“Pilat Quadrant” & �Pythagorean triples��Summary of all Pythagorean triples in relation to Pilat Quadrant. �
Building the hypothesis Pilat Quadrant is a place to learn how to cross Europe at neolithic age.
thomas.decharentenay@gmail.com +33682878350
Agenda�
What is Pilat Quadrant
Pythagoran triangles at local level (Pilat Quadrant)
Pythagoran triangles at local level (Alps, Europe)
Right angle tiangles & solticial axes (Europe level)
Speculation around neolithic navigation in Europe
See also:
Good to know about “Pilat Quadrant”
Pilat Quadrant in a neolothic site centered around a large flat and rounded stone know as “Pierre des Trois Évêques”.
This large stone is known to mark the border of 3 regions in France :
thomas.decharentenay@gmail.com +33682878350
Pierre des Trois Evêques
Good to know at “Pilat Quadrant”
“Pierre des Trois Évêques” is a large stone, that is center a vast arrangement of hundreds of stones, cromlechs and remarkable megaliths. Discovering in 2016 this setting is called the Pilat Quadrant because from “Pierre des Trois Evêques” we have alignments of standing stones targetting specific places in the landscape, as would do an orientation quadrant.
There we have specific stone triangles:
thomas.decharentenay@gmail.com +33682878350
Pierre des Trois Evêques
Good to know�about Pythagorean triangle properties
< 16.26° angle with E/W axe = we have a 7 24 25 square triangle
Initial rendering of stones positioned at Pilat Quadrant (dots) suggesting the typical angle of the 7 24 25 triangle.
16.26°
Pythagorean triples at local level
In the surroundings of Pierre des Trois Evêques
Two main settings have been explored
West side : next to Pierre des trois evêques (triangles)�East side at es Faves (cromlechs)
Setting 1 - Around Pierre des Trois Evêques, North West of the site
Setting 2 - Around Stone Circle of the Faves, South Est of the site
First findings through manual data gathering then topographic mapping
7 - 24 - 25
perfect orientated N/S triangle�Proportionate to 10 megalithic yard (MY)
November 2016 - measuring manually distances between blocks
First surprising outcome with Google map�few weeks after publication, already concrete collaboration
After circulating the link, an additional talent (architect by training, Eric Charpentier) gave a pivotal contribution.
He extracted the line from quadrant target (Pierre des Trois Eveques) to biggest megaliths called Écrin viewfinder and observed we have a 7,24 25 triangle.
His finding is based on 1 observation:
16.26° angle with E/W axe
16.26°
Écrin viewfinder.
thomas.decharentenay@gmail.com +33682878350
Pythagorean triangles at Pilat Quadrant
The “Spring” 5 12 13 triangle�See further down in this presentation
The Alps viewfinder 7 24 25 triangle
As well as PIlat Quadrant to Beigua in Italy is the same 7 24 25 triangle.
At Pilat Quadrant, we have several pythagorean triplets triangles. This presentation is about “Spring” Triangle
Discovery of the angular stone of 7-23-24 triangle
From the assumption we have a 7 24 25 triangle, (because of long line angle with E/W reference) back to the field, the angle stone of this triangle is found, at the right place it ‘should’ be :
Smaller than others, but exact position and no other stones around.
This is our first predictive model finding: defining the position of a stone before field observation from a mathematical model.
thomas.decharentenay@gmail.com +33682878350
Discovery of a megalithic yard (YM) unit
7 units = 58,06 m
Unit is 8,29 m which is 10 Megalithic Yard.
We have a 70 YM, 240 YM, 250 YM pythagorean square triangle with N/S & E/W perfect alignment, less than 0,1° to geographical North.
Finding through topographic mapping
Perfect
5 12 12 triangle
orientated N/S
Proportionate to 10 nippur cubit
October 2017 - Tacking reference point at Pierre de Trois Evêques with a profedssional GPS positionning system
Scientific mapping of main area next to the “target” �thanks to benevolent expertise of teachers of SEPR school
After a presentation of the project and first results by Th de Charentenay to teachers and program lead of SEPR professional school of Lyon,
5 Metrologists* (2 teachers & 3 students) and a 3D scanner technician for GeoTopo company, spent 3 days on the site to conduct a complete map of area next to the Pierre des Trois Evêques (East ward).
This map was then exploited in Autocad as a basis for detailed analysis.
*Metrology: the science of measurement, embracing both experimental and theoretical determinations at any level of uncertainty in any field of science and technology
Pythagorean triangles at Pilat Quadrant
The “Spring” 5 12 13 triangle�This presentation zoom
The Alps viewfinder 7 24 25 triangle
As well as PIlat Quadrant to Beigua in Italy is the same 7 24 25 triangle.
At Pilat Quadrant, we have several pythagorean triplets triangles. This presentation is about “Spring” Triangle
The “spring” pythagorean triangle�at Pilat Quadrant
At Pilat Quadrant, from Ecrin ViewFinder to “the Spring” at upper point of the entire site, the line is 22,62° vs E/W axe. We have by definition a 5-12-13 pythagorean triangle.
Surprise 1 : angle in degree equals to length in meters: 67, 38 m / 67,38°. How can it be so?�Surprise 2: Unit of this triangle 5-12-13 is of 10 NC, the Nippur cubit measure of 51,84 cm
The Spring (Autocad) (drone image)
The “spring” pythagorean triangle�at Pilat Quadrant
At Pilat Quadrant, from Ecrin ViewFinder to “the Spring” at upper point of the entire site, the line is 22,62° vs E/W axe. We have by definition a 5-12-13 pythagorean triangle.
Surprise 2: Unit of this triangle 5-12-13 is of 10 NC, the Nippur cubit measure of 51,84 cm
The Spring (Autocad) (drone image)
Hypothesis : Nippur Cubit size refers to earth size
At “Sprint” triangle, angle equal length in meters: 6 738 cm <AND> 67,38° �How can it be so? What does it mean in practical terms for people using NC.
We see that in meters and degree, that figure is proportionate : 1 metre / 67. 38 dm <> 1 degree / 67.38 degree.
We know meter is ¼ of a 10 millionth of a North South meridian.
We suppose same type of measure of distance on earth was true for antic measures and presented in our “Spring” triangle.
Then if true, on this figure : 130 NC gives 13 units <AND> 67.38 degree gives 13 “ref angle units”
Which in antic measures would mean : 1/13 th of a size of the triangle is 10 CN, as well 1/13 th of the angle is the angle unit. At that time angle are evaluated as base 60 (as it is also today thanks to mesopotamian mathematics that survived up to us!): 67.22.37 degrés (base sixty) = 67,3769 decimal = 13 x 5,1828 �
>> 1 antic reference degree = 5,1828 modern degrees (5.10’.58’’)
Then earth size = (4 x 10 000 / reference degree in degree) x (unit measure)
which is earth size = (4 x 10 000 / 1/13 of angle 5-12-13 in degree) x NC in meter.
Check: 40 009 km = 40 000 / 5,1828 x 5,184 (in fact 40 005 km is the mean value for a mean radius of 6 367 km)
Nippur Cubit artefact https://en.wikipedia.org/wiki/Cubit#/media/File:Nippur_cubit.JPG / Base 60 conversion https://www.calculatorsoup.com/calculators/time/time-to-decimal-calculator.php
Mesopotammian fractions (1/13) approximations https://en.wikipedia.org/wiki/Babylonian_mathematics#Arithmetic use of equivament to Pythagorean triangle https://en.wikipedia.org/wiki/Babylonian_mathematics#/media/File:Clay_tablet,_mathematical,_geometric-algebraic,_similar_to_the_Pythagorean_theorem._From_Tell_al-Dhabba'i,_Iraq._2003-1595_BCE._Iraq_Museum.jpg
About Nippur Cubit
Nippur Cubit artefact https://en.wikipedia.org/wiki/Cubit#/media/File:Nippur_cubit.JPG
51,84 cm
��https://www.hindawi.com/journals/janthro/2014/489757/ Nippur Cubbit of 51.85 cm �
Findings through existing digital maps
5 12 13 perfect orientated triangle from Pierre des Trois Evêques to Le Flat Standing Stone
Google Map rendering of Pilat Quadrant ‘target’ (Pierre des Trois Eveques) and Le Flat Standing stone
Pythagorean triangles at Pilat Quadrant
The “Flat Standing stone” 5 12 13 triangle
Pierre des Trois Evêques, Menhir du Flat.
Pierre des Trois Evêques
Menhir du Flat.
https://pilat-rando.fr/pedestre/le-menhir-du-flat-colombier-n-3/
Les Faves
Cromlech build following orientated pythagorean triples
proportionate to 100 MY
Guides and visitors at the cromlech des Faves, standing on each stone around the central monument
Les Faves Stone circles are studied�Description, position on the map, naming are continuously proceeded
Faves stone circles (see below detailed descriptions)
Circle 1
Baby Circle
Circle 2
Marker 1
What is remarquable in Pilat Quadrant / Faves circle 1
Large stone place North of the circle, marker 1 Marqueur au Nord du cercle
Second marker: 36, 87° from N/S line Marqueur 2
Large bassin as center of the circle : C1 Centre
Ray / Rayon : 13,2 m > perimeter is 100 MY�
Marker 2
C1
Marker 1
Ref: English & French version included https://www.facebook.com/notes/le-cadran-du-pilat/atelier-n1-cromlechs-des-faves-workshop-1-faves-stone-circle/599282673770536/
Discovery: stone circle 1 setting�Same design & methodology as in sumerian math tablets
Setting 2 of the quadrant, the Faves area, present several stone circles.
Circle 1 seem to get same standard methodology for construction as in sumerian math tablets:
Field exercise: create Faves circle #1
Following sumerian tablets instructions
In order to better explain the discovery to local population and antic mathematics amateurs, the society of guide of Natural Parc of Pilat organized a field exercise to design stone circle following sumerian tablets instructions.
Marker 1 and 2 are clearly visible and found on the field. Center of the circle is confirmed by calculation as well as 28 stones equidistant to it.
May 2018 - 60 personnes welcomed at Faves Circle 1 with step by step instructions
See details on Quadrant FB page: (English version included). https://www.facebook.com/notes/le-cadran-du-pilat/atelier-n1-cromlechs-des-faves-workshop-1-faves-stone-circle/599282673770536/
Pythagorean triples at larger scale
On the road from Beigua Italy to Carnac France
Perfect 7 24 25 pythagorean triples homothetic to the local triangle leads to jadeite axe quarry in Italy
Proportion TBD
Dec 2019 - testing the homothy from Beigua axe quarry
Discovery (thanks to R. Eschbach, PhD) - July 2019: �journey from axe quarry at Monte Beigua to Pilat Quadrant is a 7-24-25 triangle
Monte Beigua jadeite quarry is fully aligned with Quadrant target and Ecrins summit. �R. Eschbach discovered this alignment after reading this very presentation (why we added this slide). Which makes the triangle Quadrant - Beigua - Square angle, a 7-24-25 triangle (expansion of the very local triangle of Pilat Quadrant.
Monte Beigua neolithic site http://www.parcobeigua.it/Esiti-archeologici.php and pics : https://www.themodernantiquarian.com/site/11321/monte_beigua_pietra_scritta.html
Cross the Alps at neolithic age / Jadeite axes distribution from Italy https://www.persee.fr/doc/globe_0398-3412_2009_num_149_1_1555
Orange triangle: Stone of the Three Bishops - Beigua
Discovery (thanks to R. Eschbach, PhD) - July 2019: �Local triangle is by definition at scale of the larger triangle
This alignement and then proportion, lead in December 2019 to the idea of projecting the local stone survey made by the SEPR, on the larger map, taking the scale factor between the two triangles observed.
Ref : Eric Charpentier 2018 conference on Pilat Quadrant.
Overview of the statement - Ref: SEPR - Jean-Yves Lozano
Discovery (thanks to R. Eschbach, PhD) - July 2019: �Projection of local stones make sense at scale on the larger map
For the projection we make a homothety on each GPS point of the stones provided by the report of the SEPR, with as factor k:
k = distance Stone - Beigua / distance Stone of the Three Bishops - sight of the Ecrins = 1620,37037
(remark; that ratio is also the distance from Beigua to the Three bishop stone in Megalithic Yard / 250 - unit of the 25 side triangle in Megalithic Yards : 336 km / 250 in MY = 1620,37037 )
For the longest distance (Stone of the Three Bishops to Beigua) we take into account the roundness of the earth by applying the ad hoc formula in the calculation table: �
ACOS (SIN (RADIANS (C4)) * SIN (RADIANS (C5 )) + COS (RADIANS (C4)) * COS (RADIANS (C5)) * COS (RADIANS (D4-D5))) * 6371
Red dots: GPS point of the SEPR survey projected to the scale.
White dots: GPS point of the SEPR survey as of local scale at Pilat Quadrant
Local mini-map: zoom on Arc Valley
Discovery on the Arc Valley
Projections of Pilat Quadrant map make sense at very local level in the Alps
“Flat” detailed view of the central part of the statement
Raw data of this zone: each point represents a lifting point.
Reminder: Pilat Quadrant may draw a neolithic map of the Alps
Stones found at Pilat Quadrant are by our hypothesis drawing a miniature map of the Alps. Each stone you find next to Pierre des Trois Evêques in France, represent a specific location at scale in the Alps.
Conversion from Pilat Quadrant stones setting to “Alps” size is at scale of Pilat Quadrant to Beigua (Italy) where quarry of jadeite stones used to make and trade jadeite axes in Europe. (see back ups for more info)
Castle Rock
Mont Cenis Lake
Megaliths at Pilat Quadrant
Mapped over the Alps map
Match specific neolithic sites in the Alps
The Castle Rock, Arc Valley.
Neolithic engraved monument
Mapped locally
Discovery on the Arc Valley
Projections of Pilat Quadrant map make sense at very local level in the Alps
Superimposing shapes of the stones on their location confirming the possible role of visual landmarks for a walker, the landmarks being proportional to their apparent size.
Assumptions to confirm with experts on site, and taking each stone in 3D.
Remarkable elements of the landscape
Neolithic sites
Discovery on the Arc Valley
Projections of Pilat Quadrant map make sense at very local level in the Alps
Test: projections of Google Earth map on the Pilat Quadrant ‘map’
Chiomonte necropolis
Rocher du Château
Cromlech de la Toussuire
Chaine de Belledonne
Mont Thabor
Pointe de Cugne
Pointe de Tierce
Rocher des amoureux
Sapey
La Meije
Pic Bourcet
Reminder: megaliths at Pilat Quadrant, walk through as in a miniature of the Alps
Cromlech de la Toussuire
Rochetaillée
Chaine de Belledonne
Aiguilles d’Arves
Current best result of the Pilat Quadrant ‘map’
Chiomonte necropolis
Rocher du Château
Cromlech de la Toussuire
Le Grand Sorbier
La Pyramide
Chaine de Belledonne
Pointe Sommeiller
Mont Thabor
Pointe de Cugne
Pointe de Tierce
Mont Malamot
Rocher des amoureux
Sapey
Detailed map https://www.google.com/maps/d/edit?mid=1VU8as84DyW6voqvLFKzd-0eFdvM&ll=45.198814185382616%2C6.600484413035037&z=12
Mont Coburne
Rocher de la Gde Tempête
Pic du Lac Blanc
Point des Cercles�(à redresser)
Glacier Lombard�Aiguille du Galéon
La Meije
Pic Bourcet
L’Aulp du Seuil. Cromech
Montagne du Conest
Gorge des Hirondelles
Puy Gris
Pic du Frêne
Rocher du But
Mont Aiguille
Aiguille de la Grand Sassière
La Bailetta
La Transteleina
Gorges de Crossey
Pointe du Bouchet
Aiguille du Fruit
Le Grand Rocher
Rocher de Cambis
Roche Veyrand
Oppidum du Châtelard
Mont de Servelongue
Gorges de Chailles
Mont Follet
Le Gd Regardou
Pointe des Mines?
Col de Basemont
What you see on the map is what you get in landscape
Pilat Quadrant “map”
Red dot on point of view
Arrows in direction of the picture
Alps real view�from West of Rocher des Amoureux
Chiomonte necropolis
Rocher du Château
Cromlech de la Toussuire
Rochetaillée
La Pyramide
Chaine de Belledonne
Pointe Sommeiller
Mont Thabor
Aiguille du Galéon
Pointe de Tierce
Mont Malamot
Rocher des amoureux
Sapey
55 - Mont Thabor
58 - Sapey
59 - Rocher des amoureux
56 - Mont Coburne
Mont Coburne
What you see on the map is what you get in landscape
Chiomonte necropolis
Rocher du Château
Cromlech de la Toussuire
Rochetaillée
La Pyramide
Chaine de Belledonne
Pointe Sommeiller
Mont Thabor
Aiguille du Galéon
Pointe de Cugne
Pointe de Tierce
Mont Malamot
Rocher des amoureux
Sapey
Rocher des amoureux
71 - Rocher du Château
69 - Pointe de Tierce
60 -Pointe de Cugne
Pilat Quadrant “map”
Red dot on point of view
Arrows in direction of the picture
Alps real view�from top of Rocher du Chateau
Mont Coburne
Perfect 5 12 13 pythagorean triples leads from Pierre des Trois Evêques to Carnac
Proportion to�60 x 10 000 MY
Or 1.10 000 MY
Dec 2018 - evidence of alignment to the north from Pierre des Trois Evêques
Pilat Quadrant North road
Start here
North part of the route is full of key circles
Going up after Alesia, fully North we cross first Attila’s camp, and then Anvers circle (discovery thanks to this analysis)
What happens is Anvers is placed same latitude as Stonehenge, with Ringwould halfway.
Current speculation 3: Q-A-C = 5-12-13 �Triangle Carnac, Alesia, Quadrant is a 5-12-13 with square angle at Alesia
If Quadrant map principles are verified, end point of the line from Alesia to Carnac is positioned as summit of 12 side of a 5-12-13 triangle, under current sea level.
Ground distance from Quadrant (Pierre des Trois Evêques) to Alesia is about 248,6 km (Side of the triangle 5), which makes ground distance from Alesia to Carnac ‘sea’ summit as 596,64 km (side 12 of the triangle)�Carnac end point: 47.53842 -3,4612
248,832 km = 5 x 60 x 10 000 MY (0,82944 m)
5
12
13
Formula for GPS points distance (ground distance including earth curve): =ACOS(SIN(RADIANS(B2))*SIN(RADIANS(B3))+COS(RADIANS(B2))*COS(RADIANS(B3))*COS(RADIANS(C2-C3)))*6371 |
Step 1+2 optimal road: Gergovie and Poitier
Through several neolithic sites and along direct line
Route from Italy to Carnac, through Pilat Quadrant
St Pierre de Quiberon
Marlhes
Chiomonte
Proposition of route of jadeite axes at neolithic age, from quarry in Biegua Italy up to Carnac France, where jadeite axes were transported and reshaped before distribution in Europe.
Gergovie
Carnac
Beigua
Alba
Pilat Quadrant
Good to know: undersea jadeite axes in Morbihan�That support distribution of Italian axes up to Carnac, on the line from Alesia
An underwater repository Neolithic polished blades in jadeitite and sillimanite, and a book of steles submerged on the beach called Petit Rohu near Saint-Pierre-Quiberon (Morbihan)“ https://www.academia.edu/972368/Un_d%C3%A9p%C3%B4t_sous-marin_de_lames_polies_n%C3%A9olithiques_en_jad%C3%A9itite_et_sillimanite_et_un_ouvrage_de_st%C3%A8les_submerg%C3%A9_sur_la_plage_dite_du_Petit_Rohu_pr%C3%A8s_Saint-Pierre-_
“In August 2007, holidaymakers discovered two pairs of polished axeheads that had been set vertically in clayey silt on the beach of Porh Fetan,
at a place called Petit Rohu. They reported their find to the Regional Archaeology Service of Brittany and to the Carnac Museum. The shape and material of these axeheads allowed them to be identified straightaway as being of Alpine origin, in common with a number of axeheads found in the region.
[...] This palaeo-environmental context allows us to argue that the
axeheads had been deposited in a marshy environment that had developed behind a dune system, at the foot of a remarkable stone outcrop. Rises in sea level since the mid-fifth millennium BC – the likely date at which the axeheads were deposited – mean that the shore has advanced by some 500 metres since then.
Square triangles based on solticial angle for hypotenuse
On the road from Carnac France to Germany & Switzerland
Discovery : starting point of a trip across Europe
Going North from the Pilat Quadrant, on the Google map, and then from Anvers and Alesia going West, Thomas went across several noticeable sites all marked by circles, and perfectly aligned (longitude and latitude).More circles are identified on the Anvers / Stonehenge line than on this map. Noticable: Ringwould circle is exact halfway
21 June sunrise (blue line)
Constance lac is same latitude as Carnac, where the blue line is crossing that latitude. Same as for Alésia site, which is exact North of Pierre des Trois Evêques.
What is that?
We have a connection with Pilat Quadrant main hypothesis: a map enabling circulation of people at neolithic age.
https://journals.openedition.org/archeosciences/4856
Constance
Alesia
Carnac
Sunrise 21 June
21 June sunrise direction & neolithic sites
From Pierre des Trois Evêques to Sunrise 21 June, axe goes up to Constance Lake with several neolithic site on the go.
Same from Pierre des Trois Evêques to North, axe goes up to Alesia site, which is same latitude as Constance lake and Carnac.
That triangle can be explored as a basis for hypothesis: from Pilat Quadrant to Constance Lake, are there other significant neolithic sites?
Latitude of Alesia & Carnac
Sunrise 21 June from Pierre des 3 Evêques
Alesia
Gosek Circle in Germany and Antwerp circle on 3 - 4 - 5 lines from Carnac & Alesia
On the line from Stonehenge to Gosek circle...
Gosek Circle
Antwerp circle
Stonehenge circle
Constance settlement
Carnac
Alesia
Speculations
Key questions to solve, where additional expertise are expected
Practical question: why here? �Is this arrangement of stones is practical to setup here ?
The Gimel plateau is offering
1
2
3
Reasons to believe ancient trips occured in Europe
Distribution of jadeite axes supposes 6 000 years ago established ways to go from Italy to all other key places in Europe with a light but high value objects to transport.
No roads, mainly forests.
Possibly, when going further in distribution, and coming back to Pilat Quadrant, travelers would add 1 additional further destination ‘stone’ to the down scale map. For the record & expansion of capacity of others to learn the destination and distribute axes further.
Good to know about the �Long journey of the axe found next to Pilat Quadrant
Axe found seem to be carnecean type, which means first extracted in Italy (Mont Viso or Beigua), then transported and modified in Carnac (giving it its typical shape), then took back to Pilat Quadrant area.
https://anr.fr/Project-ANR-12-BSH3-0005
Page 11: Carnacean Tumiac shape (original source for this type) http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.868.9200&rep=rep1&type=pdf
Good to read for catching up on studies related to neolithic procurement of precious objects
Daily goods were made and used at a local area level�Giants in the Landscape: monumentality and Territories in the European Neolithic https://hal.archives-ouvertes.fr/hal-02022684/document
But…�precious artefacts were manufactured and then distributed in western Europe�Typology, chronology and distribution of the large Alpine axeheads in Western Europe https://www.researchgate.net/publication/281185639_Typologie_chronologie_et_repartition_des_grandes_haches_alpines_en_Europe_occidentale_Typology_chronology_and_distribution_of_the_large_Alpine_axeheads_in_Western_Europe
variety of objects of different origins (shells, jadeite axes) are found in tombs & thanks to a specific social organization in that part of Europe�Christian Jeunesse, « Hoards and elite graves in the European late Prehistory: the birth of “barbarian Europe”?», Préhistoires Méditerranéennes [En ligne], 5 | 2016, http://journals.openedition.org/pm/1299 �And long distance “trade” https://www.academia.edu/2267558/Trade_and_Exchange_in_Prehistory._A_Theoretical_Evaluation
population traveled from Iberia to British Isles at neolithic period, connected by specific families found in different megalithic tombs (DNA analysis)�Megalithic tombs in western and northern Neolithic Europe were linked to a kindred society https://www.pnas.org/content/116/19/9469
Local procurement area for daily goods
Continental diffusion areas for precious artefacts
Reasons to believe long trips issues are solved by maps
Preparation of expeditions on open sea has been proven effective from memorizing dots in a network of lines (of bambou) in the antic oceaninan culture. Pacific navigation, across Marshall islands was possible without any ground reference for the navigators.
Europe, 6 000 years ago, was not much practicable than oceans, with no routes, dense forests (“hairy Gaule”).
Memorized maps and science of following directions made it possible to keep on moving along a line, calculating well distances from angles of stars rising in the sky.
https://www.hokulea.com/education-at-sea/polynesian-navigation/