La fisica del π.
Viaggio andata-ritorno tra infinito e realtà
Maurizio Zani
Pint of science, Santeria Paladini 8, Milano 15/05/2018
bar (Brescia, Via Corfù)
scultura (Siracusa, Largo Calipari)
ristorante (Otranto, Via delle Torri)
profumo (Givenchy)
https://www.facebook.com/Pigrecobarbottegabrescia
http://carlo.gfaktor.it
http://pigreco-ristorante.it
https://www.givenchy.com
= costante
p
d
p = perimetro
=
= costante = π
2
p
p
d
r
p
Leonhard Euler**, svizzero (1748)
"Per
brevità
scriveremo
questo
numero π; così π è uguale a metà
della circonferenza di un cerchio di
raggio 1"
William Jones*, gallese (1706)
Chiama questa costante π,
iniziale
greca della parola perimetro della
circonferenza, ma soprattutto in onore
di Pitagora
** Introductio in analysin infinitorum
* Synopsis Palmariorum Matheseos
π = 3.
1415926535 8979323846 2643383279 5028841971 6939937510 5820974944 5923078164
0628620899 8628034825 3421170679 8214808651 3282306647 0938446095 5058223172
5359408128 4811174502 8410270193 8521105559 6446229489 5493038196 4428810975
6659334461 2847564823 3786783165 2712019091 4564856692 3460348610 4543266482
1339360726 0249141273 7245870066 0631558817 4881520920 9628292540 9171536436
7892590360 0113305305 4882046652 1384146951 9415116094 3305727036 5759591953
0921861173 8193261179 3105118548 0744623799 6274956735 1885752724 8912279381
8301194912 9833673362 4406566430 8602139494 6395224737 1907021798 6094370277
0539217176 2931767523 8467481846 7669405132 0005681271 4526356082 7785771342
7577896091 7363717872 1468440901 2249534301 4654958537 1050792279 6892589235
4201995611 2129021960 8640344181 5981362977 4771309960 5187072113 4999999837
2978049951 0597317328 1609631859 5024459455 3469083026 4252230825 3344685035
2619311881 7101000313 7838752886 5875332083 8142061717 7669147303 5982534904
2875546873 1159562863 8823537875 9375195778 1857780532 1712268066 1300192787
6611195909 2164201989 3809525720 1065485863 2788659361 5338182796 8230301952
0353018529 6899577362 2599413891 2497217752 8347913151 5574857242 4541506959
5082953311 6861727855 8890750983 8175463746 4939319255 0604009277 0167113900
9848824012 8583616035 6370766010 4710181942 9555961989 4676783744 9448255379
7747268471 0404753464 6208046684 2590694912 9331367702 8989152104 7521620569
6602405803 8150193511 2533824300 3558764024 7496473263 9141992726 0426992279
6782354781 6360093417 2164121992 4586315030 2861829745 5570674983 8505494588
5869269956 9092721079 7509302955 ...
∞ decimali (qui i primi 1500)
http://www.piday.org/million
Babilonesi* (2000 a.C.)
π = 3.125
1
25
π = 3 + =
8
8
* tavoletta di argilla
π = 3.160...
Egiziani** (1650 a.C.)
** papiro di Rhind, problema #50
2
16
π =
9
æ
ö÷
ç
÷
ç
÷
çè
ø
sottostima
sovrastima
π = 3
Primo Libro dei Re (600-500 a.C.)
[7,23] (Salomone) Fece un bacino di metallo fuso
di dieci cubiti* da un orlo all'altro (d), rotondo;
la sua altezza (h) era di cinque cubiti
e la sua circonferenza (p) di trenta cubiti.
* 1 cubito ≈ 50 cm
30
π =
=
10
p
d
h
d
p
0 decimali
Archimede*, greco (250 a.C.)
3.1408 < π < 3.1429
2 decimali
< <
10
3 +
10
3 +
π
71
70
* poligono di 96 lati
Claudio Tolomeo**, greco (150 d.C.)
** poligono di 120 lati
π = 3.14166
17
π = 3 +
120
Zu Chongzhi, cinese (450 d.C.)
3.1415926 < π < 3.1415927
6 decimali
π = 3.
1415926535
897932
16 decimali
Al-Kashi*, persiano (1424)
François Viète, francese (1593)
π = 3.
1415926535
8979323846
2643383279
50288
35 decimali
Ludolph van Ceulen**, tedesco (1610)
π
2
2
2
=
...
2
2
2 + 2
2 + 2 + 2
⋅
⋅
⋅
John Wallis, inglese (1655)
π
2 2 4 4 6 6 8 8
=
...
2
1 3 3 5 5 7 7 9
⋅
⋅
⋅
⋅
⋅
⋅
⋅
⋅
** Von der circle
* al-Risāla al-muḥīṭiyya
Leonhard Euler, svizzero (1748)
2
2
2
2
2
π
1
1
1
1
= 1 +
+
+
+
+ ...
6
2
3
4
5
3
3
3
3
3
3
π
1
1
1
1
1
= 1 -
+
-
+
-
+ ...
32
3
5
7
9
11
4
4
4
4
4
π
1
1
1
1
= 1 +
+
+
+
+ ...
90
2
3
4
5
Madhava*, indiano (1400)
π
1 1 1 1 1
= 1 - + -
+ -
+ ...
4
3
5 7
9 11
John Machin, inglese (1706)
1
1
π = 4 arctan
- arctan
5
239
é
ù
æ ö
æ
ö
÷
÷
ç
ç
ê
ú
÷
÷
ç
ç
÷
÷
ê
ú
ç
ç
è ø
è
ø
ë
û
* poi anche James Gregory, scozzese
e Gottfried Wilhelm von Leibniz, francese (1668)
John von Neumann*, ungherese (1949)
Fabrice Bellard*, francese (2009)
π = 3. ...
2037 decimali
π = 3. ...
2.7 miliardi di
decimali !!!
Sharma Suresh Kumar**, indiano (2015)
Akira Haraguchi**, giapponese (2016)
π = 3. ...
70 030 decimali
in 17 h 14 min
π = 3. ...
83 431 decimali
in 16 h
* calcolati
** elencati
14 marzo
Pi Day
22 luglio
03/14
π = 3.14
π = 3.14286...
22/07
π = 3. 14159...
A Bill for an act introducing a new mathematical truth and
offered as a contribution to education to be used only by the
State of Indiana free of cost by paying any royalties whatever on
the same, provided it is accepted and adopted by the official
action of the Legislature of 1897.
Section 1
[...]
Section 2
It is impossible to compute the area of a circle on the diameter as the linear unit without
trespassing upon the area outside of the circle to the extent of including one-fifth more area
than is contained within the circle's circumference, because the square on the diameter
produces the side of a square which equals nine when the arc of ninety degrees equals eight.
By taking the quadrant of the circle's circumference for the linear unit, we fulfill the
requirements of both quadrature and rectification of the circle's circumference. Furthermore,
it has revealed the ratio of the chord and arc of ninety degrees, which is as seven to eight,
and also the ratio of the diagonal and one side of a square which is as ten to seven,
disclosing the fourth important fact, that the ratio of the diameter and circumference is as
five-fourths to four; and because of these facts and the further fact that the rule in present
use fails to work both ways mathematically, it should be discarded as wholly wanting and
misleading in its practical applications.
Section 3
[...]
Edward J. Goodwin
18/01/1897
(disegno di legge #246, Senato dell'Indiana)
13/02/1897
("The Indianapolis News", pag. 11)
Fun-Making In the Senate Yesterday Afternoon
The Senate yesterday afternoon occupied Itself with House bills on second reading. The
engrossment of a number of small bills was ordered, and the bill legalising the incorporation
of Poneta, Wells county, was passed under suspension of the rules.
Representative Record's (Taylor I. Record, n.d.A.) mathematical bill legalizing a formula
for squaring the circle, was brought up and made fun of. The Senators made bad puns
about It, ridiculed it and laughed over It. The fun lasted half an hour. Senator Hubbell said
that It was not meet for the Senate, which was costing the State $250 a day, to waste its time
in such frivolity.
He said that in reading the leading newspapers of Chicago and the East, he found that the
Indiana State Legislature had laid itself open to ridicule by the action already taken on the
bill. He thought consideration of such a proportion was not dignified or worthy of the Senate.
He moved the indefinite postponement of the bill, and the motion was carried.
https://newspapers.library.in.gov/cgi-
bin/indiana?a=d&d=INN18970213-01.1.11
http://www.agecon.purdue.edu/crd/Localgov/Seco
nd%20Level%20pages/indiana_pi_bill.htm
4
π
4
3.2
5
p
=
=
=
d
1 decimale
2
1 2
e
e
q q
F = k
r
costanti matematiche vs costanti fisiche
valore & unità di misura
precisione
π = 3.141592653...
e = 2.718281828...
2
1 2
g
m m
F = γ
r
π
2
2
2
π
2
2
cos
d
ò
x
-
F = pR
θ θ
π
2
1
4
e
0
k =
ε
4
π
Δ
8
R
p
Q =
η L
6π
vis
F
= ηRv
2
5
đ
1
2
1
π
d
e
1
λ
hc
λkT
Q
hc
=
t A
λ
-
4
đ
1
d
Q
= εσT
t A
5 4
3 2
2 π
15
k
σ =
h c
√3 = 1.732050807...
φ = 1.618033988...
iπ
e + 1 = 0
Δ Δ
2
³

x
x p
2

h
=
h = 6.626070040ꞏ10-34 Js
https://physics.nist.gov/cuu
2010: in vacanza verso Rimini
205/55 R16
d = diametro del
cerchione (in)
l = larghezza del
pneumatico (mm)
h = altezza del
pneumatico (%l)
2.54
31.6 cm
2
⋅
d
r =
+ h l =
π = 3.1415
4 decimali
Δs ≈ 10 m
N = 150 000
s = 2πrꞏN ≈ 298 km
1969: missione spaziale Apollo 11 verso la Luna
memoria: 2 kB RAM, 36 kB ROM
computer: f = 2 MHz
https://spaceflight.nasa.gov/history/apollo/apollo11
http://www.ibiblio.org/apollo/listings/FP8/FP8.aea.html
1969: missione spaziale Apollo 11 verso la Luna
8 decimali
1969: missione spaziale Apollo 11 verso la Luna
computer: f = 2 MHz
computer: f = 2.6 GHz
2018: missione terrestre Maurizio verso il PoliMi
memoria: 2 kB RAM, 36 kB ROM
memoria: 16 GB RAM, 512 GB ROM
r = 21.2ꞏ109 km (= 141 UA = 20 light hour)
p = 2πr = ...
π = 3.1415926535 89
π = 3.14
Δp = 420ꞏ106 km
Δp = 42 m
12 decimali
https://voyager.jpl.nasa.gov
1969: missione spaziale Apollo 11 verso la Luna
1977: missione spaziale Voyager 1 verso l'ignoto...
π = 3.
1415926535
89793
1973: Global Positioning System (GPS)
"Relativistic effect in the GPS
are far too large to ignore"
•
costanza della velocità della luce
•
relatività della simultaneità
•
effetto Doppler
•
shift gravitazionale della frequenza
•
dilatazione dei tempi
•
termini di aggiustamento dell'orbita
c = 299 792 458 m/s
Δt = 1 ms
http://www.aapt.org/doorway/tgru/articles/ashbyarticle.pdf
15 decimali
24-32 satelliti
Δx = cꞏΔt = 300 km
alcuni link
• https://www.focus.it/scienza/scienze/ecco-perche-non-possiamo-fare-a-meno-
del-pi-greco
• https://www.wired.it/scienza/lab/2017/03/14/pi-greco-day-circonda/
• http://m.huffingtonpost.it/2016/03/14/pi-greco-10-cose-sapere-costante-
matematica_n_9457358.html
• https://www.scientificast.it/2016/03/22/quanto-lo-facciamo-lungo-sto-pi-greco/
• http://nova.ilsole24ore.com/progetti/il-mondo-che-ruota-attorno-al-pi-greco/
• https://www.jpl.nasa.gov/edu/news/2016/3/16/how-many-decimals-of-pi-do-
we-really-need/
• https://blogs.scientificamerican.com/observations/how-much-pi-do-you-need/
• http://scienzaemusica.blogspot.it/2013/03/laffascinante-storia-del-pi-
greco.html