THE RIDDLE OF NUMBERS IN THE REPUBLIC
“A city which is thus constituted can hardly be shaken; but, seeing that everything which has a beginning has also an end, even a constitution such as yours will not last for ever, but will in time be dissolved. And this is the dissolution:—In plants that grow in the earth as well as in animals that move on the earth’s surface, fertility and sterility of soul and body occur when the circumferences of the circles of each are completed, which in short-lived existences pass over a short space, and in long-lived ones over a long space. But to the knowledge of human fecundity and sterility all the wisdom and education of your rulers will not attain; the laws which regulate them will not be discovered by an intelligence which is alloyed with sense, but will escape them, and they will bring children into the world when they ought not. “Now that which is of divine birth has a period which is contained in a perfect number, but the period of human birth is comprehended in a number in which first increments by involution and evolution (or squared and cubed) obtaining three intervals and four terms of like and unlike, waxing and waning numbers, make all the terms commensurable and agreeable to one another. The base of these (3) with a third added (4) when combined with five (20) and raised to the third power furnishes two harmonies; the first a square which is a hundred times as great (400 = 4 × 100), and the other a figure having one side equal to the former, but oblong, consisting of a hundred numbers squared upon rational diameters of a square (i.e. Omitting fractions), the side of which is five (7 × 7 = 49 × 100 = 4900), each of them being less by one (than the perfect square which includes the fractions, sc. 50) or less by two perfect squares of irrational diameters (of a square the side of which is five = 50 + 50= 100); and a hundred cubes of three (27 × 100 = 2700 + 4900 + 400 = 8000).“Now this number represents a geometrical figure which has control over the good and evil of births. For when your guardians are ignorant of the law of births, and unite bride and bridegroom out of season, the children will not be goodly or fortunate … In the succeeding generation rulers will be appointed who have lost the guardian power of testing the metal of your different races, which, like Hesiod’s, are of gold and silver and brass and iron.”
–PLATO, Republic VIII, 546
THE SOLUTION OF THE REPUBLIC RIDDLE
The perfect number: was six in ancient times, because 1 + 2 + 3 = 6 and 1 × 2 × 3 = 6.
The number for human births: “in which first increments,” “3 intervals” = 3 number values, “4 terms” = 4 powers, “combined” = multiplication, the 1st power is not a multiplication.
1, 2, and 3: not possible, because with the 1, no multiplication can be performed.
2, 3, and 4: not possible, because 2 = ?4, 4 = 22
3, 4, and 5 are the first numbers that meet the requirements mentioned once in the text.
3 × 4 × 5 = 601 ( 60 and one time 3×4×5)
× × ×
3 × 4 × 5 = 602 (= 3600) 1st multiplication
× × ×
3 × 4 × 5 = 603 (= 216000) 2nd multiplication
× × ×
3 × 4 × 5 = 604 (= 12960000) 3rd multiplication, the Platonic number
The first proportion: “a square which is a hundred times as great” = the Proportion of Time.
?(12960000/100) = ?129600 = 360. The circle has been divided into 360° ever since the time of the ancient Sumerians. The 6th multiplication gives 2799360, this times seven is the Vedic great cycle of 19595520 mirror of 129591 with the 55 as the one in balance. 1959552/27=72.576
1st “circle rotation”: the day-circle, 1° = 4 minutes. 2nd “circle rotation”: the year circle,
1° = (ca.) 1 day. 3Rd: 100 years.
The “marriage number”: 60 × 4 minutes = 4 hours – the 4th hour of the morning is the most auspicious time for begetting. 60 × 4 hours = 10 days – the 10th day after beginning of menstruation is the best time for conception. 60 × 10 days = 20 months
20 months after one birth is the earliest time for the next birth. 60 × 20 months =100 years: only every 100 years, every 4th generation, the birth of a strong personality can be expected in a family.
The second proportion is that of space, because we calculate here in terms of “sides”, directions, and “diagonals”. With the shortening of a diagonal, the area must likewise decrease. The regular shortening of a diagonal yields a pyramid.
If we want to use the Platonic number (604) in a pyramid, we can use 60 (=601) as the base side. The base area is then 602. For the height, however, we must use 3 × 60 (pyramid formula = a × b × 1/3 × h) and obtain 603 as the spatial content (volume).
The fourth power of 60 (604) can only be sought in the building blocks (cuboids).
Side a = 3, side b = 4, side c (=height) = 5, because 3 × 4 × 5 = 60.
But through this, the measurements of the pyramid change: a = 60 × 3 = 180, b = 60 × 4 = 240, c (= h) = 3 × 60 × 5 = 900 . The pyramid is then equilateral in one direction (vertical section), “but longer in the other” (horizontal section).
The side ratio 3 : 4 : 5 is that of the so-called “Egyptian triangle”. A perfect example for the Pythagorean Theorem for the right triangle: a2 + b2 = c2, therefore 32 + 42 = 52, 9 + 16 = 25,
25 = 5 for the side opposite the right angle (the hypotenuse), when the short leg (a) is 3 and the long leg (b) is 4.
The two legs then form a perfect right angle.
The base of the pyramid sides a 3 × 60 (= 180) and 4 × 60 (= 240) therefore has a diagonal of 5 × 60 (= 300)
“A hundred numbers squared upon rational diameters of a square, the side of which is five, each of them being less by one” = the inclination of the pyramid edges = 1 : 3.
“Less by two perfect squares of irrational diameters” = inclination of the pyramid sides, 180 + 240 + 180 = 600 = 1 : 6. One side can be determined if three sides are given. Edge length and side height yield irrational numbers (with infinite fractions) which are elegantly described thus.
“And in the other direction (i.e. horizontally) a hundred cubes of three.”
The base area = 60 × 60 cornerstones = 3 × 4 × 3600 = 43200. 3600/100 = 36 cornerstones, 43200/100 = 432 as area. Three cornerstones are stacked and fastened with pegs in previously bored holes = 1 column; 36 columns, connected at their sides in the same manner, form one building block. Thus the pyramid becomes a step pyramid, whose calculation must follow different principles.
The calculation in terms of “columns” is performed specially for each step, and this calculation converted into building blocks (= number of columns divided by 36) yields 55 remaining columns per story for stories I through IX (for every 6 steps) which must be discarded.
55 = the sum of the cardinal numbers from 1-10, and also the sum of the squares of the numbers 1-5, that is 12 = 1, 22 = 4, 32 = 9, 42 = 16, 52 = 25. In every five steps of stories I, III, IV, VI, VII, and IX, the columns are discarded in this sequence, but in every 5 steps of stories II, V, and VIII they are discarded in reverse sequence, calculated from above 52, 42, 32, 22, 12. This can be described as a true numeric marvel.
We use the removal of these columns to create 9 huge gates, with peaks at the top in stories I, III, IV, VI,VII, and IX, and peaks at the bottom in stories II, V, and VII.
Since the number of columns can be divided evenly by 36 in every 6th step, no columns are discarded there, so that the separation into stories becomes clearly apparent. The apex of the pyramid (= story X) must yield 19 “columns.” Through the removal of 18 columns, the apex of the pyramid undergoes a decorative rearrangement. The 19th (topmost) column must be (alternately) removed from the floor of story X, because no more columns can be removed symmetrically. This fact will later be of special importance.
In order to compensate for the difference in volume between the step pyramid and ideal pyramid, even more columns must be removed from stories I through VIII.
After subtracting the 55 columns for the gates, 288 (= 24 × 12) columns remain in story I, 252 (= 21 × 12) in story II, 216 (= 18 × 12) in story III, 180 (= 15 × 12) in story IV, 144 (= 12 × 12) in story V, 108 (= 9 × 12) in story VI, 72 = (6 × 12) in story VII, 36 (= 3 × 12) in story VIII, which must be removed, meaning in each case 3 × 12 = 36 fewer columns (1 building block). We use the removal of these columns to create inner spaces in these stories, whereby the constantly returning factor 12 represents the profile in width and height, and the other factor (= 3) represents the depth in column-depths. Where the peaks of the gates point upward, meeting rooms (great halls) appear, and where the peaks of the gates point downward, lecture halls (auditoriums) appear. In the former case, the profile has three steps (3 + 4 + 5 = 12 columns), in the latter case it has two steps (9 + 3 = 12 columns), whereby at any given time, 1 column is being dismantled into its three cornerstones, thus creating seating accommodation. Staircases inside the pyramid link the inner rooms. For this, one building block is taken from the base step above the room, and dismantled into its 108 cornerstones, which are then placed in such a way that side 3 is the height of the step, side 4 the depth, and side 5 the breadth.
Number of cornerstones used:
For step A = 20, B = 19, C = 17, A + B + C = 56, for supports: D = 17, E = 35, D + E = 52, total = 108 cornerstones.
The height is therefore 30 × 3 = 90 = height of the story. The staircase follows the walls of the inner rooms and makes two 90° turns. Staircases of the above form correspond to the most extreme cases; in general far fewer cornerstones will be sufficient. Extra cornerstones are used for broadening the staircase, starting from the bottom.
The grating in the background of the picture emerges from the base surface of the pyramid with the built-on squares of the two Egyptian triangles, with the addition of a four-square. The perpendicular cross beam (= 16 + 12 + 16 + 16 = 60; 60 × 602 = 603 or 216000) = twice the cross section of the pyramid. The horizontal crossbeam (= 9 + 12 + 9 = 30; 30 × 602 = 108000) = the cross section of the pyramid. The part of a tilted cross (“St. Andrew’s cross”) visible behind the standing cross has an area of 4 × 6 × 602 = 86400, which is twice the area of the base of the pyramid (43200).
Now let’s move on to the ark of covenant. The ark is surmounted by a mercy seat, covered by the wings of two cherubim (symbol of quaternary ) and above it resides the shekinah. It is placed in a cubic enclosure (shekinah is dwelling-place of “god”) and its volume is a straight parallelepiped of the same proportions that serve as a framework for the dwelling houses constructed according the rules of the art. So what is this formula revered to? As you know that 8 times the cubit of Noah gives you this formula 5236 x 8 = formula volume 4,1888, this times the dimension of the ark of covenant 56,25 indeed the proportion of the dwelling houses 23562 as in 45 times 52,36.
And as you have seen above 55 times gives you 129591.
Moshiya van den Broek