Regrettably, for the work I’ve done on Palenque so far, I don’t think I really have anything I’d consider a finished model of any structure there, and it might be good to have such a thing to help clarify the spirit of Palenque’s architectural and astronomical mathematics. The Casa de Leon (House of the Lion, Temple of the Lion, Temple of the Jaguar) seems like it might be a likely candidate because it’s a fairly modest structure and we may have a bit of a head start on it with the possible symmetry of 2 x Pi with 2 / Pi, just as we seem to have the symmetry of 6 x Pi with 6 / Pi at El Castillo at Chichen Itza.
All of the architecture at Palenque contains lingering mysteries though, and the Temple of the Lion is no exception. It still seems difficult to definitely identify the meaning of 2.85 m = 9.350393701 ft, and there are several instances of 2.54 m = 8.333333333 ft that make one wonder if they really mean the reciprocal of 12 ( 1 / 12 = 8.333333333 / 100), or something else not yet identified. While not unheard of, expressions like that seem to be as rare as they seem unnecessary. Many simple numbers like 2 or 12 are things we can think of ourselves to try as mathematical probes rather then needing to be informed about them by the architecture.
On the other hand, we’ve seen enough to inspire a more orthodox metrological approach, wherein things 4 of this unit here and 2 of it there could be part of the package, so again some clarification may help advance understanding.
It we look at Room 2 again, the height of the room (height to springline) is 2.54 m = 8.333333333 and the height of the vault from springline to the bottom of the capstones which is 1.47 m = 4.822834646, which has been suggested to possibly be 4.810325483 or 1/100 of the height of the Great Pyramid, the total height is 2.54 + 1.47 = 4.01 m = 13.15616798 ft, very suggestive of 13.15947254.
The ratio between total height and vault height is 4.01 / 1.47 = 2.727891156, suggestive of ~27.3 days as the Tropical or Sidereal Month, and the ratio between total height and room height is 4.01 / 2.54 = 1.578740157, while the ratio between room height and vault height is 2.54 / 1.47 = 1.727891156. For some this may suggest the square root of 3. For me this might suggest a common approximation of sqrt 3 or perhaps 1.729249823, although I’m not certain offhand what connections the raw values of 1.578740157 or 1.727891156 might have with astronomy.
1.578740157 isn’t all that far from 15.88133131, which is half of 317.6266261 / 10. 317.6266261 has been found in architecture in and around Tikal and not surprisingly, demonstrated to have some relevance to astronomy and calendars. (We may find 15.88133131 at Stonehenge as a max/min ratio among the ellipses).
Interestingly, and perhaps a deliberate feature designed to offer some additional guidance, the product of 1.47 m = 4.822834646 and 2.54 m = 8.333333333 is 4.822834646 x 8.333333333 = 57.87401575, while the height to springline of Room 3 of the Temple of the Lion is given as 1.76 m = 5.774278215.
However, because there is additional involved of the two height value to gain the total, some of these numbers may not be compatible in operations of addition or subtraction. The proposed room height of 4.810325483 ft, although only a slight deviation from the raw value of 4.822834646, added to 8.33333333 = only 13.1403655882, while the proposed total 13.15947254 – 8.33333333 = 4.826139207, and 13.15947254 – 4.810325483 = 8.349147057, both of these remainders 4.826139207, and 8.349147057 being generally unfamiliar.
Perhaps they are simply using numbers whose rationale is not yet apparent to us, or perhaps I’m simply overlooking something obvious, such as 365.0200808 x 360 = 13.14072291 x 10^n, which may actually work here with the other suggestions.
The ratio might become 13.14072291 / 4.810325483 = 27.31774171 / 10. 27.31774171 does actually appear in one experimental set for the Tropical or Sidereal Month, albeit in one of the more obscure and uncertain columns.
Alternately, 13.14072291 / 4.803471728 = 2.735671958, which features more prominently in the same experimental set as the projected B value.
There do seem to be more surprises starting to show through with further inquiry in addition to the possible ~2.73 ratio noted above.
The length and width of Room 2 are 3.86 m = 12.66404199 ft and 1.94 m = 6.364829396. As previously mentioned, 6.364829396 is suggestive of 1 / (Pi / 2) or 2 / Pi and may be part of a symmetrical scheme. Numbers like 12.66514796 (125 / Pi^2), 2.533029591 (250 / Pi^2), and 250 / (Pi^4) = 2.566495564 especially have been coming into ever greater prominence because of their potential to build an approximation of the Saros Cycle from the root up, or to potentially approximate Saros / 52 = 6585.3211 / 52 = 12.66407904 / 10
2.566495564^2 = 6586.899478 / 10, textbook Saros Cycle value 6585.3211 days.
However, something else to watch out for here is that 3.86 / 2.54 = 1.519685040 = 1 / 6.580310878, also not far at all from the Saros Cycle / 10^n.
Interesting, if we look at the length and width of room 2 again, 3.86 m = ~12.66514796 (125 / Pi^2) ft and 1.94 m = 6.366197724 = (20 / Pi) ft respectively. their ratio is 3.86 / 1.94 = 12.66514796 (125 / Pi^2). If we adhere to these suggested proposed parameters, that’s a ratio of 12.66514796 / 6.366197724 = 1.989436789 and a product of 12.66514796 x 6.366197724 = 80.62883611.
Both of these are unfamiliar numbers, and 1.989436789 is disconcerting for not being 1.987998915, which is an established if subtle “wonder number” that conveniently carries with it the mathematics of the mean of the Stonehenge sarsen circle (2 / 1.987998915 = 1.006036766 = mean diamter sarsen circle in feet / 100).
1.987998915, was first discovered at the temple of Karnak in the literature cited by Algernon Berriman as his source for a Karnak Cubit in Historical Metrology. While it would make sufficient sense, particularly after what Geoff Bath has done to illuminate some of the merits of a 2:1 rectangle, if the Karnak temple’s length/width ratio happened to be 2, in the event it were actually something else, one of the better answers to what it might be instead would seem to be be 1.987998915. Surprisingly, it has some earmarks of being a worthy number.
Clearly then we may be getting into uncharted territory with the potential to learn some things we didn’t know but the ancients might have.
As it’s turning out, 12.66514796 / 6.366197724 = 1.989436789 (625 / Pi / 100, or the reciprocal of 16 Pi, or 10 Radians / 288) has at least a little pedigree of its own. This number specifically links Saturn Synodic Period B to Half Venus Cycle C, for example, and may be an intended approximation of 17 x 117. More importantly, 1.989436789 / (Pi / 3) = 18997.72194 / 10^n, the B Value for the Half Venus Cycle.
Likewise, the projected product 12.66514796 x 6.366197724 = 80.62883611 (250 / ((Pi^3), thus also part of the suggested Saros root series) may be showing some unexpected properties that may help to illuminate its nature and usefulness. It seems to be meaningful here that Tropical Month 27.3 x Lunar Month 29.53 = 80.6169 x 10.
Divided by the Anomalistic Month of ~27.55, 80.6169 / 27.55 = 29.26203267. The “RMA” or “Real Mayan Annoyance”, with demonstrated astronomical properties, is 29.26442322.
1.067438159 x 2 x 378 = 80.69832482, which might also turn out to be meaningful. The lunar Nodal Cycle in years times Jupiter’s Orbital Period in days might also be represented: 18.6 y x 4332 d = 80.5752 x 10^n. 80.62883607 may also speak to the Metonic Cycle — 6939 / 80.62883607 = 86.06099275 = ~(Pi / 365.0200808) X 10^n = 86.06629659.
Things begin to heat up as we also find that Half Venus Cycle A 18983.99126 / 1.177245771 / 2 = 80.62883607. Also, the reciprocal of 12 x (Pi^2) which links the Half Venus Cycle to the Venus Orbital Period / (Pi / 3) = 80.62883607 / 10^n.
Thus far then these equations seem to lack the exponential flamboyance of many equations seen at Tikal or elsewhere, yet they seem to deliberately recognize some neglected if more secular functions, which is in fact more reminiscent of the mathematics of Rio Bec’s structures as known so far. I often get that sense at Rio Bec of someone taking in and caring for neglected numbers as if they were neglected pets, and have mentioned it on at least several occasion in the past several years.
One more thing I would like to note about Room 2 of this structure is what happens to the width if we apply the springline offset. We would subtract the offset from each side of the width to obtain the bottom width of the vault at the springline, which in this case would be 1.94 – .13 – .13 = 1.68 m = 5.511811024, which is twice
5.511811024 / 2 = 2.755905512; Anomalistic Month ~27.55 days.
Let’s look at Room 3 of the Temple of the Lion then.
Length 2.85 m = 9.350393701 ft; Width 1.92 m = 6.299212598 ft, suggested to be 6.283185307 (2 Pi) as part of a symmetry scheme; Height to springline 1.76 m = 5.774278215 ft.
There seems to be a possible issue with the vault height of Room 3 in that in Architectural survey Palenque, Chiapas, Mexico : the temples, on page 89 it states: “Height: 1.16 m springline to bottom of capstones.” while on page 90 a repeat of the descriptions of the Temple in a different font begins wherein page 93 the vault height is given as: Height Approximately 3.49 m, springline to bottom of capstones.
Looking at the diagrams, both Andrews’ own and those of Frederick Catherwood which Andrews has included, it seems likely that 3.49 represents the figure which is error since the diagrams make it seem implausible for the height of the vault to exceed the 1.76 m height of the room.
Thus the total height would be room height 1.76 m = 5.774278215 + vault height 1.16 m = 3.805774278 ft = total height 2.92 m = 9.580052493 ft. The total height / room ratio would be 2.92 / 1.76 = 1.659090909 and the total height / vault height ratio would be 2.92 / 1.16 = 2.51741379. I’m not ready to make such a suggestion, but will note that 2 Pi x 4 = 2.513274123 x 10.
The total height could be 9.606943459 ft, it’s a very attractive and very useful number. It may be somewhat difficult to be certain since 3.805774278 is suggestive of the Venus Cycle, and as with what looks like 19 generally turning out to be some form of the 18980 day Half Venus Cycle, I expect what looks like 19 x 2 = 38 likely to represent some form of 18980 x 2 = 37960.
One interesting thing here is that Length 2.85 m = 9.350393701 ft divided by room Height 1.76 m = 5.774278215 ft = 1.619318182, suggestive of some form of Phi, most likely 1.622311470.
If we actually take 5.774278215 to be the height of the Great Pyramid in inches /10^n = 481.0325483 x 12 = 577.2390580 in my model, we might find ourselves in a bit of a predicament where 577.2390580 x 1.622311470 = (1 / 1067849508 x 10^n). This is not the customary Hashimi Cubit value of 1.067438159, and so far nothing in all this time has forced us to accept any look-alikes over the genuine item 1.067438159.
To “correct” this equation, (1 / 1.067438159) x 10^n / 1.622311470 = 5.774615035 = 1 / 1.731717169. However, for Length 2.85 m = 9.350393701, there may also be the suggestion of tapping into other series, such as the resemblance between 9.350393701 and 1/24th of the Aubrey Number, which by the way is easy to generate as 1.731717169 x 360 x 360 = 224.4305459 x 10^n. 224.4305459 / 24 = 9.351272747.
There is however also 9.348115074 which is part of a very powerful (Pi / 3) series from Tikal, and perhaps some other possibilities – but thus far, 9.351272747 may give the best responses to 1.622311470, including that if the ratio between length and room height is 1.622311479, then the best thing going may be 9.351272747, which gives a room height of 9.351272747 / 1.622311470 = 5.764166073, which recalls Munck’s “geomathematical” “Grid Point” for the Chephren Pyramid.
Thus in spite of a propensity for stray numbers, the architect of the Temple of the Lion may be proving themselves no less adept at data storage and retrieval than the prodigal architects of Tikal.
The presence of 2 Pi picks up both refined suggestions and runs two powerful series simultaneously that cover many bases when it comes to retrieval of important numbers.
We have little data for Room 4, and the data for Room 1 is also limited because of dilapidation. Room 1 and Room 4 are both largely described as fallen. “OBSERVATIONS ROOM 4 – Length: 2.59 m. Width: 1.02 m. No other data; room mostly fallen”.
2.59 m = 8.49737528 ft, which we have recently discussed being apparently the reciprocal of 1.177245771: 1 / 1.177245771 = 8.494402992 ft.
1.02 m = 3.346456693 ft, which much resembles 1.673128806 x 2 = 3.346257611 (or perhaps 1.676727943 x 2 = 3.353455886). While the first possibility is intriguing, it’s the second possibility that manages to cue into the proposed Saros root series mentioned above.
8.494402992 / 3.353455886 = 2.533029591 = 80.62883609 / Pi = (sqrt 6586.8994 / 10) x Pi^2. The combination also affords us a somewhat short but impressive series.
There are more things to check, verify, and double-check, but what we have just seen here then may be among the most successful interpretive work that has taken place at Palenque to date. Suffice it that where Apsidal Precession seemed to be a preoccupation of Oxkintok architecture, the Saros Cycle might prove to be a significant preoccupation of Palenque architecture if the Temple of the Lion is any indication.
–Luke Piwalker
Luke Piwalker 