"Mathematics is the language in which God has written the universe"
Galileo Galilei (1564 - 1642)
The Bible clearly states that the earth is designed. When Job questioned God he got the reply:
"Where were you when I laid the foundations of the earth ... who determined its measurements?" (Job 38.4,5)
In the New Testament we read that Jesus was there at the point of creation:
"All things came into being through Him" (John 1.3)
and that the hand of God is evident to all, from flowers, to human love to the starry heavens:
"His invisible attributes are clearly seen, being understood by the things that are made" (Rom 1.20)
Engineering relies upon numbers and mathematics, and so it is not surprising to find amazing numbers and mathematics embedded in God's creation (as well as in the Bible). We will examine a few of these, starting with some simple examples. In each case we see the hand of understanding and design, and often this appears to point to Christ as the Designer.
The number 8 is important in male circumcision. God instructed Abraham to circumcise a male child when he is 8 days old (Gen 17.12). Why day eight? Medical science has shown that on the eighth day the amount of blood clotting factor prothrombin present in the body is more than 100% above normal, and this is normally the only day in the male's life when this is the case, link. Today, Prothrombin Complex Concentrate (PCC) is used to accelerate blood coagulation, link. So day eight is the optimal day for minimising haemorrhaging during circumcision!
The Bible is rather brief on the design of Noah's Ark:
"The length of the Ark shall be 300 cubits, its width 50 cubits, and its height 30 cubits ... You shall make it with lower, second and third decks" (Gen 6.15,16)
Nevertheless, it turns out that this is sufficient to perform serious analysis of the Ark's stability using modern ship design techniques. Since there are many definitions of the ancient cubit, ranging from 17.5 to 26 inches, which cubit applies here? Some believe it was the Egyptian Royal cubit of about 20.6 inches, link, link. This makes the Ark about 515 feet long, or 6180 inches. Several measurements of what appears to be an ancient boat found on the mountains of Turkey appear to confirm this length, link. It is intriguing that 6180 appears in the reciprocal of the 'golden ratio' 1/Φ = 0.618034... described below.
There have been a number of naval studies based on the Ark's biblical dimensions. One 1993 study using a cubit of 20.4 inches (510 feet long) concluded that the biblical dimensions were optimal for stability, strength and comfort, and that the Ark could handle 30m waves, link. A 1977 study assumed the common cubit of 18 inches, making the Ark some 450 long, 75 feet wide and 45 feet high. This study concluded that 'the Ark was extremely stable' when four major stability features were examined, even in 210 knot winds, link. So, in major design aspects, Noah's Ark was as good as modern shipping!
Consider the footprint of Solomon's temple in Jerusalem:
"The length was 60 cubits (by cubits according to the former measure) and the width 20 cubits" (2 Chron 3.3)
We have already seen that numbers and sizes in the Bible are not random but are given for a purpose. They may be optimal in some way (as in the timing of circumcision or in the size of Noah's Ark), or they may have a deeper, spiritual significance. Here we have the footprint dimensions of a house built for the Lord (1 Kings 6.2), a holy place. When the temple was completed the Lord appeared to Solomon saying:
"I have consecrated this house which you have built to put My name there ... " (1 Kings 9.3)
So do the dimensions of the temple point to the Lord's name? Consider the perimeter of the temple, 160 cubits, where the cubit referred to is sized 'according to the former measure'. What was this measure? Some claim that the temple cubit was 18 inches, link. But others claim that in large-scale construction projects, ancient civilizations typically used the long cubit, about 19.8-20.6 inches, link. Let us assume that Solomon used the Babylonian royal cubit of 19.8 inches, link. Recent measurements on the Temple Mount in Jerusalem support the Babylonian royal cubit, link. In this case the perimeter is 19.8 × 160 inches = 3,168 inches. Now consider the phrase "Lord Jesus Christ". Applying Bible gematria, the Greek letters in 'Lord Jesus Christ' sum to exactly 3,168! So did the Lord put His name there in the very dimensions of the temple?
Fig.1: Lunar eclipse of the Sun
Fig.1: Lunar eclipse of the Sun
Scientists say 'it is a happy accident of nature', 'it is a striking coincidence', 'sheer chance' or 'a fluke of celestial mechanics'. They are referring to the amazing geometry associated with the Sun, Moon and Earth. During a total eclipse the Moon casts its umbra (shadow) upon Earth's surface so that the Sun is totally obscured, as depicted in Fig.1. The Earth, Moon, Sun geometry is such that only the beautiful corona of the Sun is left (conveniently permitting astronomers to analyse the Sun's atmosphere, link). Since the Sun's diameter is about 400 times that of the Moon, the eclipse can only happen if the distances of Sun and Moon from Earth (the geometry) are just right. It turns out that this is the case and both Sun and Moon subtend virtually that same angle Θ such that they look virtually the same size in the sky. Amazing!
Let's do the maths. Taking some average measurements, link, the Moon's diameter DM is about 2,159 miles, the Sun's diameter DS is about 864,938 miles, the Moon's average distance EM is about 238,855 miles (elliptical orbit) and the Sun's distance ES is one astronomical unit (AU) or about 92,956,000 miles. So considering the angle subtended by the Moon, Θ = 2 × arctan (0.0045195) = 0.52° and the angle subtended by the Sun is Θ = 2 × arctan (0.0046524) = 0.53°, link. Chance, or God's amazing design?
Fig.2: God's creation
In Genesis 1 we read that God created (designed) the moon and the earth:
"In the beginning God created ... the earth ... and God made two great lights ... the lesser light to rule the night"
So, as with Solomon's temple, we might expect His signature in the dimensions of these objects. Due to its rotation, the earth is an oblate spheroid with a equatorial diameter of 7,926 miles and a pole-pole diameter of 7,900 miles, link. Let us assume an earth diameter of 7,920 miles, and a moon diameter of 2,160 miles. As pointed out by Bonne Gaunt, on these figures it is interesting to observe that a circle enclosing both moon and earth (Fig.2) has a diameter D = 10,080 miles and that the Greek letters for 'Christ' (chi, rho, iota, sigma, tau, omicron, upsilon) multiply to 10,080 (adding or dropping zeros is valid in gematria). Furthermore, taking pi = 22/7, the circumference of the circle enclosing both earth and moon would be 31,680 miles, corresponding to the gematria for 'Lord Jesus Christ' (see above). The reasoning for using the approximation for pi follows from the relation between the British 'mile' and the gematria for 'Christ', link. Whilst the exact numbers are debatable, the hand of God is strongly implicated.
Nature points to a Designer
Nature points to a Designer
We now turn to what might be termed 'golden' mathematical ratios, sequences and spirals readily found in nature. It is often claimed that these can be seen for example in DNA molecules, human facial beauty, sea shells, insect populations, plants and galaxies. The 'golden ratio (designated Φ or 'Phi') is an irrational mathematical constant, approximately 1.618, and this can be used to form the 'golden rectangle' and from that we can develop the 'golden spiral'. When it comes to number sequences, the 'Fibonacci sequence' (see below) is often found in nature, and this too can be extended to form a Fibonacci spiral. Moreover, the golden ratio can be found in the Fibonacci sequence.
Many claim that these mathematical laws appear to be benefical (giving a sense of harmony and balance) and so man often tries to adopt them in his work, from art to music to building design. It is also claimed that, whilst we do not find such mathematics to be universal law, there is nevertheless a fascinatingly prevalent tendency to such laws in nature, link. The video summarises some of these concepts.
First, it is important to ask some general questions about the claims of 'golden mathematics'. Some claim they are pure conjecture and a numerical hoax:
"These beliefs are false, without substance. They have no basis in reality. They are nothing more than superstition and hoax. They are not scientific observation based on evidence; they are mystical beliefs in numerology" link
It is claimed here that whilst there are indeed some Fibonacci spirals and golden ratios in nature, this doesn't make them extraordinarily prevalent. Not all plants show the Fibonacci numbers and there are many other mathematical patterns in nature. Whilst some galaxies can be found that approximate to the golden spiral, most don't. It is argued that people have a tendency to ascribe meaning or significance to things which really mean nothing! Clearly, any counter to such claims must show how golden mathematics points to a 'Designer'.
Many see the golden ratio and related mathematics as simply a remarkable result of evolution. For example, they claim plants have evolved the optimal arrangement of leaves on the stem as they grow, and the arrangement is often described by a Fibonacci series. More generally, it is claimed that the winning design in evolution allows an organism to survive longer than other designs and that the design with a golden ratio simply helped the host organism to survive better, link. It is readily admitted that the golden ratio has a real contribution to the 'evolved' world:
"The phenomenon of the golden ratio contributes to understanding the oneness of vision, thought and movement in the evolutionary design of nature", link
Nevertheless, the idea that the golden ratio is designed is rejected:
Shapes with length/height ratios close to 3/2 are everywhere and give the impression that they are being 'designed' to match the golden ratio. (But) these shapes emerge as part of an evolutionary phenomenon", link.
That said, for these evolutionary claims to be true, evolutionary theory itself has to be verified. But to date, there is little tangible scientific support for evolution, see evolution - the truth.
The golden ratio is also known as the 'divine proportion', implying that it is real (not superstition and a hoax), it is designed (not a random result of evolution) and it is used by God in His creation. Such a claim can only be verified by taking examples of golden mathematics in nature (God's creation) and relating them back to truth, i.e. to Christ the source of truth. Does what we observe point to Christ? Let's see.
Fig.3: The Golden Ratio, Φ
The golden ratio Φ is a geometric proportion that has been known for millennia (Euclid wrote about it around 300 BC). It is an irrational number in that it cannot be written as a simple fraction, and so the decimal part goes on forever without repeating! Moreover, it is the most irrational number, link and it is claimed that this property has real significance in God's creation.
The best way of defining Φ is to segment a line as in Fig.3. From this, a quadratic solution gives Φ= 0.5[1+√(5)] = 1.6180339... . Note the perfect self-replicating property of Φ: if we fold b over a it will divide it in the golden ratio. If we then take the smaller part and fold it again it will again divide the line in the same proportion. This can be done ad infinitum without variance, link. A 'golden rectangle' is a rectangle with ratio length/width = Φ.
It is important to realise that there is a mathematical relationship between Φ and numbers in the Fibonacci sequence (each number can be derived from Φ, link):
Fibonacci sequence: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584 ... link
Here, each number is the sum of the two numbers before it, and the sequence of the ratio of successive Fibonacci numbers Fn+1/Fn converges to Φ. For example, 89/55 = 1.618181 and 2584/1597 = 1.6180338. In fact, starting with any two positive numbers, the corresponding sequence of the ratio of successive numbers formed by adding the last two numbers always converges to Φ, link. For an in-depth look, see The Divine Ratio.
We now ask: 'Is there a clear link between Φ and Fibonacci numbers to God's creation?' Consider the following.
Music is a fundamental element of God's creation. It was present at the dawn of creation (Job 38.7) and man's musical instruments are used in praise to God:
"I will sing a new song to You, O God; on a harp of ten strings I will sing praises to You" (Ps 144.9)
Musical instruments can embed Fibonacci numbers. Consider the piano keyboard. We find 5 black keys and 8 white keys making up an octave of 13 keys. And 2 black keys plus 3 black keys making up 5 black keys within the octave. These are simple progressions through the Fibonacci sequence! Fibonacci and Φ relationships are often found in the timing of musical compositions e.g. the climax of songs is often found at roughly 61.8% of the song. Fibonacci and Φ are also used in the design of violins, link. So did God embed Φ in music?
Fig.4: Φ and latitude
Fig.5: Φ in Creation
Consider a golden rectangle of sides a and b placed inside the Earth, as in Fig.4. By definition, tan(Θ) = b/a = 0.6180340, or Θ = 31.72°. So point X on the earth's surface would be at latitude 31.72°. Now, the latitude of Christ's birthplace, Bethlehem, is about 31.68°, link. So does Φ point to Christ? Is Φ the 'divine proportion'?
Let's redraw Fig.2 as in Fig.5 with triangle side b linking the centres of earth and moon. Using the same moon-earth sizes assumed in Fig.2 and scaling the sides of the triangle to give a = 1, we find c = 1.618590 i.e. very close to Φ (using other moon-earth dimensions gives c = 1.618454, link). Since the earth-moon geometry in Fig.2 has already been linked to the gematria for 'Lord Jesus Christ', does the simple geometry in Fig.5 link Φ to Christ?
Let's extend the concept of a divine proportion to the rest of the solar system. Since God created the planets we might expect Christ's signature in their orbits for example. Consider nine planets, from Mercury to Pluto. Pluto has been reclassified as a 'dwarf planet', but is, nevertheless, still there! The same goes for Ceres (the largest object in the asteroid belt between Mars and Jupiter) which is also classified as a dwarf planet, link. So in all we could consider ten 'planets' in our solar system, see also the Solar System. Each planet has an eccentric (non-circular) orbit, so we work with mean orbital distances from the sun, and normalise this distance for Mercury (set it to 1). In this case, it has been shown, link that the average of the mean orbital distances of each successive planet in relation to the one before it is 1.61874, very close to Φ.
Clearly, calculations like this are subject to further scientific discoveries, as in the dwarf planet Eris, but the close agreement with Φ is interesting, suggesting Christ's signature on the solar system. This concept is supported when we consider the period in days of each planet's revolution around the sun. When these periods are compared with that of Pluto, the ratios in round numbers come close to Fibonacci numbers, link. And since there is a close relationship between Fibonacci numbers and Φ, and between Φ and Christ, some argue that Christ's signature can be seen in the Solar System.
Let's turn from the astronomical to the molecular. God has made both and we might expect His signature on the very coding of life - our DNA.
Fig.6: B-DNA double helix
Image Public Domain
Fig.7: Axial view of ideal B-DNA
Image Loren Williams
Fig.8: Double pentagon decagon
Deoxyribonucleic acid (DNA) plays a crucial role in all living organisms because it is the key molecule responsible for storage and self-replication of genetic information. DNA can form a wide range of double helical structures and the most common form is B-DNA (Crick and Watson), Fig.6. The 'pitch' (one full 360 ° rotation of the helix) is just 34 angstroms, whilst the diameter of the B-DNA helix is variously given as 20 link or 21 angstroms, link. So the geometry of the most common DNA molecule might be seen to reflect the divine proportion: 34/21 = 1.6190.
X-ray crystallography has been used to examine the structure of the DNA molecule, link. A detailed geometric analysis (axial view) reveals five sets of concentric double pentagons, link, link. Each double pentagon can be seen as a pair of pentagons offset from each other by 36°, creating a decagon. A decagon pattern is clearly seen in the axial view of ideal B-DNA, Fig.7.
So axial analysis of DNA reveals 10 concentric pentagons. Others have constructed a helix (as in Fig.6) from 10 regular pentagons orientated about a decagon, link.
Why is the pentagon so important to DNA? Consider the double pentagon (decagon) in Fig.8. If the base of the pentagon is 1, then the span (diagonal) of the pentagon is Φ. And we know that Φ has the unique self-replication property. Some claim that DNA uses this fundamental self-replication property:
"DNA, needing to replicate itself by increasing in size at a constant ratio, ad infinitum without deviation, appears to call upon the mathematics of the Golden Section, which itself is the only mathematical configuration that can duplicate itself ad infinitum without variance.", link
"The DNA double helix molecule accomplishes (self-replication) feats by means of the very same self-replicating attributes of the very geometry upon which it is built.", link
So does DNA carry the signature of Christ via the self-replicating property of the 'divine proportion', Φ? And if not, who designed in the coded genetic information in the DNA nucleotides, link?
Let's stay with small things. In order to study nanoscale quantum effects, researchers focused on the magnetic material cobalt niobate, link, link. This has linked magnetic atoms, and the electron spins align to form chains of atoms just one atom thick which function in concert like a thin bar magnet. When a magnetic field is applied perpendicular to the spin chain, the chain of atoms acts like a nanoscale guitar string. As in the case of a guitar string, this process generates resonances, and the proportional relationship between the first two resonance frequencies corresponds to the golden ratio 1.618. So Φ is found even at the quantum level! Is this coincidence, or God's design?
Fig.9: the Golden Angle
Fig.10: Golden angle florets
Image Creative Commons
The Bible states that God created herbs and trees (Gen 1.11) and so we can expect to see Christ's signature here too. A link between Φ and plants comes from the 'golden angle', Fig.9. As in Fig.3, we make the ratio of two (arc) lengths a and b such that a/b = Φ. In this case, the golden angle Θ = 360/(1+Φ) = 137.5° approximately.
The golden angle plays a fascinating role in the arrangement of leaves on some plant stems or in the arrangement of petals on certain flowers. A common arrangement is for leaves to emerge at different nodes and at different angles on the stem, forming a spiral pattern (leaves 1,2,3...10 in Fig.10 form a spiral). A key feature is the fraction of a circle rotated between successive leaves, called the divergence angle when measured in degrees. In beech and hazel this is 1/3, in oak and apricot 2/5, in popular and pear 3/8, and in willow and almond 5/13, link. Some 250,000 plant species use such regular fraction Phyllotaxis, link. It is interesting to note that the numerator and denominator of these regular fractions are numbers from the Fibonacci sequence discussed above!
Consider beech and hazel. Here we move by 1/3 of a circle each time such that leaves emerge at angles of 120°. Ignoring lateral growth (a decreasing spiral), the third leaf will then align vertically with the first leaf, shading the first leaf from capturing sun and rain. Similar vertical alignment will happen if we move 1/4 or 2/3 round the circle for each new leaf, and there will be unused spaces in the circle. Even for 5/13 there will be gaps in the circle (try it). The problem is that we are moving by increments of a rational number (a fraction of two integers). In order to fill gaps in the circle we should progress with the help of an irrational number, and the most irrational number is Φ. So let's progress in intervals of 1/Φ = 0.618034 of a circle, corresponding to a divergence angle of 360/Φ or about 222.5°. Equivalently, two successive leaves will subtend an angle of about 137.5°. It is claimed that this angle gives optimal packing density of the leaf buds, and optimal exposure to light and falling rain. For flowers or petals it gives the best possible exposure to insects to attract them for pollination, link, link.
See 'Golden Plants' below for an example of this optimal divergence angle in a succulent:
Fig.11: Sunflower Golden Spirals
The spirals formed by a plant which uses the golden angle can exhibit another mathematical property. Consider the sunflower head in Fig.11. Clearly, the seeds form interlocking clockwise and counterclockwise spirals. The interesting point is that the number of clockwise spirals differs from the number of counterclockwise spirals and they are nearly always two consecutive Fibonacci numbers! For example, the sunflower in Fig.11 has 21 clockwise and 34 counterclockwise spirals, but it could also be 34/55, 55/89, 89/144 etc.. In fact, it is claimed the number of spirals in the centers of flowers in general, correspond to a Fibonacci number, link, link.
Sunflower Photo credit: lucapost / Foter / Creative Commons Attribution-NonCommercial-NoDerivs 2.0 Generic (CC BY-NC-ND 2.0)
We could go on. For example, the number of petals in a flower often follow the Fibonacci sequence e.g. the lily has 3 petals, buttercup 5 petals, chicory 21 petals and daisy 34 petals. Of course, not all plants follow Fibonacci numbers and the golden angle, but it is widely recognised that there is a 'fascinatingly prevalent tendency' to do so, link! Despite much analysis, the 'golden rule' in plants remains intriguing, link.
In Exodus 25.10 God instructs Moses to make the Ark of the Covenant 2.5 cubits long and 1.5 cubits wide. The ratio of these two dimensions, 5/3, is a rational approximation to Φ. There are better approximations e.g. 21/13 = 1.615, link, but clearly, 5/3 simplifies construction and visually there is little to choose between 5/3 and Φ!
Jesus was crucified in a place called 'Golgotha' or 'the place of a skull', a well-known spot and thoroughfare just outside the city gate (Matthew 27:33; Mark 15:22; John 19:17). Golgotha represents in Greek letters the Aramaic word Gulgaltha, which is the Hebrew Gulgoleth. Consider its gematria, with reference to Strong's Concordance:
Mat 27.33: Golgotha, (the skull) Gk gematria value 186, (Strong's G1115) link
Golgotha does not appear in the corresponding passage of Luke. So in Lk 23.33 (NKJV) the translators use the term 'Calvary', the name being taken from the Latin Vulgate translation 'Calvaria' of the Greek word 'kranion' or 'skull'. Consider its gematria:
Lk 23.33 (NKJV): Calvary, Gk 'kranion' (a skull) Gk gematria 301, (Strong's G2898) link.
So the two terms (Golgotha and Calvary) both refer to the place of 'the skull'. Suppose we ratio the two Greek gematria, giving 301/186 = 1.61828, which is curiously close to Φ= 1.6180339. Coincidence, or Holy Spirit inspiration? Note that some modern Bible translations omit 'Calvary', and some even appear to modify inspired scriptures, particularly those relating to Jesus, link. Better to stay with a widely trusted translation like the NKJV!
We have examined some numbers and mathematical concepts which appear to suggest a Designer, namely Christ. The simple dimensions of the Ark are surprisingly close to modern ship design, and the footprint of Solomon's Temple appears to relate to Christ, as do the relative sizes of earth and moon. Astronomical distances and sizes also appear to be just right for the perfect solar eclipse - but surely not 'a fluke of celestial mechanics', as some maintain.
Then there's the fascinating 'golden ratio' Φ = 1.6180339... which appears to relate to Christ. For example, it might even point to Bethlehem, Christ's birth place, and to the orbits of the planets. At the other end of the scale, many see Φ in the double helix molecule of our DNA, and in the leaf arrangements of many growing plants. Plants also demonstrate the mathematically related Fibonacci sequence, as does music. Φ is even found at the quantum level in atoms.
Of course, taking measurements in an analogue and fallen world will always generate uncertainty in such calculations, and there are always exceptions. But is the mathematics too close to the ideal to ignore, too close for chance? But the diversity of examples where numbers seem 'almost right', and where Φ and Fibonacci are found is surely intriguing. And especially so when they appear to point in one direction - to Christ, the Designer! Remember, the Bible says:
"All things came into being through Him" (John 1.3)
Bible quotations are from the New King James Version (NKJV)