Plato's number
I just realised something remarkable. I postulated in 2013 that the functional cycle has 4 phases, namely an organisation's
and already back then I compared it to Bismarck's theory of waxing and waning, to use Plato's words,
It is somewhat safe to assume that the above cycle of waxing and waning is the one refered to here, and although that is not safe to assume at all, it is certainly remarkable that my so called pop-cultural cycle, which is a cycle of dominating and being dominated which has 3 (or if each one is split in half 6) phases, could at least on these grounds be Plato's 1st.
Plato's 100 equal lengths forming a square refers to the 2. cycle of waxing and waning, which makes it likely that each of its phases is 25 years long and the total cycle thus 100 years. 100 would thus be formed by 8 equal lengths of 5: 100=5*5+5*5+5*5+5*5, or, because 100 isn't divisible by 8, perhaps by 4, which have to be squared. Thus in 2500 years 100 lengths of 5 would have accumulated and 2500 is indeed a square, namely that of 50. 2500 years is also a good approximation of the length of the age of works or an aeon in general.
Of course, 2500 isn't divisible by 3, but 7500 is, and if we assume that there is a cycle of 3 ages, the age of wonders, the age of watch and the age of works, 7500 years is a good approximation of the time needed to end up in the same age again. Following this kind of logic, we should divide 7500 by 4, which gives 1875 and that would have to be our oblong rectangle. The problem with this is that 1875 isn't divisible by 50, but 3750 is, meaning perhaps the time interval for total birth reset is 15,000 years, which agrees somewhat with Plato's rebirth period of common souls as stated in Phaidros, i.e. 10,000 years, or, in other words, the interval between arrivals of new human souls, like for instance Y-DNA haplogroup F 62,000 years ago and MNOPS 47,000 years - the more things change, the more they stay the same: a new breed of souls going through the birth cycle.
3750 divided by 50 is 75. 50 being the length of the square 2500 and incidentally that of one of the phases of the pop-cultural cycle, so after 25 of its passages the 3750 years would have passed. Of course, that raises the question why do all that when the least common multiple of 100 and 150 is 300 and not 15,000. 15,000 requires lcm(3,8,625). Anyway, back on track. A 100 cubes (or squares, actually) of the 1st cycle should come about as before by its repetition.
Well, the 1st cycle has 3 phases and 100 isn't divisible by three, hence κύβων τριάδος has to (and arguably does) mean of the triad of cubes, i.e. the number of cubes divided by 3, and since 75 is divisible by 3, the length of a phase of the 1st cycle has to be 50, assuming that one of the divisors of 3750 is the length of the square 2500 and one of them counts the number of phases in the rectangle. Well, there are 6 squares of 5 in 150 and hence 2 triads. And so we need 50 cycles to accumulate 100, and 50*150=7500, which means that 7500 and not 15,000 is Plato's number and the doubling of 1875 is only for convers- and commensurability.
I consider this settled. As for the pempads, they're groups of 5, and the irrational diameters the diagonals of some rectangle. Plato has to square that number then to come back to a rational one. The rectangle should be 3750, and if we want to do geometry, we might as well divide that by 625 and consider the square root of 13 as the diagonal of that rectangle, or, scaling up again, the pempad of pempads of it. Well, following Plato's description, we'll want to do something with 5*131/2-2 next, which is a little over 16, but I must admit that I don't know how to proceed from here.
More importantly and as stated before, if one cycle takes 150 years and the other 100, and that is the most likely inference, independent of the fact that I have already compared two concrete cycles of those lengths, my pop-cultural cycle and Strauss-Howe's, which is equivalent to my functional cycle, though I gave no length for it and neither can a length for organisations in general be given, it doesn't take 7500 years for them to repeat, though if we were to add a small disturbance of 150/49 years to the length of the 1st cycle and of 50/37 years to the length of the 2., and nothing's perfect after all, that would've been achieved.
Immediate postscript. Yeah, it's time again. My 4000th post.
Postscript from 30.6.2025.
I doubt that the μὲν-δὲ is referring to different dimensions, because it doesn't quite make sense: We already have 2500=50*50. If 50 is one of the dimensions, we only need to know one more, but Plato gives two formulas. Let's take τὴν δὲ ἰσομήκη as evidence that we're looking at a multiple of 2500, which is almost certainly true. I don't know what a rational diameter is, unless Plato is referring to a Pythagorean triple, but then the irrational alternative doesn't make sense. So let's just skip that as an unknown definition. Some number is to come about by 100 pempads of diameters lacking 2 each. As for the rectangle, we could consider something scaled down by factor 25 starting from 50 and an unknown multiple of 25 and further make the assumption that but also means that we're getting also either 37,500 or 7500 as the result, i.e.
- focus being on how to make use of available resources (Verwaltung, not in the sense of administration though),
- focus being on what to use them for (Selbstdarstellung),
- purpose becoming a resource for other organisations* (parasitärer Befall) and
- collapse (chaotischer Kollaps),
and already back then I compared it to Bismarck's theory of waxing and waning, to use Plato's words,
- the first generation builds the wealth,
- the second manages it,
- the third studies the history of arts and
- the fourth generation goes to seed.
It is somewhat safe to assume that the above cycle of waxing and waning is the one refered to here, and although that is not safe to assume at all, it is certainly remarkable that my so called pop-cultural cycle, which is a cycle of dominating and being dominated which has 3 (or if each one is split in half 6) phases, could at least on these grounds be Plato's 1st.
Plato's 100 equal lengths forming a square refers to the 2. cycle of waxing and waning, which makes it likely that each of its phases is 25 years long and the total cycle thus 100 years. 100 would thus be formed by 8 equal lengths of 5: 100=5*5+5*5+5*5+5*5, or, because 100 isn't divisible by 8, perhaps by 4, which have to be squared. Thus in 2500 years 100 lengths of 5 would have accumulated and 2500 is indeed a square, namely that of 50. 2500 years is also a good approximation of the length of the age of works or an aeon in general.
Of course, 2500 isn't divisible by 3, but 7500 is, and if we assume that there is a cycle of 3 ages, the age of wonders, the age of watch and the age of works, 7500 years is a good approximation of the time needed to end up in the same age again. Following this kind of logic, we should divide 7500 by 4, which gives 1875 and that would have to be our oblong rectangle. The problem with this is that 1875 isn't divisible by 50, but 3750 is, meaning perhaps the time interval for total birth reset is 15,000 years, which agrees somewhat with Plato's rebirth period of common souls as stated in Phaidros, i.e. 10,000 years, or, in other words, the interval between arrivals of new human souls, like for instance Y-DNA haplogroup F 62,000 years ago and MNOPS 47,000 years - the more things change, the more they stay the same: a new breed of souls going through the birth cycle.
3750 divided by 50 is 75. 50 being the length of the square 2500 and incidentally that of one of the phases of the pop-cultural cycle, so after 25 of its passages the 3750 years would have passed. Of course, that raises the question why do all that when the least common multiple of 100 and 150 is 300 and not 15,000. 15,000 requires lcm(3,8,625). Anyway, back on track. A 100 cubes (or squares, actually) of the 1st cycle should come about as before by its repetition.
Well, the 1st cycle has 3 phases and 100 isn't divisible by three, hence κύβων τριάδος has to (and arguably does) mean of the triad of cubes, i.e. the number of cubes divided by 3, and since 75 is divisible by 3, the length of a phase of the 1st cycle has to be 50, assuming that one of the divisors of 3750 is the length of the square 2500 and one of them counts the number of phases in the rectangle. Well, there are 6 squares of 5 in 150 and hence 2 triads. And so we need 50 cycles to accumulate 100, and 50*150=7500, which means that 7500 and not 15,000 is Plato's number and the doubling of 1875 is only for convers- and commensurability.
I consider this settled. As for the pempads, they're groups of 5, and the irrational diameters the diagonals of some rectangle. Plato has to square that number then to come back to a rational one. The rectangle should be 3750, and if we want to do geometry, we might as well divide that by 625 and consider the square root of 13 as the diagonal of that rectangle, or, scaling up again, the pempad of pempads of it. Well, following Plato's description, we'll want to do something with 5*131/2-2 next, which is a little over 16, but I must admit that I don't know how to proceed from here.
More importantly and as stated before, if one cycle takes 150 years and the other 100, and that is the most likely inference, independent of the fact that I have already compared two concrete cycles of those lengths, my pop-cultural cycle and Strauss-Howe's, which is equivalent to my functional cycle, though I gave no length for it and neither can a length for organisations in general be given, it doesn't take 7500 years for them to repeat, though if we were to add a small disturbance of 150/49 years to the length of the 1st cycle and of 50/37 years to the length of the 2., and nothing's perfect after all, that would've been achieved.
Immediate postscript. Yeah, it's time again. My 4000th post.
Postscript from 30.6.2025.
τὴν μὲν ἴσην ἰσάκις, ἑκατὸν τοσαυτάκις,5*5*100: 2500
τὴν δὲ ἰσομήκη μὲν τῇ, προμήκη δέ,in one sense equal, but also oblong,
ἑκατὸν μὲν ἀριθμῶν ἀπὸ διαμέτρων ῥητῶν πεμπάδος, δεομένων ἑνὸς ἑκάστων, ἀρρήτων δὲ δυοῖν,in one sense 100 of the pempad of rational diameters lacking 1 each or irrational lacking 2,
ἑκατὸν δὲ κύβων τριάδος.but also 100 of the triad of cubes / squares of 5: 37,500 / 7500.
I doubt that the μὲν-δὲ is referring to different dimensions, because it doesn't quite make sense: We already have 2500=50*50. If 50 is one of the dimensions, we only need to know one more, but Plato gives two formulas. Let's take τὴν δὲ ἰσομήκη as evidence that we're looking at a multiple of 2500, which is almost certainly true. I don't know what a rational diameter is, unless Plato is referring to a Pythagorean triple, but then the irrational alternative doesn't make sense. So let's just skip that as an unknown definition. Some number is to come about by 100 pempads of diameters lacking 2 each. As for the rectangle, we could consider something scaled down by factor 25 starting from 50 and an unknown multiple of 25 and further make the assumption that but also means that we're getting also either 37,500 or 7500 as the result, i.e.
500*((4+x2)1/2=37,500 => x=56211/2 orand interpreting lacking 2 as lacking the (22+...)1/2 in those formulas this leads to x=75 or x=15 and scaling up again we arrive at 1875 or 375, which is a little puzzling, because the solution for the cube leads to a quarter of the solution for the square and the solution for the square leads to the triad of the cube. Anyway, 7500 is the more practical number for birth reset, 37,500 would only be considered by people who want to be free to imagine whatever they want. By the way, a phase in the cycle of the forms of government according to the I Ching amounts to 625 years, if we accept the length of an age to be 2500 years and the length of the cycle of ages accordingly as 7500.
500*((4+x2)1/2=7,500 => x=2211/2,
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