Larry Hecht
Diophantus, who lived probably around 250 A.D., wrote a book called {On Polygonal Numbers,} of which only fragments remain. One of the famous fragments refers to his work on a definition by Hypsicles, an earlier Greek mathematician, concerning polygonal numbers. Working out what Diophantus means in this short fragment proves quite interesting, and relevant to the topics we have been discussing in these series. I will first give you a translation of the fragment from Diophantus. Don’t worry if it seems incomprehensible at first. We will construct it, and then it will all be quite clear.
Diophantus writes:
“There has also been proved what was stated by Hypsicles in a definition, namely, that `if there be as many numbers as we please, beginning from 1 and increasing by the same common difference, then, when the common difference is 1, the sum of all the numbers is a triangular number; when 2, a square number; when 3, a pentagonal number. The number of angles is called after the number which exceeds the common difference by 2, and the sides after the number of terms including 1.'”
To understand what he means, let’s take the most familiar case, that of the square numbers. Most books discussing this subject (sometimes referred to as the “figurate numbers”) draw dots as illustration; but there is a flaw in this, which you will understand after we have done the complete construction. It is far better to find some square objects, or cut them out of paper. Using these square tiles as the units, you will discover that only certain numbers of tiles go together into squares. The first grouping is 1, the second 4, and the third 9. But you should construct this for yourself, for it is already telling you something important about a certain kind of bounding condition, which interested Kepler very much.
Now, cut out some equilateral triangles and do the same thing–that is, make triangular numbers. This is a little less familiar, so I will illustrate how to count in triangles for you:
/\ 1 /\/\/\/\ 4 /\/\ 2 /\/\/\/\/\ 5 /\ /\ /\/\ 3 (2-triangled) /\/\ 6 (3-triangled) /\/\/\
You see that the first three triangular numbers are 1, 3, and 6. These have sides of lengths 1, 2, and 3, just as the first three square numbers (1,4,9) do. You might notice that there are also holes in these numbers, which the squares did not have. There is no need to worry about them. You will see by the end, why they must be there.
Finally, we come to the pentagonal numbers. Now, you must cut out at least 5 equal pentagons, although 12 would be better. Here the fun began for me: to figure out what 2-pentagoned would look like. As I don’t want to spoil it for you, I will not say right here, but let you pause and puzzle over the construction a bit. For now, I will give you the numerical values: the first three pentagonal numbers are 1, 5, and 12.
Now, it is easy to see from these constructions what Hypsicles had discovered, and described in words. We can illustrate it in the following series:
Triangular numbers (common difference = 1) Series: 1 2 3 4 5 ... Sums: 3 6 10 15 ... Square numbers (common difference = 2) Series: 1 3 5 7 9 Sums: 4 9 16 25 Pentagonal numbers (common difference = 3) Series: 1 4 7 10 13 Sums: 5 12 22 35
In each case. we start with one, and increase by the common difference, characteristic for the series. The sum of the numbers in the series is the number of tiles we had to employ to make the triangular, square, or pentagonal numbers.
This may all seem innocent enough, but there is a “fighting” matter of epistemology buried within. It is the same point which Gauss addresses from a more advanced standpoint, in his refutation of Euler, Lagrange and d’Alembert’s attempts to prove the Fundamental Theorem. Namely, do we accept any notion of number, or operations on number, that is not constructible, or subject to “constructible representation” (as Gauss once described the same issue respecting a matter in physics)? It is not only a fighting matter for us. Our enemies also get very upset over the issue. I recognized how much so, after I contemplated why the translator of the Loeb Classical Library Edition {Greek Mathematical Works, II} felt it necessary to add the bracketed phrase “[; and so on]” following the words “when 3, a pentagonal number” in the citation from Diophantus that I gave above. If Hypsicles or Diophantus had wished to say “and so on,” why would they not have done so? Sir Thomas Heath, the leading British commentator on these matters, finds it a shortcoming that Hypsicles had not gone further than the pentagonal number, and claims that what Hypsicles was really showing was how the n-th term of a series, with any common difference, could be determined.
Yet, anyone who has properly considered the significance of the Platonic solids, and stuck to the principle of mathematical rigor employed by both of Gauss and his Greek predecessors, would immediately recognize why Hypsicles stopped at the pentagon. What is being considered is not a math-class game of number series, which seem to go on forever to a bad infinity, but a process of examining the lawful constructibility of number. There is a clue to this also in the {Theaetetus} dialogue of Plato, which had been in the back of my mind, as I wondered what was getting Heath and company so worked up. Consider how Theaetetus describes there, in his examination of the problem of incommensurable numbers, the generation of the numbers 1, 2, 3 as the sides of the square numbers 1,4,9. He calls the numbers 1,2,3 “powers” (where we were taught to call them “roots”), because they have the “power” to generate squares, the singularity under examination in this case. The point in both cases, is that number must be lawfully constructed, and it is obvious that Hypsicles was doing so by examining the paradoxes generated by the Platonic solids.
So, let us now see what happens, if we take these polygonal numbers into the next dimension. The case most familiar to us is that of the square turning into a cube. Thus 1-cubed is 1, 2-cubed is 8, and 3-cubed is 27. (Remember, we are not doing a multiplication table operation, but a construction.) What, then, is the equivalent construction for the other polygons? We can see the case for the triangle most easily, if we now build ourselves four tetrahedra (that is, the Platonic solid made of four equilateral triangles), using triangles of the same size as those we cut out for the construction of the triangular numbers. Construct again the triangular number three, and place a tetrahedron atop each of those triangles. Then, place one more tetrahedron at the summit. Examining the solid so constructed, you will see that it has sides of length 2 in every direction–hence we have constructed 2-tetrahedroned. You can figure out for yourself, what 3-tetrahedroned would be [; and so on].
You might have noticed that there was a hole on the inside of the figure 2-tetrahedroned. That space in there turns out to be an octahedron, by the way. We also had those holes in the plane when we built the triangular numbers. This is telling us something interesting about the tiling of the plane, and the filling of space. Only squares and hexagons, among the regular polygons, can tile the plane, and only cubes and rhombic dodecahedra, among the regular (or quasi-regular) solids can fill space without gaps, which you can investigate for yourself, as Kepler did to his great delight. If you try to tile the plane with pentagons, you notice that when three come together at a point, there is an overlap. That is the key to constructing the figure 2-pentagoned, which I left for you to figure out earlier. You must break the unwritten rule in your mind, and allow yourself to overlap the sides.
Now, if tetrhahedrons do not quite fill space, but leave gaps, and cubes just manage to fill it up, you might expect that dodecahedra would go too far. and overfill it, just as the pentagons overtiled the plane. If you have now constructed your 2-pentagoned figure, with the overlapped sides, you can try your luck at placing a dodecahedron atop each of the five overlapped pentagons, and another dodecahedron atop each of these, to produce the number 2-dodecahedroned. You will see that, just as the pentagons had to overlap, so the dodecahedra must overlap, or interpenetrate, and so the figure 2-dodecahedroned will be of a different type than the cubic or tetrahedral numbers.
Those of you who know why there cannot be more than five regular solids, will now see why Hypsicles stopped at the pentagon. For, while the series with increasing common differences can be extended out to a bad and boring infinity, the interesting paradoxes are not going to arise, unless we have a concept of a constructive process for these numbers.*
Before closing, and since you have all the materials at hand, let us review why there can be only five regular solids. It is a famous proof, given by Kepler. The regular solids are, in fact, the plane projections of the figures produced by tiling the sphere, and they are five ways to do it. As plane solids, they must have regular polygons for their faces, so the problem is reduced to great simplicity by considering only how many of these figures may come together at a vertex. Start with the equilateral triangle. Three of these may be joined at a point, and brought together into a sort of cup, so that they could hold water. This will become the vertex of the tetrahedron. Four triangles may also be brought together, and cupped; they will form the vertex of the octahedron, which looks like two Egyptian pyramids brought base to base. Five triangles may also be brought together and cupped; they form the vertex of the 20-sided icosahedron. However, when six triangles are brought together, it is seen that they just lie flat, and cannot be made into a vertex of anything solid. Next, we try the square, and find that three can be brought together and cupped into a vertex of what becomes the cube. But four are too many; they lie flat. Three pentagons lying in the plane, and joined at a vertex, leave just enough space to be cupped into a vertex of what becomes the dodecahedron. But that is the end of the possibilities, for if we next take a regular hexagon, we find that when three are brought together at a point, they simply lie flat and cannot become the vertex of any solid.
So, you see, there is no “[and so on],” as Hypsicles and Diophantus appear to have understood better than their modern commentators of Oxford erudition. Avoiding “[and so on]” is also good advice for your speaking practice–that you not recite a series of things in sing-song fashion, as so many people do these days, as if there were no lawful cause for their being there. This is nominalism in language, as the idea of number without constructibility is nominalism in mathematics. It is part of the same disease, which we are trying to cure.
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* Of such interesting paradoxes, you might consider, as a topic for more advanced consideration, that a prime number is a constructible species in the series of numbers constructed using squares as the tiles. Following Theaetetus’s specification that we allow only square or oblong (rectangular) numbers, the prime is a number of the form that it can only be represented as a rectangle of width 1. What, then, is a prime number in the triangular or pentagonal series? What else is peculiar about the square and rectangular numbers?
Pierre de Fermat became quite fascinated with the polygonal numbers, and discovered many things about their properties of combination. His famous Last Theorem might be seen as an investigation of the constuctible properties of solid numbers of the square, cubic, and higher variety.