The Dynamics of Shadows at and below the

The D a i s of Shado s at a d elo the T opi

of Ca e

Di it ios S. De d i os

E e itus P ofesso , “ hool of A hite tu e a d U a Desig , U i e sit of

Ka sas, La e e, Ka sas, U“A.

I ‘eside e at O o d Bea h, Flo ida, U“A.

Co ta t:

f-jf@ea thli k. et

Ta le of Co te ts

A st a t

I t odu tio

Shado s a d the Te pe ate Zo e

A ief e ie of the lite atu e o shado s: the autho ’s o k A e ief e ie of the lite atu e o shado s: othe sou es Su a of ke poi ts ega di g o oliths’ ast-off shado s

Shado s at a d elo the T opi of Ca e o the No the He isphe e

The asi ifu atio The a ious phase t a sitio s

The goldilo ks effe t i the te pe ate zo e

Co lusio s

Refe e es

A k o ledge e ts

Legal Noti e o Cop ight

A st a t

This sho t pape s li ited o je ti e is to p o ide the te po al d a i s of shado s fo lo atio s o the Ea th s latitudes lose to, o a d elo the T opi of Ca e i the No the He isphe e, as ell as o the Ea th s E uato . It sets the stage fo p ese ti g the d a i s of shado s le gths i the Ea th s Te pe ate zo e, ithi the f a e o k of hi h the i te ded ole of shado s i pla ted o the Classi al G eek Te ples dis ussed i the pape of efe e e [ . ] is to e set. He e, su je ts su h a asi ifu atio , phase t a sitio s a d the o e all d a i s of

o oliths shado s ithi the  to  N of Ea th latitudes a e a al zed a d the

ualitati e featu es of thei le gths ha ges o e the ou se of a da a d a ea a e sho , fo the ase of “u -i du ed shado s ith sha pl deli eated o de s. The pape sets the stage fo highlighti g the i po ta e of the Te pe ate )o e shado ualities i the ise of Classi al G eek Te ples, to the e te t that shado s e e i o po ated i to thei desig p ope .

I t odu tio

I p io o k, the autho a al zed the shape of the shado s o oliths ast, a d the easo s h thei stud is of i po t to the stud of Neolithi Mo u e ts. I efe e e [ . ], the oad f a e o k of a Ge e al D a i Theo of “hado s as p ese ted. The a od a i s of “u -i du ed shado s le gths e e de o st ated fo ha ges du -i g the ou se of a da , a d the

a ious da s of the ea . U de a assu ptio of sha pl deli eated o de s, the d a i s e e e posed fo Ea th s latitudes a o e the T opi of Ca e . Both, ast-off a d a -o t pes of shado s e e studied fo e hi s, o oliths, pilla s, o thostats a d olu s; all of the a e o side ed to ha e a li ea geo et i shape, app o i ated -d li ea g o o , the top of hi h – the g o o s st le – a e t a ed i its shado , as ast o a pe fe tl flat a d ho izo tal g ou d. I efe e e [ . ] the a al sis as oade ed to i lude the ealit of fuzz shado s. T o se tio s a e i luded i this pape . I the fi st se tio , a sho t outli e of the p e ious o k the autho is p ese ted, that ela o ated o the asi p i iples des i i g shado s le gths a d thei d a i s fo latitudes a o e the T opi of Ca e . The , i the se o d se tio , this pape e te ds the o k the autho fou d i efe e e [ . ] to i lude the te po al a o-d a i s of li ea o oliths at a o-d elo the T opi of Ca e zo e i the No the He isphe e, as ell as o the Ea th s E uato .

Basi all , the i te t of this pape is to uild o the ealizatio outli ed i [ . ] that a o e a

e tai latitude a out N shado s get elati el too lo g th oughout the da a d th oughout

the ea to ha e ee of a p a ti al use i a hite to i desig i the Neolithi . It sho s that these le gths e o e elati el too sho t to e of a si ila use elo a e tai latitude the T opi of Ca e , thus poi ti g to the e iste e of a goldilo ks of so ts fo thei use i Neolithi A hite tu e. Mo eo e , the pape sets the stage fo dis ussi g the e ol ed ole shado s pla ed i the desig of o u e ts, f o the Neolithi to the Classi al G eek Te ples ithi the Ea th s Te pe ate zo e.

Shado s a d the Te pe ate Zo e

A ief e ie of the lite atu e o shado s: the autho ’s o k

I a p e ious set of pape s, see [ . ], [ . ], the autho a al zed the ph si al atu e p ope of “u -i du ed ast-off shado s a d thei spatial o e diffe e t latitudes a d te po al o e the

diffe e t hou s of a da u de su light, a d o e the da s of a ea d a i s. The pape s,

a o g othe thi gs, de u ked t o p e aili g ths i the lite atu e o shado s. Fi st, i [ . ] the th as de u ked that at E ui o , a d at a latitude o the Ea th s su fa e, the st le of a

g o o the tip of a o olith, e hi , o elisk, pilla , o thostat, o olu asts a shado hi h

u de the sha p shado o de s assu ptio t a es a st aight li e. The pape de o st ated that i theo it does t. I fa t, a d i theo al a s u de sha p shado o de s assu ptio s it t a es a ell-shaped u e [ . ], e ept at E ui o at the E uato . “e o d, pape [ . ]

de u ked the th that the e is a sha pl deli eated pe u a i shado s – a phe o e o

hi h si pl does t e ist. I stead the pape de o st ated that shado s a e fuzz addi g a deg ee of u e tai t i the easu e e t of thei le gths a d idths. The e edded Mathe ati s a d Ph si s La s des i i g that fuzzi ess e e e pli itl stated i [ . ].

Fu the o e, i a set of pape s see [ . ], [ . ], [ . ] , the ole of shado s i e tai Neolithi o u e ts as a al zed. The e te t to hi h these o u e ts ast-off shado s u de

su light shaped the o pholog of the o u e ts, a d the shado s possi le s oli a d

e e o ial fu tio s e e ela o ated i these th ee pape s. “pe ifi all , i [ . ], the ase of the shado s ast off the pilla s i st u tu es C a d D, La e III of Go ekli Tepe as p ese ted, as shado s ast ea h pilla ould tou h the othe at su ise a d su set du i g ea h da of the ea . I [ . ], the ase as dis ussed of shado s ei g a fa to i dete i i g the dista es et ee sto es, as ell as the dista es a o g the a ious a ed a d ot st aight li es fo i g - as e o eousl p ese ted i the lite atu e st i gs of sto es, at Le G a d Me e o u e t at Ca a i B itta . Fi all , the ole of shado s i dete i i g the spa i g of the uasi-ellipti al a d ot ho seshoe-shaped , as epo ted i the lite atu e T ilitho s E se le of sa se s at “to ehe ge as sho i [ . ]: the dista e as the le gth of “u -i du ed shado ast off

the “outhe ost sa se at the oute i ula i g of sto es at oo ti e du i g E ui o .

The afo e e tio ed pape s p ese ted a Ge e al D a i al Theo of “hado s, a d a u e of featu es f o that theo e e e pa ded i the – a d ill ot e epeated he e. Ho e e , a u e of additio al o po e ts ill e atta hed to that Ge e al D a i al Theo , ha i g to do ith e ui ale es that e ist et ee azi uths, lo ks a d ale da s. These e ui ale es ill e e pa ded i this se tio , afte a ief pa e thesis o tai i g o e ela o atio o a ious su je ts deali g ith shado s i the No the He isphe e a o e the T opi of Ca e , a d di e tl li ked to Neolithi o u e ts, is offe ed. This pa e thesis e pa ds the o k i [ . ]. Ne t, a e ief e ie of the lite atu e o shado s othe autho s is supplied.

A e ief e ie of the lite atu e o shado s: othe sou es

The su je t of shado s a d its o e tio to o u e ts has a athe e e t histo . Its ief histo is e pa ded i the o k Da id “ th, see efe e e [ . ], a d fo o e see the efe e es i oth [ . ] a d [ . ]. Nota le is also i this lite atu e the o k G. T. Meade , see fo e a ple [ . ]. A o -goi g dis ussio o these issues is taki g pla e i [ . ], ith the pa ti ipatio of a u e of a haeo-ast o o e s a d othe , i te ested i shado s of Neolithi o u e ts, i di iduals. The i te ested eade is di e ted to these sou es to o tai a ie of this apidl e pa di g lite atu e, a d of the a ious topi s add essed, hi h te d to e oth theo eti al a d also d a f o o ete e pi i al e ide e i ol i g spe ifi Neolithi sites. Although o o e all pe spe ti e has et e e ged, so e oad li es of esea h ha e ee al ead esta lished. O e of these oad li es is the ealizatio that shado s a e fuzz i atu e. The e te t to hi h the esulti g u e tai t i thei le gths a d idths has ee fa to ed i to the o u e ts use a d desig e ai s still a ope uestio . A othe li e of o k i this lite atu e is the oade uestio , of hethe shado s at all affe ted oth the desig a d fu tio s of these Neolithi o u e ts. I spe ifi , the set of uestio s, hethe the Neolithi a hite ts/ast o o e s e e a a e of the a ious shado s att i utes, he did the e o e a a e if the e e did , he e, a d ho still e ai ope esea h uestio s.

A u de sta di g of these a d elated issues, a d a a s e to these uestio s ill allo o e to o tai te po al a ke s of Neolithi o u e ts, he e pe it autious spe ulatio as to hat did the a hite ts a d ast o o e s k o a d he did the k o it at a ious Neolithi sites.

Su a of ke poi ts ega di g o oliths’ ast-off shado s

Besides the t o de u ked ths a out “u -i du ed the su je t of oo light-i du ed shado s et to e a al zed ast-off shado s o oliths o the Ea th s su fa e, see efe e es [ . ] a d [ . ], so e othe aspe ts of shado s e e esta lished i these t o pape s, hi h ea

di e tl o the su je ts to e add essed i this pape . Thus, the ill e listed i su a e t.

I the theo eti al a al sis that follo s, a u e of si plif i g assu ptio s ha e ee ade

a d eed e o i i d as the eade is offe ed this sho t su a . The o olith g o o is

o side ed to e a -d st aight li e; the sou e of light is a poi t; the g ou d is ho izo tal a d flat; a d the e s o fuzzi ess i ol ed i the shado ei g ast-off the li ea g o o . Of ou se, the o olith i ealit has a thi k ess ei g a -d o je t ; the sou e of light the “u i this ase is a dis ho e e fuzz i its e a t dia ete ; the g ou d is ot a flat, pe fe tl ho izo tal lo all su fa e, ut i stead it a ies a o alies o e i g a lo al la ds ape, topog aph a d Geog aph – a d the su fa e of the Ea th is a -d a at all poi ts o its su fa e at a la ge s ale, thus di e tl li iti g the e te t of the ho izo . Fi all , shado s a e ot sha pl defi ed, ut i stead the a e fuzz o e i g the Mathe ati s a d Ph si s outli ed i [ . ]. These si plif i g assu ptio s ill e i pa t a d to a e te t ela ed late i the te t.

Othe si plif i g assu ptio s a d app o i atio s a e also eeded to e ade, i o de to poi t out the asi p i iples of Ea th- elated Ast o o . Fo e a ple, the Ea th is t a e a t sphe e, ut a ellipsoid; the Ea th s otatio a out its a is is t pe fe tl egula the p e ise le gth of a da is t e a tl the sa e as a othe da s le gth , as it is g a itatio all affe ted ai l the elati e positio of the Moo a d to a i o deg ee the othe Pla ets positio . The

Ea th s a is of otatio is u de goi g a app o i atel K le, the e a t le gth o p e ise

t a e of hi h is la gel u k o , a d o l elati el ough esti ates e ist o a e o tai ed e o d a fe thousa d ea s. The Ea th s o it a ou d the “u is t a pe fe t i le it s a app o i ate ellipse a d the Ea th s speed of otio i that o it is ot o sta t, g a itatio all affe ted ai l the elati e positio of the “u a d to a lesse e te t the elati e positio of othe Pla ets. The pla e of the Ea th s o it the pla e efe ed to as the Celestial E uato ial Pla e is t e a tl flat, as the Ea th s e te o es a o e a d dips elo it, hile i o it a ou d the “u . The e ti e “ola “ ste , i its otio a out the Milk Wa s gala ti e te , is o sta tl affe ted g a itatio al i te a tio s e a ati g f o a host of sou es. Ho e e , all these fa to s, a d thei elated app o i atio s a d/o si plif i g assu ptio s, do ot alte the asi Ast o o a d Ph si s elated p i iples to e p ese ted he e.

The e a e th ee ajo a es of i te est i the a al sis o shado s, see Figu e , alid fo all poi ts

o the Ea th s su fa e. O e is the due East-West a is, hi h is the st aight li e that joi s the

poi ts of su ise poi t F i Figu e a d su set poi t G at E ui o assu i g fo the ti e ei g that the “u is a poi t of light, ot a dis - i the ase of a “u dis , the poi t of su ise is the poi t, a d ti e, he the e te of that dis osses the ho izo , desig ated the i le i

Figu e . A se o d a is is the st aight li e that joi s the poi t B of su ise at “u e “olsti e to

the poi t of su set at Wi te “olsti e E. A d the thi d a is is the st aight li e hi h joi sthe poi t D of su ise at Wi te “olsti e to the poi t of su set at “u e “olsti e C.

All of these th ee a es go th ough the o igi O, the poi t he e the o olith o g o o is

lo ated. The o olith is assu ed to e pla ed so that it is e ti al to the assu ed to e pe fe tl flat a d lo all ho izo tal pla e, a pla e thus ta ge t to the assu ed to e sphe i al Ea th s

su fa e at that poi t O. If a gle  is the lo atio s azi uth at “u e “olsti e hi h depe ds

o the poi t s latitude the ω= -. The o e tio et ee azi uth of “u e “olsti e

Figu e . The st i t e ui ale es a o g azi uth pla e, lo k a d ale da . At all poi ts o Ea th,

the due N-“ a is, a d the due E-W a is a e as follo s: the -  i azi uth a d id

ight-oo o the N-“ a is ; a d a - p dail hou s, -  i azi uth i the E-W a is . Off

these poi ts, the e ui ale es ollapse fo pla es hi h a e ot e uato ial pa allel that is to the

Ea th s e uato ial pla e . A gle  is the “u e “olsti e su ise azi uth at this spe ifi lo atio ,

ith -=ω. A is BE is the “u e “olsti e su ise to Wi te “olsti e su set a is; a d a is CD

is the Wi te “olsti e su ise to the “u e “olsti e su set a is. Poi t F is the due East su ise

poi t at Ve al a d Autu al E ui o , a d Poi t G is the due West su set poi t at E ui o . C iti al i these e ui ale es is the flat ess of the efe e e pla e a d the ho izo s e te t. “ou e: the autho .

U li ited ho izo a d pe fe tl flat g ou d, a d thei lo al o ditio s that is the adius of the i le i Figu e , a e i po ta t issues hi h ha e ee looked at so e detail i [ . ] a d [ . ]. i spe ifi , the su je t of the ho izo is iti al i the dis ussio ega di g oth the Ea

th-elated Ast o o , as ell as a site s A heolog a d A hite tu e. It is so, ostl e ause of the g ou d a o alies a d othe topog aphi al a d geog aphi al featu es tou hed upo i oth [ . ] a d [ . ] he e the su je t of fuzzi ess i shado s has ee add essed. I the latte efe e e, it as de o st ated i fa t p o ed that the e is t a i fi ite le gth i shado s fo a g o o s fi ite height. I additio , the Ea th s u atu e adds the ef a tio fa to , see [ . ] o

ef a tio , i shado le gths al ulatio s.

These a es of Figu e o tai fu da e tal e ui ale es as dis ussed i [ . ], the azi uth- lo

k-ale da e ui k-ale es. At a lo atio , the No th-“outh , -  , id ight – oo

e ui ale e, a d the due East-West , -  , a d the a – p e ui ale es e ist o

atte the altitude of the lo atio o the Ea th s su fa e, o the pla e of efe e e elati e to the Ea th s E uato ial Pla e, i.e. its slope o tilt elati e to that pla e. Ho e e , these e ui ale es do ot i ge e al e ist fo poi ts off these a ks, u less spe ial o ditio s hold fo the pla e of efe e e. A fi al poi t elated to these topi s: at a poi t o the Ea th s su fa e, a d o atte the g ou ds o the pla e of efe e e s slope elati e to the Ea th s E uato ial Pla e , the shado of the g o o passes th ough the e a t sa e poi t o the azi uth at a fi ed hou of the da , ut its le gth a ies depe di g o the da of the ea .

Off the poi ts he e the e is a e a t e ui ale e et ee azi uths, lo ks a d ale da s the e a e o ple elatio ships hi h o e t the th ee a ia les. These athe ati al elatio ships ill ot p eo up ou a al sis he e. It suffi es to sa that, app o i atel , a lo atio s azi uth

fo the isi g “u at “u e “olsti e ei g a o -li ea fu tio of latitude a e

app o i ated fo p a ti al pu poses, so that it a e assu ed to e a fe deg ees off that

deg ee of app o i atio ei g a fu tio of latitude the diffe e e et ee  a d the

lo atio s latitude. Mu h a e said a out this issue, ut it ill ot e add essed he e a fu the , as it does t di e tl elate to the su je t atte of this pape .

Figu e . The le gths of a g o o s shado s lo ated at the o igi poi t O at so e lo atio ith

a azi uth = -ω of the “u e “olsti e su ise fo a lo atio a o e the T opi of Ca e .

“ou e: the autho , ef. [ . ].

“hado s top e d sha p o fuzz , i.e., the -fu tio s of [ . ] see Figu e , t a e ell-shaped

ot si usoid u es, at all poi ts o the su fa e of the Ea th, No th of the  N latitude

the so- alled T opi of Ca e . It ust e oted that the -fu tio s do ot fo a o ti uous

-d su fa e; i stead, the a e dis ete paths t a ed the g o o st le o e the pe iod of the da that the “u is a o e the ho izo at a gi e lo atio o the Ea th s su fa e. This

dis ete ess has sig ifi a t i pli atio s i so fa as a dis o ti uit is i ol ed as the -fu tio s

u de go a phase t a sitio at E ui o : f o a o a e fa i g do lose to lo al oo ti e, to

a o e fa i g up set of u es. This issue ill e fu the ela o ated late i this se tio , he a u e of othe phase t a sitio s o u i g i oth spatial as ell as te po al d a i s of the shado s ill e a al zed.

Pa e theti all , all su je ts dealt ith he e add ess o ditio s i the No the He isphe e – o ditio s fo the “outhe He isphe e a e e ui ale t. O e eed o l o side a si ple o espo de e: fo a poi t u de o side atio at the No the He isphe e, a d i efe e e to the Ea th s e te , the e s a o espo di g poi t dia et i all opposite o the Ea th s su fa e at the “outhe He isphe e. Thus, o e a easil de i e e ui ale t state e ts, e ept that the

Wi te “olsti e fo the No the He isphe e ill o espo d to the “u e “olsti e fo the

“outhe He isphe e, a d the T opi of Ca e elated state e ts ill e T opi of Cap i o

lo ated at  “ elated e ui ale t o es.

These -fu tio elated u es pass th ough a poi t asso iated ith the lo al oo ti e

i i u th ough a Ea th da a d u de da light shado le gth the A , A , a d A poi ts i

Figu e . Of the th ee -fu tio s,  a d  a e ell-shaped a d thus go th ough a

i fle tio poi t; a d at su ise a d su set u de the si plif i g assu ptio s e tio ed ea lie

the a e o e o e as ptoti to a ds o e of the si ele a t st aight li e se i-a es. The si

se i-a es a e app oa hed as follo s, depe di g o the ti e of the ea , hethe the e is Ve al

o Autu al E ui o , ase a , Wi te “olsti e, ase , o “u e “olsti e, ase , these th ee

ases ei g e a i ed i tu .

I ase a , at E ui o , the shado is as ptoti to the se i-a is GO that joi s the due West

su set at E ui o poi t G ith the o igi O, a d at su set the shado e o es as ptoti to

FO , the se i-a is hi h joi s the due East su ise at E ui o poi t F ith the o igi O, i.e., the

poi t he e the o olith sta ds. It is u de s o ed that o l at E ui o , these t o se i-a es, GO a d FO ollapse o to a si gle st aight li e, FG, the due East-West a is. It is also oted that

at E ui o , a d o l at the Ea th s E uato , the “u ises a d sets at a  a gle at the ho izo .

Du i g da ti e at the E uato , the st le t a es a st aight li e, hi h is the due East-West li e. At

all othe ti es at the E uato , the “u ises a d sets at a a gle diffe e t tha , ut al a s

isi g at the poi t due East a a d setti g at the poi t due West p . Mo e o this ill e p ese ted i late su se tio s i Figu e .

E e da , a d fo a -hou seg e t, a poi t o the Ea th s E uato is a o e the Celestial E uato ial Pla e CEP , he eas at the othe -hou seg e t it dips elo CEP, see Figu e . At Ve al E ui o , a poi t o the Ea th s E uato splits its -hou da light seg e t spe di g si hou s of it a o e CEP a d the est si hou s elo CEP. E ui ale tl , the est of the -hou ight seg e t is split so that the poi t o the E uato spe ds si hou s elo a d si hou s a o e the CEP.

Figu e . The Celestial E uato ial Pla e CEP , the Ea th s a is of otatio , a d the T opi of Ca e . “ he ati all , the lo atio s of the Ea th at fou poi ts i the Ea th s o it a ou d the “u poi t “ a e sho , togethe ith the pa t of the Ea th s su fa e ei g illu i ated

su a s: at Wi te “olsti e ight , Ve al E ui o e te -left , “u e “olsti e left a d

Autu al E ui o e te - ight . “ou e: the autho .

At “u e “olsti e, a poi t o the Ea th s E uato splits the -hou pe iod spe di g the

-hou seg e t u de da light elo the CEP a d the othe half a o e it. At Autu al E ui o , the sa e poi t o the Ea th s E uato t a es a path si ila to that fo the Ve al E ui o e ept o e e se. Whe eas, at Wi te “olsti e, a poi t o the Ea th s E uato spe ds the -hou da light seg e t a o e the CEP, a d elo it at the -hou seg e t at ight. I the te po al

a ual d a i s, as the Ea th o its the “u , i te ediate o ditio s p e ail.

I ase at Wi te Solsti e, a d at su ise, the shado is as ptoti to the se i-a is that goes

th ough the poi t of su set at “u e “olsti e C o its azi uth spe ifi to e e lo atio , gi e

its latitude , a d the o igi O, i.e., the se i-a is CO ; he eas at su set, the o olith s shado e o es as ptoti to the se i-a is OB hi h joi s the poi t azi uth of the “u e “olsti e su ise, i.e., poi t B at that lo atio , ith the o igi O.

Fi all , i ase at Su e Solsti e, at su ise the shado is as ptoti to the se i-a is EO

hi h goes th ough the poi t he e the Wi te “olsti e su set o u s at that latitude, poi t E,

a d the o igi , O; a d at su set, the shado e o es as ptoti to se i-a is DO , hi h joi s

At a ious da s, et ee these th ee spe ifi da s of the ea , the efe e e a es [ CD a d BE ] a d se i-a es [ BO , CO , DO a d EO ] ha ge, although the a es CD a d BE e ai al a s

st aight li es ega dless of the da of the ea . Thei slopes ω’ ho e e depe d o the e ui ale t

azi uth of su ises a d su sets fo the da u de o side atio , keepi g i i d that these o e

i pai s. F o Figu e , a gles ’ a e al a s s et i to the East-West a d No th-“outh a es,

so that ’ is al a s g eate tha o e ual to  fo da s leadi g to “u e “olsti e, a d fo da s

past “u e “olsti e ; azi uth ’ e ai s also less tha o e ual to { -} fo a da leadi g

to a ds o o i g a a f o Wi te “olsti e.

E ui ale tl , slopes ω’ a e al a s less tha o e ual to +ω fo da s leadi g to a ds o o i g

a a f o “u e “olsti e; a d the a e g eate tha o e ual to -ω fo da s leadi g to a ds

o o i g a a f o Wi te “olsti e. All these poi ts ha e ee ela o ated i the pape s the

autho [ . ] a d [ . ]. The e, a a d idth of latitudes fa o a le fo the e ploitatio of shado s

as ide tified, as it as o side ed pa ti ula l suita le a d a e a le fo shado s elated A hite tu al desig of Neolithi o u e ts.

This is a suggested zo e i the Ea th s latitudes spe ulated to e p odu ti e i the e ploitatio of shado s, as tools fo desig . “hado s i those latitudes sta d a fa ette ha e fo ei g i o po ated i to the a hite tu al desig p ope of edifi es o st u tu es, ot o l i Neolithi o u e tal o st u tio , ut i late st u tu es as ell. This poi t is ade i the pape o the Classi al G eek Te ples, see efe e e [ . ]. This suita ilit had to do ith shado s ei g eithe too lo g o too sho t, espe iall at spe ifi hou s a d elati el lo g pe iods du i g the da th oughout the ea , to e of desig use.

We o tu to ases he e the shado s, du i g a lo g pe iod du i g the da a hou s u de su light , a e i deed too sho t a d apidl e o e too lo g to ha e a app e ia le i pa t upo a o u e t. This is the ase of the zo es elo the T opi of Ca e i the No the

He isphe e. Fo latitudes a o e app o i atel N, the shado s also e ai too lo g, o

atte the ti e of the da o the da of the ea to e of a app e ia le use i o u e tal A hite tu e at least i the Neolithi .

He e, the issue of lo atio al suita ilit e e ges, as if a goldilo ks t pe p i iple is p ese t, i.e., a opti u a d idth o Ea th zo e, he e o side a le a iatio i shado s le gths takes pla e, e de i g shado s of use i A hite tu al desig o site Pla i g. This the e is a ajo su je t this pape ad a es, a d the pape i [ . ] e ploits fo the ase of Classi al G eek Te ples.

Shado s at a d elo the t opi of Ca e o the No the He isphe e

The asi ifu atio

I Figu e , a e ui ale t diag a to that i Figu e is sho . Although Figu e as d a so that the ualitati e featu es of shado s fo latitudes g eate tha the T opi of Ca e TC e e d a , the ualitati e featu es d a i Figu e appl to lo atio s o the Ea th s su fa e

et ee the  latitude a d the E uato  latitude . At the outset, it ust e oted

that, all latitudes a o e the TC th oughout the -hou pe iod of a Ea th da a e fou d a o e

the CEP. A poi t o the TC tou hes the CEP at p at Ve al E ui o , oo at “u e

“olsti e, a at Autu al E ui o , a d at id ight o Wi te “olsti e, see Figu e .

Figu e ep ese ts a oss-se tio of the Celestial E uato ial Pla e, the pla e at hi h the Ea th o its the “u , assu ed he e to e a i ula i ealit it is a al ost ellipti al o it, so that the da ti e a d ightti e se tio s of the Ea th a e sho ; poi t “ is the lo atio of the “u , as a

poi t/sou e of light. Noted is also the fa t that at “u e “olsti e, a d at oo lo al ti e, the

“u is e ti al to the Ea th s su fa e at a poi t o the TC.

Fo a poi t at a latitude et ee the Ea th s E uato a d the TC, et ee the t o

E ui o es, the e a e t o da s of the ea that at lo al oo ti e the “u is at  a gle a o e

the ho izo . O e of these t o da s is et ee the Ve al E ui o a d the “u e “olsti e, a d

the othe et ee the “u e “olsti e a d the Autu al E ui o . The lose to the Ea th s TC

the poi t o side ed the highe the latitude, ut lo e tha  N , the lose these t o

da s a e, ollapsi g to a si gle da at “u e “olsti e at lo al oo ti e, fo the poi ts o TC.

This ifu atio e e t e ds at the E ui o es, he e the “u at lo al oo ti e is e ti al fo all

poi ts o the Ea th s E uato .

These o ditio s allo the de i atio of the e ui ale t u es to Figu e hi h e e d a fo poi ts i latitudes a o e the T opi of Ca e i the No the He isphe e to e de i ed fo latitudes elo the TC, sho i Figu e . As the t o ke a es, it is oted that a es BE a d CD ha e o a ui ed lo e a gles elati e to the EF a is, si e e a e add essi g latitudes ith

su ise at the “olsti es lose to the due East-West a is g eate azi uth fo su ise at “u e

“olsti e a d fo su set at Wi te “olsti e, a d o espo di gl lo e azi uth fo su ise at

Wi te “olsti e a d fo su set at “u e “olsti e .

Fi st, let s add ess hat takes pla e fo poi ts o the Ea th s su fa e at TC, Figu e . At TC, the

-fu tio s a e ualitati el si ila to those of Figu e ith a ota le e eptio : the 

fu tio , i.e., the fu tio depi ti g the st le s path at “u e “olsti e has a i i u le gth

depi ted poi t A at the o igi – i pl i g that at that poi t, a d at lo al oo ti e, the ast-off shado s le gth is ze o. Fo a poi ts et ee the Ea th s E uato a d TC ut ast-off oth

latitudes i the No the He isphe e fu tio  a d  etai thei ualitati e p ope ties,

he eas the fu tio  u de goes a spatiall dis o ti uous phase t a sitio to e

desig ated as PT .

This dis o ti uous phase t a sitio PT , sho i Figu e .a, ide tifies a  fu tio that

attai s a i i u at a poi t A “outh of the o igi e ause the poi t o the Ea th s su fa e is o elo the CEP . “till the le gth OA e ai s less tha OA , a d of ou se less tha OA .

As as the ase i the ualitati e p ope ties of the  a d  , ut ot  , fu tio s i

Figu e , the  a d  fu tio s o tai a i fle tio poi t. Ho e e ,  fu tio does

ot o tai a i fle tio poi t, up to e tai latitudes L elo TC.

Fo latitudes elo that th eshold L, the  fu tio does o tai a i fle tio poi t, ha i g

u de go e a othe spatiall dis o ti uous phase t a sitio , to e desig ated as PT a d to e

dis ussed i efe e e to the eha io of the shado s at a poi t at the Ea th s E uato , o e the ea , sho i Figu e .

I Figu e , the o ditio s ide tif i g the a o-d a i s ithi a da , a d o e a ea of the le gths a d lo atio s of shado s at a poi t o the Ea th s E uato a e sho . At E ui o ,

Ve al o Autu al, the shado s ast off the g o o , the  fu tio , fall o the due

East-West a is, the st le ei g at the o igi O at poi t A at lo al oo ti e. Up till oo ti e, the st le s shado o es f o poi t G to a ds O; he eas past oo ti e, it o es f o O to a ds

poi t F. At Wi te “olsti e, the st le s shado – fu tio  - t a es a ell-shaped u e ith a

a i u lo al oo ti e i i u at poi t A . E ui ale tl , at “u e “olsti e, the i o

i age fu tio  attai s a a i u i i u at lo al oo ti e at poi t A , su h that

le gths OA is e ual to OA .

Figu e .a. The te po al d a i s of shado s le gths fo poi ts o the Ea th s latitudes elo

the T opi of Ca e . The  -fu tio is o as ptoti ithout a i fle tio poi t as it

Figu e . The d a i s of shado s at the Ea th s E uato . Both  a d  fu tio o tai i fle tio poi ts the a e ell-shaped t pe u es . “ou e: the autho .

The a ious phase t a sitio s

I p e ious pape s, deali g ith poi ts o the Ea th s su fa e a o e the T opi of Ca e , Figu e , the ge e all appli a le te po al dis o ti uit i ol i g the E ui o - elated t a sitio f o a ell-shaped u e ha gi g to a st i tl o e o e as dis ussed. No , o e ela o atio o the a ious spatial a d te po al dis o ti uities ill e offe ed, o e i g all poi ts o the Ea th s su fa e i the No the He isphe e .

Poi t of dis o ti uous eha io L as ide tified ea lie . Whe e is the the poi t L o the Ea th s

su fa e he e the phase t a sitio PT o u s, he e the  fu tio fo poi ts elo

 , i.e., elo TC o tai s a i fle tio poi t? It is he e poi t A * o espo ds to a

su ise fo this pa ti ula lo atio , the o igi , a d the azi uth of the Wi te “olsti e su set hi h has i the o Ca tesia oo di ates of the N-“ a d E-W a es, a N-“ a is easu e e ual to

OA . I effe t, this is a spatiall dis o ti uous phase t a sitio .

It is f uitful he e to e tio a fe ge e al o e ts ega di g these a ious te po al o spatial

phase t a sitio s p ese t i Figu es , a d .a. The -fu tio s, i ge e al, do ot o stitute a

o ti uous -d su fa e, of ou se. The o p ise a set of dail paths sepa ated dis ete ti e i te als. E e poi t o these paths o espo ds to a pa ti ula hou of a Ea th da , a d the e t path ossi g that hou s adius f o the o igi has a -hou ti e dela i te al i it f o the p e ious da s path. I effe t, these paths o stitute a set of dis ete u es, he e te po al

ut ot e essa il spatial – that is, o e diffe e t latitudes dis o ti uities e ist.

Although, i spa e, a d o e diffe e t latitudes, s ooth a d o ti uous ha ges take pla e as the o se e o es alo g diffe e t latitudes, this s ooth ess does ot gua a tee that all spatial

phase t a sitio s o u s oothl a d a e o ti uous. A e a ple of this dis o ti uit as PT .

Thus, a d i su a , at ke poi ts, te po all dis o ti uous phase t a sitio s o u as

dis ussed i [ . ], [ . ] a d he e . These t a sitio s a e ot s ooth, ut the a e ha a te ized dis ete ju ps o e ti e ut ot e essa il o e spa e . O e spa e, s ooth spatial phase t a sitio s do o u , as the o e dis ussed a o e, that i ol ed the s it h f o a shado of a g o o o the Ea th s E uato poi ti g No th at lo al oo ti e to o e poi ti g “outh afte E ui o at lo al oo ti e.

The goldilo ks effe t i the te pe ate zo e

The e te ded a al sis of the p e ious su se tio s akes lea a d Figu es , .a, a d ake o ious that elo the T opi of Ca e ast-off shado s a e eithe too sho t at hou s lose to lo al oo ti e, si e the e ai o fi ed a d est i ted to a a o a d of g ou d lose to thei aseli e. The sudde l , a d e plosi el , the e o e e lo g keepi g i i d that the fuzz aspe t of shado s, hi h st ips the f o thei pote tiall e o ita t le gths, has ee assu ed a a . I additio , a d afte the Ve al E ui o , the alte thei di e tio s it hi g thei ast-off shado s f o a No the l di e tio to “outhe l as these lo atio s o the Ea th s su fa e egi to dip elo the CEP. The ai tai this di e tio till the Autu al E ui o , he the appea agai a o e the CEP a d ith a No the l di e tio i thei ast-off shado s. It is thus o luded that at su h latitudes, a a hite t had se e e o st ai ts to fa e i utilizi g shado s as a ea s to a ds a hie i g a hite to i goals – ai i g at li ki g a o olith to othe

ea o u e tal st u tu es.

Eithe the elati e spa i g of othe o oliths ould had ee ad e sel affe ted – so that if asti g of a shado o e o olith o to a othe as i te ded, the these o oliths ould had to e pla ed e lose to ea h othe , so ethi g that ould pote tiall egati el had affe ted o olith spatial de sities a d/o thei elati e sizes. E e f ee-sta di g o olith has

a ital effe ti e spa e su ou di g it, a d u de su h high de sit o ditio s the f ee att i ute of the f ee-sta di g ha a te izatio ould had ee egati el i pa ted. Mo u e tal s ale ould i tu had ee ad e sel affe ted, as the eed fo g eate i height o oliths ould ate ialize – se e el affe ti g o st u tio o pholog a d ost let alo e st u tu al a d esou e feasi ilit fo the e ti e o u e t .

“i ila o lusio s a e d a fo lo atio s a o e  latitude – the o oliths shado s

ould si pl had ee too lo g, fo too le gth of a ti e pe iod du i g da light. He e, these

Ea th latitudes defi itel a o e  a d lose to o fa elo the T opi of Ca e ould ot e

o du i e to o st u ti g o u e ts o tai i g o oliths so that a a hite t ould o ould take ad a tage of thei shado s. B default, the latitudes et ee the , latitudes hi h o tai

ithi the sites su h as A u “i el at a out  N to the “outh, a d Measho e at

a out  N to the No th, allo fo a oad e ough a iatio i ast-off shado s le gth

to pe it i lusio of thei effe ts upo the desig of o u e tal a d e e o ial st u tu es.

He e, the a d idth et ee the T opi of Ca e a d the  latitude o stitutes a a d idth

hi h i o i atio ith the Te pe ate zo e ithi these t o ou da ies o i e to offe a e i o e tall suita le ‘egio fo the e ploitatio of shado s i o u e tal o st u tio . Withi these t o latitudi al ou da ies, a d i fa t al ost half a et ee the t o possi le e t e es, he e shado s a e a hite tu all possi l opti all e ploita le, o e fi ds so e stella e a ples of Classi al G eek Te ples. A o g the is the Te ple of eithe He a o

Aph odite i a ase, Te ple E at “eli u te at a out  N , a i a iddle th _{e tu}

BC st u tu e i “i il ; the Fi st i a BC a d “e o d i a BC Te ples of He a at

Paestu oth lo ated at a out  N i the Ape i e Pe i sula; the Pa the o at a out

 N , desig ed a hite ts Kallik atis a d Ikti os, a d the Te ple of Epi ou ius Apollo

at a out  desig ed Ikti os, oth iddle of the th _{Ce tu BC a d Te ples o the}

G eek Pe i sula. These Te ples a e a al zed i the pape this autho titled Mo i g shado s a d the Te ples of Classi al G ee e i efe e e [ . ].

Ho e e , a ajo poi t eed e ought up o . B the ti e CGTs e e e e ted, the ole of

shado s had ee la gel t a sfo ed. F o thei Ast o o a d s olis elated fu tio s

e edded o to Neolithi o u e ts, a d thus f o a e lusi el utilita ia e ploitatio of

shado s, to o e of a aestheti use of shado s. “hado s a o-d a i s, e hi ited o e the ou se of a da u de eithe su light o oo light a d o e the ou se of a ea , a ui ed a

ole e ha i g the o e all isual effe t of thei p ese e, espe iall i o e tio ith a ke

att i ute of CGTs, a el the entasis ΕΝ Α Ι effe t, see [ . ].

This e olutio as a hie ed eithe i o po ati g the o u e ts ast-off shado s effe ts o to thei su ou di gs i the o e all desig a d site pla a d Te ple o ie tatio as the Neolithi a hite t also did, al eit fo diffe e t easo s ; a d/o th ough shapi g detailed o phologi al featu es i the Te ple s supe st u tu e o po e ts su h as the shafts of thei olu s a d the depths of thei e ta latu e s eliefs so that oth thei ast-off a d a -o

shado s effe ts upo the a hite to i detail ould add life to these st u tu al o po e ts of Classi al A hite tu e i o po ati g pe iodi a d egula ha ge, i the fo of addi g otio ,

i to the . It as a i ge ious a to g a t life to othe ise i a i ate o je ts.

These issues a e o e full add essed i efe e e [ . ].

Co lusio s

A u e of o lusio s a e d a f o the a al sis a o e, a d a u e of suggestio fo fu the esea h as ell. Fi st, shado s a e i deed o ious a d i po ta t ph si al ele e ts of st u tu es, that a a hite t ust ha e had i o po ated i to thei desig . Thei te po al as ell as spatial d a i s a e of e t e e i po ta e, oth f o a Ph si s a d Mathe ati s sta dpoi t as ell as f o a A hite tu e a d A heolog ie .

O e of the featu es hi h has ee dis ussed i a p e ious pape the autho , ut as ot i o po ated he e is that of the shado s fuzz atu e. E te di g the a o e a al sis alo g these li es ight p o ide o e pe et ati g i sights i to these iti al ele e ts of o u e ts. Fi all , the ole of the Moo -i du ed shado s ust also e add essed. The shado s u de oo light

ight e i fo ati e ega di g the atu e p ope of a Neolithi o u e t.

I the t aditio of this autho s o k o atte s of A heolog a d A hite tu e, this is a pape add essed to a ge e al audie e ith College le el e posu e to Mathe ati s a d Ast o o . Of ou se, the Mathe ati s, Ph si s a d Ast o o p ope , as ell as A hite tu e a d A heolog i ol ed topi s e ui es fu the ela o atio a ade i s ith e pe tise i these i di idual fields.

Refe e es

Autho ’s o k

[ . ] Di it ios “. De d i os, Ja ua , The Mathe ati s of Mo oliths “hado s ,

a ade ia.edu. The pape is fou d he e:

https:// .a ade ia.edu/ /The_Mathe ati s_of_Mo oliths_“hado s

[ . ] Di it ios “. De d i os, Ma h , O the Fuzz Natu e of “hado s , a ade ia.edu.

The pape is fou d he e:

https:// .a ade ia.edu/ /ON_THE_FU))Y_NATU‘E_OF_“HADOW“

[ . ] Di it ios “. De d i os, No e e , Go ekli Tepe: a th _{ille iu BC o u e t ,}

a ade ia.edu. The pape is fou d he e:

[ . ] Di it ios “. De d i os, No e e , I the “hado s of Ca a s Le Me e “to es: a Neolithi p oto supe o pute . The pape is fou d he e:

https:// .a ade ia.edu/ /I _the_“hado s_of_Ca a s_Le_Me e _“to es_A_Ne olithi _p oto_supe o pute

[ . ] Di it ios “. De d i os, Ma h , O “to ehe ge a d its Mo i g “hado s ,

a ade ia.edu. The pape is fou d he e:

https:// .a ade ia.edu/ /O _“to ehe ge_a d_its_Mo i g_“hado s

[ . ] Di it ios “. De d i os, Ap il , Mo i g “hado s a d the Te ples of Classi al

G ee e , a ade ia.edu. The pape is fou d at: https://ka sas.a ade ia.edu/Di it iosDe d i os

Othe sou es

[ . ] Da id “ th, , Taosli : a diffe e t pe spe ti e , a ade ia.edu.

[ . ] G. Te e e Meade , , D o e g sto e i le, “W I ela d: desig pla a al zed ith

espe t to su ises a d lithi shado - asti g fo the eight t aditio al ag i ultu al dates – a d

fu the alidated photog aph , fo th o i g i the Journal of Lithic Studies.

[ . ] The egalithi po tal:

http:// . egalithi . o.uk/ odules.php?op= odload& a e=Fo u &file= ie topi &topi = &fo u = &sta t=

[ . ] “i o Ne o , , A Co pe diu of Sphe i al Ast o o , Ma Milla , Ne Yo k.

A k o ledge e ts

The autho ishes to a k o ledge the o t i utio s ade to this pape th ough his i te a tio

ith e e s of the Megalithi Po tal e site http:// . egalithi . o.uk/i de .php . The

autho ishes to tha k the site a age A d Bu ha ; a d also, the e e s of the su -g oup i it Do ou look at the shado s . Withi this g oup, i spe ifi , the autho ishes to e og ize the o t i utio s Da id “ th E e g a , Neil Wise a Fea o , a d ‘i ha d Ba tosz O p it . Da id “ th has ee i st u e tal i e ou agi g the autho to pa ti ipate i that g oup, a d fo i t odu i g the autho s p io o k to the g oup e e s. A u e of othe s,

i ludi g those u de the lite a pseudo of e ig a d d olaf, ha e also o t i uted to the

fo atio of the autho s ie s a d stud of shado s.

Fu the o e, the autho ishes to e og ize his Fa e ook f ie ds a d espe iall those ho a e e e s of the se e g oups the autho has eated a d is ad i iste i g. Thei o ti uous i telle tual a d a tisti sti ulatio a d suppo t ha e ee e a ka le, a d thus a g eat se se of

g atitude is e te ded to the as ell. “pe ial e tio ust e ade to P ofesso Te e e Meade fo his o t i utio , a d espe iall his o k o the shado s at D o eg a d “to ehe ge.

But ost i po ta t a d dee to this autho has ee the o e tha ea s of e ou age e t a d suppo t he has e ei ed f o his ife Cathe i e a d thei daughte s Daph e-I is a d Ale ia-A te is. Thei o ti ui g assista e a d u de sta di g fo all those lo g hou s he spe t o doi g

esea h, he he ould ha e allotted ti e ith the , this autho ill al a s e deepl app e iati e.

Legal Noti e o Cop ight

©

The autho , Di it ios “. De d i os etai s full legal op ights to the o te ts

of this pape . ‘ep odu tio i a fo of pa ts o the hole of this pape is

p ohi ited, ithout the e pli it a d itte pe issio a d o se t the autho ,

Di it ios “. De d i os.

Figu e . The d a i s of shado s at the Ea th s E uato . Both  a d  fu tio o tai i fle tio poi ts the a e ell-shaped t pe u es . “ou e: the autho .

The a ious phase t a sitio s

I p e ious pape s, deali g ith poi ts o the Ea th s su fa e a o e the T opi of Ca e , Figu e , the ge e all appli a le te po al dis o ti uit i ol i g the E ui o - elated t a sitio f o a ell-shaped u e ha gi g to a st i tl o e o e as dis ussed. No , o e ela o atio o the a ious spatial a d te po al dis o ti uities ill e offe ed, o e i g all poi ts o the Ea th s su fa e i the No the He isphe e .

Poi t of dis o ti uous eha io L as ide tified ea lie . Whe e is the the poi t L o the Ea th s su fa e he e the phase t a sitio PT o u s, he e the  fu tio fo poi ts elo  , i.e., elo TC o tai s a i fle tio poi t? It is he e poi t A * o espo ds to a a gle of the COE a is of Figu e .a the a is hi h o tai s the azi uth of the “u e “olsti e

su ise fo this pa ti ula lo atio , the o igi , a d the azi uth of the Wi te “olsti e su set hi h has i the o Ca tesia oo di ates of the N-“ a d E-W a es, a N-“ a is easu e e ual to OA . I effe t, this is a spatiall dis o ti uous phase t a sitio .

It is f uitful he e to e tio a fe ge e al o e ts ega di g these a ious te po al o spatial phase t a sitio s p ese t i Figu es , a d .a. The -fu tio s, i ge e al, do ot o stitute a o ti uous -d su fa e, of ou se. The o p ise a set of dail paths sepa ated dis ete ti e i te als. E e poi t o these paths o espo ds to a pa ti ula hou of a Ea th da , a d the e t path ossi g that hou s adius f o the o igi has a -hou ti e dela i te al i it f o the p e ious da s path. I effe t, these paths o stitute a set of dis ete u es, he e te po al

ut ot e essa il spatial – that is, o e diffe e t latitudes dis o ti uities e ist.

Although, i spa e, a d o e diffe e t latitudes, s ooth a d o ti uous ha ges take pla e as the o se e o es alo g diffe e t latitudes, this s ooth ess does ot gua a tee that all spatial phase t a sitio s o u s oothl a d a e o ti uous. A e a ple of this dis o ti uit as PT . Thus, a d i su a , at ke poi ts, te po all dis o ti uous phase t a sitio s o u as dis ussed i [ . ], [ . ] a d he e . These t a sitio s a e ot s ooth, ut the a e ha a te ized dis ete ju ps o e ti e ut ot e essa il o e spa e . O e spa e, s ooth spatial phase t a sitio s do o u , as the o e dis ussed a o e, that i ol ed the s it h f o a shado of a g o o o the Ea th s E uato poi ti g No th at lo al oo ti e to o e poi ti g “outh afte E ui o at lo al oo ti e.

The goldilo ks effe t i the te pe ate zo e

The e te ded a al sis of the p e ious su se tio s akes lea a d Figu es , .a, a d ake o ious that elo the T opi of Ca e ast-off shado s a e eithe too sho t at hou s lose to lo al oo ti e, si e the e ai o fi ed a d est i ted to a a o a d of g ou d lose to thei aseli e. The sudde l , a d e plosi el , the e o e e lo g keepi g i i d that the fuzz aspe t of shado s, hi h st ips the f o thei pote tiall e o ita t le gths, has ee assu ed a a . I additio , a d afte the Ve al E ui o , the alte thei di e tio s it hi g thei ast-off shado s f o a No the l di e tio to “outhe l as these lo atio s o the Ea th s su fa e egi to dip elo the CEP. The ai tai this di e tio till the Autu al E ui o , he the appea agai a o e the CEP a d ith a No the l di e tio i thei ast-off shado s. It is thus o luded that at su h latitudes, a a hite t had se e e o st ai ts to fa e i utilizi g shado s as a ea s to a ds a hie i g a hite to i goals – ai i g at li ki g a o olith to othe

ea o u e tal st u tu es.

Eithe the elati e spa i g of othe o oliths ould had ee ad e sel affe ted – so that if asti g of a shado o e o olith o to a othe as i te ded, the these o oliths ould had to e pla ed e lose to ea h othe , so ethi g that ould pote tiall egati el had affe ted o olith spatial de sities a d/o thei elati e sizes. E e f ee-sta di g o olith has

a ital effe ti e spa e su ou di g it, a d u de su h high de sit o ditio s the f ee att i ute of the f ee-sta di g ha a te izatio ould had ee egati el i pa ted. Mo u e tal s ale ould i tu had ee ad e sel affe ted, as the eed fo g eate i height o oliths ould ate ialize – se e el affe ti g o st u tio o pholog a d ost let alo e st u tu al a d esou e feasi ilit fo the e ti e o u e t .

“i ila o lusio s a e d a fo lo atio s a o e  latitude – the o oliths shado s ould si pl had ee too lo g, fo too le gth of a ti e pe iod du i g da light. He e, these Ea th latitudes defi itel a o e  a d lose to o fa elo the T opi of Ca e ould ot e o du i e to o st u ti g o u e ts o tai i g o oliths so that a a hite t ould o ould take ad a tage of thei shado s. B default, the latitudes et ee the , latitudes hi h o tai ithi the sites su h as A u “i el at a out  N to the “outh, a d Measho e at a out  N to the No th, allo fo a oad e ough a iatio i ast-off shado s le gth to pe it i lusio of thei effe ts upo the desig of o u e tal a d e e o ial st u tu es. He e, the a d idth et ee the T opi of Ca e a d the  latitude o stitutes a a d idth hi h i o i atio ith the Te pe ate zo e ithi these t o ou da ies o i e to offe a e i o e tall suita le ‘egio fo the e ploitatio of shado s i o u e tal o st u tio . Withi these t o latitudi al ou da ies, a d i fa t al ost half a et ee the t o possi le e t e es, he e shado s a e a hite tu all possi l opti all e ploita le, o e fi ds so e stella e a ples of Classi al G eek Te ples. A o g the is the Te ple of eithe He a o Aph odite i a ase, Te ple E at “eli u te at a out  N , a i a iddle th _{e tu} BC st u tu e i “i il ; the Fi st i a BC a d “e o d i a BC Te ples of He a at Paestu oth lo ated at a out  N i the Ape i e Pe i sula; the Pa the o at a out  N , desig ed a hite ts Kallik atis a d Ikti os, a d the Te ple of Epi ou ius Apollo at a out  desig ed Ikti os, oth iddle of the th _{Ce tu BC a d Te ples o the} G eek Pe i sula. These Te ples a e a al zed i the pape this autho titled Mo i g shado s a d the Te ples of Classi al G ee e i efe e e [ . ].

Ho e e , a ajo poi t eed e ought up o . B the ti e CGTs e e e e ted, the ole of shado s had ee la gel t a sfo ed. F o thei Ast o o a d s olis elated fu tio s e edded o to Neolithi o u e ts, a d thus f o a e lusi el utilita ia e ploitatio of shado s, to o e of a aestheti use of shado s. “hado s a o-d a i s, e hi ited o e the ou se of a da u de eithe su light o oo light a d o e the ou se of a ea , a ui ed a ole e ha i g the o e all isual effe t of thei p ese e, espe iall i o e tio ith a ke att i ute of CGTs, a el the entasis ΕΝ Α Ι effe t, see [ . ].

This e olutio as a hie ed eithe i o po ati g the o u e ts ast-off shado s effe ts o to thei su ou di gs i the o e all desig a d site pla a d Te ple o ie tatio as the Neolithi a hite t also did, al eit fo diffe e t easo s ; a d/o th ough shapi g detailed o phologi al featu es i the Te ple s supe st u tu e o po e ts su h as the shafts of thei olu s a d the depths of thei e ta latu e s eliefs so that oth thei ast-off a d a -o

shado s effe ts upo the a hite to i detail ould add life to these st u tu al o po e ts of Classi al A hite tu e i o po ati g pe iodi a d egula ha ge, i the fo of addi g otio , i to the . It as a i ge ious a to g a t life to othe ise i a i ate o je ts.

These issues a e o e full add essed i efe e e [ . ].

Co lusio s

A u e of o lusio s a e d a f o the a al sis a o e, a d a u e of suggestio fo fu the esea h as ell. Fi st, shado s a e i deed o ious a d i po ta t ph si al ele e ts of st u tu es, that a a hite t ust ha e had i o po ated i to thei desig . Thei te po al as ell as spatial d a i s a e of e t e e i po ta e, oth f o a Ph si s a d Mathe ati s sta dpoi t as ell as f o a A hite tu e a d A heolog ie .

O e of the featu es hi h has ee dis ussed i a p e ious pape the autho , ut as ot i o po ated he e is that of the shado s fuzz atu e. E te di g the a o e a al sis alo g these li es ight p o ide o e pe et ati g i sights i to these iti al ele e ts of o u e ts. Fi all , the ole of the Moo -i du ed shado s ust also e add essed. The shado s u de oo light

ight e i fo ati e ega di g the atu e p ope of a Neolithi o u e t.

I the t aditio of this autho s o k o atte s of A heolog a d A hite tu e, this is a pape add essed to a ge e al audie e ith College le el e posu e to Mathe ati s a d Ast o o . Of ou se, the Mathe ati s, Ph si s a d Ast o o p ope , as ell as A hite tu e a d A heolog i ol ed topi s e ui es fu the ela o atio a ade i s ith e pe tise i these i di idual fields.

Refe e es

Autho ’s o k

[ . ] Di it ios “. De d i os, Ja ua , The Mathe ati s of Mo oliths “hado s , a ade ia.edu. The pape is fou d he e:

https:// .a ade ia.edu/ /The_Mathe ati s_of_Mo oliths_“hado s

[ . ] Di it ios “. De d i os, Ma h , O the Fuzz Natu e of “hado s , a ade ia.edu. The pape is fou d he e:

https:// .a ade ia.edu/ /ON_THE_FU))Y_NATU‘E_OF_“HADOW“

[ . ] Di it ios “. De d i os, No e e , Go ekli Tepe: a th _{ille iu BC o u e t ,} a ade ia.edu. The pape is fou d he e:

[ . ] Di it ios “. De d i os, No e e , I the “hado s of Ca a s Le Me e “to es: a Neolithi p oto supe o pute . The pape is fou d he e:

https:// .a ade ia.edu/ /I _the_“hado s_of_Ca a s_Le_Me e _“to es_A_Ne

olithi _p oto_supe o pute

[ . ] Di it ios “. De d i os, Ma h , O “to ehe ge a d its Mo i g “hado s , a ade ia.edu. The pape is fou d he e:

https:// .a ade ia.edu/ /O _“to ehe ge_a d_its_Mo i g_“hado s

[ . ] Di it ios “. De d i os, Ap il , Mo i g “hado s a d the Te ples of Classi al G ee e , a ade ia.edu. The pape is fou d at: https://ka sas.a ade ia.edu/Di it iosDe d i os

Othe sou es

[ . ] Da id “ th, , Taosli : a diffe e t pe spe ti e , a ade ia.edu.

[ . ] G. Te e e Meade , , D o e g sto e i le, “W I ela d: desig pla a al zed ith espe t to su ises a d lithi shado - asti g fo the eight t aditio al ag i ultu al dates – a d fu the alidated photog aph , fo th o i g i the Journal of Lithic Studies.

[ . ] The egalithi po tal:

http:// . egalithi . o.uk/ odules.php?op= odload& a e=Fo u &file= ie topi &topi =

&fo u = &sta t=

[ . ] “i o Ne o , , A Co pe diu of Sphe i al Ast o o , Ma Milla , Ne Yo k.

A k o ledge e ts

The autho ishes to a k o ledge the o t i utio s ade to this pape th ough his i te a tio ith e e s of the Megalithi Po tal e site http:// . egalithi . o.uk/i de .php . The autho ishes to tha k the site a age A d Bu ha ; a d also, the e e s of the su -g oup i it Do ou look at the shado s . Withi this g oup, i spe ifi , the autho ishes to e og ize the o t i utio s Da id “ th E e g a , Neil Wise a Fea o , a d ‘i ha d Ba tosz O p it . Da id “ th has ee i st u e tal i e ou agi g the autho to pa ti ipate i that g oup, a d fo i t odu i g the autho s p io o k to the g oup e e s. A u e of othe s, i ludi g those u de the lite a pseudo of e ig a d d olaf, ha e also o t i uted to the fo atio of the autho s ie s a d stud of shado s.

Fu the o e, the autho ishes to e og ize his Fa e ook f ie ds a d espe iall those ho a e e e s of the se e g oups the autho has eated a d is ad i iste i g. Thei o ti uous i telle tual a d a tisti sti ulatio a d suppo t ha e ee e a ka le, a d thus a g eat se se of

g atitude is e te ded to the as ell. “pe ial e tio ust e ade to P ofesso Te e e Meade fo his o t i utio , a d espe iall his o k o the shado s at D o eg a d “to ehe ge.

But ost i po ta t a d dee to this autho has ee the o e tha ea s of e ou age e t a d suppo t he has e ei ed f o his ife Cathe i e a d thei daughte s Daph e-I is a d Ale ia-A te is. Thei o ti ui g assista e a d u de sta di g fo all those lo g hou s he spe t o doi g

esea h, he he ould ha e allotted ti e ith the , this autho ill al a s e deepl app e iati e.

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