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ON THE FUZZY NATURE OF 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.

Contact: cbf-jf@earthlink.net

March 1, 2017

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Table of Contents

Abstract

Introduction

Snapshots of Fuzzy Shadows

Shadows’ discrete and continuous dynamics

Shadows’ snapshots

On the Nature of Shadows

The fuzzy nature of shadows and their micro-dynamics

Scale and distance factors: are the fuzzy borders of shadows marginal?

The standard model of Sun induced shadows

The Mathematics of Fuzzy Shadows

The General Theory model framework

Fuzzy shadows’ domains

The evidence for fuzzy shadows’ domains and the three Axioms

The Fuzzy Set Theory based shadow domains and a Lemma

Metrology and shadows

Shadows and reflectivity of materials

Conclusions

Notes

References

Acknowledgements

Legal Notice

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Abstract

This paper is a continuation of that in reference [1.1]. It has an Archeology and Archeological

Architecture focus. Obviously, the Mathematics of fuzzy shadows’ nature have broader

implications and are of a much wider interest. The original paper presented and discussed

involved a menhir’s shadows considered to be crisp with sharply defined borders. It offered a

General Dynamical Theory of Shadows and a mathematical model describing the macro spatial

and temporal dynamics of a menhir’s shadow lengths. This paper expends on the material

presented there, so that the fuzzy nature of shadows is incorporated into the General Theory

which is further refined. Evidence is presented here demonstrating that shadows’ borders are

fuzzy, and further that the shadow regime of an object (a menhir, orthostat, pillar or column

viewed as a sundial’s gnomon) contains fuzzy bands of shades with different tone intensity. Even

if, due to scale and distance, the extent of the fuzzy natured shadows’ regime might be marginal,

largely depending on the width of the monument under consideration and the distance of the

shadow cast from it being very short, the implications of the shadows’ fuzziness are major.

Whereas, the original paper in [1.1] dealt with the Sun induced shadows cast off by a menhir as

a function of its height, here the analysis focuses on the menhir’s width, and as a result it deals

with Metrology-related attempts to precisely measure the internal fuzzy nature and width of a

shadow’s regime. The Mathematics involved draw from Probability Theory, the Theory of Fuzzy

Sets, and touch issues of fuzziness in Quantum Mechanics.

In the paper’s first sub-section the discrete (snapshot type) versus continuous dynamics of

shadows are discussed. In the second sub-section, arguments are put forward documenting the

fuzzy nature of shadows, and the differences between the standard model (where umbra and

penumbra are involved with sharp borders) and the new model of shadows presented here are

laid out. The mathematical specification of a model producing shadows with fuzzy regimes is

outlined in the paper’s last sub-section, which also contains subjects of Metrology and surface

reflectivity of relevant materials involved in the casting as well as reception of shadows.

Figure A. The shadow cast by a bar in the author’s pool enclosure, from a distance of about

3.5 meters above the ground’s concrete surface.

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Introduction

In a previous paper by the author, see [1.1], the subject of shadows in Neolithic (and also in more

general terms, Archeological) Architecture was analyzed. A General Theory of Shadows and their

Spatial and Temporal Dynamics was proposed. Presenting for the first time this dynamical theory,

the mathematical specifications for the Sun-generated shadows’ lengths at various time of the

day and at various days of the year were provided. In these dynamics, the behavior of the

shadows over different latitudes was discussed (the macro-spatial dynamics of the model) and

indicated how these spatial dynamics could be incorporated in the model’s parameters.

The basic tenets of that General Theory of Shadows are as follows: (i) there are, in principle, two

types of shadows, “carry-on” and “cast-off” shadows; (ii) every archeological architectonic

structure contains both; (iii) Neolithic megalithic monuments, be those menhirs, obelisks,

orthostats, pillars or the columns of the peristyle of Greek Temples, as well as the overall

monument’s structure or edifice cast shadows under both sunlight and moonlight; (iv) these

monuments’ (carry-on and cast-off) shadows move throughout the day (under sunlight) and

through the night (under moonlight); (v) the paths these shadows trace, that is their

“choreography”, over the 1-year cycle (for sunlight generated shadows), or over the 1-month, 1-

year, and over a longer term period cycles, like for instance the so called “Metonic, 18.6-year

cycle” (for moonlight generated shadows) beyond their Astronomy related (clock or calendar)

supplied information to their users played a pivotal role in the architectural design of the

monuments; (vi) that “choreography” carried potentially very significant symbolic meaning to

the cultures that built these monuments; (vii) there are areas on the ground immediately

surrounding the menhir that experience differential shadow coverage, some never receiving a

shadow from the menhir’s top point (the style of the gnomon), thus rendering those areas useless

for calendar (but not clock) shadow indicator exploitation; (viii) along the Earth’s latitudes, there

must have been a zone, or “bandwidth” away from the Earth’s Poles and Equator, in which the

shadows played a more important role in the monuments’ performance (design as well as

ceremonial symbolism), than in areas off that bandwidth and climatic as well as topographical

factors may have been a significant factor in the determination of that zone’s latitudinal width;

(ix) the observant architect-astronomer of the Neolithic could potentially obtain significant Earth

related astronomical information from the study of the shadows’ motions, as well as information

about the Solar System; whether, or how much of, that potential information had been received

may have been among other factors a function of Geography and Topography.

All these aspects of the General Theory of Shadows are elaborated (some more extensively than

others) in reference [1.1], where the mathematical specifications of the shadows’ both spatial

(i.e., through changing a location’s latitude) and temporal (cyclical, over the course of a day and

over a year, solar generated) dynamics are presented and formally stated. In specific what were

identified as -functions were specified as describing the spatial and temporal dynamics of the

monument’s shadows. These functions are complicated functions, being neither parabolas nor

hyperbolas, as they contain in the model’s case of infinite horizon inflection points. This paper is

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a continuation of that initial paper and in numerous ways refines, complements and extends it,

by incorporating fuzziness in the monument’s lengths and widths of their cast-off shadows.

In that initial paper, the relevant assumptions regarding the horizon over which azimuths were

measured were stated (limited versus unlimited horizon), and conditions of a perfectly horizontal

and flat terrain were asserted in the formulation of the theoretical model. These theoretical

assumptions turn out to have significant implications in that General Theory, for a number of

reasons having to do with the reality of Earthly terrains. In real situations, not only the conditions

surrounding the immediate small-in-scale landscape around the monument may be far from

ideal, due to anomalies in the terrain. These small-in-scale anomalies could have been utilized by

the architect-astronomer of the monument to enhance or depress the spatially immediate

effects of the monument’s shadows. Moreover, and in general, the large-in-scale topography

surrounding a monument is seldom perfectly flat – in turn implying that broader Geography

related factors were instrumental in the design of the monument’s shadow related Architecture.

Because, the Geography of a real day or night time skyline might be considerably deviating from

perfect horizontal conditions, thus at times marginally but often significantly affecting azimuth

measurements and especially the time when specific events would occur (sunrise and sunset,

moonrise and moonset), the assumptions regarding flat, unlimited and horizontal ground are

critical. Their relaxation constitutes a sine qua non for the General Dynamical Theory of Shadows,

and thus for obtaining a more complete understanding of the roles played by a monument’s

shadows in its ceremonial functions as well as in its very design. These large in scale topographical

conditions, often affecting the monument from possibly very long distances, contain macro-

spatial influences upon the performance of the monument, as well as its design. In certain

latitudes, weather (climate actually) related conditions further add to these considerations; for

instance, amount of sunshine per year, or haze hovering over the horizon are environmental

factors interfering with the past use (and of course current study) of shadows and their effects

upon Neolithic monuments.

Snapshots of Fuzzy Shadows

Shadows’ discrete and continuous dynamics

In summary, these were the kinds of questions motivating the narrative of the paper in reference

[1.1], and to an extent are motivating this paper as well, albeit under a new angle. Taking off from

these concerns, further elaboration of this General Dynamical Theory is supplied here by a further

focus on the micro-dynamics and the very nature of shadows. In specific, two particular aspects

of shadows’ dynamics and their very nature are elaborated: their discrete perceptions and

recordings, what will be referred to as a “shadow snapshot” in their continuous dynamics; and

the quite fuzzy nature of a Sun (or Moon) induced shadow.

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Both of these aspects of shadows have some repercussions upon the way shadows were used in

Neolithic and even later (Bronze and Iron Age, Egyptian and Mesopotamian) monumental

Architecture, down to the design of Temples during the Classical Period of Greece. Concerning in

specific the case of Greece, we observe that temples were built within specific ranges of latitude,

as well as with an orientation allowing for maximum effect of shadows upon them, shadows of

both the carry-on and cast-off variety. This is particularly evident in the case of the Parthenon’s

reliefs, found on the three components of its entablature (frieze, metope and the two

pediments), a subject that will receive some special mention in a forthcoming paper by the

author. Regarding the case of Egyptian monumental Architecture (including Urban Design and

City Planning), and the role of shadows and shades in them, see Note 1.

Moreover, the close examination of and look into the nature of shadows leads one to a discovery

in their behavior which directly impacts the way shadows are measured and thus the very

Metrology of shadows. In addition to and as a direct consequence of the continuous macro-

movement of shadows, in their daily and annual (in the case of Sun related shadows) cyclical

motion, there’s a complex micro- movement occurring in the time span of minutes, seconds and

even fractions of a second (depending on atmospheric conditions, and at the angle above the

horizon the Sun is at the exact time the attempted recording is made) directly affecting the

recordings of shadows’ lengths and widths. These topics are all addressed next in some detail.

Shadows’ snapshots.

In the existing archeological literature, the role of shadows has attracted attention to a limited

extent, and this is a subject addressed in both [1.1] and a forthcoming paper on Stonehenge and

the Classical Greek Temples by this author. These have been however, “snapshot” type

excursions into the rich world of monumental Architecture shadows’ domains.

Of course, no one observes, as the expression goes, “grass grow”; and it can’t be assumed that

anyone back in the Neolithic was sitting around looking at the manner in which the shadows off

or on monuments moved continuously in space, either during the day or night. What must had

been of import is the position of the shadow at specific intervals (hour) during the day or night,

at some specific day of the year or at some specific night in a particular lunar cycle. By thus

examining the location of certain shadows at key, pivotal and critical time periods (during the day

or night), one might be able to derive some likely scenarios as to the principles employed in the

very architectural design of the monument.

To understand shadows, a basic principle governing them must be well understood by the reader,

a principle which links clocks and azimuths – a condition which will be referred to here as the

“clock-azimuth correspondence”. No matter the time of the year, at some specific hour of the

day (keeping in mind that the daily clock was devised using the sunlight produced shadows, from

a variety of sundials devised by various cultures, although clocks could have been of course

derived based on the shadows of a “moondial”, but that would had been inefficient and

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impractical on numerous counts) the vertical shadow of a gnomon is always at the same line on

the clock or azimuth.

Put it differently, the Sun is at the same azimuth (hour) but at a different angle (height above the

horizon) on any given hour of the day. For example, at 12 noon the vertical shadow cast by the

gnomon of a sundial (equatorial or not) on, say, August 10th, is the exact same spot as that of,

say, February 19th no matter the year (well, approximately so - given that all Earthly years do not

precisely have the same length and also due to the approximately 27Ky cycle in the rotation of

the Earth’s axis of rotation – but these differences are imperceptible for all practical purposes).

This condition also doesn’t depend on the point on the Earth’s surface the observer is at that

time. Moreover, the North-South axis of the plane depicting azimuths is the midnight-noon line

on sundial based clocks (again, no matter the sundial type and independently of the observer’s

position on the Earth’s surface). On the other hand, the location of the style (the top spot of a

sundial’s gnomon) varies, depending of the time of the year and the hour of the day.

Thus, the Cartesian coordinates in 2-d of the shadow cast off by a single point in space (the style

of a gnomon) is all needed to find out both the hour of the day (on the horizontal coordinate,

that being the clock part of the style’s function), as well as the day of the year (on the vertical

coordinate, that being the calendar part of the style’s function).

Consequently, one can ask an obvious question. Why would the Neolithic architect-astronomer

(in fact the various societies back then that contracted, funded and sponsored the architect of

record) construct elaborate stone circles, since both the clock as well as calendar functions

performed in them could have been largely (although not exclusively) performed just as well and

in a much more efficient manner and with far less economic resources expended, by simply

installing at some location a single post (column), rod (orthostat), bar (lintel), or menhir (obelisk)?

The answer to this basic question is also quite obvious, basic and clear. It has been elaborated in

the paper by this author on Le Grand Menec monument at Carnac, Brittany, France, see reference

[1.2].

In short, the purpose of these elaborate and complex stone enclosures and the assortment of

multiple menhirs raised on the grounds of megalithic Neolithic monuments (including

Stonehenge at all phases of it) in various configurations were far more content-rich and

numerous than simply markers of hour (clock) or day (annual calendar) or night (lunar calendar).

In effect these stone circles or enclosures (which contain stones at locations quite of no use for

any calendar or clock purpose – falling for instance on azimuths of no relevance to either solstices

or equinoxes or lunar standstills) were spatially organized (and managed) places and structures

where not only Astronomy and Mathematics were practiced, but also other social and cultural

functions were taking place, possibly having little or nothing to do with either of them.

Analyzing thus the detailed configurations and movement of shadows, both of the cast-off and

carry-on varieties would allow the student of these Neolithic (and later) monuments to discern

both the Astronomy as well as the non-Astronomy related functions of these monuments. Along

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those lines, obtaining key snapshots in the motion of the shadows would be the desirable

outcome in the spatial and temporal dynamics of shadows, since that is what the architects of

those cultures apparently did. But obtaining now a dynamic continuous movement of shadows

(possibly by computer simulations) would enhance our understanding of these monuments’

Architecture (a point made in [1.1]). Both, continuous and discrete dynamics are covered here

(to some extent, leaving quite a bit for further work).

Shadows’ snapshots were obtained through the monument’s architectonic structure (that is, by

the architectural design specifications of its components) and by the specific positioning of key

stones (or related markers) in it, to receive particular shadows from particular components of

the monument at particular hour of the day (or night), as it is expanded more in detail in the case

of Stonehenge and the Classical Greek Temples in the forthcoming paper by the author. In that

design specification and positioning of particular stones, the predictive power of the

astronomical and mathematical computing carried out in the monument (so designed by its

architect-astronomer) was in effect materializing. Obtaining thus key snapshots of shadows is

paramount for the understanding a monument’s basic design.

On the Nature of Shadows

Any snapshot in shadows isn’t sharp. Evidence presented here will demonstrate this physical fact.

Shadows consist of domains which are fuzzy. And this is an important component in the General

Theory of Shadows, to which this paper now turns. A consequence of this fuzziness in shadows’

domain is that the accuracy of predictions from the astronomical computing taking place in

megalithic Neolithic monuments had limits. In turn, the existence of such limits has profound

implications for our efforts to understand the design objectives as well as the level of

astronomical and mathematical knowledge and sophistication possessed by the Neolithic

architects-astronomers at the time of their construction, as it involves complex aspects of

Metrology, aspects which will be addressed here in this paper to some extent.

We shall see shortly, that such Metrology issues also force one to consider the possibility that

the Neolithic monuments were intended to act as places with structures to learn about

Astronomy and Mathematics by carrying out observations, and obtain information; rather than

being the product of a culture already in possession of that information, simply using the

monument for limited in range and sophistication repeated predictive purposes. In arguing that

these highly complex in structure and very resource intensive capital constructions of megalithic

Neolithic Architecture were simply to remind the chief of the time to call an annual celebration

is a highly simplistic explanation. On the other hand, these monuments could have been places

where both functions (offering and receiving information) were taking place, both to a limited

extent – as experimentation devices beyond being mechanisms for prediction. This is a topic

this author has addressed in at least two papers, see [1.2] and [1.3].

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The fuzzy nature of shadows and their micro-dynamics.

A casual look into the shadows cast by any object under either sunlight during the day, or

moonlight during a clear night (independently of the Moon’s Phase) shows that cast-off shadows

have domains where shadows are darker in tone, and domains where shadows are characterized

by a progressively lighter set of tones. These bands of varying tone intensity contain borders that

are not sharp. It is already well known that shadows indeed consist of an “umbra” and

“penumbra” – most well-known being of course the Moon’s umbra and penumbra cast off by the

Earth on the surface of the Moon during a partial or total lunar eclipse.

Although the subject of umbra and penumbra is well known in the astronomical literature, their

specific nature and most importantly the manner in which umbra and penumbra domains are

computed in space, as well as the Mathematics of their domains’ precise extent are subjects that

have been scarcely analyzed and quite erroneously estimated by existing models. Moreover, this

phenomenon of “fuzzy” borders as well as the micro-dynamics involved in shadows cast by

elements of monuments in Archeology (from the Neolithic down to Classical Greece) and their

motion have never been addressed before in the field of Archeology to the author’s knowledge.

In this regard, all material presented here, not only in revising the standard model of umbra and

penumbra type of shadows, but also in extending the General Dynamical Theory of Shadows and

their macro-dynamics from [1.1], is novel.

The topic of fuzziness in the nature of shadows and their domain as well as their micro-motion

(beyond their macro-motions in daily and annual cycles for Sun produced shadows) was first

brought up by the author in a blog hosted by the site in reference [2.1]. For more information on

this particular part of the discussion in the blog that led to the author’s work on the topic of

fuzziness in shadows, see Note 2. At the outset, it must be emphasized that there is a difference

in the way the expression “a shadow’s fuzzy domain” is used here, and the notion of “penumbra”.

It will be demonstrated that both the so called “umbra” and “penumbra” sections of the standard

model of shadows are not sharply defined, and thus contain fuzzy borders, thus they are part of

the fuzzy domain of shadows in this General Theory.

Prior models that show sharp divisions between shadow and non-shadow regimes as well as

within a shadow’s domain (including sharp divisions between umbra and penumbra type

shadows) on any type of Earth’s ground or on other manmade structures are inaccurate and

unrealistic. All objects’ spatial domain containing their shadows has fuzzy limits; that is, their

shadow’s lengths and widths on the ground or on another object when measured produce a fuzzy

number (where the term “fuzzy” is employed in its formal definition from the mathematical

theory of Fuzzy Sets). That fuzziness, it will be shown here, obeys some rules which require some

new Metrology to be put in place in order to be estimated.

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Scale and distance factors: Are the fuzzy borders of shadows marginal?

At the outset, it must be noted that depending on distance from the base of the object (that is

the point of the object closest to the ground upon which the shadow is cast) the total extent of

the shadow’s domain of fuzziness is in the general case marginal; this marginality is partially a

function of the object’s scale, i.e., its length or width (in the case of paper [1.1], the monolith-

menhir thickness). In other words, the greater the width of the gnomon (or menhir), the more

marginal the extent of the shadow’s fuzziness. This presents an interesting trade off: the thinner

the rods (typical of sundials’ gnomons) the less marginal (thus more important in relative terms)

their shadow’s fuzziness becomes – thus the more inaccurate the indication they point at on the

sundial plane’s clock indicators.

A second rule which seems to govern the extent of fuzziness in an object’s shadow regime is

distance from the base of the object (which object is assumed to be a cylinder in the case of a

menhir, obelisk or column) the shadow’s regime of fuzziness width is measured. Closer to the

cylinder’s base, the less the fuzziness one observes, which is completely eliminated at the point

of contact between the object and the ground. We shall attempt to specify the Mathematics

(Algebra and Geometry) of this relationship in a moment.

Figure 1. The diagrammatic estimation of a standing object’s umbra and penumbra according to

the standard “sharp shadow regime” model. Source: reference [2.2].

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The standard model of Sun induced shadows

In the past, a model of shadows was suggested, schematically shown in Figure 1. This can be

found in reference [2.2]. It diagrammatically shows (for illustration purposes) how the umbra and

penumbra of linear objects’ length is estimated, using the perceived Sun’s disc dimensions

(diameter, or angle) and its height above the horizon. Notice the model’s unrealistic

preconditions (assumptions): the Sun (source of light causing the shadows in question) is drawn

as an exact circle with a perfectly defined perimeter. The ensuing regimes of shadow, both umbra

and penumbra, are consequently sharp, exactly delineating spaces on the ground. That is the

crux of the model’s lack of realism.

In this General Theory of shadows, it will be asserted that, there is no sharp or real distinction

between “umbra” and “penumbra”. In reality, there is only a fuzzy domain of shadows

containing various fuzzy bands of shade with different tones, darker close to the center of the

shadow and lighter as one moves out from that center. At some area, which has unspecifiable

borders, and can only be approximately estimated, the shadow’s tone decreases dramatically

(i.e., in a discontinuous manner). That area is roughly where the standard model’s “penumbra”

would commence and the umbra would end.

This phenomenon of abrupt discontinuity in the shadow’s tone is due to the approximate disc

shape of the Sun; whereas the phenomenon of fuzziness in borders of the various tones in the

monument’s domain of shadows is due to two factors: mainly because the points of light on the

Sun’s surface do not form a precise circle or exact perimeter; and secondarily also, due to the

fact that light reflecting off the surface of neighboring objects plus the ambiance generated light

in the day’s atmosphere (in addition to other atmospheric conditions) affect the monument’s

cast-off shadows tone intensity. In addition, as noted earlier, scale of the menhir (monument)

and distance from the monument’s base also affect these tones within the shadows’ domain.

The Mathematics of Fuzzy Shadows

The General Dynamical Theory model framework

In this sub-section, we attempt to mathematically state the fuzzy domains of various tone

intensities in a monument’s cast-off shadow. It is in this estimation that certain interesting issues

of Metrology appear, involving micro-dynamics and a quantum nature of shadows’ edges. In the

estimation of “umbra” and “penumbra” shadow domains in Figure 1, height of a hypothesized

menhir was used. For all relevant elements of Figure 1, the reader is directed to Figure 2 of

reference [1.1] and their explanation offered there.

In what follows, we shall employ the width of a menhir, which as we specified already it is

assumed to have the form of a cylinder.

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The framework for the discussion that follows is set by the author in reference [1.1]. In there, a

model is presented identifying the dynamics of sharp in borders sizes of shadows cast-off from a

menhir due to sunrays during the day, over the day’s duration and over the annual cycle. In Figure

2, the scheme representing the spatial (over different latitudes) and temporal (over different

days in a year) dynamics of the model are shown. Here, an extension to this model is outlined,

which incorporates fuzzy borders to the sharp lengths (and widths) of a menhir’s cast-off

shadows due to sunrays.

Figure 2. The (sharp) lengths of shadows. Daily and annual paths of total shadow lengths are

shown in this (macro) dynamics model proposed by the author in reference [1.1].

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As a result of the author’s participation in a forum on the subject of shadows (see Note 2 for

more details on that participation), the author became aware of two prior efforts carried out

along similar lines, but with different content and configurations. One is by Waugh, see reference

[2.3]; the other is by Khavrus and Shelevytsky, see reference [2.4] on a model that shows the

shadows at Dresden, Germany. Although there are similarities between all three models, the

independent derivation of the diagram in Figure 2, and the General Theory of shadows presented

by the author is noted. There are also slight differences among the three models, which for the

purpose of this paper are not particularly relevant, thus will not be discussed any further.

The complicated true nature of the -functions presented in [1.1] is emphasized, which is neither

a parabola nor a hyperbola, as it contains inflection points when asymptotically approaching the

equinoxes and solstices’ axes at sunrise and sunset. In Figure 2, specifically, at the points D and E

on the solstices axes the function (1) approaches but never touches the axes; in fact, as the

horizon approaches infinity in theory (that is, the radius OD approaches infinity) the (1) function

attains an inflection point, asymptotically approaching the axis there, and at point E. Similar is

the behavior of the (2) function at points F and G. On the other hand, the (3) function is

parabolic (not hyperbolic), since it asymptotically approaches the solstices axes. As noted in [1.1],

at the equinoxes, a structural transformation occurs in the -functions.

Fuzzy shadows’ domains

We now proceed to extend this crisp borders based General Dynamical Theory based model of

shadows lengths (or widths) to incorporate in it fuzziness, thus render it more realistic. Again,

since this is a paper addressed to the general public (as was the reference [1.1]) the specifics of

the mathematical model will be left out, and only the overall specifications will be provided.

Although the model can be extended by a more sophisticated approach to the formal theory of

Fuzzy Sets (a huge area in modern Mathematics), the simple probability based specifications of

the model will be supplied here. Extensions are left to the interested readers and to forthcoming

papers by this author.

However, before we do so, a note must be made in reference to the shaded area in Figure 2

below, and relevant also to Figure 2 of reference [1.1]. This shaded area is a region of the ground

where at no time of the year the top of the menhir (or the style of the gnomon) reaches, thus

offering an end of the shadow’s regime. This it is an area where no matter the hour of the day or

the day of the year the style doesn’t enter. Shadow from the menhir will be present, but not its

top point (style). Thus, it would be useless as an area where the style could be employed as

calendar indicator. For some discussion on this shaded area’s meaning see Note 3.

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The evidence for fuzzy shadows’ domains and the three Axioms

Before we enter the discussion on the fuzzy nature of the menhir’s shadow, let’s take a look at

the empirical evidence for the basic propositions to be advanced. In Figure 3, the shadow of a 5-

centimeter thick horizontal support rod is shown as cast off at the ground level. It must be kept

in mind that the specific results reported here depend not only on the rod’s thickness, but also

on the angle of the Sun above the horizon. In turn this implies that the observations and results

depend on the location’s latitude, and time of the day.

Figure 3. The shadow of a 5-cm thick rod running parallel to a horizontal ground level is shown

from a distance of about three and one half meters. Noticeable is the shadow diffusion around a

fuzzy central relatively dark area. Juxtaposing this diffusion with the relatively crisp and sharp

border of the shadow cast off by the tape’s slight bending edge is informative. It is quite difficult

to visually delineate the boundaries of the rod’s shadow regime within it, as well as its external

borders. The ground is concrete, and the observations were recorded in the screened and

enclosed area at the author’s swimming pool in East-Central Florida during a crisp sunny day in

February 2017. Noticeable is also the fact that the shadows from the millimeter thin screen

panels of the enclosure do not show at all at that distance.

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The basic findings, i.e., the qualitative results from the observations, are presented as axioms,

although their specifics are subject to measurements and calibration. The findings as well as the

results reported are functions also of environmental factors as indicated in the previous sections

of the paper, as well as types of materials used in the empirical investigation. Obviously, different

materials have different surface composition and reflectivity, thus affecting shadow tones.

Axiom 1. As the distance from an object increases, the cast-off shadow’s regime becomes fuzzier.

Notice the crisp shadow regime from the tape in Figure 3, relative to the fuzzy shadow regime of

the bar located about three and one half meters above the ground. Even at the center of the

bar’s shadow, the tone of the shadow cast-off from the bar is less dark than the shadow cast

from the tape, which almost touches the ground.

Axiom 2. As the distance from the object increases, the darker in tone section of the shadow

regime shrinks. Notice in Figure 3 that the area picked up by the relatively darker region of the

shadow is less than five centimeters (the thickness of the horizontal bar). From the photo in

Figure 3, it is evident that the darker regime of the shadow is in fact at an approximate size

(keeping in mind that this is a relatively gross approximation as the shadow’s darker regime is

still fuzzy) of about no more than four centimeters. This might indicate that the decline

experienced is of the magnitude of about a quarter to a third of a percent per meter.

Axiom 3. As the distance from the object increases, the total extent of the shadow’s regime (of

all tones) increase, exceeding the initial thickness of the object. In the example depicted by Figure

3, the increase in the total (still under fuzzy external borders) regime of the bar’s shadow

approximately doubled, so that it increased by an amount between 25% and 30% per meter at a

distance of about three and one half meters. Again, it must be kept in mind, that these findings

on proportions are linked directly to the thickness of the bar, in addition to all other factors

already mentioned.

Although the Mathematics and Physics of shadows may be observed in a world governed by the

Physics of Classical Mechanics and thus be different than those of an electron’s position and field

of energy around a nucleus, thus the world of Quantum Mechanics, the qualitative morphological

properties are similar, especially when the micro-scale of the shadows are observed. The borders

of the electron’s positions in Figure A are relatively sharp, whereas the external borders of a

menhir’s shadow are fuzzy along any distance away from the menhir’s base, and position off the

center of the shadow at that location Y. Fuzziness collapses to approximately zero at the point

where the base of the menhir meets the ground. In any case, extensions along the lines of an

electron-like fuzziness in the nature of shadows are left to the interested reader and to future

research.

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Figure B. Fuzzy set and the energy cloud in an electron’s position in Quantum Mechanics

according to Niels Bohr. Source: [2.8]

The Fuzzy Set theory based shadows’ domains and a Lemma

Following the three axioms stated, we now proceed to state the mathematical specifications of

a menhir’s cast-off shadow. These specifications are indicated in an illustrative schematic

manner, in Figure 4, where the effects have been exaggerated and the case of a relatively thin

menhir is presented. The dark brown line in Figure 4 represents where the penumbra of the

models that consider shadows to have crisp limits would had been, and the light brown that of

the external shadow sharp limit (conditions and assumptions that by this “fuzzy borders of

shadows” model we relax). At any given distance Y from the menhir’s base, the probability that

at distance x from the center of the shadow there will be still shadow (no matter its tone, or

intensity) is given by a bell-shaped continuous “probability-like” distribution P(Y,x)[µ,σ^2], i.e., a

normal distribution with a mean value µ, and a variance σ^2.

The reader is directed to [2.5] for more on this type of usual probability distributions and analysis.

In the case of shadows however, the behavior of these distributions differs slightly. In Quantum

Mechanics, which does not employ strict probabilities and their conventional distributions, these

are called “amplitudes”. The author has elaborated on how methods from Quantum Mechanics

can be transferred to the study of social and spatial systems, in [1.4]. For a standard reference in

Quantum Mechanics, see Paul Dirac’s classical work in [2.7].

Their sum, discrete or continuous – in which case that would be their integral - over space x, of

these P(Y,x) is not equal to one, but some quantity K(Y,x)<1; furthermore, the sum, again discrete

or continuous – being their integral – over space Y of these quantities K(Y,x) could be equal to

some number L (equal or greater than 1), subject to estimation and calibration with observations

from a given location at some specific time period. These conditions are consistent with axioms

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1, 2 and 3. In Figure 4, notice the declining height of the top of the bell-shaped probability

distributions, reflective of the above condition obeying constant L.

Figure 4. The probability-like distributions of shadow regimes at various distances from the

menhir structure (the round base in the above figure). Parameter µ obeys a negative exponential

function of distance from the base. The effects have been exaggerated for purposes of

illustration. Source: the author.

All of these parameters are functions themselves of the factors mentioned in this paper, namely

location’s latitude, time of the day and day of the year (implying angle of the Sun above the

horizon), environmental and climatic factors and type of surfaces involved. To precisely estimate

(that is, calibrate) for these model parameters, one needs observations and a mechanism to

record data. Before we address the Metrology-related issues present here, one more element in

the above specification will be added, being central in this discussion, in the form of a lemma, a

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consequence of the above specifications for a number of the parameters mentioned, in this

specific case being the parameter µ.

Lemma 1. The value of parameter µ obeys a negative exponential distribution, D(Y)=Ae^-βY,

where A=1, and β is a parameter (obeying the above conditions as all previous parameters) and

subject to calibration by locally obtained observations. Of course, e is the base of the natural

logarithms; and as previously defined, Y is distance from the menhir’s base.

Thus, at any given distance Y from the menhir, and distance x off the center of the menhir’s cast

off shadow, the likelihood that a point in space will fall into the menhir’s shadow is given by the

combined probability distributions as stated. Membership into both constitutes membership to

a Fuzzy Set, and for more on this (and possible extensions) the reader is directed to a standard

reference, like that in [2.6].

The discussion has addressed subjects related to the width of a menhir, whereas in [1.1] subjects

related to the height of the menhir and the length of its castoff shadow were addressed. We now

turn to the issue of the menhir’s length under a fuzzy shadow’s external border conditions. At

the top of the menhir (the style of the gnomon), we have the following conditions holding under

the above three axioms and the lemma: the darkest section of the shadow regime ends, as the

schematically drawn dark brown lines in Figure 3 cross, and only the fuzzy margins of the

shadow’s regime remain. At that point, a semi-circle defines the extent of the shadow regime,

beyond which the non-shadow regime begins. This is schematically identical to a semicircle in

Figure A.

The diameter of this semicircle identifies the maximum width the shadow regime of a menhir

reaches, and it is quite greater than the width of the menhir itself, consistently with axiom 3

stated above. Thus, distance and maximum total shadow regime’s width are directly related, and

consequently directly impacting the planned interaction between the menhir and the ground (or

an adjacent menhir-target) at the distance in which the shadow was designed to fall.

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Figure 5. Time lapse and shadow displacement during the recording of data on the shadow’s

spatial regime. It is only a few minutes that have elapsed between the drawing of the two parallel

lines (indicative of the darkest tone shadow’s regime internal borders, thus its width) and the

time the photo was taken. It shows the extent of movement in the shadow’s darkest region, as a

result of the shadow’s micro-motion, and the associated (slight of course) change in its width as

a result of the time delay. No matter the amount of time it takes for a camera to record the

shadow, the amount of time will always affect the shadow’s change in its external fuzzy border

as well as its internal differential tone shadow’s dimensions during the process of the recording.

The effect was designed to be exaggerated by the relatively prolonged time period allowed to

elapse between the drawing of the two lines and the taking of the photo.

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Metrology and shadows.

An important aspect (and mathematically interesting topic) associated with the fuzzy nature of

shadows is of course the manner in which observations are made and records are kept on

shadow’s dimensions (width and length), these being the day shadows due to sunlight (and

equivalently, for night shadows due to moonlight – actually sunlight reflected off the Moon’s

surface). The problems associated with such measurements are illustrated in Figure 5. The

Metrology related issues become of importance due to the continuous movement in the

shadows, as the result of the Earth’s continuous rotation about its axis. Fuzziness in the nature

of the shadow’s regime undergoes change as the observation is made and the record is obtained.

The time elapsed during the taking of a photo-record of the shadow’s fuzzy regime borders

(internally among differential tones, as well as external borders between the shadow regime and

regime of absolutely no shadow) the sizes of these regimes will change. This interference bears

some similarity to the quantum interference in the recording of quantum states. Extensions of a

General Theory of shadows along these lines is left to future research.

Shadows and reflectivity of materials.

Notice that in the photo of Figure 5, three different surfaces are shown, off which the castoff

shadow from the bar on top of the ground is reflected. As the reflectivity of these three materials

(concrete, paper and aluminum ruler) is different in each case the width of the various fuzzy in

width bands of the castoff shadow from the bar above the ground is also slightly different.

Again, extensions into incorporating such difference when different materials are involved in

Neolithic monuments menhir structures (orthostats, pillars or columns made out of sandstone,

limestone, marble etc., and a ground composition of a different texture and surface conditions)

are all left for future research. This short remark on materials surface reflectivity and absorption

of light (or shadow) completes the mathematical analysis of the fuzzy border shadow regimes.

Conclusions

In this paper, a continuation of the paper in which a General Dynamical Theory of Shadows was

presented (found in reference [1.1]), the fuzzy nature of shadows was explored and the outlines

of a mathematical model of fuzzy shadows was stated. The model falls broadly within the scope

of the Theory of Fuzzy Sets, although it differs from it to an extent. It also borders issues covered

in the fuzzy theoretic aspects of Quantum Mechanics. The model employed empirical evidence

in substantiating its contentions, which are formulated as a set of three Axioms and a Lemma.

The micro-dynamics of shadows’ motion in space-time were discussed as being critical in the

Metrology part of the model and the recording of data on shadows’ dimensions.

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Much is left to further future research. The contents of this paper, as well as the General

Dynamical Theory of Shadows as stated in [1.1] are utilized in the discussion of the role shadows

played in the design of Stonehenge’s Phase 3 II (involving the sarsens circle and the trilithons’

quasi-elliptical stone enclosure) as well as of Classical Greek Temples in a forthcoming paper by

this author.

Notes

Note 1. Although this paper did not intend to delve into issues of Archeology, and the

forthcoming paper by the author on Stonehenge and the Classical Greek Temples does not intend

to address the role of shadows in Egyptian or Mesopotamian post Neolithic period monumental

Architecture, a few points will be made on a specific case of Amarna. It is indicative of the import

that shadows and shading played in the case of Egyptian Architecture, Design and Planning (that

is, not only in the design for specific monuments but also for larger spatial units, like cities and

sections of cities). We know that Amarna, the new capital Akhenaten established during his reign,

had a specific North-South (the Royal Road, where the Aten Temple was located) and an East-

West (the Road to Wadi and the Royal Tombs) set of axes (orientation); they were intersecting

at the very center of the city where the small Aten temple was situated, see reference by D. P.

Silverman, J. W. Wegner, J. H. Wegner, 2006, Akhenaten and Tutankhamun: Revolution and

Restoration, University of Pennsylvania Museum of Archeology and Anthropology, Philadelphia,

PA, p. 52 Figure 44. Further, we know that Akhenaten was planning to create numerous sunshade

temples in honor of prominent women in his life, (ibid, p.51). For a quick view on the role of fans

and sunshades in Egypt, see: https://www.youtube.com/watch?v=IpV6KQcIfmM It is obvious

that shadows and shades were of significant import in Dynastic Egypt on many counts. How the

Royal Wad at Amarna was designed is discussed in the reference:

http://www.touregypt.net/featurestories/royaltomb.htm

Egypt (as was the case with Mesopotamia) was falling in the special bandwidth discussed in [1.1],

that is the Earth’s zone with latitudes where shadows played a major role in the design of

monuments (and cities).

Note 2. As it can be seen in the blog cited in reference [2.1], the topic of “fuzziness in shadows”

was brought up on p. 3 of the blog. On February 1st 2017 the author brought up the subject (which

he has been studying ever since) after some very insightful comments by Neil Wiseman who

pointed out the “umbra-penumbra” aspect on shadows, which ironically and erroneously this

author initially dismissed as irrelevant for Archeology(!) but then upon reflection immediately

recognized its import and elaborated further. These exchanges were taking place as some very

pertinent comments and citations by Richard Bartosz and David Smyth – on sundials and shadows

correspondingly were offered. To these three individuals in particular, as well as to the other

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members of the group (eponymous or anonymous) the author wishes to express his gratitude for

their stimulating remarks. This site constitutes an excellent example of how cooperation and

fruitful exchange of information and ideas can lead to advancements in the analysis of scientific

subjects.

Note 3. It is noted that the description of the shaded area in Figure 2 of this paper (and in Figure

2 of reference [1.1] amends and expands on the description of the shaded area in the narrative.

The shaded area to the North of the menhir identifies a region where the style of the gnomon

never enters, and thus it is of no use for calendar indications (although it is of use for daily clock

indications).

References

The author’s related 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, 15 November 2016, “In the Shadows of Carnac’s Le Menec Stones:

A Neolithic proto supercomputer”, academia.edu. The paper is found here:

https://www.academia.edu/30164088/In_the_Shadows_of_Carnacs_Le_Menec_Stones_A_Ne

olithic_proto_supercomputer

[1.3] Dimitrios S. Dendrinos, 31 July 2016; updated in 2 August 2, 2016, “On Dowth North

Passage Tomb and K51”, academia.edu. The paper is found here:

https://www.academia.edu/27511804/On_Dowth_North_Passage_Tomb_and_K51_update_2

[1.4] Dimitrios S. Dendrinos, December 1991, “Methods in Quantum Mechanics and the Socio-

Spatial World”, Socio-Spatial Dynamics, Vol. 2, No. 2; pp: 81-108.

Other sources

[2.1] The specific blog is that by the name “Do you look at shadows” found here:

http://www.megalithic.co.uk/modules.php?op=modload&name=Forum&file=viewtopic&topic=

7230&forum=1&start=100

It is hosted by the site “Megalithic Portal” found here: http://www.megalithic.co.uk/index.php

[2.2] https://takethisskepticalblog.wordpress.com/2015/10/21/how-to-disprove-the-flat-earth/

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[2.3] A. E. Waugh, 1973, Sundials: their theory and construction, Dover Publications, New York.

[2.4] V. Khavrus and I. Shelevytsky, undated, “Introduction to solar motion geometry on the basis

of a simple model” and drawn from a publication in the Journal Physics Education, 2010, Volume

45, p. 641. Section 4 which contains the model proposed by the two authors is here:

https://sites.google.com/site/khavrus/public-activities/SolarEng/solengshadd

[2.5] http://mathworld.wolfram.com/NormalDistribution.html

[2.6] H.-J. Zimmermann, 2001, Fuzzy Set Theory and its Applications, (4th edition) Kluwer

Academic Publishers, The Netherlands.

[2.7] P. A. M.Dirac, 1958, Principles of Quantum Mechanics, 4th edition, Oxford University Press,

Oxford.

[2.8] http://conocimientosquantummechanics.blogspot.com/2010/06/quantum-mechanical-

model-of-atom.html

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; 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.

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 of the monuments at

Drombeg and Stonehenge.

But most important and deer to this authors has been the more than 20 years of encouragement

and support he has received from his wife Catherine and our daughters Daphne-Iris and Alexia-

Artemis. Their continuing assistance and understanding for all those hours spent on doing his

research, this author will be always grateful.

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© 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.