AUTHOR’S FOREWORD

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




QUOTATIONS

Plato. “The Master [Socrates] passed on the flame of Prometheus. In Plato it grew to gigantic brightness.”

–Frank Thiess, Das Reich der Dämonen

Jesus and Plato. “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

Plato and geometry. “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

The annihilation of ancient authoritative works. “...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

The Platonic riddles of numbers are glass bead games. “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 science with art, and to reconcile art or science with religion.”

–Herman Hesse, The Gass Bead Game.3

“Thanks to the glass bead game player!”

–Herman Hesse, to the Author.

 

INTRODUCTION

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 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

× × ×

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.

 

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 (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.