![The lengths of a gnomon's shadows located at the origin (point O) at some location with an azimuth =90-ω of the Summer Solstice sunrise for any location above the Tropic of Cancer. Source: the author, ref. [1.2].](https://www.researchgate.net/profile/Dimitrios-Dendrinos/publication/317416497/figure/fig1/AS:503154770362368@1496972853709/The-lengths-of-a-gnomons-shadows-located-at-the-origin-point-O-at-some-location-with_Q320.jpg)
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1
The Dynamics of Shadows at and below the Tropic
of Cancer. Update #1
Dimitrios S. Dendrinos
Emeritus Professor, School of Architecture and Urban Design, University of
Kansas, Lawrence, Kansas, USA.
In Residence at Ormond Beach, Florida, USA.
Contact: cbf-jf@earthlink.net
April 7, 2017; revised on April 13, 2017
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Table of Contents
Abstract
Introduction
Shadows and the Temperate Zone
A brief review of the literature on shadows: the author’s work
A very brief review of the literature on shadows: other sources
Summary of key points regarding monoliths’ cast-off shadows
Shadows at and below the Tropic of Cancer on the Northern Hemisphere
The basic bifurcation
The various phase transitions
The goldilocks effect in the temperate zone
Conclusions
References
Acknowledgements
Legal Notice on Copyright
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Abstract
This short paper’s limited objective is to provide the temporal dynamics of shadows for locations
on the Earth’s latitudes close to, on and below the Tropic of Cancer in the Northern Hemisphere,
as well as on the Earth’s Equator. It sets the stage for presenting the dynamics of shadows’
lengths in the Earth’s Temperate zone, within the framework of which the intended role of
shadows implanted on the Classical Greek Temples (discussed in the paper of reference [1.6]) is
to be set. Here, subjects such a basic bifurcation, phase transitions and the overall dynamics of
monoliths’ shadows within the 0 to 2326’13”N of Earth latitudes are analyzed and the
qualitative features of their lengths’ changes over the course of a day and a year are shown, for
the case of Sun-induced shadows with sharply delineated borders. The paper sets the stage for
highlighting the importance of the Temperate Zone’ shadow qualities in the rise of Classical Greek
Temples, to the extent that shadows were incorporated into their design proper.
Introduction
In prior work, the author analyzed the shape of the shadows monoliths cast, and the reasons why
their study is of import to the study of Neolithic Monuments. In reference [1.1], the broad
framework of a General Dynamic Theory of Shadows was presented. The macro-dynamics of Sun-
induced shadows’ lengths were demonstrated for changes during the course of a day, and the
various days of the year. Under an assumption of sharply delineated borders, the dynamics were
exposed for Earth’s latitudes above the Tropic of Cancer. Both, cast-off and carry-on types of
shadows were studied for menhirs, monoliths, pillars, orthostats and columns; all of them can be
considered to have a linear geometric shape, approximated by 1-d linear gnomon, the top of
which – the gnomon’s style – can be traced in its shadow, as cast on a perfectly flat and horizontal
ground. In reference [1.2] the analysis was broadened to include the reality of fuzzy shadows.
Two sections are included in this paper. In the first section, a short outline of the previous work
by the author is presented, that elaborated on the basic principles describing shadows’ lengths
and their dynamics for latitudes above the Tropic of Cancer. Then, in the second section, this
paper extends the work by the author found in reference [1.1] to include the temporal macro-
dynamics of linear monoliths at and below the Tropic of Cancer zone in the Northern Hemisphere,
as well as on the Earth’s Equator.
Basically, the intent of this paper is to build on the realization (outlined in [1.1]) that above a
certain latitude (about 60N) shadows get relatively too long throughout the day and throughout
the year to have been of any practical use in architectonic design in the Neolithic. It shows that
these lengths become relatively too short to be of a similar use below a certain latitude (the
Tropic of Cancer), thus pointing to the existence of a goldilocks of sorts for their use in Neolithic
Architecture. Moreover, the paper sets the stage for discussing the evolved role shadows played
in the design of monuments, from the Neolithic to the Classical Greek Temples within the Earth’s
Temperate zone.
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Shadows and the Temperate Zone
A brief review of the literature on shadows: the author’s work
In a previous set of papers, see [1.1], [1.2], the author analyzed the physical nature proper of
Sun-induced cast-off shadows and their spatial (over different latitudes) and temporal (over the
different hours of a day under sunlight, and over the 365 days of a year) dynamics. The papers,
among other things, debunked two prevailing myths in the literature on shadows. First, in [1.1]
the myth was debunked that at Equinox, and at any latitude on the Earth’s surface, the style of a
gnomon (the tip of a monolith, menhir, obelisk, pillar, orthostat, or column) casts a shadow which
(under the sharp shadow borders assumption) traces a straight line. The paper demonstrated
that in theory it doesn’t. In fact, and in theory (always under sharp shadow borders assumptions)
it traces a bell-shaped curve [1.1], except at Equinox at the Equator. Second, paper [1.2]
debunked the myth that there is a sharply delineated “penumbra” in shadows – a phenomenon
which simply doesn’t exist. Instead the paper demonstrated that shadows are fuzzy adding a
degree of uncertainty in the measurement of their lengths and widths. The embedded
Mathematics and Physics Laws describing that fuzziness were explicitly stated in [1.2].
Furthermore, in a set of papers (see [1.3], [1.4], [1.5]), the role of shadows in certain Neolithic
monuments was analyzed. The extent to which these monuments’ cast-off shadows under
sunlight shaped the morphology of the monuments, and the shadows’ possible symbolic and
ceremonial functions were elaborated in these three papers. Specifically, in [1.3], the case of the
shadows cast off by the pillars in structures C and D, Layer III of Gobekli Tepe was presented, as
shadows cast by each pillar would touch the other at sunrise and sunset during each day of the
year. In [1.4], the case was discussed of shadows being a factor in determining the distances
between stones, as well as the distances among the various arced (and not straight lines forming
- as erroneously presented in the literature) strings of stones, at Le Grand Menec monument at
Carnac in Brittany. Finally, the role of shadows in determining the spacing of the quasi-elliptical
(and not “horseshoe-shaped”, as reported in the literature) Trilithons Ensemble of sarsens at
Stonehenge was shown in [1.5]: the distance was the length of Sun-induced shadow cast off by
the Southernmost sarsen at the outer circular ring of stones at noon time during Equinox.
The aforementioned papers presented a General Dynamical Theory of Shadows, and a number
of features from that theory were expanded in them – and will not be repeated here. However,
a number of additional components will be attached to that General Dynamical Theory, having
to do with equivalences that exist between azimuths, clocks and calendars. These equivalences
will be expanded in this section, after a brief parenthesis containing more elaboration on various
subjects dealing with shadows in the Northern Hemisphere above the Tropic of Cancer, and
directly linked to Neolithic monuments, is offered. This parenthesis expands the work in [1.1].
Next, a very brief review of the literature on shadows by other authors is supplied.
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A very brief review of the literature on shadows: other sources
The subject of shadows and its connection to monuments has a rather recent history. Its brief
history is expanded in the work by David Smyth, see reference [2.1], and for more see the
references in both [1.1] and [1.5]. Notable is also in this literature the work by G. T. Meaden, see
for example [2.2]. An on-going discussion on these issues is taking place in [2.3], with the
participation of a number of archaeo-astronomers and other, interested in shadows of Neolithic
monuments, individuals. The interested reader is directed to these sources to obtain a view of
this rapidly expanding literature, and of the various topics addressed, which tend to be both
theoretical and also draw from concrete empirical evidence involving specific Neolithic sites.
Although no overall perspective has yet emerged, some broad lines of research have been
already established. One of these broad lines is the realization that shadows are fuzzy in nature.
The extent to which the resulting uncertainty in their lengths and widths has been factored into
the monuments’ use and design remains still an open question. Another line of work in this
literature is the broader question, of whether shadows at all affected both the design and
functions of these Neolithic monuments. In specific, the set of questions, whether the Neolithic
architects/astronomers were aware of the various shadows’ attributes, when did they become
aware (if they ever did), where, and how still remain open research questions.
An understanding of these and related issues, and an answer to these questions will allow one to
obtain temporal markers of Neolithic monuments, hence permit cautious speculation as to what
did the architects and astronomers know and when did they know it at various Neolithic sites.
Summary of key points regarding monoliths’ cast-off shadows
Besides the two debunked myths about Sun-induced (the subject of moonlight-induced shadows
yet to be analyzed) cast-off shadows by monoliths on the Earth’s surface, see references [1.1]
and [1.2], some other aspects of shadows were established in these two papers, which bear
directly on the subjects to be addressed in this paper. Thus, they will be listed in summary next.
In the theoretical analysis that follows, a number of simplifying assumptions have been made
and need be born in mind as the reader is offered this short summary. The monolith (gnomon) is
considered to be a 1-d straight line; the source of light is a point; the ground is horizontal and
flat; and there’s no fuzziness involved in the shadow being cast-off the linear gnomon. Of course,
the monolith in reality has a thickness (being a 3-d object); the source of light (the Sun in this
case) is a disc (however fuzzy in its exact diameter); the ground is not a flat, perfectly horizontal
(locally) surface, but instead it carries anomalies obeying a local landscape, topography and
Geography – and the surface of the Earth is a 3-d arc at all points on its surface at a large scale,
thus directly limiting the extent of the horizon. Finally, shadows are not sharply defined, but
instead they are fuzzy obeying the Mathematics and Physics outlined in [1.2]. These simplifying
assumptions will be in part and to an extent relaxed later in the text.
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Other simplifying assumptions and approximations are also needed to be made, in order to point
out the basic principles of Earth-related Astronomy. For example, the Earth isn’t an exact sphere,
but an ellipsoid; the Earth’s rotation about its axis isn’t perfectly regular (the precise length of
any day isn’t exactly the same as any other day’s length), as it is gravitationally affected mainly
by the relative position of the Moon and to a minor degree by the other Planets’ position. The
Earth’s axis of rotation is undergoing an approximately 26Ky cycle, the exact length or precise
trace of which is largely unknown, and only relatively rough estimates exist (or can be obtained)
beyond a few thousand years. The Earth’s orbit around the Sun isn’t a perfect circle (it’s an
approximate ellipse) and the Earth’s speed of motion in that orbit is not constant, gravitationally
affected mainly by the relative position of the Sun and to a lesser extent the relative position of
other Planets. The plane of the Earth’s orbit (the plane referred to as the Celestial Equatorial
Plane) isn’t exactly flat, as the Earth’s center moves above and dips below it, while in orbit around
the Sun. The entire Solar System, in its motion about the Milky Way’s galactic center, is constantly
affected by gravitational interactions emanating from a host of sources. However, all these
factors, and their related approximations and/or simplifying assumptions, do not alter the basic
Astronomy and Physics related principles to be presented here.
There are three major axes of interest in the analysis on shadows, see Figure 1, valid for all points
on the Earth’s surface. One is the due East-West axis, which is the straight line that joins the
points of sunrise (point F in Figure 1) and sunset (point G) at Equinox (assuming for the time being
that the Sun is a point of light, not a disc - in the case of a Sun disc, the point of sunrise is the
point, and time, when the center of that disc crosses the horizon, designated by the circle in
Figure 1). A second axis is the straight line that joins the point B of sunrise at Summer Solstice to
the point of sunset at Winter Solstice E. And the third axis is the straight line which joins the point
D of sunrise at Winter Solstice to the point of sunset at Summer Solstice C.
All of these three axes go through the origin O, the point where the monolith (or gnomon) is
located. The monolith is assumed to be placed so that it is vertical to the (assumed to be perfectly
flat and locally horizontal) plane, a plane thus tangent to the (assumed to be spherical) Earth’s
surface at that point O. If angle is the location’s azimuth at Summer Solstice (which depends
on the point’s latitude) then ω=90-. The connection between azimuth of Summer Solstice
sunrise and latitude is of further interest and it will be discussed more extensively momentarily.
7
Figure 1. The strict equivalences among azimuth plane, clock and calendar. At all points on Earth,
the due N-S axis, and the due E-W axis are as follows: the 0-180 in azimuth and 12midnight-
12noon (on the N-S axis); and 6am-6pm daily hours, 90-270 in azimuth (in the E-W axis). Off
these points, the equivalences collapse for planes which are not equatorial (parallel that is to the
Earth’s equatorial plane). Angle is the Summer Solstice sunrise azimuth at this specific location,
with 90-=ω. Axis BE is the Summer Solstice sunrise to Winter Solstice sunset axis; and axis CD
is the Winter Solstice sunrise to the Summer Solstice sunset axis. Point F is the due East sunrise
point at Vernal and Autumnal Equinox, and Point G is the due West sunset point at Equinox.
Critical in these equivalences is the flatness of the reference plane and the horizon’s extent.
Source: the author.
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Unlimited horizon and perfectly flat ground, and their local conditions (that is the radius of the
circle in Figure 1), are important issues which have been looked at some detail in [1.1] and [1.2].
in specific, the subject of the “horizon” is critical in the discussion regarding both the Earth-
related Astronomy, as well as a site’s Archeology and Architecture. It is so, mostly because of the
ground anomalies and other topographical and geographical features touched upon in both [1.1]
and [1.2] where the subject of “fuzziness” in shadows has been addressed. In the latter reference,
it was demonstrated (in fact proved) that there isn’t an infinite length in shadows for any
gnomon’s finite height. In addition, the Earth’s curvature adds the refraction factor, see [2.4] on
“refraction”, in shadow lengths’ calculations.
These axes of Figure 1 contain fundamental equivalences as discussed in [1.2], the azimuth-clock-
calendar equivalences. At any location, the (North-South), (0-180), (12 midnight – 12 noon)
equivalence, and the (due East-West), (90-270), and the (6am–6pm) equivalences exist no
matter the altitude of the location on the Earth’s surface, or the plane of reference relative to
the Earth’s Equatorial Plane, i.e. its slope or tilt relative to that plane. However, these
equivalences do not in general exist for points off these marks, unless special conditions hold for
the plane of reference. A final point related to these topics: at any point on the Earth’s surface,
and no matter the grounds (or the plane of reference’s slope relative to the Earth’s Equatorial
Plane), the shadow of the gnomon passes through the exact same point on the azimuth at a fixed
hour of the day, but its length varies depending on the day of the year.
Off the points where there is an exact equivalence between azimuths, clocks and calendars there
are complex relationships which connect the three variables. These mathematical relationships
will not preoccupy our analysis here. It suffices to say that, approximately, a location’s azimuth
for the rising Sun at Summer Solstice (being a non-linear function of latitude) can be
approximated for practical purposes, so that it can be assumed to be a few degrees off (that
degree of approximation being a function of latitude) the difference between 90 and the
location’s latitude. Much can be said about this issue, but it will not be addressed here any
further, as it doesn’t directly relate to the subject matter of this paper.
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Figure 2. The lengths of a gnomon’s shadows located at the origin (point O) at some location with
an azimuth =90-ω of the Summer Solstice sunrise for any location above the Tropic of Cancer.
Source: the author, ref. [1.2].
Shadows’ top end (sharp or fuzzy), i.e., the -functions of [1.1] see Figure 2, trace bell-shaped
(not sinusoid) curves, at all points on the surface of the Earth, North of the 2326’13”N latitude
(the so-called “Tropic of Cancer”). It must be noted that the -functions do not form a continuous
3-d surface; instead, they are discrete paths traced by the gnomon’ style over the period of the
day that the Sun is above the horizon at any given location on the Earth’s surface. This
10
discreteness has significant implications in so far as a discontinuity is involved as the -functions
undergo a phase transition at Equinox: from a concave (facing down) close to local noon time, to
a convex (facing up) set of curves. This issue will be further elaborated later in this section, when
a number of other phase transitions occurring in both spatial as well as temporal dynamics of the
shadows will be analyzed.
Parenthetically, all subjects dealt with here address conditions in the Northern Hemisphere –
conditions for the Southern Hemisphere are equivalent. One need only consider a simple
correspondence: for any point under consideration at the Northern Hemisphere, and in reference
to the Earth’s center, there’s a corresponding point diametrically opposite on the Earth’s surface
at the Southern Hemisphere. Thus, one can easily derive equivalent statements, except that the
Winter Solstice for the Northern Hemisphere will correspond to the Summer Solstice for the
Southern Hemisphere, and the Tropic of Cancer related statements will be Tropic of Capricorn
(located at 2326’13”S) related equivalent ones.
These -function related curves pass through a point associated with the local noontime
(minimum through an Earth day and under daylight) shadow length (the A1, A2, and A3 points in
Figure 2). Of the three -functions, (1) and (2) are bell-shaped and thus go through an
inflection point; and at sunrise and sunset (under the simplifying assumptions mentioned earlier)
they are or become asymptotic towards one of the six relevant (straight line) semi-axes. The six
semi-axes are approached as follows, depending on the time of the year, whether there is Vernal
or Autumnal Equinox, case (a), Winter Solstice, case (b), or Summer Solstice, case (c), these three
cases being examined in turn.
In case (a), at Equinox, the shadow is asymptotic to the semi-axis (GO) that joins the due West
sunset at Equinox point G with the origin O, and at sunset the shadow becomes asymptotic to
(FO), the semi-axis which joins the due East sunrise at Equinox point F with the origin O, i.e., the
point where the monolith stands. It is underscored that only at Equinox, these two semi-axes,
(GO) and (FO) collapse onto a single straight line, FG, the due East-West axis. It is also noted that
at Equinox, and only at the Earth’s Equator, the Sun rises and sets at a 90 angle at the horizon.
During daytime at the Equator, the style traces a straight line, which is the due East-West line. At
all other times at the Equator, the Sun rises and sets at an angle different than 90, but always
rising at the point due East (6am) and setting at the point due West (6pm). More on this will be
presented in later subsections (in Figure 5).
Every day, and for a 12-hour segment, any point on the Earth’s Equator is above the Celestial
Equatorial Plane (CEP), whereas at the other 12-hour segment it dips below CEP, see Figure 3. At
Vernal Equinox, any point on the Earth’s Equator splits its 12-hour daylight segment by spending
six hours of it above CEP and the rest six hours below CEP. Equivalently, the rest of the 12-hour
night segment is split so that the point on the Equator spends six hours below and six hours above
the CEP.
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Figure 3. The Celestial Equatorial Plane (CEP), the Earth’s axis of rotation, and the Tropic of
Cancer. Schematically, the locations of the Earth at four points in the Earth’s orbit around the
Sun (point S) are shown, together with the part of the Earth’s surface being illuminated by
sunrays: at Winter Solstice (right), Vernal Equinox (center-left), Summer Solstice (left) and
Autumnal Equinox (center-right). This is a corrected Figure from the one in the 4/7/17 version of
the paper. Source: the author.
At Summer Solstice, any point on the Earth’s Equator splits the 24-hour period by spending the
12-hour segment under daylight below the CEP and the other half above it. At Autumnal Equinox,
the same point on the Earth’s Equator traces a path similar to that for the Vernal Equinox except
on reverse. Whereas, at Winter Solstice, any point on the Earth’s Equator spends the 12-hour
daylight segment above the CEP, and below it at the 12-hour segment at night. In the temporal
(annual) dynamics, as the Earth orbits the Sun, intermediate conditions prevail.
In case (b) at Winter Solstice, and at sunrise, the shadow is asymptotic to the semi-axis that goes
through the point of sunset at Summer Solstice C on its azimuth (specific to every location, given
its latitude), and the origin O, i.e., the semi-axis (CO); whereas at sunset, the monolith’s shadow
becomes asymptotic to the semi-axis (OB) which joins the point (azimuth) of the Summer Solstice
sunrise, i.e., point B at that location, with the origin O.
Finally, in case (c) at Summer Solstice, at sunrise the shadow is asymptotic to the semi-axis (EO)
which goes through the point where the Winter Solstice sunset occurs at that latitude, point E,
and the origin, O; and at sunset, the shadow becomes asymptotic to semi-axis (DO), which joins
the Winter Solstice sunrise azimuth D point at that location (latitude) and the origin O.
At various days, between these three specific days of the year, the reference axes [(CD) and (BE)]
and semi-axes [(BO), (CO), (DO) and (EO)] change, although the axes (CD) and (BE) remain always
straight lines regardless of the day of the year. Their slopes ω’ however depend on the equivalent
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azimuth of sunrises and sunsets for the day under consideration, keeping in mind that these come
in pairs. From Figure 1, angles ’ are always symmetric to the East-West and North-South axes,
so that ’ is always greater than or equal to (for days leading to Summer Solstice, and for days
past Summer Solstice); azimuth ’ remains also less than or equal to {180-} for any day leading
towards or moving away from Winter Solstice.
Equivalently, slopes ω’ are always less than or equal to +ω for days leading towards or moving
away from Summer Solstice; and they are greater than or equal to -ω for days leading towards
or moving away from Winter Solstice. All these points have been elaborated in the papers by the
author [1.1] and [1.2]. There, a bandwidth of latitudes favorable for the exploitation of shadows
was identified, as it was considered particularly “suitable” and “amenable” for shadows related
Architectural design of Neolithic monuments.
This is a suggested zone in the Earth’s latitudes speculated to be productive in the exploitation
of shadows, as tools for design. Shadows in those latitudes stand a far better chance for being
incorporated into the architectural design proper of edifices or structures, not only in Neolithic
monumental construction, but in later structures as well. This point is made in the paper on the
Classical Greek Temples, see reference [1.6]. This suitability had to do with shadows being neither
too long nor too short, especially at specific hours and relatively long periods during the day
throughout the year, to be of design use.
We now turn to cases where the shadows, during a long period during the day (many hours under
sunlight), are indeed too short and rapidly become too long to have any appreciable impact upon
a monument. This is the case of the zones below the Tropic of Cancer in the Northern
Hemisphere. For latitudes above approximately 60N, the shadows also remain too long, no
matter the time of the day or the day of the year to be of any appreciable use in monumental
Architecture (at least in the Neolithic).
Hence, the issue of locational suitability emerges, as if a “goldilocks type principle” is present,
i.e., an optimum bandwidth or Earth zone, where considerable variation in shadows lengths takes
place, rendering shadows of use in Architectural design or site Planning. This theme is a major
subject this paper advances, and the paper in [1.6] exploits for the case of Classical Greek
Temples.
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Shadows at and below the tropic of Cancer on the Northern Hemisphere
The basic bifurcation
In Figure 4, an equivalent diagram to that in Figure 2 is shown. Although Figure 2 was drawn so
that the qualitative features of shadows for latitudes greater than the Tropic of Cancer (TC) were
drawn, the qualitative features drawn in Figure 4 apply to locations on the Earth’s surface
between the 2326’13” latitude and the Equator (0 latitude). At the outset, it must be noted
that, all latitudes above the TC throughout the 24-hour period of an Earth day are found above
the CEP. Any point on the TC touches the CEP at 6pm at Vernal Equinox, 12noon at Summer
Solstice, 6am at Autumnal Equinox, and at 12midnight on Winter Solstice, see Figure 4.
Figure 4. The temporal dynamics in the lengths of a gnomon’s shadows at the Tropic of Cancer.
14
Figure 3 represents a cross-section of the Celestial Equatorial Plane, the plane at which the Earth
orbits the Sun, assumed here to be a circular (in reality it is an almost elliptical) orbit, so that the
daytime and nighttime sections of the Earth are shown; point S is the location of the Sun, as a
point/source of light. Noted is also the fact that at Summer Solstice, and at noon local time, the
Sun is vertical to the Earth’s surface at any point on the TC.
For any point at any latitude between the Earth’s Equator and the TC, between the two
Equinoxes, there are two days of the year that at local noontime the Sun is at 90 angle above
the horizon. One of these two days is between the Vernal Equinox and the Summer Solstice, and
the other between the Summer Solstice and the Autumnal Equinox. The closer to the Earth’s TC
the point considered (the higher the latitude, but lower than 2326’13”N), the closer these two
days are, collapsing to a single day at Summer Solstice at local noontime, for the points on TC.
This bifurcation event ends at the Equinoxes, where the Sun at local noontime is vertical for all
points on the Earth’s Equator.
These conditions allow the derivation of the equivalent curves to Figure 2 (which were drawn for
points in latitudes above the Tropic of Cancer in the Northern Hemisphere) to be derived for
latitudes below the TC, shown in Figure 4. As the two key axes, it is noted that axes (BE) and (CD)
have now acquired lower angles relative to the (EF) axis, since we are addressing latitudes with
sunrise at the Solstices closer to the due East-West axis (greater azimuth for sunrise at Summer
Solstice and for sunset at Winter Solstice, and correspondingly lower azimuth for sunrise at
Winter Solstice and for sunset at Summer Solstice).
First, let’s address what takes place for points on the Earth’s surface at TC, Figure 4. At TC, the
-functions are qualitatively similar to those of Figure 2 with a notable exception: the (1)
function, i.e., the function depicting the style’s path at Summer Solstice has a minimum length
depicted by point A1 at the origin – implying that at that point, and at local noontime, the cast-
off shadow’s length is zero. For any points between the Earth’s Equator and TC (but off both
latitudes in the Northern Hemisphere) function (2) and (3) retain their qualitative properties,
whereas the function (1) undergoes a spatially discontinuous phase transition to be
designated as PT1.
This discontinuous phase transition PT1, shown in Figure 4.a, identifies a (1) function that
attains a minimum at a point A1’ South of the origin (because the point on the Earth’s surface is
now below the CEP). Still the length (OA1’) remains less than (OA2), and of course less than (OA3).
As was the case in the qualitative properties of the (2) and (1), but not (3), functions in
Figure 2, the (1) and (2) functions contain an inflection point. However, (1)’ function does
not contain an inflection point, up to certain latitudes L below TC.
For latitudes below that threshold L, the (1) function does obtain an inflection point, having
undergone another spatially discontinuous phase transition, to be designated as PT2 and to be
discussed in reference to the behavior of the shadows at any point at the Earth’s Equator, over
the year, shown in Figure 5.
15
In Figure 5, the conditions identifying the macro-dynamics (within a day, and over a year) of the
lengths (and locations) of shadows at any point on the Earth’s Equator are shown. At Equinox,
Vernal or Autumnal, the shadows cast off the gnomon, the (2) function, fall on the due East-
West axis, the style being at the origin O (at point A2) at local noontime. Up till noontime, the
style’s shadow moves from point G towards O; whereas past noontime, it moves from O towards
point F. At Winter Solstice, the style’s shadow – function (3) - traces a bell-shaped curve with a
maximum (local noontime minimum) at point A3. Equivalently, at Summer Solstice, the mirror
image function (1) attains a maximum (minimum at local noontime) at point A1, such that
lengths (OA3) is equal to (OA1).
Figure 4.a. The temporal dynamics of shadows’ lengths for points on the Earth’s latitudes below
the Tropic of Cancer. The (1)-function is now asymptotic without an inflection point as it
becomes tangent to the OD and OE semi-axes over extended spatial horizons. Source: the author.
16
Figure 5. The dynamics of shadows at the Earth’s Equator. Both (1) and (3) function contain
inflection points (they are bell-shaped type curves). Source: the author.
The various phase transitions
In previous papers, dealing with points on the Earth’s surface above the Tropic of Cancer, Figure
2, the generally applicable temporal discontinuity involving the Equinox-related transition (from
a bell-shaped curve changing to a strictly convex one) was discussed. Now, more elaboration on
the various spatial and temporal discontinuities will be offered, covering all points on the Earth’s
surface (in the Northern Hemisphere).
Point of discontinuous behavior L was identified earlier. Where is then the point L on the Earth’s
surface where the phase transition PT2 occurs, whereby the (1) function (for points below
2326’13”, i.e., below TC) obtains an inflection point? It is where point A1* corresponds to an
angle of the COE axis of Figure 4.a (the axis which contains the azimuth of the Summer Solstice
17
sunrise for this particular location, the origin, and the azimuth of the Winter Solstice sunset)
which has in the now Cartesian coordinates of the N-S and E-W axes, a N-S axis measure equal to
(OA1”). In effect, this is a spatially discontinuous phase transition.
It is fruitful here to mention a few general comments regarding these various temporal or spatial
phase transitions present in Figures 2, 4 and 4.a. The -functions, in general, do not constitute a
continuous 3-d surface, of course. They comprise a set of daily paths separated by discrete time
intervals. Every point on these paths corresponds to a particular hour of an Earth day, and the
next path crossing that hour’s radius from the origin has a 24-hour time delay (interval) in it from
the previous day’s path. In effect, these paths constitute a set of discrete curves, where temporal
(but not necessarily spatial – that is, over different latitudes) discontinuities exist.
Although, in space, and over different latitudes, smooth and continuous changes take place as
the observer moves along different latitudes, this smoothness does not guarantee that all spatial
phase transitions occur smoothly and are continuous. An example of this discontinuity was PT1.
Thus, and in summary, at key points, temporally discontinuous phase transitions occur (as
discussed in [1.1], [1.2] and here). These transitions are not smooth, but they are characterized
by discrete jumps over time (but not necessarily over space). Over space, smooth spatial phase
transitions do occur, as the one discussed above, that involved the switch from a shadow of a
gnomon on the Earth’s Equator pointing North at local noontime to one pointing South after
Equinox at local noon time.
The goldilocks effect in the temperate zone
The extended analysis of the previous subsections makes clear (and Figures 4, 4.a, and 5 make
obvious) that below the Tropic of Cancer cast-off shadows are either too short at hours close to
local noontime, since they remain confined and restricted to a narrow band of ground close to
their baseline. Then suddenly, and explosively, they become very long (keeping in mind that the
fuzzy aspect of shadows, which strips them from their potentially exorbitant lengths, has been
assumed away). In addition, and after the Vernal Equinox, they alter their direction switching
their cast-off shadows from a Northerly direction to Southerly as these locations on the Earth’s
surface begin to dip below the CEP. They maintain this direction till the Autumnal Equinox, when
they appear again above the CEP and with a Northerly direction in their cast-off shadows. It is
thus concluded that at such latitudes, an architect had severe constraints to face in utilizing
shadows as a means towards achieving architectonic goals – aiming at linking a monolith to other
nearby monumental structures.
Either the relative spacing of other monoliths would had been adversely affected – so that if
casting of a shadow by one monolith onto another was intended, then these monoliths would
had to be placed very close to each other, something that would potentially negatively had
affected monolith spatial densities and/or their relative sizes. Every free-standing monolith has
18
a vital effective space surrounding it, and under such high density conditions the “free” attribute
of the “free-standing” characterization would had been negatively impacted. Monumental
‘scale” would in turn had been adversely affected, as the need for greater in height monoliths
would materialize – severely affecting construction morphology and cost (let alone structural and
resource feasibility for the entire monument).
Similar conclusions can be drawn for locations above 60 latitude – the monoliths’ shadows
would simply had been too long, for too lengthy of a time period during daylight. Hence, these
Earth latitudes (definitely above 60 and close to or far below the Tropic of Cancer) would not be
conducive to constructing monuments containing monoliths so that an architect would or could
take advantage of their shadows. By default, the latitudes between them, latitudes which contain
within them sites such as Abu Simbel (at about 2220’13”N) to the South, and Meashowe (at
about 5859’48”N) to the North, allow for a broad enough variation in cast-off shadows length
to permit inclusion of their effects upon the design of monumental and ceremonial structures.
Hence, the bandwidth between the Tropic of Cancer and the 60 latitude constitutes a bandwidth
which in combination with the Temperate zone within these two boundaries combine to offer an
environmentally suitable Region for the exploitation of shadows in monumental construction.
Within these two latitudinal boundaries, and in fact almost halfway between the two possible
extremes, where shadows are architecturally possibly optimally exploitable, one finds some
stellar examples of Classical Greek Temples. Among them is the Temple of either Hera or
Aphrodite (in any case, Temple E) at Selinunte (at about 3734’59”N), a circa middle 6th century
BC structure in Sicily; the First (circa 550 BC) and Second (circa 470 BC) Temples of Hera at
Paestum (both located at about 4025’20”N) in the Apennine Peninsula; the Parthenon (at about
3758’18”N), designed by architects Kallikratis and Iktinos, and the Temple of Epicourius Apollo
(at about 3725’47”) designed by Iktinos, both middle of the 5th Century BC and Temples on the
Greek Peninsula. These Temples are analyzed in the paper by this author titled “Moving shadows
and the Temples of Classical Greece” in reference [1.6].
However, a major point need be brought up now. By the time CGTs were erected, the role of
shadows had been largely transformed. From their Astronomy and symbolism related functions
embedded onto Neolithic monuments, and thus from an exclusively utilitarian exploitation of
shadows, to more of an aesthetic use of shadows. Shadows macro-dynamics, exhibited over the
course of a day (under either sunlight or moonlight) and over the course of a year, acquired a
role enhancing the overall visual effect of their presence, especially in connection with a key
attribute of CGTs, namely the entasis (ΕΝΤΑΣΙΣ) effect, see [1.6].
This evolution was achieved by either incorporating the monuments’ cast-off shadows effects
onto their surroundings in the overall design and site plan and Temple orientation (as the
Neolithic architect also did, albeit for different reasons); and/or through shaping detailed
morphological features in the Temple’s superstructure components (such as the shafts of their
columns and the depths of their entablature’s reliefs) so that both their cast-off and carry-on
19
shadows’ effects upon the architectonic detail would add life to these structural components of
Classical Architecture by incorporating periodic and regular change, in the form of adding motion,
into them. It was an ingenious way to grant life to otherwise inanimate objects.
These issues are more fully addressed in reference [1.6].
Conclusions
A number of conclusions can be drawn from the analysis above, and a number of suggestion for
further research as well. First, shadows are indeed obvious and important physical elements of
structures, that an architect must have had incorporated into their design. Their temporal as well
as spatial dynamics are of extreme importance, both from a Physics and Mathematics standpoint
as well as from an Architecture and Archeology view.
One of the features which has been discussed in a previous paper by the author, but was not
incorporated here is that of the shadows’ fuzzy nature. Extending the above analysis along these
lines might provide more penetrating insights into these critical elements of monuments. Finally,
the role of the Moon-induced shadows must also be addressed. The shadows under moonlight
might be informative regarding the nature proper of a Neolithic monument.
In the tradition of this author’s work on matters of Archeology and Architecture, this is a paper
addressed to a general audience with College level exposure to Mathematics and Astronomy. Of
course, the Mathematics, Physics and Astronomy proper, as well as Architecture and Archeology
involved topics requires further elaboration by academics with expertise in these individual
fields.
References
Author’s work
[1.1] Dimitrios S. Dendrinos, 24 January 2017, “The Mathematics of Monoliths’ Shadows”,
academia.edu. The paper is found here:
https://www.academia.edu/31101997/The_Mathematics_of_Monoliths_Shadows
[1.2] Dimitrios S. Dendrinos, 1 March 2017, “On the Fuzzy Nature of Shadows”, academia.edu.
The paper is found here:
https://www.academia.edu/31671102/ON_THE_FUZZY_NATURE_OF_SHADOWS
[1.3] Dimitrios S. Dendrinos, 25 November 2016, “Gobekli Tepe: a 6th millennium BC monument”,
academia.edu. The paper is found here:
https://www.academia.edu/30163462/Gobekli_Tepe_a_6_th_millennium_BC_monument
20
[1.4] Dimitrios S. Dendrinos, 15 November 2016, “In the Shadows of Carnac’s Le Menec Stones:
a Neolithic proto supercomputer”. The paper is found here:
https://www.academia.edu/30164088/In_the_Shadows_of_Carnacs_Le_Menec_Stones_A_Ne
olithic_proto_supercomputer
[1.5] Dimitrios S. Dendrinos, 14 March 2017, “On Stonehenge and its Moving Shadows”,
academia.edu. The paper is found here:
https://www.academia.edu/31884455/On_Stonehenge_and_its_Moving_Shadows
[1.6] Dimitrios S. Dendrinos, 10 April 2017, “Moving Shadows and the Temples of Classical
Greece”, academia.edu. The paper is found at: https://kansas.academia.edu/DimitriosDendrinos
Other sources
[2.1] David Smyth, 2017, “Taoslin: a different perspective”, academia.edu.
[2.2] G. Terence Meaden, 2017, “Dromberg stone circle, SW Ireland: design plan analyzed with
respect to sunrises and lithic shadow-casting for the eight traditional agricultural dates – and
further validated by photography”, forthcoming in the Journal of Lithic Studies.
[2.3] The megalithic portal:
http://www.megalithic.co.uk/modules.php?op=modload&name=Forum&file=viewtopic&topic=
7230&forum=1&start=160
[2.4] Simon Newcomb, 1906, A Compendium of Spherical Astronomy, MacMillan, New York.
Acknowledgements
The author wishes to acknowledge the contributions made to this paper through his interaction
with members of the Megalithic Portal website (http://www.megalithic.co.uk/index.php ). The
author wishes to thank the site manager Andy Burnham; and also, the members of the sub-group
in it “Do you look at the shadows”. Within this group, in specific, the author wishes to recognize
the contributions by David Smyth (Energyman), Neil Wiseman (Feanor), and Richard Bartosz
(Orpbit). David Smyth has been instrumental in encouraging the author to participate in that
group, and for introducing the author’s prior work to the group members. A number of others,
including those under the literary pseudonym of cerrig and drolaf, have also contributed to the
formation of the author’s views and study of shadows. Special thanks to cerrig for pointing out
the partially incorrect shading of the Earth at the Sosltices in the previous version of the paper,
and to David Smyth for comments and suggestions.
21
Furthermore, the author wishes to recognize his Facebook friends and especially those who are
members of the seven groups the author has created and is administering. Their continuous
intellectual and artistic stimulation and support have been remarkable, and thus a great sense of
gratitude is extended to them as well. Special mention must be made to Professor Terence
Meaden for his contribution, and especially his work on the shadows at Drombeg and
Stonehenge.
But most important and deer to this author has been the more than 20 years of encouragement
and support he has received from his wife Catherine and their daughters Daphne-Iris and Alexia-
Artemis. Their continuing assistance and understanding for all those long hours he spent on doing
research, when he could have allotted time with them, this author will always be deeply
appreciative.
Legal Notice on Copyright
© The author, Dimitrios S. Dendrinos retains full legal copyrights to the contents
of this paper. Reproduction in any form of parts or the whole of this paper is
prohibited, without the explicit and written permission and consent by the author,
Dimitrios S. Dendrinos.
