**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

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

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.