The 120-cell

Note: This demonstration is not difficult to follow but requires more explanation than the earlier ones. Thus, it is intended mainly for enthusiasts of the BKS theorem!

The 120-cell is the generalization of the regular dodecahedron in a space of four dimensions. Like the 600-cell and the two-qubit system, it has billions of simple patterns that give instant proofs of the Bell-Kochen-Specker theorem. However, the table of numbers in which the patterns are embedded is too large to be shown on a computer screen and so the patterns are presented here in a compressed form that nevertheless allows them to be recognized as being genuine.

You should be familiar with the exhibit for the 600-cell before looking at this one, as that will make it easier to follow.

The 120-cell has 600 vertices lying on the surface of a sphere in four-dimensional (Euclidean) space. The vertices come in antipodal pairs, and the line through any antipodal pair of vertices will be termed a ray. The rays are numbered from 1 to 300 according to a scheme to be laid out shortly. A set of four rays in mutually orthogonal directions from the center of the 120-cell will be termed a basis. The 300 rays of the 120-cell form 675 bases, with each ray occurring in exactly nine bases. The 120-cell can therefore be described by the ray-basis symbol 300_9 – 675_4, where the subscript on the left indicates the number of bases to which each ray belongs and that on the right the number of rays in each basis; note that 300 x 9 = 675 x 4, so that the total ray count is the same whether done by rays or bases.

The numbering of the rays is based on the “triacontagonal projection” of H.S.M.Coxeter, which is a way of casting a shadow of certain higher dimensional figures (of which the 120-cell is one) on to a two-dimensional plane in such a way that their vertices lie on a number of concentric regular triacontagons (or 30-gons). Modifying Coxeter's construction to apply to rays rather than vertices, the rays are associated, in sets of fifteen, with the vertices of twenty concentric regular pentadecagons (or 15-gons) that are labeled by the letters A through T. Rays 1-15 are associated with the vertices of A in counterclockwise order beginning from a chosen vertex, rays 16-30 with the vertices of B in a similar manner, and so on, and rays 286-300 with the vertices of T. The association of the rays with the 15-gons is shown below.

15-gon	A	B	C	D	E	F	G	H	I	J
Rays	1-15	16-30	31-45	46-60	61-75	76-90	91-105	106-120	121-135	136-150

15-gon	K	L	M	N	O	P	Q	R	S	T
Rays	151-165	166-180	181-195	196-210	211-225	226-240	241-255	256-270	271-285	286-300

With this numbering of the rays, the bases become easy to specify because it turns out that adding one to each ray of any basis always yields another basis, provided the following rider is kept in mind: if a ray is a multiple of fifteen, then instead of adding one to it, one subtracts fourteen from it. To understand what this means, consider the basis consisting of the rays 19, 120, 224, 249. Adding one to each of the rays gives 20, 121, 225, 250, but this is not a basis because the ray 120 is a multiple of fifteen and must be replaced by 106 rather than 121; thus the proper successor to the basis 19, 120, 224, 249 is 20, 106, 225, 250. Adding one to each ray of this new basis gives 21, 107, 211, 251 (note the drop in number of the third ray!), and one can continue in this way to obtain a chain of fifteen bases whose final member goes back into the initial one when its rays are incremented. The chain can in fact be generated from any of its members, and the one chosen for the purpose is termed the generator. The 120-cell has 45 generators and they give rise to its 45 x 15 = 675 bases.

The generators of the 120-cell are shown below, with a lower case letter attached to each. Clicking on a generator will show the fifteen bases associated with it in a pop up window. If you run your eye down any column of numbers in the window, you will see all the rays associated with the 15-gon whose label is indicated at the top of the column. The rider mentioned in the previous paragraph ensures that the rays in any column are confined to those of a particular 15-gon. It is quite remarkable (and far from obvious!) that the bases of the 120-cell can be built up in this way from a set of generators. The existence of the chains of bases suggests another way in which a generator, and indeed the entire chain of which it is a part, can be labeled: it can be labeled by the capital letters of the four 15-gons to which its rays belong. Thus, generator a can be labeled as ABST, generator b as ACRT, and so on. This dual labeling will be made use of below.

The patterns of the 120-cell that prove the BKS theorem can be written as “words” made up of an odd number of distinct letters, each representing one of its 45 generators. A valid pattern (i.e., one that proves the BKS theorem) must consist of an odd number of letters because each letter represents fifteen bases and the total number of bases in the pattern must be odd. However, not all odd-letter words represent patterns because of the additional requirement that each ray occur an even number of times over all the bases of the pattern. Since only complete 15-gons of rays occur in a word (via its letters), the requirement that each ray occur an even number of times over the bases of the pattern is the same as the requirement that every capital letter occur an even number of times over all the lower case letters of the word representing the pattern.

This makes it easy to tell if an odd-letter word represents a pattern or not. One need only replace each lower case letter in the word by its set of four capital letters and see if each capital letter occurs an even number of times over the word; if this is the case, the word represents a pattern, otherwise not.

As an example, consider the 7-letter word abegkri'. The application of the above test to it is shown in the flowchart below:

             abegkri' → (ABST)(ACRT)(AFPT)(AHMT)(BEPS)(CFOR)(EHMO) → (BCEFHMOPRS)₂(AT)₄ → 150₂30₄ – 105₄

In the first step, each lower case letter is replaced by its set of four capital letters (which can be looked up in the pop up windows of the generators). In the second step, all the capital letters occurring the same number of times over the word are collected together, with the number of occurrences indicated as a subscript. At this stage one can tell if the word represents a pattern or not simply by seeing if all the subscripts are even. In the final step, the ray-basis formula of the pattern is obtained by replacing each string of letters within brackets by fifteen times the number of letters and adding the number of bases (which is fifteen times the number of letters in the word) after a dash.

The task of picking out all odd-letter words that represent patterns is not straightforward, but it has been made easy for you. Simply enter a 0 or 1 in each of the 29 square boxes below (there are 15 boxes in the first row and 14 in the second) and click the set button to their right. This will cause the letters of the pattern to appear in the round boxes. Clicking the second set button will show how many times each of the capital letters occurs over the letters of the pattern, with even counts shown on a green background and odd counts on a red one. This allows one to confirm at a glance that the pattern is a valid one simply by seeing that all the boxes are green. Instead of clicking the second set button, you could click on the letters of the pattern one by one and see it build up gradually. If you do this, you will see red boxes occurring frequently during the buildup, but when all the letters have been clicked only green boxes will remain.

Sometimes, a word representing a pattern may have a smaller word inside it that is also a pattern. For example, the word cduvy represents a pattern with the formula 150₂ – 75₄ and dropping two of its letters gives the shorter word cdy which is also a pattern with the formula 90₂ – 45₄. After you have called up a pattern, you can drop some of its letters (by clicking on them and turning them red) and see if the green letters that remain make up a pattern. You will see that this happens more frequently as the patterns get longer. It is a challenge to find long word representing patterns that doesn't have any smaller patterns inside them. An example is provided by the 9-letter word fghilmsup', which represents a pattern with the formula 180₂45₄ – 135₄. You can check that none of the 7-,5-,3- or 1-letter words in it are patterns.

In closing, it should be mentioned that the method described above gives only patterns of a special type, namely, those involving both rays and bases that are multiples of fifteen. The 120-cell has a large number of other patterns that prove the BKS theorem, but they have been ignored here.