** Number π and Solomon's molten sea**

As
stated in Ch.7 of the book of Kings I, King Solomon ordered Hiram from Zor to make, among other things, a molten sea. "And he
made a molten sea, ten cubits from the one brim to the other: it was round all
about, and its height was five cubits: and a line of thirty cubits did circle
it round about" (Kings I, Ch. 7:23) (see the side view and __the view from above__). Since the diameter of the vessel was 10
cubits and the circumference 30, their ratio was 3.
Quite a poor approximation of the number π=3.1415…! Perhaps the diameter
of 10 cubits included the thickness of the walls of the vessel, while the
perimeter was measured from inside? Indeed the Talmud in Iruvin
14a makes such a suggestion. However it is rejected
immediately. The Talmud cites the verse ^{2}∙5=375 cub. cubits, i.e. 5/6
times less than required. Then Talmud
quotes an external source ("braita") saying
that the Solomon sea for the first 3 cubits of its
height was square and for the next 2 cubits was round. Hence the volume of the
sea was 10^{2}∙3+3∙5^{2}∙2=450, exactly as
stated in the book of Kings!

Is it possible that the sages of Talmud
thought that π is exactly 3? Even a rough measurement would show that
π is larger than 3! The approximation of π by the ratio 22/7 was
known in the ancient time. It is impossible to imagine that King Solomon, the
wisest of all men, did not know this elementary fact. And at the same time he
was able to manufacture a gigantic copper vessel of a given size and
volume.

Somebody drew attention to the fact
that the word "line" in the verse

The number
π as any number could be expanded into continuous fraction a+1/(b+1/(c+1/(d+…)). In the
case of π a=3, b=7, c=15, d=1. The ratio 333/106=3+(1/(7+1/15))
is a second approximation of π. The fist approximation is 22/7, the third
is 355/113=3.1415929… It is remarkable that the first three approximations of
π are extremely efficient, i.e. they are very close to π while having small denominators.

It
appears to me that the correction ÷å/÷åä (qava/qav) has not merely
numerical meaning. The word ÷åä (qava) is feminine (in
Hebrew the feminine words almost always end with ä) while ÷å (qav) is
masculine. The way the word is spelled is called "masoret"-îñåøú and is
feminine, the way it is pronounced is called "mickra"-
î÷øà and
is masculine. On the other side, in the pair circle-diameter, the circle
represents a feminine, material notion (e.g. the mother Earth) while the
straight line represents the masculine, spiritual notion (e.g. the rain that
fertilizes the earth). Hence the word ÷åä (qava) is
related to the circle while ÷å (qav) to the diameter. With
this correspondence the
verse

Notice that that all objects in the tabernacle where
straight. May be this is the reason why Rambam draw
the Menorah with
straight branches? If in the "heavens", in the spiritual world, there
are no curved lines, the circle is perhaps represented there by a polygon. In
case of the perimeter of the circle, the hexagon could serve as a model. In
case of the area, the dodecagon could be the model. In the first case the
perimeter is equal 2∙3∙radius of the surrounding circle; in the
second case the area equals 3∙square of the radius, as if π=3. That
is why the Sages considered the equality π=3 not as an acceptable
approximation but as a reflection of a certain spiritual truth.

The idea that π represents the ratio of feminine to
masculine could be demonstrated by the number 135. Doubling of its digits
113355 and separation 113:355 leads to the ratio 355/113 that approximates
π with seven significant digits!
The analogy is the double faced Adam and his separation into a man and a
woman. The same idea could be traced in the volume of kaporet-
the cover of the arc. Its length was 2.5 cubits or 15 palms, the width was 1.5
cubits or 9 palms and the thickness was one palm. Thus
its volume was 135 cub. palms. Two cherubim, a boy and
a girl, where growing out of it, separated by a tiny gap between their wings.
The same was the volume of the sockets under the pillars of the tabernacle
(their dimension was 4.5x6x6 palms from which one should subtract the volume of
the wooden legs of the pillars 1.5x3x6 palms). Two sockets were put on two legs
of a pillar and were united by the pillar. We encounter again the number 135 in
the plan of the

Let us discuss now the volume of the molten sea. As already
mentioned above, this volume according to Kings I ^{2}∙5=392.7 cub. cubits. If one assumes according to the Talmud Eruvin 14a that the lower three cubits of the sea were a
square of 10 by 10 and the upper two cubits were a circle with diameter 10,
then one obtains the volume 300+ π5^{2}∙2=457.08 cub. cubits. One can suggest that the Scriptures rounded down
this number to 450, or adjusted it to the approximation of π by 3. But, since a hidden deep meaning was found in
the perimeter of the sea, we will try to find such meaning also in the volume
of the sea. For that sake one should
first understand by what cubits the sea was measured. It is written in Talmud Iruvin 83a that the unit of volume "seah" (equal to 3/40 cub. cubits) increased in ^{3}=1.19946... . For practical matters this number is
equal to the ratio 6/5=1.2 between the unit of volume in

After this discourse let us return to the ^{3} by 1.2. If we use the
correct value of π and of (51/48)^{3 }then the volume of the sea
will be 1.04673 times bigger. Let us return to the description of the walls of
the sea in the verse **and it was a hand breadth thick, and brim was
wrought like the brim of a cup, like the petals of a lily**". We will
assume that the sea was a cylinder of diameter 10 cubits __from outside__ so
that the thickness of the walls decreased its inner volume. The wall was hand
breadth (1/6 of the cubit) thick at the bottom of the sea and practically zero
at the top. What is the form of lily? Certainly not straight. The simplest
curve next to the straight line is an arc of a circle. It is natural to assume
that this arc was perpendicular to the bottom of the sea (see the side view). It is not difficult to calculate the volume of such
wall. Denote by r the radius of the exterior cylinder (r=5), by h its height
(h=5) and by d the thickness of the wall at the bottom of the cylinder (d=1/6).
The radius of the arc is R=(h^{2}+d^{2})/(2d)=75.0833.
The wall is the body of rotation around the axis of the cylinder. Its volume is

V_{1}=2π∫(R+r-d-x)(R^{2}-x^{2})^{1/2}dx, R-d≤x≤R

=2π((R+r-d)(R^{2}arcsin(h/R)-(R-d)h)/2-h^{3}/3)=17.224386.

The volume of the sea is
V=πr^{2}h-V_{1}=375.474696 cub. cubits
of 51 cm. If we translate them into cubits of 48 cm, we obtain 450.36796 cub. cubits. It is almost the requested number. But we did not
pay attention to the fixtures of the sea. The verse 7:24 reads "and under
the brim of it round about there were knops (peqaim
in Hebrew) compassing it, ten in a cubit, compassing the sea round about; the
knops were in two rows, cast with it in the same casting" (see __the view from above____)__. Perhaps
these knops were cast in the walls from inside and thus decreased the volume of
the sea? The word peqaat in Hebrew means something
round, apparently a ball. If ten of them were in one cubit then the diameter of
each one was 1/10 cubit. Interestingly enough, the ancient Aramaic translation
of Jonathan ben Uziel calls these peqaim- eggs. By definition, the volume of an egg is 1/1920
of a cub. cubit. The volume of a ball of diameter 1/10
is 4/3π(1/20)^{3}=1/1909.86.
If one approximates π by the ratio 25/8=3.125 then one obtains exactly
1/1920. How many balls were in a row? If we calculate the perimeter of the brim
by the correct value of π, we obtain 314.5 balls. Most likely the number
of balls was "round", 300. The perimeter of 30 cubits as stated by
the verse

Let us summarize the results of our investigation. Obviously,
King Solomon knew not only the value of π with high accuracy but was also
able to calculate the volume of a body of rotation with 6 digits. But we are
talking here not about human wisdom but of divine wisdom. No human mind could
design a vessel of such simplicity and harmony and at the same time arrive at a
"round" volume with 6-digit accuracy. Also the play of words with the
line ÷åä/÷å (qava/qav) which corresponds to
the approximation of π by a continuous fraction, should have been built in
the Hebrew language.

Our story would be incomplete if we will not bring the
reaction of critics. What do they say? The pair of words ÷å/÷åä (qava/qav) appears in the Scriptures in two more places: in Jeremiah
31:38 and Zachariah 1:16. In both cases the text does not talk about the length
of a circle and the number π. Hence the difference between the spelling
and the reading of this word does not have the meaning we attributed to it.
What can we reply? The same word or group of words could be used in different
places to code different things. The fact that we don't understand the coding
in one place does not diminish our understanding in another place. Yet, I will
suggest an interpretation of this pair of words in the above two places. In
both cases the text talks about the future expansion of

Now let us calculate the perimeter of the strip of Messiah.
The ** 31.57074°**. The cross-section of the Earth on this
latitude is a circle with the radius of 5439.132 km. The perimeter of this
circle is 34175.075 km or

All this is remarkable, but what has it to do with the
ratio ÷å/÷åä (qava/qav)? The strip that surrounds the future oblation of ** 31999.12** mils. A
remarkable accuracy!

The inevitable conclusion is that the one, who designed
this equality, had a full control of the size of the Earth and of the geography
of