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Abstract and Figures

The paper presents both an introduction to the theory of shadows and their mathematics, with a concentration in the cast-off shadows from megalithic Neolithic monuments. It is addressing lengths of shadows' dynamics above the Tropic of Cancer at the Northern Hemisphere. A companion paper presents the length of shadows' dynamics at and below the Tropic of Cancer here:
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The Mathematics of Monoliths’ Shadows
Dimitrios S. Dendrinos
Emeritus Professor, School of Architecture and Urban Design, University of
Kansas, Lawrence, Kansas, USA.
In Residence at Ormond Beach, Florida, USA.
January 24, 2017
Table of Contents
Introduction: On Neolithic Shadows
The Geometry of cast-off shadows’ lengths: the -functions
Cast-off shadows’ lengths at Solstices and Equinoxes
Cast-off shadows’ locational dynamics
Legal Notice
Irunnari Menhir (Elazmuno) at Navarra, Spain
A central feature of this paper, the role of shadows in Archeology and Neolithic Architecture, has
been first recognized and elaborated by the author in two previous papers. One was on Carnac’s
Le Grand Menec monument, cited in [1]; the second was on Gobekli Tepe’s structures C and D
(Layer III) cited in [2]. These two papers constituted a dual effort: first, to analyze the role that
shadows apparently played in the design of Neolithic monuments and demonstrate that shadows
of standing stones in specific and structures in general were an integral part of the monuments’
architectonic design; and second, to bring the subject to the forefront of archeological research
and make it a branch of the studies in Neolithic Architecture. Here, a more complete coverage of
the underlying Mathematics (in both Algebra and Geometry) of monoliths’ cast shadows and
their daily choreography is attempted. In specific, the length of menhirs and free standing stones
(columns, pillars or orthostats) cast shadows during the day (under sunlight, but it can also be
equivalently applicable for the night under moonlight), and over the various days of the year (and
month, in the case of moonlight resulting shadows), are plotted and the derived mathematical
functions are analyzed. The author discovered that in the case of sunlight derived shadows the
mathematical functions describing the monuments’ shadows differ (and at times significantly)
over different periods of the year, the four seasons. A particularly critical transformation in these
mathematical functions occurs during the Spring and Autumnal Equinoxes.
To the author’s knowledge, this is the first attempt ever undertaken to comprehensively analyze
and measure standing stones cast shadows’ lengths and positions and their daily (and nightly)
movements. Of course, in the past, isolated cases involving snapshots of various standing stones
(and/or gnomons) cast shadows’ lengths and their motions have been recorded, and in some
instances their study has been extensive. The subject of sundials is just an example of such
extensive work on shadows by archaeo-astronomers. How shadows have been recorded and
utilized in a number of contexts over the past two and a half millennia has also been sporadically
mentioned in conventional archeological and mathematical literature. In that regard,
Eratosthenes’ computation of the Earth’s circumference (using Classical Geometry’s basic
properties of triangles) in the 3rd century BC is also an example.
The study reported in this paper presents a new mathematical family of functions, designated as
the set of -functions, to study shadows. The full and complete exposition of the mathematical
properties of this set of functions, and their implicit spatial and temporal dynamics, is left to
further future research. Only the elements and essential basics of this set of -functions are
presented here, since this is a paper addressed to the general public, and not to mathematicians
or astronomers.
Although the paper deals with the simple case of a menhir (reduced to a gnomon), the concepts
are general and could apply to more complex forms of Neolithic monumental structures. The
propositions advanced here are testable, and commercial computer programming applications
of the suggested theory and mathematical formulations are possible.
Introduction: on Neolithic shadows
The subject of cast shadows is a topic initially brought to the attention of contemporary analysts
(with interest in Archeology) by William Chappel in 1778, see Michell [3]. The role of shadows is
also mentioned (although sparingly) in the Astronomy seen implanted in the megalithic
monuments of the British Isles by Alexander Thom [4]. For a more complete survey of the
literature and an introduction to the subject of the role shadows play in the study of certain
archeological sites and specifically on the shadows cast by a number of megaliths in those sites,
see the paper by David Smyth [5].
Of course, the role of free standing stones (be those isolated menhirs, formations of megaliths in
stone enclosures or cromlechs, dolmens and the like) in the performance of the monuments’
various functions (be those ceremonial, or in their positioning associated with various
astronomical alignments, or simply Metrology related) has been recently covered by numerous
archeologists, mathematicians, archaeo-astronomers etc., in the context of numerous Neolithic
and Bronze as well as Iron Age monuments. References [3] [4] and especially [5] and their cited
references contain a rather complete set of citations on the subject. Since the objective of this
paper is not to supply a comprehensive survey of this topic, not much will be discussed or added
here regarding these case studies. The reader is directed to the references in [5] for more
information on certain isolated efforts to deal with shadows in monumental Architecture. In so
far as how the expected shadows cast by monoliths have directly affected the architectural
design of the monuments themselves during the Neolithic (and beyond, down to the design of
monuments in Classical Greece) the reader is referred to the papers in [1] and [2] written by this
author. It is an extension of these two specific papers that this paper is written.
One may approach the subject of shadows on megalithic structures from numerous angles. A
number of those angles are discussed in this paper at various lengths. Every monolith, be that an
isolated menhir, an exposed orthostat in the context of a stone enclosure (a cromlech), or an
orthostat at a partially filled masonry structure (like structures A, B, C and D at Gobekli Tepe or
one of the central pillars of these four structures); all these cases of standing stones contain on
them carry-on as well as cast-off them shadows, either under direct sunlight or moonlight.
If a part of a monolith’s total surface is (and only a part can be at any given point in time) exposed
to sun or moon light, then an equal part of it (projected on a surface, discounting for topical
irregularities or stone surface anomalies) is in darkness. This is a fundamental duality in
monoliths’ shadows. To every surface section that is in light, there corresponds an equal surface
section that is in darkness, the smoother the surface the more so. During the day (or night) these
carry-on lighted or dark areas undergo change. Thus, the monolith undergoes morphological
transitions dependent on the stone’s form. The shadows’ dynamics, in effect imprinted on them,
render the stones living organisms. Undoubtedly, these transitions and their cultural symbolism
(whatever that might have been at any particular point in space-time) didn’t go unnoticed and
certainly were accounted for as to their expected effects by the architects of these structures.
But no matter how important to the architect of the monolithic structure the carry-on light and
shadow dynamics manifested on each and every freestanding stone, a much more important role
is played by the stone upon its immediate environment, and the surrounding ground supporting
it: the overall stone’s shadow as cast on the ground and on the potentially neighboring monoliths
or structures around the monolith in question. In the daily and nightly cyclical motion, these
shadows exhibit both a changing form, as their surface area on the ground or the surrounding
structures changes through the day (or night); as well as through their motion in space, a
repeated choreography. The choreography’s daily dynamics are qualitatively equivalent but
never identical in the course of a year. They undergo subtle and smooth transitions on a daily
basis, but at some critical days of the year as we shall see, these transitions are sudden and
significant. From all of these angles one can approach shadows, a specific element of the
shadows’ architectonic performance and role in the monuments’ design is to be analyzed in this
paper, namely the shadows’ changing lengths during the day, and over the year.
In two specific cases examined in some detail by this author, it was found that the positioning of
monoliths (orthostats and pillars in the case of Gobekli Tepe’s structures, see [2], and strings of
free standing stones in the case of Carnac’ Le Menec and its allied monuments, see [1]) the
architect wanted to convey information to the relevant public at the time, i.e., pass on cultural
messages associated with the various social functions performed by these monuments.
Moreover, due to these shadows, the architect designed the monument to accommodate the
shadows’ movements, their daily as well as the annual dynamics. It is of little doubt that the sizes
and form of structures C and D (as well as all other structures, either already uncovered or waiting
to be unearthed at Gobekli Tepe) were determined by the dynamics in the choreography of these
monoliths’ shadows. As is of little doubt that a critical part in the determination of distances
between monoliths at the strings of stones at Le Grand Menec, Kermario, Kerlescan, and Le Petit
Menec at Carnac, as well as distances among the strings themselves, were the shadows’ lengths
at some critical hour and day of the year, and daily as well as annual changes of cast-off shadows’
daytime (possibly nighttime as well, under moonlight obeying lunar cycle) lengths.
But more than this direct effect upon the very design of these monuments, shadows played
through that daily dynamic movement and by the yearly cycle in those daily motions another
important role: they were senders of astronomical information. The observant, intelligent and
alert architect-astronomer of the monument received information from the shadows’ daily and
yearly motions. And that information transcended the specific locale and time period. The
valuable information obtained had mathematical and astronomical worth of greater importance
and held more profound significance than local narrow socio-cultural conditions, functions and
purposes. It was information of significant durability and relevance over a far greater in space-
time framework. In an effort to identify that information and depict its import, going beyond
what has already been discussed in [1] and [2], we now turn to examining the Mathematics
(Geometry, Algebra and Calculus) of these shadows. For sure, that offered them information
regarding the length of the seasons in a year. Potentially, it could offer them information
regarding the Astronomy of the Earth, and certain celestial bodies – the Sun and the Moon.
The Geometry of cast-off shadows’ lengths: the set of -functions
Before entering the subject matter and prior to presenting the results that ensue, a number of
points need be made. At the outset, it must be noted that no matter the provisions locally made
at any point in space and time of the past, to produce smooth, flat and perfectly horizontal
surfaces to raise either menhirs or freely standing stones and to accommodate as well as measure
their shadows’ varying lengths and surface area, no place on the surface of this Earth affords such
perfect conditions over any spatially and temporally extended area or horizon. Local ground
anomalies and imperfections, as well geological changes over millennial time scales, constantly
and at times abruptly and dramatically alter the local shapes of the ground.
This set of constraints could be a blessing as well as a curse to the local in space-time architect.
A curse, because it could not offer ideal ground conditions to obtain the theoretical results
mentioned in the following subsections of the paper. At the same time, these ground anomalies
could be a blessing, as they would either accentuate and enhance or diminish and de-emphasize
the lengths of shadows at particular points in time (day, month or year). Moreover, these ground
anomalies could add cultural and architectural information to these shadows as well as to the
monument itself. In this context, these metamorphoses could potentially be subject to design
exploitation by the architect of the monument, as they could either elevate or subdue the
shadows’ import (for instance, length) at certain times and spaces.
In analyzing and discussing shadows, a fundamental point must be kept in mind at all times.
Shadows appear as a result of an apparent fast movement of the Sun and the Moon over the
location’s horizon. In actuality, the shadows’ daytime motions are due to the Earth’s daily fast
rotation (from West to East) about its axis, as well as the Earth’s slow annual orbiting of the Sun.
Nightlight shadows are due to the Earth’s daily rotation about its axis, as well as the relatively
slow (monthly) dynamics of the Moon’s orbiting the Earth. Thus, the moonlight derived motion
in shadows are more complex in their daily, monthly and annual choreography than the sunlight
due daily and annual motions. Here, only sunlight derived shadow motions will be analyzed.
As to when in space and time, it became known to the architect-astronomer of the Neolithic (if
it actually ever did) that the arcs traced by the Sun and the Moon over the horizon are “apparent”
and not actual, is a subject still under investigation. It is the author’s hunch that it became
“apparent” long before we do think so, namely at the time of Aristarchus, who is considered to
be the founder of the Heliocentric System. It has been argued by the author, see [7], that the
heliocentric system was known potentially a century before Aristarchus, possibly before Callippus
of Cyzikus and even at the time of Philolaus the Pythagorean (470 385 BC), see [7] p. 17 for
more documentation. Whether however, such a proto knowledge of an “Earth orbiting the Sun”
and a “Moon orbiting the Earth” and “an Earth rotating” system was understood by the Neolithic
architect is very unlikely, although not impossible. Moreover, when the realization of an angle to
the Earth’s rotational axis became reality, let alone estimated and finally measured is also subject
to further research, and not entirely settled either. See [7] for some reference on this subject.
Although, in the following subsections, a static view of the daily cycle in recording the changes of
a standing stone’s cast shadow length is presented, at some reference (abstract) location, the
reader should infer (and the paper will discuss to some extent) annual dynamics in certain
parameters of the three functions (to be referred to as -functions) depicted here. Moreover,
the spatial dynamics of these functions can be analyzed, as the point of reference for deriving
them moves in space by changing its latitude. Their full analytical treatment is left to future
research. Of course, these dynamics are picked up by the changing values in the parameters of
the mathematical functions presented. These variations depend on the underlying Earth related
Astronomy, and in particular on the latitude of the monument on the Earth’s surface, and the
angle of the Sun (or the Moon) above the location’s horizon at a particular hour of a particular
day of the month or the year, all simple functions of the location’s latitude also. Although these
angles are quite straightforward and simple to obtain in the case of daylight Sun based shadows,
as they repeat on a yearly cycle, they get far more complex for the case of nightly moonlight
based shadows, as the Moon obeys various short term (moon phases related) as well as very long
term (Metonic type) cycles. More on this in the Conclusions.
The reader of this paper is assumed to possess some astronomical information. For instance,
what is the plane describing the azimuth of a location’s sun/moonrise or setting during Sun-
related Solstices and Equinoxes, and Moon’s so-called Standstill. Their connections are assumed
to be commonly possessed knowledge. Not much more will be asked by the reader to know, in
order to follow the basic tenets of the paper. Some of the discussion here is related to the sundial
literature. The reader should be somewhat familiar with the elementary aspects of that rich and
extensive body of work, and some of these essentials are found in [1]. The rod used in sundials is
referred to as a “gnomon”, and in this paper for simplicity all megaliths (be them menhirs or
orthostats, pillars, columns etc.) will be considered to be simply a rod, that is a thin linear
gnomon. The tip of the gnomon is referred to as a “style”, and this is the point at the very top of
a megalith that we shall be concerned with in this paper.
The discussion will assume that the ground’s plane is perfectly smooth and flat, locally horizontal,
but not infinite in extent along both of the two surface axes. These two axes, (X and Y, see Figure
2) will be considered also as the basis of the Cartesian space over which two dimensional vectors
are positioned and their lengths measured. As for the spatial extent of the surface, it is a critical
assumption, in that it is not assumed here to be infinite in extent – a fact which bears directly on
the form of the identified -functions. This is a strong assumption and its full implications are
discussed in the next subsections of the paper.
Surface smoothness implies that the horizon is unobstructed by any physical imperfections or
other obstacles. This is of course a weak assumption. Further, the monolith (rod-gnomon) will be
assumed to be located vertically to the ground’s plane and to be linear (one dimensional). Vertical
positioning of course implies that the monolith-gnomon points towards the Earth’s very center,
and that the flat (locally horizontal but not infinite) plane of reference (over which the various
locations’ azimuth angles are stated) is at an angle to the Earth’s equatorial plane and to the
Celestial equatorial plane, what astronomers refer to as the “ecliptic”. These two different planes
intersect at a line going through the spherical (this is an approximation of course, since the Earth
is an ellipsoid not an exact sphere) Earth’s center. The locally horizontal plane, tangent to the
(assumed to be) spherical Earth has an angle in reference to the Earth’s equatorial plane, which
is a function of the specific location’s latitude. Of course, variations or deviations from these ideal
conditions can be dealt with as extensions to this work, especially the relaxation of a finite plane
assumption and the establishment of some curvature about it (obviously, the Earth’s curvature).
All such complexities are left to future work and to the interested reader.
In the text that follows, the distinction between the two meanings of the term “day” must be
kept in mind: the 24-hour long “day” (to be designated as “24h-d”); and the day (versus “night”)
duration, from sunrise to sunset (to be designated as “Srs-d”); obviously then, the night’s
duration, to be designated as “N-d”, is simply {N-d = 24 – (Srs-d}}. By the term “daily” cycle, the
24-hour cycle will be implied.
Earth related Astronomy is an extensively covered and complex field, to which obviously not all
readers can be assumed to have had full exposure and knowledge. Some essential elements, for
following the exposition of this paper, are presented in Figure 1. The diagram in that Figure
schematically shows how the angle of the Sun from the horizon, at noon time, varies over the
year, and its connection to the Equinoxes (Spring and Autumnal) as well as the Solstices (Winter
and Summer) for both the Northern and Southern Hemispheres. All this of course, as well as the
associated issue regarding the presence of seasons in a year are related to the approximately
23.5 from the perpendicular in the Earth’s axis of rotation. The Earth’s axis of rotation is
undergoing an approximately 27,000-year cycle, but the real effect of this cycle is not of import
at this stage for the discussion of the monuments’ shadows. It has been determined that it hasn’t
seriously affected the monuments’ various related azimuths thus their shadows since the
Neolithic times. The 23.5 tilt, what astronomers refer to as “obliquity”, is behind seasons, and
its effects are recorded in Figure 1. They are depicted by the sinusoidal in form function shown.
The sinusoid function in Figure 1 is location-specific (that is, it depends on the position of a
specific site at a certain latitude on the Earth’s surface). Next to it, the site at the exact opposite
location (symmetric in reference to the, assumed to be perfectly spherical, Earth’s center) is
shown. Line E depicts the day of Equinoxes, both Spring and Autumnal; line S above line E depicts
the Winter Solstice day for the specific location at the Northern Hemisphere (the segment in red)
and the Summer Solstice for its corresponding opposite site on the Southern Hemisphere (the
segment in green); correspondingly, the E line below the S line depicts the Summer Solstice day
of the Northern Hemisphere site (line segment in red), whereas the green segment of the line
depicts the Winter Solstice day for the opposite point at the Southern Hemisphere. The cycle
repeats to both left and right of this section of the sinusoidal function.
Notice a fundamental connection linking the points of the two Solstices. The Summer Solstice of
a point at the Northern Hemisphere on the sinusoidal function coincides with the point of the
Winter Solstice at the opposite point on the Earth’s surface at point K. Of course, there is an
equivalent one at L, Figure 1. The equivalences existing between these two points (opposite in
reference to the Earth’s center, on the Earth’s surface) is essential in understanding the lines as
drawn depicting the sunrises and sunsets at Solstices in the Figure 2 which follows, associated
with the azimuths of any location. Of course, this is due to the fact that sunset at a location on
Earth’s surface is sunrise at its opposite point, and vice versa for its sunrise.
In Figure 1, the point H (red) picks up the (highest) position of the Sun from the horizon in the
Northern Hemisphere; whereas point H (green) depicts the highest position of the Sun from the
horizon in the opposite point at the earth’s surface at the Southern hemisphere occurring at noon
time. Obviously, through the year, as the Earth orbits the Sun, the position of the horizon’s
surface of the site in question moves up or down between the two S lines. Notice, the curves (AE,
H, SE) and (AE, H, SE) are arcs, and as drawn the Northern Hemisphere is represented by a convex
arc, whereas the Southern Hemisphere by a concave one.
In general the points of inflection, designated as AE (the point of Autumnal Equinox for the
Northern Hemisphere, shown in red) and SE (the point of the Spring Equinox for the Northern
Hemisphere, also in red) and AE (the point of the Autumnal Equinox for the opposite point at the
Southern Hemisphere, shown in green) and SE (the point of the Spring Equinox for the Southern
Hemisphere, in green, and also coinciding with the point of the Northern Hemisphere’s Autumnal
Equinox) is where the tangents to these two arcs are at an angle and not vertical; they become
close to vertical (that is, close to 90) when the arcs become close to semicircles at points close
to the Earth’s Equator. This up and down movement of a location’s horizon (surface) represents
the temporal dynamics of the location, extended over the course of a year. As the horizon moves
up and down between the two S lines of Figure 1, representing the change in seasons for that
location (and the corresponding changes for its opposite point on the Earth’s surface) the rest of
the lines remain fixed.
But there’s also another dynamic involved in the diagram of Figure 1: the spatial dynamic – that
is the changes in the shape of the sinusoidal function as one moves North-South on the Earth’s
latitude (at a fixed longitude). By moving up or down the latitude on the Earth’s surface, of course
the amplitude of the sinusoidal function, that is the distance separating the two lines where
points H (red) and H (green) lie, and thus its form, changes. As one approaches the Earth’s
Equator, the two S lines closely approach the horizontal line E. Moving close to the Equator, the
amplitude of the sinusoidal increases reaching a maximum at the Equator and the two segments
of the sinusoidal function in Figure 1 become almost semicircles with the point of inflection
having a tangent almost vertical, at a 90 angle. At the days of the two Equinoxes, at noon time
at the Earth’s Equator, the Sun is precisely overhead, at a 90 angle. At the Earth’s Equatorial
plane the seasonal variations are minimal.
Figure 1. The Sun’s angle on the Celestial Sphere above the horizon of a reference location at the
Northern Hemisphere and at its opposite location at the Southern Hemisphere: the four seasons.
Source: the author.
On the other hand, at the Northern (and Southern) most points (the two points where the,
assumed perfectly spherical, Earth’s axis of rotation are located) the sinusoidal almost collapses
(and so are the two S lines) to the E line. At these two points, the Sun hovers slightly above the
horizon throughout the 24h-d through half of the year (between the Spring and Autumnal
Equinoxes). It never sets at Summer Solstice 24h-d, moving on a circle above the horizon.
Meanwhile the Sun is below the horizon for most of the days within the six-month period
between the Autumnal and Spring Equinoxes, without rising at Winter Solstice at all. These
conditions are reversed for the Southern polar point in reference to seasons.
All that is well discussed and analyzed in the Earth-related Astronomy literature. What follows
however has neither been addressed by astronomers or archeologists, and that is how all this
can be translated into the dynamics of shadows for structures of archeological and architectonic
interest. Next, we take a look at the forms and lengths of shadows for a typical middle Northern
latitude point on the Earth’s surface. It must be noted in concluding this subsection, and primarily
due to the Astronomy-related factors mentioned, that very likely the monuments shadows’ role
was far more important in these middle latitude regions of the earth’s surface than under the
extreme conditions governing both the Norther (and Southern) most as well the Equatorial places
of the Earth’s surface.
Cast-off shadows’ length at Solstices and Equinoxes
What this section of the paper establishes is that the shapes of the cast-off type shadows (their
lengths and locations) are of course different during the various seasons of the year; but most
importantly, the mathematical properties describing these shapes differ from season to season,
and at times considerably. These significant transitions occur at Equinoxes. The section offers
testable hypotheses, which could be commercially exploited through appropriate computer
programming applications to derive the daily cast-off shadows of monuments over a year’s time.
In Figure 2, the horizon is shown for a particular monument-site containing a menhir (monolith
or pillar) at point O (also the origin of a Cartesian 2-d space). The azimuth (angle from the Y-axis)
is given of the site’s location at sunrise (B, D) and sunset (C, E) during the two (Summer and
Winter) Solstice 24h-d’s. The azimuth’s angles are formed by the two green lines BE and CD. What
was mentioned in the previous subsection becomes central now in understanding this diagram.
Points B, O, E lie on straight line BE; whereas points C, O, D lie on straight line CD. These two
lines identify the directions of the two Solstices for the location in question. Horizontal line FG
identifies the direction (and in this case also the points) of the two (Spring and Summer)
Equinoxes’ sunrise and sunset. Angle identifies the northernmost point the Sun rises at this
location during the Summer Solstice. This angle is of course a function of the location’s latitude.
This angle is also the angle that identifies the southernmost point the Sun rises at Winter Solstice.
Equivalent statements apply for sunsets at C and E. Azimuths for all four points can be expressed
as functions of this angle .
More specifically, point B is the sunrise of the Summer Solstice, point D is the sunrise for the
Winter solstice, point E is the sunset for the Winter Solstice, and point C is the sunset for the
Summer Solstice. The horizontal axis X defines the line of the two (Spring and Autumnal)
Equinoxes’ sunrise (point F) and sunset (point G). The line’s length defines this circle’s diameter
thus horizon’s extent, which in turn defines the azimuths for this typical Northern Hemisphere,
middle latitude location. This circle defines and limits the horizon of this specific location on the
Earth’s surface, and the Mathematics which follow use this limitation.
Axes X and Y, besides their role in depicting geographical features of the location in question, can
also be used as the basic two-dimensional coordinates in a Cartesian system where vectors can
be assigned and their lengths be measured, as it will be seen in a moment.
Assume now a menhir located at point O, of some height R. As already mentioned, this menhir
will be assumed to be a one-dimensional (linear) gnomon with its tip (the style) at height R above
point O, O being a point on the surface of this circle’s plane. The diagram in Figure 2 identifies
the various (and differing) trajectories of this style’s shadow on the plane, during different days
of the year. It is these trajectories’ innovative (in both statics as well as spatial and temporal
dynamics) Mathematics, the -functions of Figure 2, what this paper is all about. This way of
analyzing and approaching the role and choreography of shadows has never been attempted
before to this author’s knowledge.
The shadows’ length and motion during the Srs-d at Summer Solstice. This motion is depicted
by the (1) function of Figure 2. At the limits, and from point B in the direction BE at the point
before sunrise, there are no shadows, that is, their length is zero. At the point of sunrise, the
length extends to the end of the horizon, at E in Figure 2. This is one of the interesting
mathematical features of this set of -functions. What follows is of equal interest. The shadow
of this gnomon gradually decreases, at a decreasing rate (the slope, or the function’s first
derivative with respect to x, of the function’s tangent is decreasing, reaching zero at the point
A(1) in Figure 2. This is of course the point at which at noon time during the SrS-d at Summer
Solstice the length {OA(1)} is at minimum minimorum, i.e., not only of the minimum length for
this specific day of the year, but for all days of the year.
Figure 2. The (parabolic) shapes of the shadows’ length during Solstices: Summer Solstice (1),
Equinoxes (2), and Winter Solstice (3), at the reference location. Source: the author.
Some mathematical properties of this (1)-function (and in effect of all three -functions
discussed here and shown in Figure 2). The first property involves lengths of shadows: in the
Cartesian two-dimensional space, for any point M(x,y) on the (1)-function, representing the
monument’s shadow length, according to the Pythagorean theorem measured as the square root
of (x^2+y^2), is greater than the distance OA(1). This function allows one to see how the relative
lengths (in absolute value) x and y decrease as one moves closer to point A(1) from the left of
the Figure, that is as the Sun apparently moves in the pre-noon time period of the day at right.
Equivalently, as the Sun apparently moves in the post-noon time period of the day at the left of
the Figure, the shadows lengths increase to the right of the Figure. A second important
mathematical property is that the -functions are symmetric in reference to the Y-axis. A third
property is that since the height of the gnomon and the distance to the Sun do not change
through the day, the only variable determining the shadow’s length is the angle . It’s rate of
change is exactly equal to the rate of change in the monument’s shadow length. If the apparent
trajectory traced by the Sun in the Celestial Sphere is a parabola’s arc, then the -function is also
a section of a parabola. And vice versa, a point probably reflected in the shapes carved on the
orthostats of the monument (among many other Neolithic petroglyphs at many location in
Western Eurasia) shown in Figure 3.
The -functions have been drawn schematically in Figure 2. The length of the gnomon’s shadow
during any specific point in time during the Srs-d is given on the Cartesian coordinates drawn
vector, linking the origin (point O) with the corresponding point on the -function of the point
on the horizon the Sun is located (at its azimuth). The one-to-one equivalence between a point
on the horizon representing the Sun’s position in the Celestial Sphere and a point on the -
function depends on only two variables: the gnomon’s height (or the position of the style) and
angle the Sun is at above the horizon. Parenthetically, this simple relationship enables the
architect-astronomer of the time to directly compute the angle since both the height and the
base of the right angle involved are easily now measured. In its apparent motion in the Celestial
Sphere, the Sun traces the southern arc BC on the azimuth circle (or the horizon’s circle) for this
monument located at O. For each point on that arc, there is a corresponding point on the (1)
function. In effect, there is a mathematical translation of the Summer Solstice apparent Sun-
traced arc onto the (1) function by way of point O. Whatever mathematical properties
characterize the Sun’s apparent Celestial trajectory in the Summer Solstice Srs-d time is exactly
the mathematical properties which are transferred onto the (1) function.
The shadows’ length and motion during the Srs-d at (Spring or Autumnal) Equinox. Equivalent
statements apply to this case as well, as those of the previous paragraphs. The limit conditions
are identical. However, the shape of the (2) functions is slightly different. It is less convex, but
its minimum attained at noon on (Spring or Autumnal, they are equal in length) Equinox day is
greater than the minimum minimorum attained during Summer Solstice {A(2)>A(1)}. Of course,
this is due to the fact that the Sun at noon during the Equinoxes is not as high in the sky as during
the Summer Solstice case. In other words, the angle is less than under (1) conditions at the
same hour of the day. At these two days of the year (during Autumnal and Spring Equinox), a
morphological transformation takes place as to these -functions: from a parabolic shape facing
down, we now have a parabolic shape facing up.
The shadows’ length and motion during the Srs-d at Winter Solstice. In this case, we have a
significant change in the morphology of the (3) function from the morphology of these
functions in the previous two cases. Now, the curve is concave. Moreover, the minimum shadow
length achieved during the Srs-d is the greatest of all minima (maximum minimorum). This again,
is due to the fact that at noon, the Sun is at its lowest point above the horizon than in any other
day of the year a condition that applies to any location, anywhere on the Earth’s surface, as are
the qualitative properties characterizing Solstices outlined earlier.
Cast-off shadows’ locational Dynamics
Having presented the three cases involving the intra-Srs-d dynamics (i.e., the motion) of cast-off
shadows’ lengths on the (limited) surface plane, one can easily now derive the transitions in their
annual dynamics (put differently, the temporal dynamics), based on the Srs-d snapshots offered
above. As it is obvious from the exposition and specifically the diagram in Figure 2, beyond the
shadows’ lengths that undergo change over the course of a Srs-d, as well as the course of a year,
the location of the gnomon’s style (the place where the tip of the megaliths’ shadows is found)
changes as well. It is this specific feature of the shadows that the Neolithic architect must had
exploited at both Gobekli Tepe and Carnac. See [1] and [2] for more on this angle of the story.
Yearly dynamics at a fixed location (latitude). As the yearly cycle goes on at the specific location
in hand, and the sunrises move on the arc BD, moving from point B at Summer Solstice, to point
F at Autumnal Equinox, to point D at Winter Solstice, back to F at Spring Equinox, and finally
coming back closing the cycle at B during the following Summer Solstice, the -functions go from:
(1) to (2) to (3) to (2) to (1) and thus they complete the cycle. Consequently, the
transitions are reversed in the course of a year. It seems that this rhythmic choreography in the
dynamical motion in space of the monument’s shadows was the underlying frequency recorded
in the design of these monuments. The allegory of the double movement, reversing itself, and
revealing a dual pattern of symmetry is a point not gone unnoticed by the architect of the
Neolithic who exploited this feature of shadows and went ahead and incorporated into the
symbolism (beyond of course the design) of the structures. This is particularly evident in the
symmetric duality depicted in the monolith with the male-female figure symmetry at Gobekli
Tepe, an element extensively discussed in [2].
Another locational aspect of the shadows cast is the feature depicted by the shaded area in the
diagram of Figure 2. It shows area on the ground that surrounds the monument (gnomon), where
shadows from different seasons of the year overlap. There’s a particular area to the north of the
monument, the shaded area of Figure 2, where shadows from the monument exist no matter the
24h-d of the year (although not continuously throughout the Srs-d). The presence of season-
independent shadows’ surfaces and their total area, seem to have played a role in the location
of standing stones at the monuments at Carnac [1].
On the other hand, there are also areas identified by the diagram which are shadow free, no
matter the hour of the Srs-d or 24h-d of the year. Moreover, there are surface areas which
contain only two overlapping sets of shadows, shown by the areas immediately surrounding the
shaded area in Figure 2. And finally, there are areas on the surface plane only covered by a single
set of shadows. The presence of all these differently shaded areas are features of course that
could have been (and might have been) exploitable by the Neolithic architect. Searching for such
features in Neolithic Architecture, and their connection to the design of such structures might be
a very fruitful avenue to future archeological research.
As a parting comment, one must also note that the delineated areas on the monument’s ground
surface plane where not shadows from the monument are cast throughout the year have shapes
of interesting geometric non-linear design. Again, these are shapes naturally produced, and thus
of interest to the Neolithic architect and beyond. No doubt, since these shapes are produced by
the apparent motions in the Celestial Sphere of both the Sun and the Moon, they could (possibly
must) have played some role in religious and cultural symbolism, beyond architectural design.
Locational dynamics of cast-off shadows by changing latitude. Now let us switch to the
locational dynamics of shadows from a broader perspective, that is, how do shadows’ paths
change (for a similar menhir, of similar size and specifications) as one moves up and down the
earth’s latitude. In other words, let us review what happens to the shapes of the three sets of -
functions as one changes latitude. To study these dynamics, one can reflect on the
transformations in the three types of functions shown in Figure 2.
Moving closer to the Earth’s Equator, the BE and CD lines converge towards the FG line (the X-
axis). In effect, the three -functions are compressed towards the GF line. A(1), A(2) and A(3)
move closer to the origin O. At the Summer Solstice 24h-d at noon, the gnomon casts no shadow.
As one goes up the latitude of the Earth’s surface, towards the North Pole (equivalent statements
can be made for the Southern Pole) the BE and CD lines converge towards the Y-axis. Points B
and C converge towards the northernmost point (where the Y-axis intersects the horizon’s circle,
where azimuths are measured); whereas, the points D and E converge towards the equivalent
southernmost point in Figure 2. During six months of the year, between the Autumnal and Spring
Equinox, the gnomon casts no shadow; whereas during the rest of the year (the Spring and
Autumnal Equinox) the gnomon casts very long shadows almost continuously during the SrS-d,
which is close to the 24h-d. At Summer Solstice 24h-d, the very long shadow cast by the gnomon
rotates 360 around the gnomon. Cast-off shadows are more prominent at certain bandwidths
of the Earth’s latitude. Thus, they may have been more prominent and important in the design
of monuments in those bandwidths of both the Northern and Southern Hemispheres.
What this analysis alludes to is that the cast-off shadows may have been more important for the
architect at Brittany’s Carnac monuments than for the Pyramids at the Giza Plateau, or the
Nubian pyramids. This might also explain why we come across many more menhirs in Western
Eurasia that any other place on Earth, including places where obelisks were raised.
Infinite plane and the -functions. The manner in which the paper dealt with the three -
functions is by appropriately truncating them to meet the finite horizon restriction. An extension
of this work could be the theoretical, from a purely mathematical viewpoint possibly without any
direct or real implications for Archeology and/or Architecture, to examine the possibility of an
infinite in extent plane, (i.e., along the X and Y axes) and incorporating a complete (rather than
truncated) set of -functions.
Figure 3. Arcs carved on the orthostats of the passage tomb at Carvinis, Brittany, France. One
may speculate that these arcs are the perceived by the architect-astronomer of the Neolithic
trajectories of the Sun and the Moon over the location’s horizon during different times of the
year (for the Sun) and different days of the month (for the Moon). Also noted is the carry-on
nature of the shadows encountered here (and in all other similar petroglyphs from the Neolithic).
Source: reference [6].
This extension would necessitate creating three -functions, that may be asymptotic to the BE,
CD lines and the FG lines of Figure 2. In this case, function (1) will be asymptotic to the lines OD
and OE; (2) will be asymptotic to the X-axis; and (3) will be asymptotic to lines OC and OD.
With the exception of function (3) which contains point A(3) – a parabolic function in effect –
the other two functions, (2) and (1) must contain two points of inflection each. The
mathematical specifications of these three functions are left to future research and to the
interested reader. A more detailed approach to the Sun’s apparent motion above the horizon
(i.e., the Srs-d dynamics of angle formed by the rising Sun on the way to noon hour peak and
then to the point of sunset) during Summer Solstice may necessitate projecting the horizon as
shown in Figure 2 to points B(1) and C(1). This is a topic left for future research.
The apparent Celestial arcs traced by the Sun (and the Moon) and Archeology. In Figure 3, the
case of a monument is shown where repeated arcs are carved on its monoliths. The rendering of
a set of arcs could very well be the recording of the Sun (or the Moon, or both) apparent paths
in the Celestial Sphere by the Neolithic artist. These arcs (in number and sizes) might represent
specific perceived paths during critical days of the year (and they could be referring to the
apparent Sun’s motion at Solstices and Equinoxes); and/or, they could be referring to lunar
apparent motions at different days of a month. For instance, at standstills, a Metonic type cycle
occurring every 18.6 years, where the declination of the Moon reaches a maximum. The
literature in archaeo-astronomy has focused extensively on astronomical alignments within
monuments’ structures. However, the subject of shadows cast by those very structures during
important points in lunisolar cycles has been largely unexplored. It seems it is argued here that it
is high time for this omission to be corrected.
In this paper, a number of issues were covered and some topics were presented and elaborated
for the first time in the archeological as well as architectural literatures. Novel findings regarding
the shapes of “carry-on” and “cast-off” shadows were discovered. The subject of shadows in
Archeological Architecture is indeed enormous. However, not much interest has been directed
towards it. It was suggested that by looking at these monuments and artifacts (especially of the
Neolithic) in the manner they cast shadows might be very productive. This paper presented an
initial attempt to dig into this relatively new field of Architectural and Archeological interest, and
hopefully to elevate to the prominent position it deserves in both Archeology and Architecture.
A set of functions were uncovered, and their basic mathematical properties established, which
depict the megalithic cast-off shadows’ motion during daylight. The temporal as well as spatial
dynamics of these functions were explored. Continuing from citations [1] and [2], how these
dynamics could have played into the design (site plan and structures’ form), as well as its
symbolism, regarding Neolithic megalithic monuments were subjects the paper dealt with and
touched upon at an introductory and indicative level.
It was discovered that the cast-off shadows obey different functions during the various periods
of the year, the four seasons, and that a significant morphological transformation occurs at Spring
and Autumnal Equinoxes regarding these functions. It was also argued, as a result of the
locational dynamics explored regarding changes in shadows as one moves up or down the Earth’s
latitude, that the role of shadows might have been more important at middle Earth’s latitude
than the locations either close to the Equator or the Earth’s Pole(s). This presence of such
“bandwidths” in Earth’s latitudes of course does not intend to diminish the role or importance of
cast-off shadows in latitudes away from such (fuzzy in essence) bandwidths.
As the paper dealt with shadows being the result of apparent arcs formed by the Sun (and the
Moon, by extension and to an extent) over the horizon, one might ask when did it become known
to the architect-astronomer of the Neolithic (if indeed it ever did) that these shadows’ motions
are the outcome of “apparent” and not actual arcs traced by the Sun (and the Moon). But instead,
that the monuments’ moving shadows and their choreography were the result of the Earth’s
rotation about its axis (the daily) and the Earth’s orbit around the Sun (the annual, seasonal)
motions. A more careful look into some of these monuments from the perspective brought about
in this paper, as well as in [1] and [2] and [7], might provide clues as to potential answers.
Although most of the material covered here dealt with shadows cast by monuments during
sunlight, equally applicable (although far more complex) are the general principles outlined here
for monuments under moonlight. Lunar phases and cycles are very complex subjects in Lunar
Astronomy. Moon dynamics contain lunar phases, the monthly (synodic) cycle (approximately
29.53 Earth days), among other monthly cycles, and the Metonic cycle (approximately a 19-year
cycle) over a longer haul. In that complexity, one may add the Moon’s differing distances from
Earth (due to the Moon’s elliptical orbit around the Earth).
All these factors are of import here, where cast-off shadows are in focus: they affect the amount
of moonlight available, thus the resulting intensity and delineated sharpness of the shadows.
Monuments that were designed to accommodate such moonlight derived shadows must have
paid attention to these ‘very short’, ‘short’ and ‘long term’ cycles their time scales and dynamics.
In that context, the moonlight during a Full Moon 24h-d’s N-d must had played a major role on a
monument’s design, it must be concluded, especially for monuments which clearly had a lunar
cultural or astronomical (and possibly both in essence) focus. Conversely, one (through the study
of such cycles in the monuments’ shadows cast) might be able to determine whether or not a
Neolithic monument had a lunar focus.
Many of the concepts presented here, in both their expressly archeological implications but also
(and possibly especially) in their mathematical as well as astronomical terminology, statements
and exposition, have been kept at an introductory level. These subjects are of course to be further
explored by this author, and anyone else interested in this broad and fruitful subject for research.
It is hoped that such a continuous study might ultimately result in obtaining a deeper
understanding of Neolithic monuments and their Architecture. It must also be noted, in these
concluding section’s remarks that all the above formulations of the -functions can be
computerized and the propositions suggested here empirically tested. There is the potential for
commercial computer applications, where the effects of cast-off shadows from megalithic
monuments can be simulated, and their import evaluated – without in situ measuring the
transformations in shape and length over the day or year (for sunlight derived cast-off shadows)
or month (for moonlight derived cast-off shadows). The hypothesis regarding the presence of a
“bandwidth” of cast-off shadows import can also be tested, under such an application. It is
suggested that both Carnac as well as Gobekli Tepe fall under this “bandwidth” in Earth’s
Northern latitudes.
The study of shadows in Archeology and Architecture (especially Neolithic Architecture) has been
underdeveloped (not to say, undeveloped) and largely unappreciated thus far. It is hoped that
this study may contribute to a more concerted effort by archeologists, architects,
mathematicians, computer programmers and archaeo-astronomers towards a greater
involvement in this new venue of archeological work.
Note 1: The menhir of the cover photo is the El Cabezudo standing stone, at Cantabria, Spain.
Note 2. The menhir shown in page 2 of this paper (the Irunnari Elazmuno standing stone from
Navarra, Spain) is quite clearly a case of a menhir acting as a gnomon of a sundial.
Note 3. On January 24, 2017 my daughter Daphne-Iris turned 20 years of age. I wish with great
love to dedicate this paper to her.
[1] Dimitrios S. Dendrinos, 2016, “In the Shadows of Carnac’s Le Menec Stones: A Neolithic
Supercomputer”, The paper is found here:
[2] Dimitrios S. Dendrinos, 2016, “Gobekli Tepe: a 6th Millennium BC Monument”,
The paper is found here:
[3] J. Michell, 1989, A Little History of Astro-Archeology, Thames and Hudson, London.
[4] Alexander Thom, 1967, Megalithic Sites in Britain, Clarendon Press, Oxford.
[5] David Smyth, 2017, “Taoslin, A Different Perspective”, The paper is found here:
[7] Dimitrios S. Dendrinos, 2016, “The Earth’s Orbit Around the Sun and the Tumulus at Kasta:
update 1”, The paper is found here:
The author wishes to sincerely acknowledge the support and inspiration received from posts by
all his Facebook friends, especially those who are members of his seven Facebook groups.
Moreover, the author wants also to acknowledge the productive interaction he has had with
David Smyth on the topic of shadows in Archeology.
Most importantly, the author wishes to acknowledge the deep debt he owes to his family for
their continuous support and understanding they have shown over the past few years while the
author spent time away from family commitments for the purpose of devoting a large part of his
time to conducting his research.
Legal Notice
© The author, Dimitrios S. Dendrinos, retains all legal copyrights to this paper. No part or the
whole paper can be reproduced by any means at any medium, without the explicit and written
consent by the author, Dimitrios S. Dendrinos.
Full-text available
The paper addresses the fuzzy nature of shadows cast by Neolithic monuments. It presents a mathematical theory of fuzzy shadows, and extends a previous paper by the author of a General Dynamical Theory of Shadows. That original paper is found at the site here: and also here at here:'_Shadows
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The paper analyzes the Stonehenge Phase 3 II Architecture, modular structure, and their connection to the sun-induced cast-off shadows from its sarsens, and the motion of the shadows over the course of a day and throughout the year. It establishes that a direct link exists between the size of certain shadows and the design of the monument. It further documents that besides the summer solstice sunrise alignment, the vernal and autumnal equinox alignment was a major one in the monument's design. Moreover, the paper demonstrates that the Trilithons sarsens ensemble's quasi-elliptical form is a type belonging to an extension of the Alexander Thom's classification of stone enclosures.
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The paper explores the use of shadows in the very design of Classical Greek temples. It also analyzes the role of cast-off and carry-on shadows in the Temples' morphology by utilizing the author's General Dynamical Theory of Shadows. Further, the paper produces a comparative analysis of five Classical Greek Temples. It also supplies an new angle to the Parthenon Marbles' inappropriate housing in Museums
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With this paper a significant discovery and a possible but equally significant preliminary finding are reported. Close examination of Kasta Tumulus’ exterior morphology reveals that the mathematics and astronomy at the Great Amphipolis Tomb, as imprinted on the Tumulus exterior wall, are far more advanced and complex than previously thought. Specifically, we document the significant discovery that on Kasta Tumulus’ exterior wall Earth’s elliptical motion around the Sun is depicted. We also document the potential significant finding that Kasta Tumulus’ overall shape as a solid, is that of an oblate ellipsoid, representing Earth. Both of these findings overturn what is currently thought of about both subjects, namely that Earth’s elliptical orbit about the Sun, as well as Earth’s oblate ellipsoid shape were intellectual achievements of the late 2 nd millennium AD. In a Note at the end of the paper (Note 2), more on the possible intellectual tradition behind the elliptical motion of the Earth is provided. It traces back to astronomer and mathematician Philolaus the Pythagorean. Here we take an analytical approach, albeit algebraically simplified, to describe and understand the morphology of the Great Tomb at Amphipolis. It’s a follow up study to previous work undertaken by the author. That prior work is both mildly revised and considerably extended with this paper. Work reported here now presents the Tumulus’ form as potentially depicting significantly innovative ripples in the mathematics and astronomy of that Era (the last quarter of the 4 th Century BC) in Greece. The paper advances the thesis that on Kasta’s perimeter wall we “read” the Greeks’ understanding of Earth’s ellipsoid shape and the Earth’s elliptical motion around the Sun, beyond the recognition that this wall was shaped as well in the form of an annual calendar (a prior finding by the author). It is again suggested that mathematician and astronomer Callippus, or possibly one of his students with prior understanding of (what was later attributed to) Aristarchus heliocentric system was the intellectual force behind Kasta. The impetus behind this partial modification and considerable extension of the author’s previous work, presented here, was the site plan of the monument, as presented to the public by the head of the archeological team Mrs. Katerina Peristeri on January 29, 2016
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The paper analyzes the monument of Le Menec at Carnac, in Brittany, France. It advances a number of propositions, key among them being that the strings of stones at Le Menec are not linear parallel alignments but converging arcs. These stones and strings performed a variety of cultural functions. Arcs acted as a Theme Park for celestial objects and their orbits. It is suggested that Le Grand Menec, Kermario, Kerlescan, and Le Petit Menec represented the four seasons. The paper also suggests that the stones' shadows were used as part of sundials. Each stone was used as a computing device, and collectively they constituted a proto Neolithic supercomputer.
Full-text available
The paper documents the date for the initial construction phases of Layer III of structures D (middle 6th millennium BC) and structure C (end of 6th millennium BC - beginning of 5th millennium BC) at Gobekli Tepe. It is a sequel to the author's September 19, 2016 paper "Dating Gobekli Tepe". It uses comparative Architecture and Design analysis from Catalhoyuk and Nevali Cori as well as Jerf El Ahmar for the dating process. It also employs Alexander Thom's schema of classifying stone enclosures, by appropriately expanding it and applying it to Gobekli Tepe. The paper also traces linkages between Gobekli Tepe, Carnac, Malta, Stonhenge and Menorca.
Taoslin, A Different Perspective
  • David Smyth
David Smyth, 2017, "Taoslin, A Different Perspective", The paper is found here: