One ought to exercise great caution when taking a scientific approach to the subjects of the Torah and we must bear in mind what the last Lubavitcher Rebbe said: the Torah is the absolute truth, whereas truth in science is relative. The Torah is not aTruth in science is relative physics textbook. Except in the rare cases when the Torah quotes figures explicitly, one should not search for any definitive scientific formulae. One should instead use the Torah to find the fundamental ideas of our world order as the basis for scientific formulae.

Let us give an example. The famous equation by Albert Einstein (E = mc2) united matter and energy. One cannot find this equation anywhere in the Torah. Yet a close reading of the sacred texts of the Bible will lead us to the conclusion that both matter and energy are derived from one and the same Divine Light and thus, as a matter of fact, are essentially one and the same.

During the 19th and 20th centuries some brilliant theoretical predictions were made (“on the quill’s edge”) regarding natural phenomena that were later confirmed via scientific experimentation. Thus, in 1846 Urbain Le Verrier predicted the existence and position of Neptune (a planet that had not yet been discovered at that time) using only mathematics. In 1915 Albert Einstein predicted the existence of gravitational waves in the universe and just recently, in 2016, this phenomenon was scientifically proven. In 1929 the English physicist Paul Dirac predicted the existence of an antiparticle to the electron (the positron), which was then experimentally observed in 1932. In the 1960s Peter Higgs and others predicted the existence of the Higgs boson, the reason that all elementary particles have mass. Fifty years later the Higgs boson was discovered through experiments at the Large Hadron Collider.

To my mind, all these facts provide direct evidence of the fact that the law given by G‑d to our universe upon its creation was written in terms that can be understood mathematically. (Galileo Galilei expressed a similar idea when he said that “the laws of nature are mathematical.”) Mathematical laws are universal and not affected by time. Mathematics is based on numbers, and zero and one play a very special role.

The tenets of Judaism and the Torah include that G‑d is one and that the universe was created by Him out of absolute nothing, i.e. ex nihilo. In mathematical terms, these concepts correspond to one and zero. It is extremely difficult to overestimate the concept of Absolute Nothing. In mathematics zero is considered the number most difficult to understand. Our mind is incapable of embracing the essence of Absolute Nothing, and our universe has nothing in it that could resemble Absolute Nothing. Even if we consider a vacuum we should still bear in mind that it exists in space and time and has a residual level of energy (referred to as vacuum energy), so it cannot be considered the same as Absolute Nothing.

The notion that the universe was created ex nihilo may be derived from the Torah itself and is stated explicitly in other sacred texts. This concept is first mentioned in the book of Sefer Yetzirah traditionally ascribed to Abraham, our forefather. It states that G‑d “created existent from non-existent.” Later, the concept of ex nihilo was more fully developed by Saadia Gaon (in his Book of Beliefs and Opinions) and Maimonides (in his Guide for the Perplexed), and later still by the leaders of Hasidism.

One of the proofs proposed by the Judaic philosophers who believed the universe was created out of absolute nothing is as follows: If we assume that our universe was created out ofIt is difficult to overestimate the concept of Absolute Nothing something, then the question arises, what was this “something” made of? And this line of reasoning continues ad infinitum. If we assume that the universe was created out of some pre-matter that had always existed, then we would have to conceded that this pre-matter was never created and is thus beyond G‑d’s control, which is inconceivable.

Mathematics does not exist without zero being considered as its own number (and consequently without negative and complex numbers). The numeral systems used by the Egyptians and Babylonians implied that zero was a sign used to separate positions, but was in itself not a separate number. The ancient Greeks, specifically the Pythagoreans as the first to say that our universe is composed of numbers, started their numerical series from the monad, i.e. one. They did not consider zero a separate number either. Then the ancient Greeks asked themselves: “How can it be that something can appear out of nothing?” We will return to answer this question later. It was only in the 6th century A.D. that Brahmagupta, an Indian mathematician, for the first time in human history spoke of zero as a separate number and of negative numbers. Thus, from all the above it emerges that it is in the Torah that we may find the fundamental idea of world order, i.e. the Absolute Nothing (zero).

But let us get back to the question that the ancient Greeks asked themselves: “How is it that something can appear out of nothing?” The laws of mathematics state that zero may exist on its own or be the sum of a negative and positive number, e.g. –5 + 5 = 0. Today, many influential physicists share the idea that the total energy of our universe is equal to zero. In other words, positive mass energy is precisely balanced with negative gravitational potential energy. Edward Tryon was the first person to propose this theory in 1973. Today it is adhered to by a number of eminent academicians, including Alan Guth (the man behind the inflationary universe theory) and Stephen Hawking. As Stephen Hawking put it: “The amount of positive energy in the form of matter exactly balances the negative energy in the form of gravity.” And Alan Guth wrote that "It is possible to prove that the energy of the gravitational field is clearly negative. This is true both in the context of the Newtonian theory of gravity and also in the context of general relativity developed by Albert Einstein". Obviously, this is just a hypothesis. However, if we proceed and accept it we may answer the question the Ancient Greeks put forth: Absolute Nothing (zero) generates zero, but in the current case it is because the result of one minus one equals zero.

When reading the Holy Scriptures we are guided by the fact that the Torah, which is the Word of G‑d Himself, does not contain even a single letter that is unnecessary. Every letter and evenHow can something appear out of nothing? every punctuation mark is meaningful. The L‑rd clearly instructed Noah about the exact spatial dimensions of the Ark: "This is how you are to build it: The ark is to be three hundred cubits long, fifty cubits wide and thirty cubits high. You shall make a (tzohar) window for the ark, and narrow it to a cubit at the top". The picture below shows how this appears in geometerical terms.

 Figure 1. Geometry of the Ark
Figure 1. Geometry of the Ark

Figure 1 shows that the Ark was shaped like a truncated pyramid. The angle of elevation at its side equals 50.76°. The angle of elevation of the Great Pyramid of Giza is 51.52°. The angle of elevation of the Pyramid of Khafre is 52.2°. The angle of elevation of the Pyramid of Menkaure is 50.47°. Thus, it is obvious that the angle of elevation of the Ark’s side matches the angles of elevation of the three pyramids of Giza within the bounds of observational error.

Figure 2. Geometry of the Great Pyramid of Giza
Figure 2. Geometry of the Great Pyramid of Giza

Based on what we shared above, we may make the bold supposition and say that the Egyptian Pyramids were constructed in the likeness of Noah's Ark and they were ‘modelled’ on it.

If we divide the sum of width and height of the Ark by its width, i.e. 50 + 30.612/50, the result is 1.612. If we divide the width of the Ark by its height (50/30) the result is approximately 1.667.

As for the Great Pyramid of Giza, this ratio is approximately 1.631. The Golden Ratio (φ) is approximately 1.618. Just like π, φ is a universal irrational number in mathematics. In this case, the insignificant deviation of the Ark’s dimensions from φ can be explained by the fact that the L‑rd instructed Noah only in integers.

When the Jews constructed the tabernacle in the Wilderness, G‑d commanded Moses: “Let them make an Ark [of covenant], of cedar wood, two and a half cubits long, one and a half cubit wide, and one and a half cubit high,” (figure 3).

Figure 3
Figure 3

If we take ratio (2.5 + 1.5)/2.5 = 1.6

If we take ratio 2.5/1.5 = 1.(6)

Here the difference between ratios of Ark of Covenant and Golden Section (φ) is due to the fact that G‑d gave the size of the Ark in integers.

The Golden Ratio is a special number found by dividing a line into two parts, so that the longer part divided by the smaller part is in the proportion approximately of 68/32. The first mathematician to study the Golden Ratio was Euclid, who did so around 300 B.C. Euclid demonstrated that the Golden Ratio can be found in various geometric figures. The Golden Ratio was also studied in the Middle Ages, and even today mathematicians continue researching thisThe Golden Ratio is often utilized in painting, music and architecture phenomenon. Among those who have studied the Golden Ratio we can name Leonardo of Pisa, astronomer Johannes Kepler, and Roger Penrose. Fibonacci, the great Italian mathematician of the 12th century who brought algebra to Europe, showed that the ratio between any two adjacent numbers in a series named after him (where every following number is the sum of the two preceding ones) tends towards φ. The Golden Ratio is often utilized in painting, music and architecture, and we can often observe it in nature (in the structure of leaves or parts of the human body), as well as on the atomic level. Some researchers compare the Golden Ratio to the structure of the human DNA genome.

The fact that we find in Torah a universal mathematical number constitutes irrefutable evidence that the L-rd has written the Law of our universe in a way that it can be also read in mathematical language.

It is also not by accident that the Ark was shaped like a truncated pyramid. Some hypotheses (although not yet confirmed academically) say that the space inside a pyramid-shaped structure acquires special properties in terms of energy. Famous Torah commentators have also written about the shape of the Ark.

Commenting on Bereshit 6:16

Ibn Ezra stated that the Ark was of triangular shape with a sharp vertex and acute angles to prevent it from overturning.

Commenting on Bereshit 8:4

Ramban said: And what is more: on the seventeenth day of Elul he sent a dove, and there was water all around the earth and the trees were beneath the water. But twelve days passed and everything dried up. We may conclude that if the Ark had eleven cubits beneath the water level (which is more than one third of its height) it should have sunk, as it would be too wide at its bottom part and have just one cubit at its top part. This is not the way one should build a ship, as such a design makes it too heavy.

Commenting on Bereshit 6:16

Abravanel wrote: the Holy Scripture says that He instructed him to build the Ark in a triangular shape, leaving at the top a length of just one cubit and six cubits wide formed by four beveled facets so that the falling raindrops stream down the walls of the Ark.

Thus, we see once more that the Torah contains the fundamental mathematical principles of our universe's structure.

With my deepest gratitude to Berel Lazar, Chief Rabbi of Russia, and Alexander Boroda, Chairman of the Board, the Federation of Jewish Communities of Russia (FJCR).