15TH INTERNATIONAL CONFERENCE ON GEOMETRY AND GRAPHICS ©2012 ISGG1–5 AUGUST, 2012, MONTREAL, CANADA

Paper #75

HOW TO PRECISELY MEASURE ASTRONOMIC PERIODS OF TIMEBY MEANS OF STONE AGE GEOMETRY

Georg GLAESERUniversity of Applied Arts Vienna, Austria

ABSTRACT: Astronomy always used to be a typical application field of geometry. Monumentalstructures like Stonehenge and the Egyptian or Central American pyramids were also giant observa-tories for the evidently strange motions in the sky. The author will show that the alignments of thosestructures not only had to do with sun positions, but might also have been influenced by hard-to-comprehend extreme moon positions (lunar standstills) and special positions of the Pleiades. Somespecial constellations allowed different archaic societies on three continents to precisely determinelong periods of time with Stone Age methods, e.g., the length of a year or the time in-between lunarstandstills.

Keywords: Archaeo-astronomy, solsticial alignments, geometric astronomy, lunar standstill

1. INTRODUCTIONThe alleged rotation of the sky around the Earthwas the source of many myths and whole reli-gions. It inspired archaic societies in the en-tire ancient world to invent methods to exactlymeasure periodically occurring phenomena andtherefore be able to predict them in the future.

Orion belt? 4 stars!

Pleiades

Figure 1: The famous “hall of the bulls”: Notonly bulls, but also star clusters like the Pleiadeswere depicted very impressively.

As a strong evidence of how long people haveaccurately observed the sky we start with one ofthe oldest examples: The 17000 years old Upper

Paleolithic cave drawings in Lascaux (Figure 1,[8]). A bull is painted which might fit into theconstellation Taurus, since over the shoulder ofthe bull one can find a star cluster which lookssimilar to that of the Pleiades.

Orion belt (3 visible stars)

Figure 2: The corresponding constellations onthe sky fit quite well - except the fourth star inthe Orion belt.

To the left of Taurus we would then expect thevery significant constellation Orion with its im-mediately recognizable three belt stars. On the

painting, one can indeed see such aligned starsnoticeably exact in the correct direction towardsthe Pleiades. The artist, however, painted fourstars in a row.

When we take the map of the night sky andplace it over the painting (Figure 2), we can seethat the artist indeed might have meant the Orionbelt1.

In Section 2, we will see how it may have beenpossible to determine the length of year up totwo digits after the comma.

The major examples focus on the CentralAmerican pyramid towns of Teotihuacan andChichen Itza [5], [1] which have denoted andvery precisely given directions that allow to pre-dict the night when the Pleiades faded exactlyat the zenith above their pyramids at sunrise ofthe 13th of August (see Figure 1). This worked,however, only for two or three respective cen-turies when the pyramid towns were at theirheights, and got more and more inaccurate at thetime the places were deserted for unknown rea-sons.

In Section 3, we provide the reader with sometheoretical and mathematical considerations inorder to better understand the measurements incontext with Section 4. There, we mainly focuson the pyraimds of Giza and Stonehenge in con-text with lunar standstills.

2. HOW TO MEASURE THE LENGTH OFA YEAR WITH STONE AGE METHODS

In astronomy we distinguish the “tropical year”(the time from vernal equinox to vernal equinox)and the “sidereal year” (the time taken for a360-rotation of the Earth around the sun). To-day we know that a sidereal year lasts ys ≈365,2564 days. Due to the precession move-ment of the Earth (period ≈ 26000 years), the

1A fourth star would be something very unusual, al-though it is theoretically possible that a visible cometcrossed the area (this happened occasionally in the past,e.g., in 2009, when Comet 217P Linear crossed the belt ofOrion, M42 / M43, magnitude 9.8). However, the latterstays pure speculation since it is impossible to recalculateall comet paths so far back in the past.

tropical year is about 20 minutes shorter (yt ≈(1−1/26000)ys ≈ 365,2422 days).

2.1 The tropical yearHow accurately could (theoretically) Stone Agepeople measure yt or ys? If one measures thelength of the shadow of a large and stable object(e.g., of an obelisk), yt is the time difference oftwo subsequent points of time where the shadowlength is minimal. In countries like Egypt with“guaranteed sunshine” in June this would imme-diately lead to the result yt ≈ 365d. Measuredover 4 years by a priestly caste, the result wouldalready be yt ≈ 365,25d, and this result wouldbe more and more confirmed over the decades.The ancient Egyptians knew yt very well and hadleap years every fourth year ([4]).

2.2 The sidereal yearThe sidereal year seems a bit harder to be de-termined. It can be measured as the time differ-ence of two subsequent points of time where sunand stars are in the same relative position. Sunand stars can only be seen together at sunrise orsunset. In these few minutes stars begin to glowor fade. The sun rises and sets every day at an-other position on the horizon and the stars beginto glow or fade “somewhere” above the horizonon their orbit which makes a comparison inaccu-rate.

At certain locations, however, some stars orstar clusters begin (or began) to glow or fadeat very special positions on specific days of theyear.

2.3 A special constellation of the PleiadesIn Teotihuacan (≈ 50 km northeast of MexicoCity, 19 41’ 52.56¨ N 98 50’ 40.46¨ W, Fig-ure 3) with some of the largest pyramidal struc-tures built in the pre-Columbian Americas, thewell-visible Pleiades went through the zenith ev-ery day at around 400 A.D. (today this is not thecase anymore, due to the precession movementof the Earth). This phenomenon was visible tothe spectator during August 13 and January 19 –the rest of the year the sun was already above the

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Figure 3: The temple town of Teotihuacan hadits golden age 450-650 A.D. The city name“birthplace of the gods” was given by the Aztecswho discovered the deserted town centuries later.

horizon when the Pleiades passed the zenith.

Figure 4: This Google Earth image of Teoti-huacan is rotated 90◦ and thus the “Street of theDead” (name given by the Aztecs centuries later)has exactly the direction of the sunset on Au-gust 13 in northwesterly direction.

So people only had to wait for the day whenthe Pleiades first “made it” up to the zenith atsunrise2 (this was the case on August 13). In or-der to predict the day, it was a good idea to markthe direction where the sunset was the eveningbefore. This sunset was in northwesterly direc-tion, with a deviation of 15,3◦ from West. Sur-prisingly, the whole temple town of Teotihuacan

2The durance of a complete rotation of the Earth is23h56’. Therefore the stars appear / disappear in the aver-age four minutes earlier on the horizon every day.

is adjusted precisely 15,3◦ from North in north-easterly direction (over a length of 2,4 km, com-pare Figure 4). According to [6] the exactly per-pendicular direction to the setting sun on Au-gust 12 was marked by two stone crosses in 3km distance. When observing from the east-erly cross the setting sun disk coincided with thewesterly cross on August 12. In the next morn-ing the spectacle might have been celebrated,however, there are no written documents aboutTeotihuacan at all. The only coincidence is thatthe Maya calendar starts on an August 133.

S

Pleiades

zenith

horizon

ecliptic

N

ϕ

Figure 5: The Pleiades fade at sunrise exactlyin the zenith above Teotihuacan (ϕ = 19,7◦)on August 13, 400 A.D. The archaic societiesof Mesoamerica believed that the present worldwas created on a 13th of August. This date isperfectly suitable for precise measurement of thelength of a sidereal year (365,26 days).

Even without any speculation, the specific

3The people of Teotihuacan were not Mayas. Never-theless, the Maya calendar has its roots in the historicaldevelopment of the Maya’s predecessors in Mesoamer-ica. Furthermore, celebrations in context with the Pleiadeshave been reported and still happen in Mesoamerica.

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date August 13 was perfectly suitable for themeasurement of the length of a sidereal year.Once the exact direction of the sunset on Au-gust 12 was marked, one only had to wait untilthat would happen again. Since a given sunsetdirection is ambiguous, it happens two times ayear that the sun sets in such a direction – inour case (August 12 is approx. 50 days afterthe summer solstice) around May 1 (50 days be-fore the summer solstice). This date had to beskipped, of course.

So, within years, the astronomers of thepriestly caste could count again and again thedays in-between and come to a more and moreaccurate result for the value ys of the siderealyear. On average by counting only 365 days thenevery fourth year was no longer accurate. Atleast from the Maya calendar we know that theMayas inserted one day every four years [4]4.

2.4 The precession movement must have beena problem

There was one fact, however, the Stone Ageastronomers did not know about: The axis ofthe Earth – in Teotihuacan under elevation an-gle (=latitude angle) ϕ = 19,7◦ – is subject tothe precession movement (Figure 6) and there-fore the celestial north pole rotates about the nor-mal n of the ecliptic plane π1. In the special caseof Teotihuacan this meant that the declination δ

of the Pleiades (the elevation with respect to thecelestial equator c) changed from 19,7◦ in 400A.D. – with the useful coincidence ϕ = δ , sothat the Pleiades were in the zenith – to 20,7◦ in1000 A.D. (currently it is ≈ 24◦). Although onedegree seems to be comparatively little: It is ap-proximately the double diameter of the Pleiades.The measuring of the length of the sidereal yearwould still have worked somehow, but the townwas deserted for unknown reasons.

Interestingly enough, the new center of

4The Maya calendar had 360 days (18 months with20 days each) plus a short month – usually five “unluckydays” with corrections every fourth year and probably alsoadditional corrections within larger periods of time.

an⊥ π1

cS

sun

V

π1

N

E

N. . . North celest. poleV . . . vernal equinoxc. . . celestial equatorE. . . Pole of π1

Figure 6: For investigating Stone Age celestialgeometry the precision movement of the Earthhas to be taken into account.

⊥ to sun in zenith

⊥ to Pleiades in zenith

⊥ and ‖ to equinoxes (Kukulkan)

‖ to summer solstice

Figure 7: Some temples and platforms inChichen Itza very precisely indicate importantsunset directions (Google Earth image).

Mesoamerica became Chichen Itza at ϕ = 20,7◦

in approx. 1000 A.D., where the measuringagain worked perfectly. The angle to the spe-cific sundown the evening before the Pleiadesfade in the zenith at sunrise changed slightly inChichen Itza and was marked very precisely and

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similar to Teotihuacan by means of the sides ofa long platform (Figure 7) – again perpendicu-lar to the direction. Another platform directionis perpendicular to the direction of the sunset onthe days when the sun is in the zenith. Two otherbuildings indicate accurately the direction of thesunset at the summer solstice and the equinoxes(Temple of Kukulcan).

2.5 Venus cycles

Figure 8: The inner planets (with Venus closestto the Earth). The proportions of the diameters,the distances from the sun and the obliquitiesfrom the ecliptic plane are correct. In the de-picted constellation, Venus would be visible asmorning star.

Planet observations are much more compli-cated than the relatively simple rotation of thestars, since we have an additional rotation of thecorresponding planet around the sun (Figure 8).Coincidentally, a Venus-year is almost exactly5/8 of the sidereal year5. Therefore, when theMesoamerican astronomers watched the “Au-gust 13- phenomenon” (Pleiades in zenith at sun-rise), they realized that every eighth year Venuswas in the same relative position on the sky. Inthe average, in 50% of the observations at thatspecial sunrise, Venus was already above thehorizon (“morning star”), in the rest of the cases

51 Earth year equals ≈ 1,599 Venus years.

it was already visible on the 12th of August be-fore sunset (“evening star”).

In consequence, the astronomers developed avery accurate calendar for the Venus cycles. Ev-ery once in a while, i.e., sometimes even afterdecades, they had to insert or delete a day in thecalender which got more and more refined ([4],[7]). For the Mayas, Venus positions were of ex-treme importance, and each king had to ask theastronomers first before he decided anything im-portant.

3. A CONTEMPORARY GEOMETRICALAPPROACH

Today sun, moon, planets and stars are notgods anymore, which they certainly were con-sidered to be by the archaic societies. Wenow know very much about the rather compli-cated movements in our solar system. Besidesthe Earth’s uniform spin (at least within several10000 years) around its axis and its non-uniformrotation along an ellipse – which is neverthelessoptically hard to distinguish from a circle – thereis the precession movement and the nutation.

All these variables have to be consideredwhen we judge the very precise markings of an-cient societies. Half a degree and less is alreadya major error when we consider that directionswere set by stone markers with a distance of upto several kilometers.

Despite the complexity of the composedmovements, we can use a well approximatingformula for the rising or setting sun. It is themore accurate, the closer we come to the equa-tor, but it can easily be used up to a latitude of52◦ (which includes most popular sites includingStonehenge).

If we consider all sun rays during the daythrough a fixed point, they will generate a gen-eral cone which is in a good approximation acone of revolution with an axis parallel to theEarth axis, especially around the solstices. Thehalf opening angle σ varies in between 90◦±ε atthe solstices and is exactly 90◦ at the equinoxes.The Earth’s axial tilt ε , also called the obliquity

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a

c

sun

Vh

NZ

ϕ

S

N. . . north celestial poleV . . . vernal equinoxc. . . celestial equatorZ. . . zenithh. . . horizon

Figure 9: The axis through the celestial poles isparallel to the Earth axis a and inclined under theelevation angle ϕ .

of the ecliptic plane, is currently≈ 23,44◦, how-ever this value is varying due to the so-called nu-tation in a 41000 year cycle between 24,5◦ and22◦.

ϕ

ψ

σ

h

h

xy

z

1

3

2

O

~s

~a

Figure 10: The coordinate system for the calcu-lation of sunrise and sundown.

In a Cartesian coordinate system (origin O)with horizontal (x,y)-plane and y-axis in Northdirection (Figure 10), the axis has the direc-tion ~a = (0,cosϕ,sinϕ) (where ϕ is the lati-

tude). Let ψ be the angle between the directionto the setting sun and the North direction, then~s = (sinψ,cosψ,0) is the corresponding vectorto the setting sun. The angle between the bothunit vectors is σ . With cosσ =~a ·~s we have thesimple relation

cosσ = cosϕ cosψ (1)

which allows to calculate ψ for given σ and ϕ

(the formula is exactly the “spherical Pythago-ras” in the rectangular spherical triangle 123with side lengths σ , ϕ and ψ). The value of σ

is “trivial” at the solstices (σ = 90◦± ε) and theequinoxes (σ = 90◦), else it has to be calculated[2].

Figure 10 also illustrates how an observer Owould have seen the top of the Pyramid ofKhafre (Chephren) under elevation angle (=lati-tude) ϕ = 30◦ from a viewpoint exactly in dis-tance h south of the pyramid, where h is theheight of the pyramid. Looking in this direc-tion, the sky seems to rotate around the top ofthe pyramid. On the night sky, behind the topof the pyramid, there was the former pole starThuban in the constellation Draco.

4. HOW TO MEASURE LUNAR PERIODSWITH STONE AGE METHODS

The tilt of the moon’s axis induces an additionalprecession movement of the moon with a periodof 18,60 years. It leads to the so-called lunarstandstills where the winter full moon rises ex-tremely high or low respectively. It has to betaken into account to understand the height ofthe moon relative to the horizon. The timing ofeclipses – which was of great importance for an-cient societies – is also part of such considera-tions: A solar eclipse never happens at the timesof lunar standstills. It rather occurs (like the lu-nar eclipses, Figure 11) in between the stand-stills.

An adaption of Formula (1) holds in good ap-proximation for the rising and setting of the lu-nar standstills since then the elevation angle onlychanges marginally during one day. We just have

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S

Mecliptic

c

zenith

horizon

ecliptic

N

ϕ = 51,2◦

Figure 11: Winter full moon in Stonehenge atmidnight: The sun culmination 12 hours beforewas very low (≈ 15◦) – the opposite full moonM, however, makes it high up – comparable tothe sun at summer solstice. The “snap shot”was taken in between two lunar standstills: Themoon’s orbit plane contains the sun: A potentialcandidate for a total lunar eclipse!

to add or subtract the moon’s axial tilt εM withrespect to the ecliptic plane (currently ≈ 5,2◦,5000 years ago ≈ 6◦).[1], [3]

4.1 The moon and the Great Pyramids of GizaThe Great Pyramids are located at a latitudeof almost precisely ϕ = 30◦. When they werebuilt 4500 years ago, the obliquity of the eclipticplane was ε = 24,0◦, and the additional tilt of thecarrier plane of the moon’s orbit was εM = 6,0◦.Thus we had the coincidence

ϕ + ε + εM = 90◦

4500 years ago in Giza. (Today the value of thesum is ≈ 1,5◦ less.) There is no doubt that theancient Egyptians noticed the following care-fully:

Every 18,6 years, the winter full moon stoodin the zenith above the pyramids of Giza.

The problem is that the winter full moonwould almost never be at its maximal height ex-actly at the winter solstice which was knownvery well. If such a constellation happens, it willtake hundreds of years to repeat again. Thereforeit will have taken a long period of time to figureout the duration of the cycle at least to one digitafter the comma.

4.2 Stonehenge and the length of a full lunarcircle

Figure 12: Stonehenge: The inner horseshoepattern contains of 19 bluestones. The symme-try axis points in northwesterly direction to thesunrise at summer solstice and in southwesterlydirection to the sunset at winter solstice.

Stonehenge (latitude ϕ = 51,2◦) was builtwithin centuries, but some parts were built at thesame time as the Pyramids of Giza. Thereforewe can use comparable numbers for the tilts ε

and εM and we have a maximum elevation an-gle of the winter full moon ϕ + ε + εM ≈ 67,8◦

(major lunar standstill). This is quite impres-sive for the latitude (the sun only made it upto six degrees less at the summer solstice, Fig-ure 11), and the duration of the full moon nightcomes close to 16 hours. Without any furtherspeculation we give some facts: Stonehenge hasseveral rings of stones and holes. The inner

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horseshoe pattern consists of 19 bluestones (Fig-ure 12). 56 = 3 ·18,67 Aubrey holes lie outsidethe sarsen ring but inside the circular ditch (Fig-ure 13, [3]).

Figure 13: Stonehenge (Google Earth): Fromthe satellite one can clearly see another directionin northwesterly direction. This direction leadsaccurately to the moon rise at minor lunar stand-still.

The symmetry axis of Stonehenge preciselyindicates the solstices (Figure 12). The mini-mum elevation of the winter full moon is ϕ +ε − εM ≈ 55,8◦ and might as well have playeda role for the builders of Stonehenge, since fromthe air one can see clearly another direction devi-ating from the symmetry axis (Figure 13). It ac-curately directs in northeasterly direction to therising moon at minor moon standstill ([9]).

5. CONCLUSIONUnderstanding astronomy geometrically has al-ways been fundamental for deeper insights – es-pecially when it comes to archaeo-astronomy.Archaic cultures had no computers, but with thehelp of accurately marking of directions, i.e.,drawing straight lines, they could achieve results

that are amazing even for our society.

REFERENCES[1] G. Glaeser. Himmelskunde anhand von

Monumentalbauten fruher Zivilisationen.Informationsblatter der Geometrie (IBDG),pages 28–33, 2009.

[2] G. Glaeser. Geometry and its Applicationsin Arts, Science and Nature. Springer - Edi-tion Angewandte, to appear in Spring 2013.

[3] D. Greene. Light and Dark: An explorationin science, nature, art and technology. In-stitute of Physics Publishing, 2003. ISBN0750308745.

[4] B. I. Gutberlet. Der Maya-Kalender.Luebbe Verlag, 2009. ISBN 3431037909.

[5] C. R. I. Morley. The Archeology of Measure-ment. Comprehending Heaven, Earth andTime in Ancient Societies. Cambridge Uni-versity Press, 2010. ISBN 0521135885.

[6] B. Steinrucken. Zur astronomis-chen Orientierung von Bauwerkenund Stadten der Azteken, Mayasund Inkas im prakolumbischenAmerika. pages http://www.sternwarte--recklinghausen.de/files/aztek_maya_inka.pdf,2007.

[7] Wikipedia. Maya calendar. pagehttp://en.wikipedia.org/wiki/Maya_calendar, 2012.

[8] G. Wolfschmidt. The pleiades and hyadesas celestial spatiotemporal indicators in theastronomy of archaic and indigenous cul-tures. Beitrage zur Geschichte der Natur-wissenschaft 3, pages 12–29, 2008.

[9] J. S. Young. Moon teachings for themasses at the umass sunwheel and aroundthe world: The major lunar standstillsof 2006 and 2024-25. page http:

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//www.umass.edu/sunwheel/pages/moonteaching.html, 2010.

ABOUT THE AUTHORGeorg Glaeser is full professor of Geometry atthe University of Applied Arts Vienna (Instituteof Art and Technology) since 1998. He studiedmathematics and geometry at the Technical Uni-versity of Vienna and habilitated there in com-putational geometry. He is the author of severalbooks about mathematics, geometry and com-puter graphics. He can be reached by e-mail:georg.glaeser@uni-ak.ac.at. Moreimages and videos to this paper can be found onthe his website http://www1.uni-ak.ac.at/geom/staff_gg.php.

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