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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.

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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

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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.

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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.

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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.

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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.

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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.