Paolo Di Pasquale - Academia.edu (original) (raw)

Papers by Paolo Di Pasquale

L'opera che fonde arte e geometria in un unico linguaggio è stata esaminata attraverso la sovrapp... more L'opera che fonde arte e geometria in un unico linguaggio è stata esaminata attraverso la sovrapposizione di una griglia 24x24. Le linee di questa griglia sezionano la posa dell'uomo eretto e ne evidenziano la corrispondenza con le proporzioni che Vitruvio definì "dettate dalla natura". In questo contesto, viene messo a fuoco anche il parallelismo tra il busto dell'Uomo Vitruviano e lo schizzo del piccolo busto nel disegno anatomico, dove Leonardo illustra il coito tra uomo e donna.
Una seconda sovrapposizione eseguita con una griglia 28x28, consente di chiarire il ruolo della proporzione 1/14 introdotta da Leonardo, necessaria a ridurre l'altezza dell'uomo eretto per adattarlo al cerchio. Questa proporzione risulta anche essere la chiave per dimostrare l'equivalenza geometrica emersa tra il disegno dell'Uomo Vitruviano e il foglio 307 V del Codice Atlantico.
Infine, la proporzione di 1/14 è stata fondamentale per tracciare la Divina Proporzione che Leonardo incorpora nella relazione tra la posizione delle mani e dei piedi dell'uomo nel cerchio.

Paolo Di Pasquale, 2023

The work that fuses art and geometry into a single language was examined through the superimposit... more The work that fuses art and geometry into a single language was examined through the superimposition of a 24x24 grid. The lines of this grid dissect the pose of the upright man and highlight its correspondence with the proportions that Vitruvius defined as 'dictated by nature'. In this context, the parallelism between the bust of the Vitruvian Man and the sketch of the small bust in the anatomical drawing, where Leonardo illustrates the coition between man and woman, is also focused.
A second superimposition performed with a 28x28 grid, allows us to clarify the role of the 1/14 proportion introduced by Leonardo, necessary to reduce the height of the upright man in order to fit him in the circle. This proportion also turns out to be the key to demonstrating the geometric equivalence that has emerged between the drawing of Vitruvian Man and Folio 307 V of the Codex Atlanticus.
Finally, the proportion of 1/14 was fundamental in tracing the Divine Proportion that Leonardo incorporates in the relationship between the position of hands and feet of the man in the circle.

Göttinger Miszellen 258, 2019

This study compares the arch generated with the measurements indicated on the Saqqara ostracon, c... more This study compares the arch generated with the measurements indicated on the Saqqara ostracon,
currently kept in the Saqqara Imhotep Museum under code JE50036, with the arch of the blue faience
tiles panel created during the reign of Djoser under the step pyramid, and currently on public display in
the Egyptian Museum in Cairo under code JE68921.
Applying an equidistance of 41 fingers to the intervals between the vertical lines of the ostracon
diagram, the generated curve is a part of the circumference of a circle.
Digitally analyzing the curve of the arch supported by the 11 Djed pillars of the blue faience tiles panel,
this curve is also a part of the circumference of a circle.
The size and position of the blue faience tiles arch in relation to its panel, highlights a direct and
unambiguous proportionality with the arch generated by the ostracon diagram.
There is also a relationship of magnitude between the two arches taken into account. In fact, the size
of the one made on the panel is exactly 1/4 of the size of the arch developed with the diagram of the
ostracon.

This research analize the proportions and architectural features of Osireion Hypogeum in order to... more This research analize the proportions and architectural features of Osireion Hypogeum in order to determine the level of knowledge of the geometric rules achieved by Egyptian Priests and Harpedonaptae before the birth of Euclidean Geometry.

This study, demonstrates in four simple steps how the section of the Bent Pyramid was developed. ... more This study, demonstrates in four simple steps how the section of the Bent Pyramid was developed. Its geometric proportions share, as in the equilateral triangle, the relationship established between √2 and √3.
As well as the grid formed by rectangles, described on page 7 is a classic Egyptian method with which it is possible to indicate in scale the location of the entrances of the corridors leading to the inner chambers.
That' one of the reason why more than 4500 years ago, at the edge of the Dhashur desert, Pharaoh Snefru intended to create an impressive architectural work of extreme beauty, making it a witness to the high scientific thought reached at that time by Egyptian Architects.

Explanation of the Royal Cubit on the Great Pyramid profile by way of the Flower of Life

This study aims to find a solution to problem 53 of the "Rhind Mathematical Papyrus" housed in th... more This study aims to find a solution to problem 53 of the "Rhind Mathematical Papyrus" housed in the British Museum since 1864. To date, the problem has not yet been solved and is considered incomplete by Egyptologists who have dealt with it. My hypothesis, completely new, starts from the assumption that the data contained in it are sufficient to identify the procedure and to obtain the result. The triangular shape drawn on the papyrus was divided into three overlapping isosceles triangles having the same vertex, and then the position and size of each of the hieratic numbers indicated on the triangular drawing of the problem 53 was analyzed. The calculations below the drawing seem to have a mathematical correspondence with the measurements of the three triangles having the same vertex, and in this context the numbers indicated by the scribe suggest that the three isosceles triangles superimposed are directly proportional to each other. This solution would confirm what Eric Peet had noted about the Egyptians' ability to calculate "direct proportionality" in Problem 11 of the Moscow Papyrus. The mathematical result is also confirmed by the correspondence between the freehand triangle drawn by the scribe on the papyrus and the geometric drawing made by computer with the measurements resulting from the calculations described in RMP 53. Finally, the geometric drawings resulting from the hypotheses formulated by Egyptologists Chace, Clagett and Miatello were also compared with Ahmose's drawing on Rhind's papyrus.

RHIND PAPYRUS and BASE 7, Jul 20, 2012

"The main point is that when one takes into account what we know about ancient Egyptian mathemati... more "The main point is that when one takes into account what we know about ancient Egyptian mathematics (based primarily on the Rhind Papyrus) and what we know about the system of measuring lengths in terms of "cubits," "palms," and "fingers" , everything involves the number 7 and its multiple.

Several Astronomical Events occurring in the sky of Giza that took place between the IV and XVIII... more Several Astronomical Events occurring in the sky of Giza that took place between the IV and XVIII Dynasty were examined and then associated with the Egyptian Calendar consisting of 365 days divided into 12 months of 30 days (plus 5 epagomenal days),

Coming Soon

This update must be considered a sheet necessary to make what has already been reported in my art... more This update must be considered a sheet necessary to make what has already been reported in my article 'The three Pyramids of Giza and the Bent Pyramid shape' more understandable. To verify if the relationship √3 and √2 exists in the plan position of the three Pyramids of Giza it is necessary to start from the calculations that indicate the presence of this relationship between the upper and lower part of the Bent Pyramid.

The geometric analysis carried out in this study is essentially based on a comparison between the... more The geometric analysis carried out in this study is essentially based on a comparison between the three pyramids of Giza and the Bent Pyramid of Dahshur. By examining the two survey data plans drawn up by Sir W. Petrie at the respective locations, it appears that there is a proportional relationship between the distances separating the three pyramids and the profile of the Bent Pyramid. By rotating the plan of the three pyramids by 3.14 degrees, we obtain exactly the profile of the Bent Pyramid's double-slope shape. Furthermore, it turns out that, by multiplying by 9/7 the measurements resulting on the rotated plan, we obtain the measurements of the base, the height, as well as the measurements of the two slopes that characterise the rhomboidal profile of the Dahshur pyramid.

Since its discovery by Battiscombe Gunn, it is believed that Saqqara's Ostrakon is able to improv... more Since its discovery by Battiscombe Gunn, it is believed that Saqqara's Ostrakon is able to improve our understanding of how Egyptians designed curved elements, but the geometric significance of the hieratic values is still uncertain. Indeed, in order to obtain an indisputable result, the distance between the vertical ordinates is required, and this distance is not explicitly indicated on the limestone artefact.
But since there were no algebraic formulas at the time of the Pharaohs, a simple grid-based procedure was developed such as those used by the Egyptians to reproduce the scale drawings, to try to decipher the geometric meaning of the curve drawn on the find.
To conclude, this simple process permits to understand that the five hieratic values of the Saqqara Ostrakon describe a segmental arch whose span and rise correspond to the architectural proportions of several Egyptian tombs and arches, and that the numerical sequence of the artefact could be considered the most "ancient" mathematical formula of the circle.

RMP 54 - Analisis of Calculations and results, 2018

The translation of the Second Step of RMP 54 by Prof. Peet in his the book 'The Rhind Mathematica... more The translation of the Second Step of RMP 54 by Prof. Peet in his the book 'The Rhind Mathematical
Papyrus'1 is incorrect. Eric Peet, in the first two lines confuses the Cubit Strip believing they were Cubits-of-Lands, while in the fourth line confuses 1 Thousand of Land with 10 Cubits of Land.
For this reason its results are wrong, and does not correspond to what the Scribe indicated in the
RMP 54.

The solution to the problems 56, 57, 58, 59 and 59B proposed in this study shows that the Egyptia... more The solution to the problems 56, 57, 58, 59 and 59B proposed in this study shows that the Egyptians knew the properties of the right triangle from the time of the Rhind Papyrus that, as the scribe Ahmes indicates, the original text dates back to the XIIth dynasty, 1300 years before Pythagoras.
Problem 56 shows which are the calculation procedures to obtain the Seked (in palms) in relation to a cubit of 7 palms. In this first problem a pyramid is drawn with double lines next to the description, different from those related to the following four problems that don't have the double line on the sides but only on the base.
The four problems 57, 58, 59 and 59B all refer to the Seked of 5+1/4 which reproduces a right-triangle with the same proportions as the Pythagorean Triple 3-4-5.
In fact in RMP 57 and 58 the right triangle has a base 70 which is at its height 93+1/3 as 3 is at 4. In RMP 59 and 59B is indicated the right triangle with base 6 and height 8 whose hipotenuse corresponds to 10.
These right triangles with Pythagorean Triple 3-4-5 also correspond to the measurements of the Khafre's Pyramid which has a seked of 5+1/4 since the base is 215.5m. and the height should be 143.5m.

Questo studio è dedicato alla ricerca della soluzione del Problema 53 del "Rhind Mathematical Pap... more Questo studio è dedicato alla ricerca della soluzione del Problema 53 del "Rhind Mathematical Papyrus" ospitato nel British Museum dal 1864. Ad oggi il problema non è stato ancora risolto ed è considerato incompleto dagli egittologi che se ne sono occupati. La mia ipotesi, del tutto nuova, parte dal presupposto che i dati in esso contenuti siano sufficienti a comprendere la procedura e a raggiungere il risultato. In questo articolo, la forma triangolare disegnata sul papiro viene prima suddivisa in tre triangoli isosceli sovrapposti aventi lo stesso vertice, e successivamente viene analizzata la posizione di ciascuno dei numeri ieratici che indicano le dimensioni della forma triangolare. I calcoli sotto il disegno sembrano avere una corrispondenza matematica con le misure dei tre triangoli aventi lo stesso vertice, e in questo contesto i numeri contrassegnati suggeriscono la soluzione che dimostra che i tre triangoli isosceli sovrapposti sono direttamente proporzionali tra loro. Questa soluzione confermerebbe ciò che Eric Peet aveva notato sulla capacità degli egiziani di calcolare la proporzione nel Problema 11 del Papiro di Mosca. Il risultato matematico è confermato anche dalla corrispondenza tra il triangolo disegnato a mano libera dallo scriba sul papiro e il disegno geometrico graficamente realizzabile con le misure che risultano dai calcoli descritti nel RMP 53. Infine, i disegni geometrici risultanti dalle ipotesi formulate dagli egittologi Chace, Clagett e Miatello sono stati confrontati con il disegno realizzato da Ahmose sul papiro di Rhind.

For a better understanding of this article you should also read "Bent Pyramid and its relationshi... more For a better understanding of this article you should also read "Bent Pyramid and its relationship with √2 and √3" [ https://www.academia.edu/40961858/Bent_Pyramid_and_its_relationship_with_2_and_3 ].
In this article we analyze the Planimetry of Bent Pyramid drawn up with the measurements indicated by Sir William F. Petrie in 1887.
Focusing on its geometric dimensions, it can be said that this monumental work of great mass and grandeur, combined with the meticulous choice of distances between the elements that make up the Pyramid Complex, also represents the will of Snefru's Architects to pass on to posterity the essence of the «Egyptian Mathematical System»

In questo articolo si analizza la Planimetria della Bent Pyramid redatta con le misure indicate d... more In questo articolo si analizza la Planimetria della Bent Pyramid redatta con le misure indicate da Sir William F. Petrie nel 1887 .
Nel focalizzare le sue dimensioni geometriche si puo’ affermare che questa monumentale opera di notevole massa e di grande imponenza, unita alla meticolosa scelta delle distanze tra gli elementi che compongono il Complesso Piramidale, rappresenta anche la volonta’ degli Architetti di Snefru di tramandare ai posteri l’essenza del «sistema matematico» egizio.

Bent Pyramid, relationship √2 - √3, Equilateral Triangle, and Entrance of Corridors

This study, demonstrates in four simple steps how the section of the Bent Pyramid was developed. ... more This study, demonstrates in four simple steps how the section of the Bent Pyramid was developed. Its geometric proportions share, as in the equilateral triangle, the relationship established between √2 and √3.
As well as the grid formed by rectangles, described on page 7 is a classic Egyptian method with which it is possible to indicate in scale the location of the entrances of the corridors leading to the inner chambers.
That's one of the reason why more than 4500 years ago, at the edge of the Dhashur desert, Pharaoh Snefru intended to create an impressive architectural work of extreme beauty, making it a witness to the high scientific thought reached at that time by Egyptian Architects.

Se sull’Ostrakon è descritta una sorta di formula con la quale si ottiene una determinata curva u... more Se sull’Ostrakon è descritta una sorta di formula con la quale si ottiene una determinata curva utilizzabile per la progettazione di un manufatto, la chiave necessaria per decifrarla risiede nell’individuare la distanza tra le nervature verticali, perché è proprio questa distanza che ne determina il risultato. La presente ricerca parte da questo presupposto, le indicazioni forniteci attraverso i numeri ieratici impressi sull’Ostrakon devono essere in grado di contenere anche il valore di tale distanza, altrimenti la sua formula lascerebbe spazio ad infinite soluzioni invalidandone il suo contenuto geometrico.

Paolo Di Pasquale, 2023

Göttinger Miszellen 258, 2019

Explanation of the Royal Cubit on the Great Pyramid profile by way of the Flower of Life

RHIND PAPYRUS and BASE 7, Jul 20, 2012

Coming Soon

RMP 54 - Analisis of Calculations and results, 2018

Bent Pyramid, relationship √2 - √3, Equilateral Triangle, and Entrance of Corridors

This study, demonstrates in four simple steps how the section of the Bent Pyramid was developed. ... more This study, demonstrates in four simple steps how the section of the Bent Pyramid was developed. Its geometric proportions share, as in the equilateral triangle, the relationship established between √2 and √3.
As well as the grid formed by rectangles, described on page 7 is a classic Egyptian method with which it is possible to indicate in scale the location of the entrances of the corridors leading to the inner chambers.
That's one of the reason why more than 4500 years ago, at the edge of the Dhashur desert, Pharaoh Snefru intended to create an impressive architectural work of extreme beauty, making it a witness to the high scientific thought reached at that time by Egyptian Architects.

This study analyses the Bent Pyramid measurements resulting from the Survey-Data Plan carry out b... more This study analyses the Bent Pyramid measurements resulting from the Survey-Data Plan carry out by Sir William Flinders Petrie. First, these measures suggest that Egyptian builders have planned the characteristic dual slope design, the hypothesis derives mainly from the correspondence between the measurement of the slope relative to the lower part and the height of the upper part of the Pyramid. But a second hypothesis emerges, very interesting, since the obtuse angle deriving from the two slopes of the Bent Pyramid corresponds to that resulting from the position of the three pyramids on the Gizeh plain.