GALILEO GALILEI'S LOCATION, SHAPE AND SIZE OF DANTE'S INFERNO AN ARTISTIC AND EDUCATIONAL PROJECT
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GALILEO GALILEI’S LOCATION, SHAPE AND SIZE
OF DANTE’S INFERNO
AN ARTISTIC AND EDUCATIONAL PROJECT
Alessandra Angelini
Corso di Grafica d’Arte dell’Accademia di Belle Arti di Brera
Paola Magnaghi- Delfino Tullia Norando
Laboratorio Didattico FDS -Politecnico di Milano
Aplimat -Bratislava February 4 – 6 , 2014
1
http://fds.mate.polimi.it
2
Text Presentation QR code
3
The lectures on Dante’s Inferno
Autograph manuscript of Galileo’s lectures
4
The Artistic and Educational Project
5
Martina RIZZATI Rubinia DI STEFANO
6
Marta FONTANA
7
Poster of the project 8
Dante’s Memorial
9
Bergamo’s Science Festival
101586 The Little Balance
11Università degli Studi di Pisa
Guidobaldo Del Monte
121540 Cosimo I de’ Medici
131633 Two New Sciences
14The structure of the Inferno by Antonio di Tuccio Manetti
Paolo dal Pozzo Toscanelli
1397 -1482
PERSPECTIVE GEOMETRY
ARITHMETIC
Filippo Brunelleschi
1377 - 1446
COSMOGRAPHY
Antonio di Tuccio
Manetti
Florence 1423 -1497 ASTRONOMY
Leon Battista Alberti
1404 -1472
VITA DI FILIPPO
BRUNELLESCHI
DIALOGO CIRCA IL SITO, FORMA ET
MISURA DELLO INFERNO
15Alternative structures of the Inferno
Cristoforo Landino
1481
Antonio di Tuccio
Manetti
Girolamo Benivieni
Accademia
1506 Fiorentina
Commedia’s
Florentine Editions
Galileo Galilei’s
lectures
Alessandro Vellutello Commedia’s Venetian
1544 Editions 1587 -1588
1617
Map ( XII century)
Map T - O (1472)
18Cape of Ptolemy
19The shape of the Inferno
Jerusalem is in the middle of the arc.
The angle at the center is 60 degrees.
20The funnel of the Inferno
Giovanni Stradano (Jan van der Straet)
Bruges 1523-Florence 1605
21Traditional pattern of the Inferno
22The First Six Levels
• Distance from the Earth’s center
The various levels of Manetti’s Inferno are regularly spaced, in fact the first six
levels are equidistant with 1/8 the radius of the Earth between each level and
the next.
Level Distance from the Earth’s center
Limbus 2839 17/22
Level 2 2434 1/11
Level 3 2028 9/22
Level 4 1622 8/11
Level 5 1217 1/22
Level 6 811 4/11
23Antonio Manetti’s plan
24Grand Old Man of Crete
25Grand Old Man
26Dante’s path
E io a lui: «Se 'l presente rigagno
si diriva così dal nostro mondo,
perché ci appar pur a questo vivagno?».
Ed elli a me: «Tu sai che 'l loco è tondo;
e tutto che tu sie venuto molto,
pur a sinistra, giù calando al fondo,
non se' ancor per tutto il cerchio vòlto:
per che, se cosa n'apparisce nova,
non de' addur maraviglia al tuo volto».
Inferno XIV , 121 - 129
27Dante’s path
And I to him: "If so the present runnel
Doth take its rise in this way from our world,
Why only on this verge appears it to us?“
And he to me: "Thou knowest the place is round,
And notwithstanding thou hast journeyed far,
Still to the left descending to the bottom,
Thou hast not yet through all the circle turned.
Therefore if something new appear to us,
It should not bring amazement to thy face."
Inferno XIV , 121 - 129
28Dante’s path
29Thales’ Similarity Theorem
30• Widths of the first six levels
Manetti divided the length of the arc on the surface from Cuma to Jerusalem into
two parts: 1000 miles + 700 miles
In the first 1000 miles he marked 10 spaces, each one of 100 miles, beginning
from the mouth; from these he deduced the widths of the first six levels
widths on the surface
Limbus 87 1/2 100
Level 2 75 100
Level 3 62 12 100
Level 4 50 100
Ring 1 37 1/2
Level 5 112 1/2 300 Ring 2 37 1/2
Ring 3 37 1/2
Ring 1 25
Level 6 75 300 Ring 2 25
Ring 3 25 31Malebolge
Tu non hai fatto sì a l’altre bolge;
pensa, se tu annoverar le credi,
che miglia 22 la valle volge.
Thou hast not done so at the other Bolge;
consider, if to count them thou believes,
that two – and – twenty miles the valley winds.
Inferno XXIX , 7 - 9
32Malebolge
Cercando lui tra questa gente sconcia,
con tutto ch’ella volge 11 miglia,
e men d’un mezzo di traverso non ci ha.
Seeking him out among this squalid folk,
although the circuit be eleven miles,
and be not less than half a mile across.
Inferno XXX , 85 - 87
33Malebolge
Dante says that the ninth bolgia turns through 22 miles, and, in consequence,
the diameter must be 7 miles. Then Dante also says (Inferno, XXX, 82-87) that
the tenth bolgia turns through 11 miles, and, in consequence, the diameter
must be 3 1/2 miles. Manetti thus supposed that the radii of the bolge were in
arithmetic progression and obtained
Bolgia Arc lenght Diameter Radius
10 11 3 1/2 1 3/4
9 22 7 3 1/2
8 33 10 1/2 5 1/4
7 44 14 7
6 55 17 1/2 8 3/4
5 66 21 10 1/2
4 77 24 ½ 12 1/4
3 88 28 14
2 99 31 ½ 15 3/4
1 110 35 17 1/2
34(17 1/2 : 700) (3245 5/11) = 81 3/22
Distance of Malebolge from the center of the Earth
2/8 (3245 5/11) - 81 3/22 = 730 5/22
The depht of Geryon’s ravine
35The Well of Giants
36The width of Malebolge and Well
width on the Earth’s surface
Bolgia 1 1 3/4 70
Bolgia 2 1 3/4 70
Bolgia 3 1 3/4 70
Bolgia 4 1 3/4 70
Bolgia 5 1 3/4 70
Bolgia 6 1 3/4 70
Bolgia 7 1 3/4 70
Bolgia 8 1 3/4 70
Bolgia 9 1 3/4 70
Bolgia 10 1/2 20
Land Malebolge-Well 1/4 10
Well 1 40
37In the Divina Commedia from these verses
Facemmo adunque più lungo viaggio,
Volti a sinistra; e al trar d’un balestro
Trovammo l’altro assai più fiero e maggio.
Therefore a longer journey did we make,
Turned to the left, and a crossbow-shot oft
We found another far more fierce and large.
Inferno, XXXI, 82 -84
We can argue that “Dante and Virgilius turn around the well” and so
the well must have a circular or polygonal shape, and that the
distance from one Giant to the other is about 300 braccia
(a crossbow-shot).
38The size of Lucifer and the spheres of ice
Lo ‘mperador del doloroso regno
da mezzo ‘l petto uscia fuor de la ghiaccia;
e più con un gigante io mi convegno,
che i giganti non fan con le sue braccia
The Emperor of the kingdom dolorous
from his mid-breast forth issued from the ice,
and better with a giant I compare
than do the giants with those arms of his
Inferno , XXXIV, 28 - 31
39The size of Lucifer and the spheres of ice
40The size of Lucifer and the spheres of ice
La faccia sua mi parea lunga e grossa
come la pina di San Pietro a Roma,
e a sua proporzione eran l’altre ossa
His face appeared to me as long and large
As is at Rome the pine-cone of Saint Peter's,
And in proportion were the other bones
Inferno , XXXI, 58 - 60
41Pinecone is bronze artefact of Roman origin, which is now in the
Belvedere’s Garden (Città del Vaticano, Rome)
42Height of a man = 8 times the face
Height of a man = 3 times the arm
Height of a man = 4 distance from the navel to the middle of the chest
43braccia
Pinecone 5 ½
Nembrot 44
Dante 3
Arm of Lucifer 645 1/3
Lucifer 1936
Navel- middle of the breast 484
braccia
Fourth sphere 500
Third sphere 1000
Second sphere 1500
First sphere 2000
44We can assess the huge size of Lucifer if we compare his height with that
of the tallest buildings in the world
45Students
46FEDERICA AMORUSO
47FEDERICA AMORUSO
48CARLO BARONI
49CARLO BARONI
50ANNA BASSI
51ANNA BASSI
52ANDREA BERTOLETTI
53ANDREA BERTOLETTI
54CLAUDIA CARIGLIA
55CLAUDIA CARIGLIA
56RUBINIA DI STEFANO
57BIANCA FASIOLO
58BIANCA FASIOLO
59MARTA FONTANA
60CAMILLA GUERRA
61CAMILLA GUERRA
62ELENA MAFFIOLI
63ELENA MAFFIOLI
64MARTINA RIZZATI
65But you have disposed all things by measure and
number and weight.
Holy Bible, The Book of Wisdom, 11 - 20
66Alessandra Angelini
Artist and Graphic Art professor
Accademia di Belle Arti di Brera
www.alessandraangelini.org
Thank you for your attention
Paola Magnaghi-Delfino
Tullia Norando
Department of Mathematics
FDS Laboratory
Politecnico di Milano
www.mate.polimi.it
67Social Network
MostraGalileoPolimi MostraGalileoPolimi MostraGalileoPolimi
68Manutius edition-1515
69Alessandro Vellutello’s Inferno
Galileo Galilei’s Life
Magnaghi & Norando – FDS Main Projects
70Alternative funnels of the Inferno
Stradano 1523 - 1605 Vellutello 1544
71Alessandro Vellutello’s plan
72Alessandro Vellutello versus Antonio Manetti
73Galileo Galilei’s life
Galileo Galilei was born on February 15, 1564, in
Pisa in the Duchy of Florence, Italy. He was the first
of six children born to Vincenzo Galilei, a well-
known musician and music theorist, and Giulia
Ammannati. In 1574, the family moved to
Florence, where Galileo started his formal
education at the Camaldolese monastery in
Vallombrosa.
74Galileo Galilei’s life
1581 – Enrols as medical student at University of Pisa
1582 – Attends mathematics lecture by Ostilio Ricci and decides to study math
and science
1585 – Leaves University of Pisa without degree and works as tutor
1586 – Invents hydrostatic balance; wrote La Balancitta (The little balance)
1589 – Appointed to Mathematics Chair, University of Pisa
1590 – Partially completes De Motu (On Motion), which is never published
1591 – Death of his father, Vicenzo Galilei
1592 – Appointed professor of mathematics at University of Padua, remains
18 years
~1593 – Invents early thermometer that unfortunately depended on both
temperature and pressure
~1595 – Invents improved ballistics calculation geometric and military
compass, which he later improves for surveying and general calculations and
earns income from tutoring on its use
1600 – First child, Virginia is born; ~1600 Le Meccaniche (Mechanics)
75Galileo Galilei’s life
1610 – Publishes Siderius Nuncius(Starry Messenger); views our moon's
mountains and craters and brightest 4 of Jupiter's moons
1611 – Discovers phases of Venus; granted audience with Pope; made
member of Lincean Academy
1616 – Officially warned by the Church not to hold or defend the Copernican
System
1616 – The Catholic Church places De revolutionibus orbium coelestium on
the List of Prohibited Books
1616 – Private letter Discourse on the Tides
1617 – Moves into Bellosguardo, west of Florence, near his daughters'
convent; observes double star Mizar in Ursa Major
1630 – Completes Dialogue Concerning the Two Chief World Systems and
subsequently receives approval of Church censor
1632 – Publishes Dialogue Concerning the Two Chief World Systems
76Galileo Galilei’s life
1633 – sentenced by the Inquisition to imprisonment, commuted to house
arrest, for vehement suspicion of heresy
1633 – Catholic Church places Dialogue Concerning the Two Chief World
Systems on the List of Prohibited Books
1638 – Publishes Dialogues Concerning Two New Sciences
1642 – death in Arcetri, Italy
77FDS - Magnaghi & Norando - Main Projects
78Dante’s Commedia
Luca Pacioli’s Capital Letters in progress
Jonathan Swift’ Laputa Island
Alessandro Mazzucotelli, the iron and fire of Art
Through the looking-glass in progress
79Stage 2009-2010
Analisi della struttura dell’Isola di Laputa
Jonathan Swift’ Laputa Island
Our project’s aim is the study of the structure of Laputa Island, the floating island which
appears in the third chapter of Jonathan Swift’s novel “Gulliver’s Travels”. The students
conjecture that this island can really float thanks to the magnetic field, created by the
material which constitutes magnetic field, created by the material which constitutes the
core.
80Stage 2010-2011
Il sacro fuoco ( e il ferro ) dell'arte
Alessandro Mazzucotelli, the iron and fire of Art
Alessandro Mazzucotelli was born in Lodi not far from Milan, his family were dealers in
iron and he worked as blacksmith. He also designed jewellery and fabrics for the weaving
factory at Brembate. He is best-known for his wrought ironwork, in a vigorous Art
Nouveau, the style he not only followed but which he managed to exceed thanks to his
thorough studies from life of nature inspiration to the artistic movement, from which he
discovered also geometric -mathematical formulas.
The students , inspired by his works , decided to create a frieze.
81TeatroInMatematica
82I Numeri Primi e la Crittografia Il Dilemma del Prigioniero
Prime Numbers and the Cryptography Prisoner’s Dilemma
Topics: Prime Numbers, Cryptography Topics: Games Theory
Parallelismi: Geometrie Euclidee e Non L’Irrazionale leggerezza dei
Straight Line and Geometry that it describes Numeri
Topics: Euclidean a The Irrational Number Lightness
Non-Euclidean Geometry Topic: Irrational numbers
Il Caso Probabilmente: la partita a dadi Metti, una serie a cena
The chance: a game of dice One night, a series at dinner
Topics: The roots of the Probability’s Theory Topics: Fibonacci’s series, Golden Ratio
I 7 ponti e il mistero dei Grafi Appuntamento al limite
The seven bridges and the mystery Appointment to the Limit
of Graph Theory Topics: Function, Limit, Derivatives
Topics: Graph Theory
83Stage 2007-2008
La cicloide: nuovi orizzonti per lo sci
The Cycloid: a new way of skiing
In this project, the students apply the properties of the cycloid to the study
of special and giants slalom.
84Stage 2008-2009
La teoria martolemaica
The Marptolemaeus’ solar system theory
Marptolemaeus, an hypothetical Mercian astronomer, has defined the mathematic model
of the cosmologic system. This is the aim of these research: building the Mercian system,
supposing Mars to be at the centre of the universe. The choice of an astronomic theme
has been influenced by the fact that 2009 has been proclaimed the year of astronomy
because for the first time four thousand years ago Galileo observed the sky with the
telescope. Besides the Sun moves around Mars following an ellipse. The other planets,
Earth, Mercury and Venus, instead, describe orbits which don’t appear in our earthly
geometric books and that we have imaginatively called “epiclissoidi”.
85Stage 2009-2010
Analisi della rete delle farmacie di Monza
Analysis of the chemists network in Monza
This paper deals with a research carried out in Monza to analyse the efficiency of the
network of chemists through the study of minimum paths and Voronoi tessellation of the
city map. In the first part, we give an in-depth explanation of the nature and purpose of
Voronoi diagrams and we briefly discuss Fortune’s algorithm for computational
construction of V.d. and how they can be applied to our study case. The second part of
the paper relates how we enforced our mathematical model by means of a statistical
inquiry and how we came to set up a working simulation.
86Stage 2010-2011
Sulle orme di Keplero
A study about Jupiter’s mass
This project was finalized at calculating the mass
of Jupiter through observing the same four
satellites (Io, Europa, Ganymede and Callisto)
which both Galileo and Kepler used to follow
with their means almost 400 years ago. This
project implied several on-the-ground
experiences at the Astronomical Observatory of
Merate (AOM) which greatly enriched our
knowledge about some astronomical related
subjects that had been studied at school only
under their theoretical aspect.
87Stage 2011-2012
Operazione meridiana
Sundial
The main aim of this project is to complete the
mathematical and geometrical planning as well as
the construction of a fully working sundial, equipped
with a solar calendar The position of the hour-lines
and date-lines has been calculated and laid out
through the application of some theorems about
spherical trigonometry in order to sort out a spatial
geometry problem. An important part of the project
consists in planning a spreadsheet which calculates
the equations of hour-lines and date-lines for a
sundial working in Central Europe.
88Stage 2012-2013
Il suono delle campane
The song of bells
A systematic study regarding bells sound
requires the knowledge of three
important features: the theoretical model
about sound characteristics, the technical
aspect of the instrument and the
historical-artistic one. The students
contacted the Italian Campanology
Association, then, they applied the
Fourier analysis to examine the sound
produced by two different bell concerts:
Lodi Cathedral and Wilten Abbey in
Innsbruck.
89Stage 2012-2013
Sunshine project: let’s roll!
The purposes of this project are the following: studying the differential rotation of
the Sun and making three-dimensional images of the star. This project allowed the
students to develop abilities in taking pictures of the Sun through a solar dedicated
telescope and to improve their knowledge about the Sun. It was carried out on two
complementary sides: the direct observations of the Sun were made in the Brera
Astronomic Observatory in Merate (LC) and a study about the differential rotation of
the Sun conducted, following the motion of solar spots, analyzed using our
knowledge of Kinematiks and pictures of the satellite (SOHO).
90Learning Week
91In Action with Math
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