However, regardless of whether we have a good idea of the extent of the perfect numbers (now at 47 and counting), or there possible oddity, not enough attention has been paid to the meaning of the “perfect” numbers themselves. 496 is just an example of this more general phenomena of “perfection” which is an ancient notion.
These numbers are known as “perfect” because the parts (factors) add up to the number themselves. They are perfect because in some sense they are supervenient. They are neither excessive or deficient. The wholes are not greater nor less than the sum of the parts. In other words they are a possible violation of the idea that the whole is greater than the sum of the parts, which is our definition of the gestalt. Rather they make us consider the possibility that the whole might be less than the sum of its parts (a whole full of holes like a sponge) or exactly equal to the sum of its parts, which I call a special system, due to the rarity of this kind of system.
What we do not ask ourselves enough is why these kinds of transparent numbers where the whole is equal to the sum of the parts were considered perfect by the ancients.
Mathematics tends to concentrate on form alone and neglect meaning. Why is this understanding of these numbers intrinsic to our understanding of “perfection” within our essentially Greek tradition, and why has that lasted up until today?
One thing to understand is that the number six plays many roles. http://en.wikipedia.org/wiki/6_(number)
What I would like to bring out, following B. Fuller is that the number 6 is the pathways in a tetrahedron which is the minimal solid in the third dimension. The number six represents primarily perfection of interconnection.
Six is the only number that is both the sum and the product of three consecutive positive numbers (wikipedia)
Notice that these numbers are 1, 2, and 3 which are the first numbers.
Chapter 42 (Meaning of Life the Universe and everything in this paragraph of the Tao Te Ching)
Tao produces one
One produces two
Two produce three
Three produce myriad things
Myriad things, backed by yin and embracing yang
Achieve harmony by integrating their energy
What the people dislike
Are alone, bereft, and unworthy
But the rulers call themselves with these terms
So with all things
Appear to take loss but benefit
Or receive benefit but lose
What the ancients taught
I will also teach
The violent one cannot have a natural death
I will use this as the principal of all teachings
The very next number is four, what B. Fuller calls the minimal system.
The lattice of the tetrhedron is 1-4-6-4-1
Pascals triangle goes
1 null — ultra one (cf. Badiou)
1 0 1 point
1 0 2 0 1 line
1 0 3 0 3 0 1 triangle
1 0 4 0 6 0 4 0 1 tetrahedron
Notice that in Pascal’s triangle prior to the tetrahedron are 1, 2, and 3.
The tetrahedron has four points and three triangles and six lines.
These six lines have perfect inter-relation because they contain 1, 2, and 3 as their parts, and nothing else. Add them or multiply them and you get the same result. So six is fundamental because it is agnostic with regard to addition or multiplication of its parts. And the parts precisely express the whole without lack or excess.
Now there are a lot of ways to interpret 6 but I would argue that transparent interconnection is definitely one of the most significant of them.
And this brings us to the point that there are a lot of strange things about mathematics but one of the strangest is how upon approaching the low numbers the mathematical system is forced into perfection generally. And this is one of the expressions of that general tendency toward fusion as the number system approaches the limit of one.
Now if we augment this line of argument by going on to the next perfect number which is 28 then the key thing about 28 is that it is the number of relations between eight things (n^2-n/2).
Now in order to understand this we must go up to the next level of solids which are the octahedron and cube which have the lattice 1-6-12-8-1.
Now notice that this is the next harmonic level of thought where two minimal systems either fuse or interpenetrate according to B. Fuller in Synergetics.
But we notice immediately that in terms of their lattice the central six (points or faces) is one of the ends of this lattice, while the other end is eight elements (points or faces). Twice 6 or 12 is the center, and 12 is known as the most easily dividable small number, i.e. the dozen. But the perfection is in the relations between the eight. Six is already perfect. So the other end of the lattice has its perfection in the relations between the eight elements (trigrams). So here perfection is once removed rather than central, and what is central is dividablity.
The divisors or factors of 12 are
— Wolfram Alpha
Notice that 12 has six divisors, and thus is perfect in that. Its divisors are the first four numbers and six as well as itself.
8 on the other and is 2^3 which is a stage in the binary progression which relates the number 2 to the power of three.
28 has many associations in history http://en.wikipedia.org/wiki/28_(number)
It is a composite number, its proper divisors being 1, 2, 4, 7, and 14.
Twenty-eight is the second perfect number. As a perfect number, it is related to the Mersenne prime 7, since 22(23 – 1) = 28. The next perfect number is 496, the previous being 6.
28 is a perfect number expressible as the sum of first five prime numbers i.e., 2 + 3 + 5 + 7 + 11 = 28.
28 is the number of Chinese constellations (“Xiu” or “mansions”) in their zodiac
28 is the number of Arabic letters
I take these associations to be significant, a recognition of the significance of this number for the organization of things, like the kosmos, like the alphabet
It is a physical magic number http://en.wikipedia.org/wiki/Magic_number_(physics)
It also has 6 divisors and is perfectly transparent and neither excessive nor lacking.
Now that we know a little about 6 and 28 we can hypothesize that the first is associated with the first level of synthesis within the Platonic Solids of the minimal system, i.e. the tetrahedron. When we are dealing with the dynamics of four things in relation to each other we have a system. Anything below that is not really a system or synthesis and is thus a product of analysis of a synthesis like tertiary relations and binary oppositions. Kant for instance is always bringing in a third thing to resolve a dualism (apriori synthesis [spacetime] for instance). But his philosophy is systematic because he sees that there are four basic kinds of categories/judgements. He was the first philosopher to be concerned with the architectonic of this philosophy as a system itself and he introduced the concept that synthesis comes before analysis.
If the tetrahedron is the first synthesis, then the perfection of its relations its lines that mediate between the points and faces is crucial because it makes the tetrahedron transparent to our conceptualizations due to the perfection of its relations between the points or between the sides.
Note that the Tetrahedron gives form to the quadralectic as developed in my dissertation on Emergent Design at http://about.me/emergentdesign
Now when we go to the next transparent threshold of thought that is precisely where two tetrahedra interact either in terms of fusion or interpenetration taking the form of cube or octahedra with the same lattice 1-8-12-6-1. We can assign point, line, surface in either direction to get the dual solids.
But what was relations at the first level become elements at the second level, and what was 2^2 elements are transformed into the next binary unfolding as 2^3 and the relations between these trigrams are perfect in themselves just as the relations between 2^2 was perfect. However we see that this perfection has shifted from being central to a counterbalancing as the relations between the trigrams. And we see that the central element in the lattice is twelve the most dividable number. Ease of division has taken central place between the perfection of the elements that is counterbalanced by the perfection of the relations at the next level up with 28 which is of course 4*7, thus introducing the next stage beyond perfection.
What we need to think about deeply is how these numbers tell us something about the relation between our concepts. If the tetrahedron is the first synthesis then the octahedron/cube is the first interaction between two syntheses and that can either be by fusion or interpenetration, but regardless the next level turns what were perfect relations into a set of elements, and couter-balances that with the next binary unfolding and the perfections of relations between those trigrams, and it paces as central ease of dividablity. This has implications for the ease of thinking at the cubo/octa level of synthesys.
These syntheses are forged by the fusion of the numbers as they head toward one, and so these fusions that are transparent can be seen as stages of the unfolding of One, once it is posited over what Badiou calls the multiple. Each threshold of efficaciousness of thought (effective and efficient) is part of the unfolding of the ultra-one. Given in Geometry these are part of the apriori synthesis that we project upon experience.
So finally we get to 496 which the original question was about. Notice how much is within the simple question as to the number of the next perfect number. What constitutes perfection is always a deep question.
(32^2-32)/2 = 496
496 is the relations between 32 things and this is 2^5, i.e. the fifth level of binary progressive articulation.
Here is where things get interesting, and we will have to summarize because the path is long and winding.
The next level of syntheses of thought is the icosa/dodecathedron which has a lattice 1=12-30-20-1. Notice that it starts with twelve the central number in the last lattice. This is in effect the interaction of five tetrahedral systems, one more than merely the interaction of two cubo/octo transparent synthetic systems. And by a miraculous twist of mathematical fate these two platonic solids have the same group A5 as the four-dimensional pentachora which is the minimal solid in the fourth dimension.
The pentachora has a lattice of 1-5-10-10-5-1 (I am used to calling this the pentahedron of four-dimensional space).
And it is the pentachora that represents the 2^5 progressive bisection threshold. Notice that we have skipped the 2^4 threshold. It did not get represented in relation to the icosa/dodeca-hedron threshold. This is very significant, because the unfolding of the unfolding of the geometric thresholds is veering away from the progressive bisection seen in the Pascal Triangle.
In fact, everything revolves around the introduction of A5 as a blindspot in the unfolding, because polynomials cannot be solved which are of degree 5 or higher which is a problem for the application of mathematics and calls for special analysis of higher polynomials. A5 is like a closed-door that prevents variables being rotated out into manifestation for equations of degree 5 or higher making it difficult to dissect complex systems described by polynomials.
What we see is that in actuality the third thought threshold of synthesis has two manifestations as the most complex platonic solids in the third dimension and the simplest platonic solids at the fifth dimension. And this has implications for the relation between quadraletics and pentalectics that I discuss in my dissertation on Emergent Design.
Quadralectics naturally gives rise to Pentalectics through this interface between the third and fourth dimensions via A5 group, and also between the tetrahedron and pentachora minimal solids of these respective dimensions.
It is strange that I cannot find precursors who have developed the dialectics and trialectics (work) of Hegel to the quadralectical and pentaletical levels. But the synergetic systems theory of B. Fuller when extended into the fourth dimension gives this result for thought and opens up further thresholds of synthesis in the fourth dimension associated with the 8cell/16cell regular polytopes, the 24 cell polytope, and the 120/600 cell polytopes in the fourth dimension.
Since these are the thresholds where thought becomes transparent at least mathematically speaking, we need to pay special attention to them and their deviation from the unfolding of the progressive bisection. The Pascal Triangle is the unfolding of the progressive bisection and at the same time the unfolding of the minimal solids in each dimension. So the migration away from these structures into asymmetries is particularly significant.
What is striking is that the 8cell and 16cell polytopes which are duals embody the relation between 2^3 and 2^4 and combines that with the 24 cell lattice that mediates the two.
When we look back at the Cubo/octa-hedral lattice 1-6-12-8-1 we see that the 24 cell has doubled the 12 at the center of that lattice, it has incorporated the 8 and extended it to the 16. The 8cell/16cell is the analog of the cubo/octahedral form in the fourth dimension, where it is expressed as three all space filling lattices which complement each other. So where we thought we skipped the 2^4 level we actually only deferred its appearance so that it comes back to us in an even more complex form deeply embedded in the fourth dimension.
This lattice is 1-8-24-32-16-1
Now here we see that the 2^3, 2^4 and 2^5 progressive bisection levels are related to the 24 which is the doubling of the 12 at the lower level.
Note that the lattice fo the 24 cell is 1-24-96-96-24-1. It is self-dual like the minimal solids. It is unique in all dimensions, as all higher dimensions have only three platonic solids. It forms an all 4-space filling lattice that mediates between the lattices of the 8 and 16 cell polytopes.
So while 12 as an easily divisible number mediated between the perfect set 6 and the 8 perfected by 28. Hee we have 8 being mediated by 24 and 32, and the 32 is perfected by the 496 which is the third perfect number. 24 in this case intervenes as the 24 cell polytope which is unique. And what is being countervailed is the 2^4 which is the 16 points/cells of the 16 cell dual of the 8cell. While the 8cell is composed of cubes the 16 cell is composed of tetrahedrons. Just as the 5 cell is composed of tetrahedrons.
So now we can see that the first perfect number 6 expresses the relations between 4 entities. It appears as 6 entities in the cubo/octahedron (fused or interpenetrated tetrahedrons). But at that point it is counterbalanced by the 3^4 or trigrams. The trigrams are perfected by the 28 paths between them. But intervening between the perfect elements and the next higher progressive bisection threshold is 12 the dozen an immanently dividable number.
But 2^4 does not come next, rather we get 2^5 as the threshold of the penta-chora//icosa/dodacahedron duality. We can see the pentachora as the Husserlian essence of the icosa/dodacahedral noematic nucleus. In it we see synergy at work in the reuse of parts because only 5 points and ten lines gives us 5 tetrahedra.
However, this synergetic core signified by the pentahedron is surpassed by the unique 24 cell polytope that relates the 2^3 to the 2^4 seen as 4d polytopes. But 2^5 of the pentahedral combinatorics intervenes as well. The 32 permutations of the pentahedron is perfected by the 496 paths between them.
The 24 cell has the property of unimpeded flow (jing chi) that only exists in octahedrons of which it is made. So the octahedrons are displaced here and are not the dual of the cubs in the 8cell. Rather tetrahedrons appear in the 8cell. This is a fascinating asymmetry at this analogous level in the fourth dimension to the cubo/octohedron.
Part of that displacement is the intervention of the pentahedron which has intruded into the unfolding of the bisections by introducing 2^5 prior to 2^4 in the series of geometrically unfolding thresholds of comprehension.
To account for this intrusion and asymmetry we need to think hard about what the 4-d synergetic math is telling us. And one thing it might be pointing toward is Integrity which is the next principle up according to B. Fuller. A principle you see in tensegrity. The asymmetry and the intrusion of the dynamic of the pentahedron represents a dynamic internal to the core of the fourth dimensional figures. 2^5 and the 24 intervenes between 2^3 and 2^4. In both cases these are excluded elements coming back. In the case of the 24 it is the octahedron coming back into relation with the cube at a higher dimensional threshold of understanding and synthesis. In the case of the 32 it is the structure of the pentachora that has an abundance of an extra tetrahedron over what it should have by rights due to the way the mathematics is distorted as it moves toward the ultra-one.
So ultimately we see that the 496 is again the perfection of the relations but this time of the combinatorics of the 5 elements of the pentachora. This perfecting element of the relations becomes more and more distant from the central role it played in the frist minimal solid. And this has something to do with the rarity of these numbers. Fusion into perfection becomes more and more rare as we get away from the ultra one.
The next such number is 8128 in case you are about to ask.
It is the relation between 128 elements. That is 2^7 http://en.wikipedia.org/wiki/128_(number)
Something to do with the Hepteract.
“In geometry, a 7-cube is a seven-dimensional hypercube with 128 vertices, 448 edges, 672 square faces, 560 cubic cells, 280 tesseract 4-faces, 84 penteract 5-faces, and 14 hexeract 6-faces.”
Which has a dual called the 7-orthoplex
“In geometry, a 7-orthoplex, or 7-cross polytope, is a regular 7-polytope with 14 vertices, 84 edges, 280 triangle faces, 560 tetrahedron cells, 672 5-cells 4-faces, 448 5-faces, and 128 6-faces.”
We notice here that 2^6 (64) is skipped. So the asymmetry at the heat of number theory and geometry continues to what end and why we do not know. Mathematicians seem unconcerned. So we will not worry about it either.
But our point has been made that something veers off course in the progressive bisection of the Pascal triangle at the level of the Pentachora. And that veering does not stop but continues as we go higher and higher into the rarer and rarer perfect numbers.
Somehow perfection is bound up with the asymmetries that appear in the math as we move toward the ultra-one. It is as if the true and ultimate heterogeneity of the Multiple gets expressed as a series of symmetry breaking Events as we move from infinity toward the One. These tend to revolve around the primes 7 and 5. The perfection of the 6 between them seems to be counter-balancing these asymmetries giving extra fusion and transparency to counterbalance for instance the opacity of A5.
Plato had above the door of the academy that only those who knew geometry should enter. Sometimes I wonder if anyone in our tradition really entered that door. Philosophers seem unconcerned with the oddities of math, and mathematicians appear to have no concern with the meaning of the structures they discover for thought. That is why B. Fuller’s work is important. He systematizes three-dimensional geometry as expressed in platonic solids and so all we have to do is extend his work into the fourth dimension to get a deeper insight into the structure of the thresholds of thought which is necessary to have even an inkling as to what the perfection 496 may mean.
Notice how its divisors veer off from the progressive bisection
In the number 496 the Perfect number factors have veered off the progressive bisection by exactly the quantity of the progressive bisection.
What we should ask is not what is the next perfect number after 6 and 28, but what does it mean for our thought that it is perfect in itself, but its factors are exactly deficient with respect to the progressive bisection by the progressive bisection up to that point. But that deficiency only starts after 16. Thus 5 factors are true to the progressive bisection, while 5 are deficient by the progressive bisection, and that deficiency is precisely the numbers that remain true in the first part of the series of factors.
An interesting quandary the number of your question provides.