The Phases of Ithil, the Cycles of Arda, and the Hidden Numbers of the ‘Stars’: A Mathematical and Elvish Connection

The Moon (Ithil) has always been integral to life on Earth, influencing timekeeping, agriculture, and mythology. Its phases and orbit reveal significant mathematical patterns, especially in relation to Lucas primes—prime numbers in the Lucas series, a sequence closely related to the Fibonacci series. Lucas numbers, like Fibonacci numbers, follow a very similar pattern but differ in their starting values (2 and 1). While Fibonacci numbers frequently appear in natural growth patterns, Lucas primes hold their own unique significance, particularly when connected to the Moon’s cycles. This article explores how specific Lucas primes—2, 3, 7, 11, 29, 199, and 521—intertwine with the Moon's phases, cycles, tides, and eclipses.

Lunar Phases and Cycles

The Moon’s most visible movement is seen in its phases, created by the changing angles between the Earth, Moon, and Sun as the Moon orbits Earth. A full cycle, from new moon to full moon and back, known as the synodic month, lasts about 29.53 days—a period that corresponds closely to the Lucas prime number 29.

The Moon’s phases can be divided into eight stages:

1. New Moon
2. Waxing Crescent
3. First Quarter
4. Waxing Gibbous
5. Full Moon
6. Waning Gibbous
7. Third Quarter
8. Waning Crescent

Each Waxing and Waning phase lasts approximately three days, while the New and Full moon phases last one day each. Together, they comprise the eight stages in the full 29.53-day cycle, which forms the foundation of many ancient calendars. In addition to the synodic month, the Moon follows two other key cycles:

1. Draconic Month: The time it takes for the Moon to return to the same node—points where its orbit crosses Earth's path (the ecliptic)—which lasts 27.21 days. This cycle is key for predicting eclipses.
2. Anomalistic Month: The duration between two perigees, the points where the Moon is closest to Earth, lasting 27.55 days.

Though the numbers 7 and 29 aren't directly reflected in draconic or anomalistic months, they hold significant roles in lunar phenomena and calendars. The number 29 is crucial for approximating the synodic month, with its average duration of about 29.53 days, making it essential in constructing lunar-based calendars across many ancient cultures. The number 7 plays a major part in the division of the week into 7 days, a cycle that mirrors the Moon's phases, each lasting roughly 7.4 days.

The number 7 is also relevant in the Metonic cycle, a 19-year period after which lunar phases repeat on the same days of the year. This cycle is used in various calendars, including the Jewish and ancient Greek ones, and is highly accurate due to the close match between 19 solar years (approximately 6,939.60 days) and 235 lunar months (approximately 6,939.69 days). The Golden Number, calculated as (year mod 19) + 1, identifies the year within this cycle, and the modulo operator (used here) simply gives the remainder when dividing one number by another.

Additionally, 7 plays a role in the Epact, which tracks the Moon's age and resets in 7-year increments to keep calendars aligned with lunar phases. The number is crucial for making periodic corrections.

The Saros cycle, covering 242 draconic months—which factorizes into 2 × 11 × 11, both Lucas primes—connects draconic months to lunar eclipses. While 7 doesn't appear in the Saros cycle, it complements the Metonic and Saros cycles by structuring weeks and synchronizing with cosmic cycles. Furthermore, the number 27, which features prominently in both the draconic and anomalistic months, equals 3 cubed—a mathematical pattern aligning with Lucas primes, showing how these numbers interweave with cosmic and calendrical cycles.

These patterns hint at deeper mathematical structures underlying the cycles of the solar system.

Tides and Lunar Influence

The Moon's gravitational pull significantly impacts Earth's oceans, resulting in tides—the periodic rise and fall of sea levels. Tides occur due to the gravitational attraction between the Earth, Moon, and to a lesser extent, the Sun. As the Moon orbits Earth, it pulls water towards it, causing a bulge on the side of Earth facing the Moon. Simultaneously, a second bulge forms on the opposite side due to water's inertia, creating two high tides on opposite sides of the planet.

Earth's rotation every 24 hours means most places experience two high tides and two low tides daily. These are known as semidiurnal tides and are primarily driven by the Moon's gravitational pull, though the Sun's influence is also significant. At the new moon and full moon phases, when the Earth, Moon, and Sun align, their combined gravitational forces create spring tides, which are the highest and lowest tides of the month, occurring roughly every 14.75 days or half a synodic month.

During the first and third quarter phases, when the Sun and Moon form a right angle relative to Earth, their gravitational forces partially cancel out, leading to neap tides, the smallest tidal ranges, occurring every 7 days after a spring tide. For the Elves of Middle-earth, particularly those living in coastal regions such as the Grey Havens, tides would have been a powerful and observable reminder of the Moon’s influence on the world.

The regularity of tidal cycles mirrors the consistent mathematical structure observed in lunar periods. Tides are also influenced by factors such as coastline shapes, ocean depths, and Earth's rotation, yet the Moon remains the dominant force. Its connection to Lucas primes, especially in cycles related to the 12 hours between successive high and low tides, resonates with the Eldar’s preference for numbers like 6 and 12 in their time-keeping systems.

Eclipses and the Saros Cycle

Eclipses occur when the Sun, Moon, and Earth align, casting shadows on one another. Solar eclipses happen when the Moon passes between Earth and the Sun, while lunar eclipses occur when Earth moves between the Sun and the Moon.

For an eclipse to take place, the Moon must be near one of its nodes—points where its orbit crosses the ecliptic. These nodes gradually drift over time, completing a cycle every 18.6 years. Consequently, eclipses don’t occur monthly but follow cycles like the Saros cycle, which repeats every 18 years, 11 days, and 8 hours, or 223 synodic months. After one Saros cycle, the Sun, Earth, and Moon return to similar relative positions, leading to similar eclipses. This cycle is vital for predicting both solar and lunar eclipses.

In Tolkien’s mythology, solar and lunar eclipses carry deep symbolic meaning. The solar eclipse is reflected in the words of The Silmarillion, where Varda commands the Moon to rise only after the Sun has descended. However, Tilion, the Maia steering the Moon, is drawn toward Arien, who guides the Sun:

"Varda commanded the Moon to journey in like manner, and passing under Earth to arise in the east, but only after the Sun had descended from heaven. But Tilion went with uncertain pace, as yet he goes, and was still drawn towards Arien, as he shall ever be; so that often both may be seen above the Earth together, or at times it will chance that he comes so nigh that his shadow cuts off her brightness and there is a darkness amid the day." – The Silmarillion

This mythological explanation mirrors the occurrence of solar eclipses, as Tilion’s "uncertain pace" leads him to block Arien's light, bringing darkness during the day.

Lunar eclipses, on the other hand, are attributed to the forces of Morgoth, who sends spirits of shadow against Tilion in an attempt to disrupt the light of the Moon:

"But Morgoth hated the new lights and was for a while confounded by this unlooked-for stroke of the Valar. Then he assailed Tilion, sending spirits of shadow against him, and there was strife in Ilmen beneath the paths of the stars, and Tilion was the victor: as he ever yet hath been, though still the pursuing darkness overtakes him at whiles." – Morgoth’s Ring, HoME Vol. X

This narrative echoes the periodic nature of lunar eclipses, where Earth’s shadow temporarily overtakes the Moon, just as Morgoth’s darkness pursues but never fully conquers Tilion.

Despite the Saros cycle being an approximation and not aligning perfectly with whole numbers, it demonstrates a remarkable natural precision, much like how Lucas primes closely approximate the Moon’s cycles. This connection between mythological and astronomical cycles weaves a deeper harmony into the fabric of the cosmos, echoing the mysteries that bind both realms.

The Golden Ratio and Planetary Orbits

A particularly fascinating mathematical concept related to the Moon and other celestial objects is phi (ϕ), or the golden ratio—an irrational number roughly 1.61803 $(\frac{(1+\surd 5)}{2})$. Known for its aesthetically pleasing properties since ancient Greece, the golden ratio also appears in mathematical sequences like the Fibonacci and Lucas series, where successive term ratios converge to phi:

$\frac{L_{n+1}}{L_n}\approx \varphi$

For example, dividing consecutive Lucas numbers such as 521 by 322 gives a ratio close to 1.61801, approximating phi. This pattern mirrors that of the Fibonacci numbers and hints at a deeper mathematical harmony in the universe. While the Moon’s orbit doesn’t follow phi exactly, its geometric ratios show intriguing approximations. For instance, a line from Earth’s center to the Moon’s center forms the hypotenuse of a right triangle, with the base being the sum of Earth’s and the Moon’s radii. The hypotenuse-to-base ratio approximates the golden ratio (ϕ), a configuration known as the Kepler triangle, first described by Johannes Kepler.

The golden ratio's influence may extend beyond lunar cycles to the broader solar system. Some propose that planetary orbits, including the Moon's, may naturally align with phi, contributing to their stability.

For instance, the full moon cycle (FMC) refers to the period during which the Moon’s phases and perigee (its closest approach to Earth) align. This cycle is determined by the beat period between the synodic month (SM) and the anomalistic month (AM), calculated as:

$\frac{(SM\times AM)}{(SM-AM)}=FMC=411.78443$ days

The beat period is the time interval between two periodic cycles when their frequencies are slightly different. It represents the duration over which the cycles align and create a repeating pattern of constructive and destructive interference. The Saros cycle is equal to 19 FMCs.

The FMC shows an intriguing relationship with Jupiter’s orbit, which lasts approximately 4,332.82 days. When compared to the FMC, the ratio approximates the golden ratio (ϕ), at about 1.61803. Simplifying this, one Jupiter orbit is roughly equal to ϕ × (13/2) FMCs—only about 2 days short, representing a discrepancy of approximately 0.00046%.

The FMC is also connected to other planetary cycles. Five FMCs are roughly equivalent to eight Venusian cycles (each lasting 584 days), while three FMCs approximate seven Martian cycles (each lasting 687 days). These relationships underscore the mathematical elegance of celestial patterns, emphasizing the role of numbers like the Lucas primes in understanding cosmic cycles.

Finally, the FMC affects phenomena on Earth, such as tidal patterns, which are strongest when the Moon is at perigee and weakest at apogee. The FMC also influences seasonal variations due to the changing positions of Earth and Moon. The Moon’s perigee and apogee can subtly affect gravitational forces, potentially influencing weather patterns and ocean currents. Some research suggests a link between the FMC and solar activity, such as sunspot cycles and solar flares, though these connections are still under investigation.

The concept of celestial bodies following golden ratio resonances is reflected in the orbital periods of various planets. The table below demonstrates approximate correlations between planetary orbital periods and powers of φ.

Planet	Power of φ	Calculated Period	Actual Period
Mercury	-3	0.24	0.24
Venus	-1	0.62	0.62
Earth	0	1.0	1.0
Jupiter	5	11.1	11.9
Saturn	7	29.0	29.5

_{These values are approximate and not exact matches. Information derived from Phi and the Solar System [link].}

These values are approximations rather than exact matches, but they indicate that planetary orbital mechanics may also align with the golden ratio. NOTE: In Tolkien's Legendarium, the planets are not distinguished from the stars, which is why that term is enclosed in single quotes in the title.

The Inex Cycle and Lucas Primes

The Inex cycle, spanning 358 synodic months or around 29 years, is another significant lunar cycle. Notably, 18 Inex cycles cover 521 years, with both 18 and 521 being Lucas numbers—521 also being a Lucas prime. This connection hints at a deeper relationship between lunar cycles and Lucas primes.

In 521 years, the number of sidereal months totals 6965, which equals 199 (a Lucas prime) multiplied by 35. Interestingly, 199 is the sum of two Fibonacci numbers (144 + 55), and 521 is the sum of two larger Fibonacci numbers (377 + 144). Dividing 521 years by 199 yields approximately 2.618, close to φ² (the square of the golden ratio), suggesting a natural order within these cycles.

When a Saros series concludes, a new one often begins exactly one Inex cycle later, further demonstrating the Inex cycle’s role in organizing eclipse patterns. This interplay between Saros and Inex cycles, along with the Callipic cycle, underscores a system that aligns with Lucas numbers and the golden ratio. The 521-year period, sometimes called the Hyper Saros, has even been described as the “basic period of the solar system” [link], underlying many recurring celestial patterns.

In this context, the appearance of Lucas primes like 199 and 521 in the structure of eclipse cycles is as remarkable as the well-known presence of Fibonacci numbers in nature. While Fibonacci numbers are associated with growth patterns in living organisms, Lucas primes appear to govern celestial mechanics, especially in the lunar cycle. The Callipic cycle, spanning roughly 76 years (or 4 × 19 years, equaling 940 synodic months), provides another example of a cycle connected to Lucas numbers. The fact that 76 is a Lucas number suggests a strong link between lunar and eclipse cycles and Lucas primes, particularly 199 and 521.

Elvish Numerology and the Lucas Primes

In Tolkien’s Elvish numerology, numbers hold symbolic and mystical significance. As a philologist fascinated by language, myth, and hidden patterns, Tolkien embedded numerical structures into his works, reflecting natural cycles and celestial events. The Elves, in particular, are portrayed as inherently connected to the stars and the passage of time.

In addition to numbers like 7, 3, 2, 12, and 6, which are already significant in the Legendarium, the recurrence of lesser-known Lucas primes such as 29 and 521 would have resonated with the Elvish understanding of the universe. These primes could represent cosmic harmony, fitting into the Elves' connection to time and the stars.

The concept of Valian time, based on the rhythms of the Two Trees in Valinor, embodied this sense of harmony. A Valian Year consisted of 1,000 Valian Days, each divided into twelve Valian hours. It is unclear whether this '1,000' was in the duodecimal system the elves preferred, which would equal 1,728 in decimal. The Trees had a radiance cycle of seven hours, with an additional hour of overlap between them (“In seven hours, the glory of each tree waxed to full and waned again to naught; and each awoke once more to life an hour before the other ceased to shine”). This overlap was not counted, preserving a precise duodecimal system that reflected a deeper cosmic order. Each Valian Day followed the waxing and waning of the Trees' light, allowing the Valar to track the world's passage through time.

Some Tolkien scholars, though, believe that time flowed more slowly in Aman, making a Valian Year seem to pass like a solar year. This effect was due to the fact that each Valian hour was roughly equivalent to 7 solar hours. Therefore, while a Valian Year might have measured out to a much longer span in solar terms, the inhabitants of Aman would have perceived it as a single year's passage. This difference in perception highlights how the timeless quality of Aman contrasted with the more fleeting and changeable world outside.

The destruction of the Two Trees and the subsequent creation of the Sun and Moon disrupted the perfect, harmonious system of time that had been established in Valinor. The Valar, seeking to replicate the lost rhythm of the Trees, designed the Solar Year to mirror the balance of Telperion and Laurelin. They purposed that each Solar Year would contain 700 cycles of Moonlight and Sunlight, with each cycle divided into two 12-hour periods—one for the Moon and one for the Sun—reminiscent of the alternating radiance of the Two Trees. This created a 24-hour Solar Day and a 350-day Solar Year, composed of exactly 8400 hours (24 hours per day multiplied by 350 days per year).

However, despite their efforts to preserve this celestial order, the Sun and Moon proved less stable than the Two Trees, leading to discrepancies in timekeeping. The actual Solar Year, rather than the intended 350 days, extended to 8766 Solar hours. This increase signified a more erratic flow of time in the mortal realms, contrasting the perfect consistency of the Valian calendar. The new system, though designed with great care, could not fully recapture the stability of the original, marking a shift in the balance of time and reflecting the increasing complexity and unpredictability of the world after the Trees' destruction.

In this new world order, 7 remained a crucial number, particularly in auxiliary lunar calculations like the Epact and the seven-day week, both of which echoed the lost harmony of the Valian calendar. Similarly, the Lucas prime 29, which plays an essential role in lunar calendars due to the average length of a synodic month being approximately 29.53 days, symbolized the continuing connection between timekeeping and celestial cycles.

Thus, the transition from the Valian Year to the irregular Solar Year reflects a cosmic shift. The number 7—while central to the passage of time in both Valinor and Middle-earth—became a reminder of a lost balance, while 29 attained new significance as a measure of the lunar cycle. These numbers illustrate Tolkien's intricate blending of myth, mathematics, and the flow of time, where numbers were more than abstract entities; they represented the pulse of the solar system itself. And so, those who walk under the stars remember that, though much is hidden, the patterns of the heavens are the language of the Valar, proceeding from the past and continuing into the future.

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References:

• MathWorld - Lucas Primes. [link]
• R Knott - Fibonacci & Lucas Numbers. [link]
• NASA - Lunar Phases and Eclipses. [link]
• Tallbloke. (2015, December 28). Why phi, Jupiter, and the full moon cycle? Tallbloke’s Talkshop. [link]
• Golden Number. (2012, June 8). The solar system and the golden ratio. GoldenNumber.net. [link]
• Menichelli, M. (2018). Fibonacci & Lucas numbers, Moon-Sun cycles, financial timing [Conference session]. ResearchGate. [link]
• Wilde, T. (2015, January 27). Why Phi? The inex eclipse cycle, Fibonacci, and Lucas. Tallbloke's Talkshop. [link]
• Tolkien, J.R.R. The Lord of the Rings, Appendix D.
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