Only with reluctance does the Author bring to public light the results of a 9-year search for the solution to the Platonic number riddles. He is aware that, not being an expert in this field, he is in an awkward position from the viewpoint of the orthodox sciences. So many people, specialists and laymen alike, have fantasized about these number riddles, that nowadays any dealing with this material is often dismissed with a shrug. The Author would never have attained the results that follow if he had wasted his time studying authoritative sources and allowing them to lead him down the wrong path, especially (for reasons that will become obvious later) because in the literature at our disposal, anything that could have facilitated the task of solution had been completely destroyed. The only practicable method was that of logical thought, applying mathematical and especially geometric formulae, most of which had to be rediscovered first. Only after the preliminary completion of the solution did the Author find corroborations and references useful for the clarification of the remaining mysterious points in authoritative publications, especially those of the church fathers.

Nevertheless, the Author would never have succeeded in achieving these results had he not been constantly led forward, step by step, in a way incomprehensible to him. He remembers certain decisive turning points at which, almost without his participation, he was guided around obstacles which would otherwise have led him to fail miserably.

The Author is far from being above receiving suggestions for improvements —the better is the enemy of the good here as well— because hardly a week passes by without the necessity for major or minor changes. However, it is assumed that one observes those rules of the game which are known indisputably to be valid. Here it should be mentioned that the Platonic number riddles are really games, logical games of thought, “glass bead games,” as the great writer and great man Hermann Hesse calls this kind of mental occupation. May this publication bring to wider circles something of that which made the Author enjoy his task so greatly: revelation by means of number, and thereby the certainty of the existence of the divine!

**- Eberhard Wortmann**

The Master [Socrates] passed on the flame of Prometheus. In Plato it grew to gigantic brightness.– Frank Thiess, Das Reich der Dämonen

Jesus has not eaten from the tree of knowledge. He who has done that will not have peace within himself. He will strive onward, he needs the mediator to the divine, Eros. Jesus is not qualified to be such a mediator, because he does not know this longing. He cannot replace Plato.– Ulrich von Wilamowitz-Moellendorf, Der Glaube der Hellenen

This was the state of affairs when Plato came into Sicily ... There was a general passion for reasoning and philosophy, insomuch that the very palace, it is reported, was filled with dust by the concourse of the students in mathematics who were working their problems there. [Geometric figures were drawn in the dust.]– Plutarch, Dion.1

...But those works were thought devoid of interest or even dangerous by the devout Middle Ages, and they are not likely to have survived the fall of paganism.

–Franz Cumont, The Oriental Religions in Roman Paganism.2

In the end it depends on the choice of the historian how far back he wants to put the beginning and prehistory of the Glass Bead Game ... As an idea we find it ... already performed in earlier ages, e.g. by Pythagoras. ... The same eternal Idea was the basis of every movement of the mind towards the ideal goal of a Universitas Litterarum, of every Platonic Academy, of any congregation of a spiritual elite, every attempt to approximate the exact and the free sciences, every attempt to reconcile science with art, and to reconcile art or science with religion.– Herman Hesse, The Glass Bead Game.3

Thanks to the glass bead game player!– Herman Hesse, to the Author.

The work of solution began with the riddle in the Republic. At an advanced age, the Author came across Plato’s Republic—known as Der Staat in the German-speaking world, Politeia in the original Greek. In reading it, he got only as far as the number riddle in book VIII, and then did not rest until he had found the solution. Since he had just “coincidentally” been studying the right triangle with sides of ratio 3 : 4 : 5, the so-called “Egyptian triangle”—an astonishing entity, if one considers that any flat surface can be divided into isosceles triangles, and these further divided into right triangles, of which the smallest is the Egyptian triangle, which also has numerous unique features—he immediately saw that the solution was within reach, since the text of the riddle makes it quite clear that it concerns a pyramid. Only after the final solution, taking into account all the subtleties—which took him a year—did the Author realize the great importance of the solution and hear of a well-endowed competition, which had caused a great stir at the time and which, despite lively participation, had not brought about a solution satisfactory to all sides.

The resulting disenchantment caused interest in the solution of the riddle to fade. Many people even became convinced that there was no solution. Plato, they said, only wanted to make people interested in him, by virtue of his supposedly possessing a secret formula for running a state, so that he would be remembered favorably and recommended for high office. But in fact, Plato had already given up his political ambitions at that time. He “squatted in the corner with a few youths,” as his opponents said, and devoted himself exclusively to the task of setting up an ideal that could never be realized and is therefore eternally valid, the Ideal State, the first of many such utopias.

One will see that this riddle concerns ancient thoughts, and that every detail, even the smallest, emerges of itself.

## 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 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

× × ×

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.

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 (see Figure 1a).

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 (Figure 1b).

But through this, the measurements of the pyramid change: a = 60 × 3 = 180, b = 60 × 4 = 240, c (= h) = 3 × 60 × 5 = 900 (Figure 1c). 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 (Figure 1d).

The base of the pyramid sides a 3 × 60 (= 180) and 4 × 60 (= 240) therefore has a diagonal of 5 × 60 (= 300) (Figure 1e).

“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 (see Figure 1f).

Thus the pyramid becomes a step pyramid, whose calculation must follow different principles (see page 10). 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 of 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 (see Figure 1g and Plate 2).

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 (Figure 1h).

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 (Figure 1l and Plate 2). 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.

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 (Figure 1j).

The grating in the background of the picture (see Plate 3) 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).

Measuring units might be 1/2 foot, 1/4 ell = ca 15 cm, therefore height = 135 m, step = 2.25 m, stair = 45 cm.

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### Hans Kayser - Textbook of Harmonics

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