which a 4-cube can execute planar rotation and ought not each to have its own set of pictures?

Thus there are six different sets, each of four parallel, plane faces, about which the 4-cube can pivot. So rather than my measly 4 pictures, actually it would seem to take 6x4 = 24 of them to show the full range of a 4-cube’s rotations from the start through ¼ turn, ½ turn and ¾ turn positions.

By chance, a 4-cube has 24 square faces. Coincidence?

Well, no, not really. Because it wouldn’t need 24 pictures. It would need 6 sets of 4 of them. But the openers of each set would all be the same, namely the starting position. That leaves 24 – 5 = 19 different pictures.

Am I right?

John Scott
johnscott.hyperspace@gmail.com

4D rotation without distortion (more or less)

I still never cease to be impressed by Mark Newbold’s animations of a rotating 4-cube.   See http://dogfeathers.com/java/hypercube2.html.

Question:  (to Mark or anyone skilled enough to organise animations of this kind):  is there some way to start from my “fig 9 view” of a 4-cube (see my original blog) and show an animation of it as it rotates in 4D?   Preferably without distorting it out of its original fig 9 shape?

Reverting to simple, planar (i.e. non-composite, or non-Clifford) rotation, one can readily draw still pictures of the fig 9 4-cube in its start position and its positions after a ¼ turn, ½ turn and ¾ turn.   I may add these later.   If we identify the 4-cube’s corners by the letters ABCDEFGHJKLMNPQR, the four pictures will all be the same with only the corner-lettering changed.   (I choose to leave out I and O;  it’s what I was taught to do.)
For my part, I think these 4 pictures would give a good (albeit limited) account of what happens to the 4-cube as it spins or rolls.   At least you could see exactly where all 16 corners were at each quarter-turn.

Perhaps someone may be skilled enough to animate this view of a 4-cube as it spins or rolls?

By comparison, I have trouble following or analysing the animated rotary 4-cube at the top of the blog.   Partly this is because as it spins it keeps distorting its component 3-cubes into trapezoids.   Mainly though my problem is that one can’t shout “STOP” and see just where all its 16 corners have got to.   Mark Newbold’s Dogfeathers animation is brilliant in that respect however and I must analyse it further.

The two pivotal planes of a composite-rotating 4-cube

I have had some very helpful contributions by email explaining how a 4-cube can rotate (or, presumably, roll) at one and the same time about two separate planes. These two pivotal planes, as I understand it, must meet only at a point, not along a line. Thus once you have chosen one face from the 24 square faces of the 4-cube to be a pivotal plane, that leaves only one set of four other faces (all parallel to one another) to choose from for the other pivotal plane. I shall be able to explain this better (probably?) with diagrams in a later post.

John Baez very helpfully directed me to http://eusebeia.dyndns.org/4d/vis/09-rot-1.html which goes into 4D rotation in some detail, with helpful animations. Also to http://en.wikipedia.org/wiki/SO%284%29 which is altogether more advanced mathematically. Too advanced for me right now I am afraid, but I shall try to learn enough to follow it. Mark Newbold also gave that wikipedia reference. Plus he has, on http://dogfeathers.com/java/hypercube2.html, an extremely helpful animation of a rotating 4-cube, which you can stop mid-spin in order to probe into its subtleties. I am in the process of doing this, with great interest.

Once you have settled on a pair of pivotal planes, on both of which the 4-cube is required to pivot, it does seem to take a considerable feat of mental gymnastics to be able to follow the 4-cube as it executes its double rotation (or rolling). However – just possibly – and with the aid of the above sites, I am getting there. Slowly. Perhaps. I think.

I shall keep trying. Meanwhile, can anyone help me to get to grips with the “too advanced” website referred to above? Is there a site which could gradually work me through the math on which it is based, matrices and so on?

Some of the animations I have looked at of a rotating 4-cube show its various component 3-cubes distorting themselves into trapezoidal shapes during the rotation. I find this slightly distracting, for the same reason that I prefer the fig 9 view of the 4-cube (from my original blog) rather than the fig 3 view. In real life (if that means anything) the 8 component 3-cubes of a 4-cube will perfectly retain their 3-cubical shapes during rotation, won’t they?

They may well appear, what with perspective and whatnot, to distort themselves into trapezoids as the 4-cube spins. But, surely, this is just a consequence of the 2D paper on which we poor benighted 3D humans are forced to depict them? If only we could see properly into the fourth dimension, we would surely find that all 8 component 3-cubes retain their 3-cubical shapes throughout. Wouldn’t we?