The form of the Menorah
and the weight of the shekel
by Daniel Michelson
Department of Computer Science
and Applied Mathematics
Weizmann Institute of Science
The form of the Menorah is reconstructed mathematically using the available data from the Torah and Talmud with some additional “natural” constraints. The crucial piece of information is the total weight of 3000 shekels. Several arguments are brought to establish a simple relation between the weight of the shekel and the basic unit of volume. As a result, four different shapes of the Menorah are obtained. The fact that they have almost the correct weight despite very few choices of parameters, and the fixtures have simple rational dimensions, should present a puzzle to a skeptical mind.
Of all vessels in the Tabernacle and the Temple, the Menorah appears to have been the most complicated one. Our sages say that Moses found it very difficult to make the Menorah. According to one opinion (Bemidbar Raba Ch. 15:10), Moses could not understand or remember how to make it, until Bezalel finally constructed the Menorah. By another opinion (Tanchuma, Shemini, Ch. 8), even after all explanations Moses still could not make it, until G-d told him to throw the gold into fire and the Menorah was made by itself. Thus there were both theoretical and practical difficulties in manufacturing the Menorah. Everyone is familiar with the bulky Menorah shown on the Titus gate in Rome. We will show that the true Menorah was much leaner. The aim of this article is to reconstruct mathematically the true structure of the Menorah with all its parts and fixtures. A crucial piece of information is the weight of the Menorah. We will see that this single constraint serves to recover all the necessary details of the Menorah. Since the weight of the Menorah is given in shekels, we include in this paper a discussion of the issue of the weight of the shekel and its relation to the units of volume.
1. What is known about the Menorah?
It is written in the Torah “And thou shalt make a Menorah of pure gold: of beaten work shall the Menorah be made: its shaft, and its branches, its bowls, its bulbs, and its flowers, shall be of the same” (Exodus, Ch. 25, 31). The nine verses following, give farther details of the Menorah. Even with this description, it is impossible to reconstruct the Menorah. Thus we do not know the precise shape of the bowls, bulbs and flowers. Moreover, there is a major controversy about the shape of the branches. According to Rambam and apparently Rashi, the branches were straight. According to Even Ezra they were round. There is also a controversy concerning the form of legs. According to Rashi there was a box-like basis with three legs on its bottom. In picture drawn by Rambam it appears that the basis was dome-like standing on three legs. The Talmud (Menachot, p. 28b) presents additional data relating to the height of the Menorah and its parts. “The height of the Menorah is 18 palms. The legs and the flower 3 palms; 2 palms plain; a palm with bowl, bulb and flower; 2 palms plain; a palm with bulb and two branches going out of it - one there and another there, extending and rising against the height of the Menorah; a palm plain; a palm with bulb and two branches etc.; a palm plain; a palm with bulb and two branches etc.; 2 palms plain; there are left three palms with 3 bowls, bulb and flower”. The widths of the Menorah is not explicitly stated. However, since the Menorah stood against the length of the table which was 12 palms, one can infer that the width of the exterior branches was 12 palms. It seems natural to divide this width into six equal parts, so that the distance between the upper ends of neighboring branches is two palms. Yet, there is one piece of information that was never used in reconstructing the shape of the Menorah; to wit it’s weight. “Of a talent of pure gold shall he make it, with all the vessels” (Exodus, Ch. 25, 39). Talent of Torah is 3000 shekels, while the shekel is either approximately 14 gr. (according to Rashi) or approx. 17 gr. (according to Rambam and Geonim). The average density of hammered gold is 19.3 g/cm3. Based upon these figures, one can calculate the volume of the Menorah. It was either approx. 2176 cm3 or approx. 2642 cm3. This is a relatively small volume. The kind of Menorah shown on the Titus gate in Rome or exhibited today in Cardo in Jerusalem, is several times more massive. A heavy box-like basis is also excluded. On the other hand, the branches could not have been thin since the pure gold will easily bend. It seems that the design of the Menorah was optimal, with no “extras”. Our hypothesis was that the single weight restriction would provide the necessary clue for the several unknown parameters relating to the form of the Menorah. Before proceeding, we should establish the exact relation between the weight of the shekel and the unit of length- the palm.
2. The cubit and shekel of the Torah.
It is impossible in this short article to fully discuss the Torah measures and weights. We wrote on this subject a long and detailed paper "The cubit and shekel of Torah". Our premise is that the Torah cubit is 48 cm. This is the average length of arm in our time and the accepted length by Chalacha, attributed to Rabbi Haim Nae. (There is another wide spread opinion, supported by Hazon Ish, according to which the cubit is 6/5 longer). The corresponding length of a palm of 8 cm survived in the circumference of the Roman coin Sela of Neron. In a paper Borders of the Temple Mount we demonstrated that the cubit of the Temple and of the Land of Israel was 51 cm. The relation between these two cubits is such that the ratio of the corresponding units of volume is (51/48)3=1.19946…≈1.2. This is the reason why the unit of volume in Jerusalem increased 6/5 times from seah of the desert to the seah of Jerusalem.
As for the shekel, it is agreed that its weight increased by ratio 6/5 sometime in the beginning of the second Temple. The final weight, as we already mentioned, according to Rashi was about 14 gr. and according to Rambam and Geonim was about 17 gr. The actual shekel coins from the time of the destruction of the second Temple and from the time of Bar-Kohba's revolt weigh about 14 gr. Alternatively, the stone weights from the time of the First Temple (see Raz Kletter,Economic Keystones: The Weight System of the Kingdom of Judeah (JSOT Supplement 276), Sheffield, 1998) indicate that the shekel was approx. 11.5-12 gr. The ratio of these two weights is about 6/5. Thus, it seems that Rashi was right and Rambam and Gaoinim were not! The difficulty with this supposition is that Geonim’s view was based upon tradition of a heavier shekel. Their weight also agrees with four Attic drachmas, which was the weight of shekel according to Josephus. How this contradiction between Rashi and the Geonim can be reconciled?
Our claim is that the shekel of Torah satisfied the equation: 5 shekels=cup of water, where a cup=seah/96 and 40 seah equal 3 cubic cubits. With a cubit of 48 cm, the cup equals 86.4 cm3 and the shekel 17.28 gr. When the Children of Israel entered the Land, the shekel was reduced by the ratio 2/3 and became 11.52 gr. In the second Temple it increased by the ratio by 6/5 and became 13.824 grams. Simultaneously with the standard weight, there existed increased weights with so called “Kalbon” of 1/48 and 1/24. The corresponding weights for the first Temple period were 11.76 and 12 gr. and for the second Temple 14.112 and 14.4 gr. This latter weight relates to the shekel of Torah as the ratio of 5 to 6. The Geonim kept the tradition of the original shekel, while Rashi retained the weight of the shekel of the second Temple. It is possible also, that subsequent to the Bar-Kohba revolt, the sages again increased the weight of a shekel by a ratio 6/5, from 14.4 to 17.28 gr. , thus reverting to the original shekel of Torah. We find indication of it in the fact that the volume of seah was increased 6/5 times from the seah of Jerusalem to the seah of Zippori. The increased shekel was not a real coin, but rather a measure of weight, since Jews at that time were not allowed to mint coins. Yet, they could “translate” the shekel into local currency or a number of grains.
We arrived at the above described relation of the shekel and cup from hints in several sources. One was hinted to by Joseph when he placed the silver cup in the bag of Benjamin. Joseph was sold by 20 silver dinars or 5 shekels (see Yerushalmi, Shekalim, 9b). According to Yerushalmi this is the reason why the firstborn are redeemed by 5 shekels. The volume of a standard cup is thus equal to this amount.
Another hint we found in the identity “loaf of bread”= “loaf of silver”. Talent of silver is called in Hebrew “kikar” which is the same as the loaf. There is a holy kikar of 3000 shekels and a regular kikar of 1500 shekels. On the other hand, the loaf of bread in the desert was an omer of manna. Now, if one fills the omer with silver, one obtains the regular kikar of silver. Indeed, omer is equal 3/10 of seah or 28.8 cups. The specific gravity of pure silver is 10.5 gr/cm3. When filled with silver, omer should weigh 28.8x5x10.5=1512 shekels, very close to 1500 (Remark: In order to obtain 1500 shekels one should assume that the specific gravity of silver is 10.4166. This number lies in the range 10.4-10.6 given, for example, by specific gravity. It is also possible that one should use the specific gravity of silver coins of the time of Second Temple. These coins were minted in Tyre and had on the average 94.56% of silver (see Metrology of Roman Silver Coinage, B.A.R. (British Archaelogical Report) 5, 1976, p. 58 by D.R. Walker). Taking the rest to be copper (density of 8.96) one obtains the average density of 10.416).
Another source for the above relation is in Ezekiel Ch. 4, 10 “And thy food which thou shalt eat shall be by weight twenty shekels a day”. The standard quantity a man ate per day, was 8 eggs of grain of wheat, 4 eggs per meal (e.g. see Mishna Para, Ch. 1, Mishna 1). The specific weight of wheat grain is about 0.77 gr/cm3. Since by definition, one egg equals 2/3 of a cup, eight eggs of grain weigh 4.1 cups of water. The verse says that this weight was 20 shekels, e.g. one cup equals approx. five shekels. One may argue that Ezekiel did not eat pure wheat but mixture of “wheat and barley and beans and lentils and millet”. Also there is another smaller standard of a meal, 3 eggs instead of 4 (see Iruvin 62b). Yet we assume that Ezekiel was not requested to suffer from hunger for 430 days, but rather demonstrate the type of food they will eat under the siege. The smaller standard of meal was apparently compensated by addition of legume and figs (see Ketubot 64a), while Ezekiel did not eat but the above mixture. Hence, it is reasonable to assume that Ezekiel ate food of a standard weight, only of a poor quality.
The most explicit relation between the weight and volume appears in MishnaTrumot 10:5 and Yerushalmi, Trumot 53a. According to this relation, 2 cups=25 sela of Judea. With the cubit of 48 cm, two cups equal 2x86.4 gr. and sela of Judea equals 6.912 gr. The only unit of weight in time of the second Temple that fits this amount is one half of the shekel of 13.824 gr. This was the coin every Jew had to contribute annually to the Temple. This is the reason for its importance. One quarter of this amount was called “zuz of Jehuda” and was equal to the amount of silver in the basic coin of “selah of the land” or “Astira” (see Kidushin 11b). On the other hand, it is impossible that the Judea sela was the standard shekel of the second Temple of about 14 gr, since the resulting cubit would be more than 60 cm.
The unit sela of Judea is mentioned in connection with the commandment of the first of the fleece. The minimal amount of wool one should give to a priest is five selas of Judea, enough to make a small garment (Chulin 135a). On the other hand, the Talmud (Shabat 10b) states
that because of two shekels weight of garment Jacob gave to Josef, more than to the rest of the sons, our forefathers went into exile in Egypt. It seems that the commandment of first of the fleece is related to these additional two shekels. Namely, the minimal amount of five selas of Judea is equal to two shekels of Torah. Hence, a cup is equal to five shekels of Torah.
The basic unit of weight above the shekel is maneh. In the time of the second Temple it was 25 shekels. According to the relation between the shekel of Torah and the shekels of the first and second Temple we proposed, this maneh equals 30 shekels of the first Temple and 20 shekels of Torah (Remark: its weight is thus 345.6 grams, in between the English tower pound of 350 gr. and 12 ounces of 28.35 gr.). The future maneh according Ezekiel 45:12 will consist of 60 shekels and will be partitioned into 20, 25 and 15 shekels. It seems that this partition hints to the old units, 20 - the maneh of Torah, 25- the maneh of the second Temple and 15- half of the maneh of the first Temple (but the shekel will be the one of Torah).
One last observation regarding the weight of the shekel. The total amount of gold used in the Tabernacle was 29 kikar and 730 shekels (Exodus 38:24). The covering of the arc was 2.5 by 1.5 cubits (Exodus 25:17). According to Talmud (Succa 5a) the covering was a palm thick. Thus, its volume was 135 cubic palms or exactly 800 cups. Since it was made of pure gold, its weight was 800x5x19.3=77200 shekels or 25 kikar and 2200 shekels. We should reserve also kikar for Menorah. The amount of gold for the rest of the vessels is not specified, so it may have been very small. For example, the wood may have been covered by few microns of gold. Thus, the total amount of gold could suffice. But if the shekel of Torah was the same as the shekel of the Second temple (namely, about 17.28/1.2 gr), while the cubit was 48 cm, than the covering would weigh 77200x1.2 shekels, which is more than total amount of gold.
3. The approximate calculation of the Menorah
Our calculation will begin with an estimate of the weight of the body of the Menorah : the central branch, the side branches and the legs. The thickness of these parts is not found in the Bible or later sources. The basic unit of length below the palm is a thumb, 1/4 of a palm, i.e. 2 cm. This was actually the smallest unit used in the Tabernacle (the thickness of the walls of the arc according to Rabbi Yehuda in Baba Batra 14a). Thus, we will start with the assumption that all branches and all legs were 2 cm thick. According to Talmud Menachot, p. 28a as we quoted in Ch. 1, the Menorah was 18 palms high. The three legs separated from the body at h=3 palms (Remark: it is possible that one should subtract from h=3 the height of the flower), the first pair of branches separated at h=9, the next at h=11 and the last at h=13 palms. Their span was correspondingly 6, 4 and 2 palms. The span of the legs is not known but we will assume that it was 2 palms (from the central branch) as the span of the upper branches. According to Rambam all branches were straight. We will therefore assume that the legs were also straight. Thus, the total length of branches and legs is
(3.1) 3√(32+22)+2√(92+62)+2√(72+42)+2√(52+22)=59.345 palms
The length of the central branch is 18-3=15 palms. Hence the total length is 74.345 palms. The cross section of all parts is a circle of diameter 2 cm. Hence the volume of the body of the Menorah is 74.345x8p=1868.49 cm3 and its weight is 1868.49x19.3/17.28=2086.9 shekels. The rest of 913.1 shekels should be divided between 42 fixtures - 22 bowls, 11 bulbs and 9 flowers, and 7 lamps. If we make all these 49 details equal, their weight would be approx. 18.6 shekels. It would be nice if the weight would be exactly a maneh of 20 shekels. The lamps had also handles to support the wicks. If we assume that their total weight was also 20 shekels, then the body would weigh 100 maneh and the rest 50 maneh. It would be an esthetic solution. However, the weight of the body we obtained is apparently minimal! It came to my mind that the number p in the Talmud was always counted as 3. If we replace p by 3, the weight of the body would be approx. 1993 shekels, close to the required 2000. If, for example, we assume that the three legs formed a perfect tetrahedron, then the length of a leg would be 3√1.5 palms and total weight would become 1998.38 shekels. By some sources (e.g. see specific gravity) the density of a pure gold is 19.32. Then the weight of the body of the Menorah would be 2000.44 shekels.
How can we make p equal 3? By assuming that the cross-section of the branches and legs was not a circle with radius 1 cm but a perfect dodecagon (12 side) inscribed in this circle! Then the area of the cross-section is exactly 3 cm2, while the exterior diameter of the cross-section is still a thumb. Another possibility is to have a circular cross-section with the radius r0=√(3/π). However, in such case introduce an irrational measure into the design of the Menorah. We will see in the next section that the first choice is the natural one. Of course, our simplified calculation of the volume of the body was not exact. The side brunches do not originate from the center but from the surface of bulb. Hence, they are shorter then we calculated. The legs too have mutual intersection. We will do the precise calculation later. The main lesson we learned is that one should separate the body from the fixtures. The body weights 2000 shekels and each fixture 20 shekels. We will start with the fixtures.
4. The bulb
According to the Talmud Menachot 28b, the bulbs look like apples from Krit (or Kritim?). Their length is a palm since the Talmud says “a palm with bulb and two branches going out of it”. On pictures of Menorah bulbs are usually displayed as kind of ellipsoids. We will define it exactly as ellipsoid of rotation around the axis of the Menorah, cut by two horizontal planes at the distance 4 cm from its center (see fig. 1). Thus, its length is exactly 8 cm. The weight of the ellipsoid should be 20 shekels of 17.28 gr. plus the weight of the main branch of length 8. Notice that the cross-section of ellipsoid is a circle while the cross-section of the branch is a dodecagon of external radius 1 cm. Denote the radius of the circle by r0. We have two natural choices. One is to make r0=1 so that the dodecagon is inside the circle. Another one is to make r0=√(3/π), the circle of the same area as the dodecagon. We shall consider both cases. With r denoting the horizontal distance from the axis and z the vertical coordinate measured from the center of the bulb, the equation of the ellipsoid is
(4.1) r2/a2+z2/b2=1; where r02/a2+42/b2=1
The volume of the ellipsoid between two cuts is
(4.2) V=p∫a2(1-z2/b2)dz, -4≤ y ≤ 4; V=2π(4a2(1 -16/b2/3)
On the other hand, this volume is equal to the volume of a segment of central branch of length 8 plus the volume of 20 shekels of gold. Thus we obtain the equation
(4.3) V=2π(4a2(1 -16/b2/3)=8x3+20x17.28/19.3=41.9067 cm3
(4.4) a=√(1.5(V/8/p-r02/3)); b=4/√(1-(r0/a)2).
In the first case
(4.5) a=1.41461…, b=5.65526… , r0=1,
in the second case
(4.6) a=1.42255…, b=5.50420…, r0=√(3/π) .
There is negligible difference in the thickness of the bulb in two cases. Indeed, the “apple” is quite narrow. Notice that in the first case a≈√2, and b≈4√2 (the last follows from the first). The slope of the bulb at its bottom is thus very close to 4. For density of gold 19.32, the approximation is near exact (up to 6 digits). Could it be a hint that we should choose the first solution?
(Remark: The density of the hammered gold depends on the amount of pressure applied. In most handbooks they use the number 19.3. In some, 19.32. It is possible that the fixtures were a little more dense than the branches, since a large pressure was applied to a thin layer of gold to produce a proper form. In this paper the weight of the body of the Menorah was calculated always using the density 19.3. The fixtures were always assumed to have the constant weight of 20 shekels regardless of the density)
5. The bowl
The Talmud in Menachot 28b states that the bowls look like the ones from Alexandria. Rashi explains that they are long and narrow. On the picture drawn by Rambam they look as cones with a narrow base. Therefore, we will make the bowl as a cone with top cut by horizontal plane. The walls will be of a constant thickness. The bowl is defined by four parameters: the inner radius of the lower base r1, the inner radius of the upper base r2, the thickness d and the height h. We will make the upper base minimal (it will be shown that the bowls are turned upside down!), just to encircle the dodecagon. Thus r2=1. What is the height of the bowl? The Talmud says that there is “a palm with bowl, bulb and flower”, and also “there are left three palms with 3 bowls, bulb and flower”. Since the bulb itself is a palm long, how could these two sentences be correct? Some commentators explain that in first sentence the bowl and flower are attached to two sides of the bulb, each one a palm long, while in the second sentence the bowls are partially one inside another and the bulb is partially inside a bowl. This solution breaks the symmetry of the Menorah. It is also not clear how one can produce the attachment by hammering only.
There is, however, very simple and natural solution to the puzzle. The flower contains the bottom half of the bulb and the bowl contains its upper half (see fig. 1). Hence, the height of the flower and of the bowl is half a palm. In the second sentence, three bowls, bulb and flower occupy 3x0.5+1+0.5=3 palms in successive order! Indeed, in Rambam’s drawing, the bowls are turned upside down. Instead, one could overturn the flowers. However, the upper flowers form the base of the lamps. Hence, their openings should turn up. This explains Rambam’s strange picture.
(Remark: There is a deep meaning in the position of the fixtures mentioned in the first sentence. See the end of the article in Hebrew http://truthofland.co.il/hebrew/chanuka.htm).
We are now left with two parameters r1 and d. The only restriction is the given mass of the bowl. We conjecture that the volume of the bowl is exactly the volume of a standard cup, 86.4 cm3. There are two possibilities: 1) this volume includes the thickness of the walls; 2) it does not. We will consider both of them. Now we have two equations
where r2=1, h=4.
In the first case V0=86.4-V1, in the second one V0=86.4. From the first equation we find r1, and from the second d. The solution is in the first case
(5.2) r1=3.44988…, d=0.30000016,
and in the second case
(5.3) r1=3.9583…, d=0.2724234…
Notice that in the first case r1»3.4500, d is extremely close to 0.3 and the exterior diameter of the bowl 2(r1+d)»7.5 cm. We will see that the first solution besides being "almost" rational, also perfectly fits the size of the lamps. (Remark: the "rationality" is with respect to the basic unit of a thumb and is independent of the metric system). As one can see on the picture, the half of the bulb sits inside the bowl
According to our solution, the inverted bowl does not have a solid top and is attached to the branches only through 12 vertices of the dodecagon. In reality, it might have a very thin top which does not affect the weight of the Menorah.
6. The flower
According to the quoted above Talmud, the flowers are like the flowers of columns. As we concluded in the last chapter, their height was half a palm. Since the lower bulb is covered from the bottom by a flower and from the top by a bowl, we will make the top of the flower identical to the bottom of the bowl. As for the form of the flower, we will define it as a body of rotation around the central axis. The vertical cross-section of it is shown on fig. 1 . The inner boundary of the section is formed by two identical circular arcs AB and BC, which are tangent at the point of contact B. The secant ABC is the inner boundary of the vertical cross-section of the bowl (with the narrow base at the bottom). Hence the radius EA=r2=1 and FC=r1 . The exterior boundary A1B1C1 is obtained by shifting the inner boundary to the right by distance d, where the thickness d is the same as of the bowl. Denote by r=f(z) the curve ABC, r=g(z) the secant ABC, where z is the vertical coordinate and r is the horizontal radius. The volume of the wall of the flower is equal to the volume of the wall of the bowl
(6.1) π∫d(2f(z)+d)dz= π∫d(2g(z)+d)dz , 0≤z≤4,
since the areas bounded by the curve ABC and the secant ABC are the same.
Hence, the mass of the flower is equal 20 shekels. To define the flower uniquely we request the tangent at the contact point B to be vertical. As a result, the radius of the arcs (in the case of r1 as in (5.2)) is
As seen on fig. 1, the bulb lies in the interior of the flower and the bowl.
7. The lamp
The only thing we know about the lamps is their volume: they contained half a log or two cups of oil (Mishna Menachot Ch. 9, 3). The simplest form of a vessel is a cylinder. Interestingly enough, there is a cylinder with ideal measures, which has almost a required volume. Namely, one with diameter and height equal to three thumbs or 6 cm (see fig. 2). Its volume is 169.646 cm3 while two cups are 172.8 cm3. The extra 3.154 cm3 could be absorbed into the overflow of a height 1.1 mm, which is about the usual overflow of olive oil.
We will assume that the walls and the bottom of the lamp have the same thickness d (but not necessarily the same as the bowl). The volume of the walls and the bottom is
(7.1) V=π((r+d)2(h+d)-r2h ), where r=3, h=6.
There are two cases to consider. In the first, the thickness of the bottom is included into the 18 palms height of the Menorah. In such a case, one should subtract from V the volume of the branch inside the bottom. If the branch would enter the bottom vertically, this volume would be exactly 3d. The total height of the Menorah with the lamps would be a “nice” number of 150 cm (or 75 thumbs). In the second case, the bottom is not included. Then the total height of the Menorah with the lamps would be 150+d. In the first case, we solve the equation
d=0.12510 cm, or 1/8 cm up to 1 micron. For the density of gold 19.32 d=0.125-0.00002.
In the second case, d=0.12261…, (and for the density 19.32, d=0.12249…) not close to a rational measure. The total height of Menorah is not a “nice” number too.
Now notice how well the lamp fits into the flower. The radius of the arc of the flower is given in (6.2). The upper (inner) arc of it passes trough the point r=3.45, z=144, and its center lies at the height z=142. Hence the equation of this arc is
At height z=144-0.125, r=3.235, i.e. distance of 1.1 mm only between the wall of the flower and the bottom of the lamp. Should we use solution (5.3), the gap would be much bigger.
8. The handles for the wicks
We are left with a total of 20 shekels. Our suggestion is that they were used for handles to support the wicks, which were pulled out from the lamps (see fig. 2 (. Indeed, the Talmud Menachot 98b says: “The seven lamps shall give light towards the body of the Menorah. This teaches us that the faces of the lamps where leaning towards the central lamp”. This sentence is understood (see Rashi, Bemidbar 8:2) in the following sense. Six wicks were extending out of the lamps and leaning towards the central lamp, while the central wick is leaning westward, to the Holy of Holies. However, these wicks need some support. The same Talmud on p. 88b says “The lamps in the Temple where of segments”. Farther, the Talmud explains: “Like a tray (tas) of gold was on it (on a lamp). When he was cleaning it, he was pushing (the tray) towards its mouth (of the lamp), when he was putting oil in it, he was pushing it towards its head (of the lamp)”. The word “tas” is also mentioned in Talmud Succa 5a “Mitre (of the High Priest) is alike tas of gold, two fingers wide and surrounding from ear to ear”. Clearly, this tray is not a cover of the lamp but an extension of it. These are the segments mentioned in the Talmud. The mouth of the lamp is the place where the handle is attached to the lamp. The head is the other end of the handle. In normal position the handle is lifted up, so that the wick that lies on it, will point diagonally to the central lamp. Therefore, the flame is also leaning diagonally towards the central lamp. The (normally) lifted end of the handle is called the head of the lamp. When he (the priest) was cleaning the lamp, he was pushing the handle down to the level of the mouth and little below. The wick was than pumping the leftovers of the oil from the lamp to a vessel. After filling the lamp with fresh oil, the handle was lifted up towards the position of the head, and accommodated a new wick.
In designing the form of the handles, we will use the standard measures, a palm and a thumb. Recall that the distance between the centers of the lamps is two palms. Since the lamps are 6.25 cm wide, the distance between the lamps is 9.75, a little more than a palm. We will assume that the handles were a palm long. In order to accommodate the wick we will assume that they were a half cane. The diameter of the cane will be a thumb like the branches of the Menorah. If the thickness of the cane is d, then
(8.1) 8πd=20x17.28/19.3/7=2.55810…, d=1.018 mm
We will round it to a rational measure of thumb/20=1mm. (Remark: This is apparently the standard minimal measure called “the thickness of a golden denarius”). Notice that the left hand side of (8.1) gives the volume of a cane with radius 1 measured from the center of its thickness. If, for example, we would measure the radius r=1 from 1/3 of the thickness, then the volume of the handle with d=0.1 will be
almost the requested one. (We get even better approximation for density 19.32). The exact form of attachment of the handle to the lamp requires additional deliberations. On fig. 2 we presented a simplified picture of the attachment in a case of a flat handle.
9. The side branches and legs
Now with all fixtures set, we will return to the issue of the form of the branches. Our simplified solution of straight branches was not exact because of the leftovers, as was mentioned at the end of Ch. 3. Since there are two conflicting opinions, of straight and of round branches, we suggest a compromise: let the branches be parabolas! With x being horizontal coordinate measured from the axis of the Menorah and z –the vertical coordinate measured from the bottom of the Menorah, the equation of the central line of a side branch is
(9.1) z=zi+aix2, where z1=72, z2=88, z3=104.
The coefficients ai are found from conditions
(9.2) zi+aixi2=144, where x1=48, x2=32, x3=16.
The length of a parabola
(9.3) li=∫ √(1+4ai2x2) dx, 0≤x≤xi
is given by the formula
(9.4) li=[t√(1+t2) +ln(t+√(1+t2)]/4/ai , where t=2aixi .
At each point of a parabola, we build the dodecagon with center at this point and perpendicular to the tangent to the parabola (the normal to the parabola will be one of the axis of the dodecagon). The branch is defined as the set of such dodecagons. It is easy to see that the volume of the branch is equal the length of the parabola times the area of cross-section 3 cm2. From this volume we should subtract the part of the branch that is common with the bulb. In the first approximation we consider the side branch as a horizontal cylinder of radius r=√(3/p). The corresponding volume is given by the integral
(9.5) I=∫∫ a(1-y2/a2-z2/b2)1/2dzdy, y2+z2≤r2,
where a and b are defined in (4.5). This integral is transformed into polar coordinates, integrated analytically with the respect to the radius and numerically with respect to the polar angle. Its value is 3.9807 cm3. In the second approximation, we add to it the difference between the actual length of the parabola (9.1) inside the bulb and the semi-axis a, multiplied by the area of the cross-section 3. To the leading order it is
(9.6) ΔIi=2a3ai2 .
The legs require special attention. Recall that according to the Talmud “legs and the flower three palms”. Since the flower is above the legs and is half-palm high, the legs are 2.5 palms or 20 cm high! Since they are 2 cm thick, the central line of the leg starts at the height z4=19. Its equation is
(9.7) z=z4+a4x2, where z=0 at x=16.
The length of this parabola l4 is given by formula (9.4). As above, one should subtract from it the common volume of the leg and the central branch. Both are considered cylinders of radius r, the leg being horizontal. The corresponding volume is given by the integral
(9.8) I4=∫∫ (r2-y2)1/2dzdy, y2+z2≤r2
Its value is found analytically, I4=8r3/3. As in (9.6), one should add to it
(9.9) ΔI4=2r3a42 .
There is, however, mutual intersection of the legs outside the central branch. Again, for simplicity, we assume that the legs are horizontal cylinders of radius r with 120° angle between their axes. If we cut them by a horizontal plane z=z4+h, we obtain three semi-infinite strips of width 2√(r2-h2). Their intersection could be split into three cusps shown of fig. 3. The area of the cusp ABC is approximated by the area of triangle EFC. The height of it is √(4/3(r2-h2))-r and the top angle C=120o. Thus, the said volume is given by the integral
(9.10) ΔV=3tan60°∫ (√(4/3(r2-h2))-r)2dh , -r/√3≤h≤ r/√3
The central branch starts at the height of 18 palms=144 cm and goes down till the point where the legs depart completely from the central branch. If the legs would be horizontal cylinders of radius r, this height would be 19-r. Since the legs are parabolic, the height becomes 19-r-a4r2. The length of the central branch is
The total volume of the body of the Menorah is
(9.12) Vtotal=πr2(l0+2l1+2l2+2l3+3l4)-6I-3I4-2(ΔI1+ ΔI2+ ΔI3)-3ΔI4- ΔV.
We calculated it by computer (the program was later modified to apply to other forms of the Menorah as described in the following sections). Vtotal is 1793.38…cm3 and the corresponding weight in shekels is Wtotal=19.3/17.28Vtotal=2003.023 shekels. We have extra 3 shekels! It turns out that if we make the span of the bottom branches 48-0.75 cm instead of 48 cm, than the weight will be Wtotal=1999.987 shekels! But what is a justification for such a span? Recall that there are flowers at the top of the branches, of exterior diameter 7.5 cm. The total width of the Menorah will be
(9.13) 2(48-0.75) + 7.5=102 cm.
This number is exactly two cubits of the Temple!
Yet, there is one very serious drawback in our design. The side-branches are not vertical at the top! If we attach there the lamps in direction of the branches, the oil will spill out! We can instead cut the branches with plane z=144 cm. Their volume will not change since we cut from inner part of the branch and add at the outer part. We can also turn the upper flowers vertically and attach the lamps to the branches as exhibited on fig. 4. However, the intersection of the top of the branches with the bottom of the lamps will be not 3d as in Ch. 7 but a little larger, and will be different for different branches. This will diminish the total weight of the Menorah by 0.095 shekel. Since the upper flowers are sitting at the top of the bulbs, their axis will not align with the top of the branches but will shift inwards. For the most left and right flowers the shift is 4/3 cm. Thus, the width of the Menorah will be not 102 but 99.333… cm. The bottom of the legs, too, should be cut by the plane z=0, so that they stand firmly on the ground. This, however, does not affect the weight of the legs. (Remark: We have drawn on fig. 4 and elsewhere the three legs schematically. One leg in front is seen as continuation of the main branch. The side legs are shown as if they lie in the vertical plane of the section).
Another possibility is to bend the branches by hand, to make them vertical at the top (at least the upper 4 cm inside the flower) and yet preserving the old position of the very top. This introduces a “human” factor in the otherwise mathematical design.
10. Circular branches
The Menorah described above, fits the opinion in the Talmud Menachot 86b, that the lamps were included in the 3000 shekels. This is also the ruling of Rambam in “Halachot Beit Habechira, Ch 3:6. However, there is another opinion in Talmud, that the lamps were separated from the Menorah and were not included in 3000 shekels. With fixtures defined as in previous chapters, there are left 160 shekels more gold for the body. Hence, the branches could be made longer. It might be even possible that the branches were circular.
Namely, we will make the bottom part of each branch a quarter of a circle and the upper part vertical. Notice that the origin of the branches 9, 7 and 5 palms below the top of the Menorah versus their span of 6, 4, and 2 palms correspondingly, implies that the vertical part of the branch is the same for all branches. Exactly three palms as the space occupied by three bowls, a bulb and a flower! We will make the legs too of the same form- upper part a quarter of a circle and the bottom part vertical. Since the central line of the leg originates at z=19 and since its span is 16 cm, its vertical part will be 3cm. Now the Menorah stands firmly on its legs and the lamps are vertical and fit the branches. The Menorah is exhibited in fig. 5. As in fig. 4 the legs are shown schematically. It was easy to modify our computer program. Since the cross-section of the branches is the same as before, the fixtures of the Menorah are unchanged. The length of the branches and of the legs is calculated trivially. The formulas (9.5), (9.8) and (9.10) are still valid. In (9.6), (9.9) and (9.11) one should replace the coefficients ai of the parabolas by the corresponding coefficient for the circle. It is 1/(2R), where R is the radius of a circle. We obtained the total volume of the body of the Menorah 1936.561 cm3 and its weight 2162.9416 shekels. Thus, we have extra 2.9416 shekels. However, we forgot to make one adjustment. The top of the branches have a common volume with the bottom of the lamps. This volume is 3d for a branch, 21d=2.625 cm3 total. Since the lamps are not part of the amount of 3000 shekels, the above volume should be not included in the volume of the body of the Menorah. Now the weight diminishes by 2.625x19.3/17.28=2.932… shekels. We are left with a negligible excess of 0.01 shekel. This is indeed a miracle! We had no parameters to play with. Thus, there is controversy between the parabolic solution which includes the lamps in the 3000 shekels and which requires a human adjustment, and an ideal circular solution, which does not include the lamps.
11. The Menorah of Rambam
In Section 3 we made approximate calculation of the weight of the Menorah of Rambam, namely the Menorah with straight branches. This calculation does not take into account the mutual intersections of branches and legs. Since these intersections will decrease the weight of the Menorah, we can balance it by increase of the thickness of the branches. Namely, we will assume that the branches have circular cross-section of radius r0=1. The intersection of branches with the bulb depends on the size of the bulb. The corresponding semi-axes a and by are given by the formula (4.4) where the volume V=8π+20x17.28/19.3=43.0395. Their values are
(11.1) a=1.4383, b=5.5652 cm.
The equation of a side branch is
(11.2) (x-(z-zi)/tanα)2/a12+y2/r02=1, where a1=r0/sinα, α=atan((144-zi)/xi),
and xi, zi are defined as in (9.1),(9.2). The equation of a bulb is
with a and b satisfying relations (4.3), (4.4) with r0=1 (i.e. V=43.0395).
For given z, the intersection of these two bodies is intersection of two ellipses. One can easily write the formula for the area of the branch outside the bulb. We integrated this area with respect to z numerically. The volumes of three couples of branches till the very top z=144 are correspondingly
(11.4) Vi= 528.6045, 388.4730 and 249.9793 cm3, i=1,2,3.
It turns out that the two upper branches (i=3) intersect also the central branch above the bulb (see fig. 6). The corresponding volume is
(11.5) ΔV3=0.401 cm3
for each branch. In addition, we should take into account the intersection of the branches with the bottom of the lamps. Its volume is
(11.6) πr02/sinα ∙0.125, α is in (11.2).
However, the part 3x0.125 was already subtracted from the volume of the lamp in (7.2). Thus, we should subtract from the volume of the branches the difference
(11.7) (πr02/sinα -3) ∙d, d as in (7.2)
If we assume that the weight of 20 shekels of the lamp does not include the intersection of a vertical branch of cross-section pr02 with the bottom of the lamp, then one should replace the number 3/π in (7.2) by 1, and (11.7) should be replaced by
(11.8) ΔVli= (πr02/sinα -π) ∙d» 0.08, 0.06, 0.03 , d as in (7.2).
The most difficult is the calculation of the volume of the legs. As in sections 9 and 10, we assume that the center of the bottom of the leg is two palms away from the axis of the Menorah. If we place this center on the x axis, the equation of the leg becomes
(11.9) (x-16+z/tan α)2/a12+y2/r02=1, a1=r0/sinα
The other two legs are obtained from the above, by rotation around the axis of the Menorah with angle 120° and 240°. The leg touches the central branch at
(11.10) z1=(16-r0-r0/sinα)∙tan α
and the axis of the Menorah at
(11.11) z2=(16-r0/sinα)∙tan α .
The exterior side of the leg touches the central branch at
(11.12) z3=(16-r0+r0/sinα)∙tan α .
This is the top of the leg (see fig. 7). Since the flower sits at the level z=20, we obtain the equation for the angle α
(11.13) (16-r0+r0/sinα)∙tan α = 20, α»50.84°
We will first assume that the central branch extends down to the level z=z1. One should calculate the volume of the legs from z1 till z3 outside the central branch. A care should be taken for the mutual intersection of the legs. We did all these computations analytically and compared them with numerical integration. The resulting integral is
(11.14) I13=21.5472 cm3
(out of it, the mutual intersection of the legs is 3x0.0438 cm3). The volume of three legs from bottom till z1 is
(11.15) I1=204.6116 cm3.
The volume of the central branch from z1 till z=144 is
(11.16) Icent=399.5068 cm3
(under the assumption that (11.8) is valid, the volume Icent needs no correction). The final volume of the body of the Menorah is
(11.17)Vbd= I1+I13+Icent+∑(Vi-2ΔVli)-ΔV3=1791.983 cm3
and the corresponding weight 2001.463 shekels. Notice that a part of the central branch from z1 till z2 is unnecessary. The three legs meet together at the axis of the Menorah at z=z2 (see fig. 7), and surround a cone of the height z2-z1 with the basis πr02. The volume of this cone is 1.2857 cm3 and its weight is 1.436 shekel. If we remove this cone, the weight of the body of Menorah will be 2000.027 shekels. In case we use the formula (11.7), the weight will be 2000-0.112 shekels. Practically speaking, this is identically 2000 shekels. Remark that, as in the case of circular branches, we had no free parameters to play with. The only choice was to cut out the cone surrounded by the legs. This way the legs are extending from a dome-like basis, as on the picture drawn by Rambam. The only problem with this Menorah is the adjustment of the lamps to the upper flowers or the flowers to the upper bulbs. We suggest here two solutions. One is shown in fig. 6 and another one in fig. 8. In the first solution we placed the center of the bottom of all 7 upper bulbs at the same level z=132. Notice, that due to different slopes of the brunches, the flowers are at different levels. The axes of the lamps pass through the centers of the top of the brunches. The two flowers at the top of the two upper branches almost touch the lamps but do not intersect with them. The total weight of this Menorah is 2999.9 shekels. The only disadvantage of this solution is that the lamps do not sit on the top of the flowers. In the second solution, the lamps are aligned with branches. The center of the top of the flowers is set at the same level z=144. The lamps are in the form of inclined cylinders cut by a horizontal plane that passes through the center of the top of the original lamps ( z=144+6cosα, α- the slope of the branch). Notice that the lamps contain the same volume of oil and their walls are of the same volume too. The total weight of this Menorah is 3000.27 shekels. The problem with this solution is that the lamps are different. There is however no indication in the Talmud that the lamps should be all the same. In the future, as a source of water will open up in the Holy of Hollies that will flow against the gravity force, it is possible that the lamps will be inclined cylinders (see fig. 9) and yet the oil will not spill out.
12. Menorah of “Chabad”
The drawback of the parabolic Menorah and the one of Rambam is that the side branches are not vertical at the end. Hence, the words of Talmud “there are left three palms with 3 bowls, bulb and flower” (see Section 1) apply literally only to the central branch. At the side branches, if we place the fixtures on the same height, there are left more than 3 palms. On the other hand, the circular Menorah does not fit the ruling of Rambam that the lamps are included in 3000 shekels. Yet, everyone is familiar with the Menorah of Chabad, lighted on Chanukah in public places. It has straight legs and branches but the upper parts of the branches are vertical. The dimensions of the Menorah suggest a unique design of that kind. Namely, the side branches have a slope of 45° up to the height of 15 palms and then rise vertically for three palms (see fig. 10). The width of the Menorah is 12 palms as requested. Since the length of the branches is bigger than in the case of the Menorah of Rambam, we should make it thinner. However, we assumed that the branches are 2 cm thick! The only possibility is to make their cross-section dodecagon inscribed in circle of radius 1.
We will leave the same form of legs as with Rambam’s Menorah. For simplicity, we will do all calculation as if the cross-section is circular of area 3 cm2, i.e. of radius r0=√(3/π). We can use the same formulas (11.9)-(11.13). As above, we remove the cone enclosed by the legs. The calculation of the volume of the branches is a little easier, since we do not have the intersections in (11.5) and (11.8). Remark, that the bulb has the same dimensions as in (4.5). It turns out that the total weight of the body of the Menorah is 2020.832 shekels. We have extra 20 shekels! Perhaps for this Menorah, the handles are not part of the lamps but could be detached from them. Indeed, the words of the Talmud in Manachot 86b that the lamps were of segments, could mean that the handles were removable. There is also a practical reason for it, since constant lifting and lowering of the handles, could break their connection to the lamps. If the handles are removable, they are not included in the total 3000 shekels. In such case, we have additional 20 shekels for the body. The extra 0.832 shekels could be reduced by smoothing out the joint between the diagonal and vertical parts of the branches.
13. Strength of the Menorah.
The last test is whether the Menorah can sustain its own weight. The most problematic case is the one of circular branches. Out of all branches, the lowest one produces the largest angular momentum at the point where it joints the bulb (see fig. 5). Recall that the branch consists of a quarter of a circle with radius R=48 cm and vertical part of length l=24 cm. The cross-section of the branch has area of S=3 cm2. The angular momentum of the weight of the branch is
(13.1) M=(R2+Rl)Sdg ,
where d=19.3 g/cm3 is density of gold and g=980 cm/sec2 the Earth gravity. In addition, we have the momentum of the fixtures: 3 bowls, bulb, flower and lamp with handle and oil. Their total weight is about 130 shekels and their angular momentum is about 130x17.28gR. This momentum is balanced by the momentum of stress at the joint with the bulb. We may assume that the cross-section of the joint with a vertical plain is a circle x2+z2≤r2=3/π. To the first approximation, the stress σ is a linear function of the vertical coordinate z, σ=σ0z/r. The momentum of it is
(13.2) σ0/r ∫z2dxdz =σ0πr3/4.
(13.3) σ0»44x107din/cm2=44 MPa.
The yield point of annealed 24 carat gold is 70 MPa. If the stress does not exceed this value, the material will return to its origin shape as the stress is removed.
Another critical point is the joint of legs and the central branch. The force acting on the bottom of the leg is 1000 shekels. One should add to it the weight of lamps, handles and oil since they are not included in 3000 shekels. The total force is about 1076x17.28g and its arm is 16 cm. Its angular momentum is about 97% of the angular momentum for the lower branch. Thus the legs will not deform too. Notice that for cold worked gold, the yield point is higher (144 MPa for 20% cold work, 205 MPa for 60% cold work). Indeed, the Menorah was manufactured by hammering, and hence was much stronger. Yet, the Talmud in Menachot 29a describes an event that the Menorah of the Temple had excess of a dinar of gold and was put 80 times into furnace until the weight was right. After a heating in furnace, the gold becomes annealed. Hence, the strength of the Menorah should be calculated for the annealed gold.
14. Manufacturing the Menorah
According to Beraita Melechet Hamishkan, the Menorah as whole was hammered out from a single piece of gold. No casting was used. The instruction written in the Torah "Miksha Ahat" means "one piece", but the word "Miksha" also means "beaten work". This poses a formidable difficulty in manufacturing the Menorah. However, the form of the original piece of gold is not specified. So one may suggest casting the Menorah close to the desired form, and then finishing it with few strokes of a hammer. Yet, it does not seem to be the intention of Torah. As for the form of the original piece, I suggest a ball- a simplest body of a given volume. Perhaps, this is hinted in Zekharya 4:2 “a candlestick all of gold, with a ball (in Hebrew “Hagula”) upon the top it”. Since we know the weight of all parts of Menorah, we could solve the problem, provided we can make every part of the Menorah while separating it from the rest. The separation should be done by bands. One should only find a way to prevent a flow of gold through the band under pressure. It is also allowed to heat the Menorah or any part of it to reduce the rigidity. Many special tools need to be designed and many small scale experiments should be carried out until the necessary experience will be gained.
We have reconstructed in this article Menorahs of four different kinds: parabolic, straight, round and piecewise straight. Which is the correct one? It is stated in the book of Kings I, Ch.7:49 that King Solomon made ten menorahs in addition to the Menorah of Moses. It is possible that they were of 11 different shapes and represented 11 different solutions to the problem. If we count three variants of the straight Menorah in fig. 6, fig. 8 and fig. 9 as three different shapes, we have already 6 Menorahs. In the parabolic case too, the lamps could be placed as in fig. 8 and 9. In addition there is a parabolic Menorah with branches that have a circular cross-section of radius 1 (see fig. 11). It is based an a little different interpretation of the Talmud Manachot 28b (we will not discuss it in this article). It has also the correct weight of 3000 shekels. With three possibilities for the lamps, we have three more solutions. Altogether, 11 shapes. These could be the eleven Menorahs of King Solomon. Which of them was the Menorah of Moses? My guess is that this was the straight Menorah of Rambam. It has the simplest form, fits the ruling of Rambam and has the exact weight without any adjustment. The wicks converging to the central branch form an upper triangle of fire that was compliment to the lower diverging triangle of golden branches. If the conjecture stated at the end of Ch. 11 will come true, perhaps the Menorah in fig. 9 will be chosen.