3.141592653589793238462643 38327950288419716939937510 ... · – It’s the rao of the...

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3.141592653589793238462643 38327950288419716939937510 58209749445920899862803482 53421170679821480865132823 06647093844609550582231725 35940812848111745028410270 19385211055596446229489549 30381964428810975665933446 12847564823378678316527120 190HelpI’mTrappedInAUniverse Factory914564856692346034… In search of π Aidan Randle‐Conde 22/7 (/2009) 1

Transcript of 3.141592653589793238462643 38327950288419716939937510 ... · – It’s the rao of the...

  • 3.1415926535897932384626433832795028841971693993751058209749445920899862803482534211706798214808651328230664709384460955058223172535940812848111745028410270193852110555964462294895493038196442881097566593344612847564823378678316527120190HelpI’mTrappedInAUniverseFactory914564856692346034…

    In
search
of
π


    Aidan
Randle‐Conde


    22/7
(/2009)


    1


  • Why
talk
about
π?


    •  In
the
states
March
14th
is
designated
π
Day
(3‐14)
•  In
the
UK
July
22nd
is
wriLen
22/7.

So
today
is
π
ApproximaRon
Day!


    –  Although
today
is
all
about
approximaRon,
it’s
ironically
more
accurately
named
than
π
Day!


    •  Depending
on
your
taste
there
are
a
few
other
days
that
can
help
you
celebrate
π:
–  November
10th
is
the
314th
day
of
the
year.
–  At
1:13am
on
December
21st
the
day/Rme
is
335/113,
an
ancient
Chinese


    approximaRon.


    –  By
April
26th
the
Earth
travels
through
1/π
of
its
orbit
since
the
start
of
the
year.
–  In
2015
we’ll
have
Super
π
Day,
3‐14‐15!
–  00:37
on
July
21st
2069
will
be
the
π
epoch
(3141592654
seconds
since
the
Unix


    epoch)

Less
than
a
day
away
from
π
ApproximaRon
Day!


    2


  • What
is
π?


    •  π
doesn’t
really
need
an
introducRon
–  It’s
the
raRo
of
the
circumference
of
a
circle
to
its
diameter.
–  It’s
easy
to
understand,
and
yet
its
exact
value
has
eluded
mathemaRcians
for


    millenia.


    –  It
pops
up
in
the
most
unusual
and
unexpected
of
places.
•  What
we
know
about
π:


    –  It’s
not
an
integer
–  It’s
not
raRonal
–  It’s
not
algebraic


    •  What
we
don’t
know
about
π:
–  Is
it
normal?
–  For
that
maLer,
is
our
obsession
with
it
normal?!


    3


  • •  One
of
the
first
methods
used
(by
Archimedes)
was
the
method
of
exhausRon.
•  Sandwich
a
circle
between
two
regular
polygons,
measure
their
perimeters
and


    place
limits
on
the
value
of
π.


    •  This
method
also
gives
the
familiar
limits:


    θ
½


    pS


    pL


    ExhausRon


    •  The
length
of
the
side
of
the
smaller
polygon,
pS,
is:


    •  And
for
the
larger
polygon,
pL,
is:


    •  So
for
an
n‐sided
polygon
the
approximaRon
becomes:
€

    pS = sinθ

    pL = tanθ

    n sinθ < π < n tanθ

    θ = π /n

    ;

    sinθ →θ

    lim θ → 0[ ]

    tanθ →θ

    tanθ > sinθ4


  • ExhausRon


    •  Since
θ
=
π/n,
for
an
arbitrary
value
of
n
the
argument
will
become
circular
(ie
we
would
need
to
know
the
value
of
π
to
esRmate
π.)


    •  But
we
know


















































so
we
can
use
the
half‐angle
formulae:


    •  Square
roots
weren’t
easy
to
calculate
for
the
ancient
Greeks.
•  Using
various
tricks
Archimedes
got
as
far
as
a
96‐sided
polygon


    before
moving
on
to
other
calculaRons.


    •  He
calculated:


    •  All
that
computaRon
got
π
correct
to
three
decimal
places.

Not
bad…


    cos π /4( ) = sin π /4( ) = 12

    cos(θ /2) = 1+ cosθ2

    sin(θ /2) = 1− cosθ2

    3.1410 < π < 3.1427

    5


  • •  The
first
few
iteraRons
improve
the
approximaRons
quite
rapidly:


    ExhausRon


    n
Lower
limit


    Upper
limit


    4
 2.828427 4

    8
 3.061467 3.313708

    16
 3.121445 3.182597

    32
 3.136548 3.151724

    64
 3.140331 3.144118

    128
 3.141277 3.14222

    256
 3.141514 3.141750

    512
 3.141573 3.141632

    1024
 3.141588 3.14160

    2048
 3.141591 3.141595

    6


  • RaRonal
approximaRons


    •  Over
the
coming
centuries
various
mathemaRcians
approximated
π
using
the
following
fracRons:
–  Archimedes
used
the
areas
of
his
96‐gons
to
find
the
raRonal
limits:


    31071

    < π < 31070

    –  He
used
the
laLer
approximaRon,
22/7,
which
is
sRll
occasionally
used
today.


    –  Ptolemy
got
slightly
closer
using
the
same
method
with
377/120
=
3.1416…


    –  Tsu
Ch’ung‐Chi
used
355/113
=
3.141593…
but
we
don’t
know
a
great
deal
about
him
or
his
methods.


    –  Nothing
much
changes
unRl
the
development
of
infinite
series.


    7


  • Infinite
series


    •  Using
discoveries
involving
infinite
series
and
calculus
various
expressions
for
π
were
found.

Most
of
them
are
very
preLy:


    π 2

    6=112

    +122

    +132

    +142

    +152

    +162

    +172

    π − 34

    =1

    2 × 3× 4−

    14 × 5 × 6

    +1

    6 × 7 × 8…

    π4

    =11−13

    +15−17

    +19−111

    +113

    −115

    π2

    =21×23×43×45×65×67×87×89…

    8,553,103× 224 ×π 26

    3× 26!=1126

    +1226

    +1326

    John
Wallis


    Leonhard
Euler


    Gokried
Leibniz


    Leonhard
Euler


    8


  • Infinite
series


    •  Although
the
infinite
series
are
very
elegant
they
can
be
slow
to
converge:


    n


    π2

    =21×23×43×45×65×67×87×89…

    π 2

    6=112

    +122

    +132

    +142

    +152

    +162

    +172

    π − 34

    =1

    2 × 3× 4−

    14 × 5 × 6

    +1

    6 × 7 × 8…

    π4

    =11−13

    +15−17

    +19−111

    +113

    −115

    8,553,103× 224 ×π 26

    3× 26!=1126

    +1226

    +1326

    9


  • Example:
π2/6
•  Out
of
these
(and
many
other)
series
one
of
the
most
interesRng
to
derive
is
π2/6.
•  Euler
considered
the
funcRon


    •  This
equaRon
has
roots
at
x=±nπ,
so
it
can
also
be
wriLen:


    •  EquaRng
the
terms
in
x2
gives:


    •  The
same
method
can
be
used
to
get
higher
order
terms
in
π2n.
•  You
can
get
the
same
result
using
Fourier
Series,
Reinmann
Zeta
funcRons
etc…


    sin xx

    = 1− xnπ

    n=−∞

    ∞,x≠0

    ∏ = 1− x2

    n2π 2

    n=1

    ∏€

    sin xx

    =−1( )n x 2n

    2n +1( )!n= 0

    −x 2

    n2π 2n=1

    ∑ = −x2

    6⇒

    π 2

    6=

    1n2n=1

    10


  • Machin’s
method
•  Archimedes’
method
could
be
extended
to
polygons
of
arbitrary
numbers
of
sides


    and
the
known
series
expansions
could
be
extended
to
arbitrary
number
of
terms.


    •  Each
step
requires
more
computaRon‐
some
people
spent
their
working
lives
churning
out
more
digits!


    •  In
1706
Machin
discovered
a
more
efficient
method
using
the
differences
arctan
funcRons:


    π4

    = 4arctan 12

    − arctan

    3117

    π4

    = 4arctan 13

    − arctan

    1731

    π4

    = 4arctan 14

    − arctan

    79401

    arctan(x) = x − x3

    3+x 5

    5−x 7

    7+x 9

    9…

    •  Proof
is
in
the
backup
slides.
•  The
second
term
converges
rapidly.
•  People
oren
made
mistakes:


    •  Shanks
got
the
last
180
of
707
digits
incorrect.


    •  This
was
discovered
when
the
digit
7
didn’t
occur
“oren
enough”


    11


  • Geung
irraRonal
•  For
those
who
didn’t
have
Rme
to
sum
hundreds
of
terms,
or
calculate
26!,


    Ramanujan
gave
a
number
of
approximaRons,
using
geometrical
arguments
and
series:


    92 + 192

    22

    14

    = 3.14159265262…

    632517 +15 5( ) 7 +15 5( ) = 3.141592652…992

    2206 2= 3.14159265…

    = 8 (1103+ 26390n)(2n −1)!!(4n −1)!!994n+232n (n!)3n= 0

    •  The
last
two
come
from
a
remarkable
collecRon
of
equaRons
published
in
1914,
such
as


    12


  • IrraRonal,
transcendental,
normal?
•  In
1761
Lambert
showed
that
π
is
irraRonal.
•  All
aLempts
to
find
an
exact
value
of
π
in
the
form
π
=
a/b
are
doomed
to
failure.
•  Worse
sRll,
in
1882
Lindemann
showed
that
π
is
transcendental,
so
all
expressions


    of
the
form:


    where
there
are
a
finite
number
of
terms
and
all
an
and
n
are
raRonal
and
non‐zero
are
also
doomed
to
failure.


    •  One
of
the
remaining
problems
is
to
determine
whether
or
not
π
is
normal‐
does
every
digit
appear
with
equal
frequency
in
all
bases?
–  StaRsRcally,
the
answer
is
yes.
–  As
far
as
mathemaRcians
are
concerned,
the
quesRon
is
unresolved.
–  Perhaps
we
can’t
use
the
frequency
of
the
digit
7
to
find
errors
arer
all…


    •  None
of
these
facts
stopped
people
trying
to
find
ever‐more
precise
approximaRons.


    0 = anπn∑

    13


  • Buffon’s
needle
•  Towards
the
end
of
the
18th
century
some
experiments
were
done
dropping


    needles
onto
lined
paper.


    •  The
probability
that
a
needle
of
length
l
crossing
lines
of
spacing
s
is:


    •  By
dropping
needles
and
counRng
the
how
many
cross
a
line
you
can
approximate
the
value
of
π.

(One
poor
student
of
deMorgan
got
π
≈
3.317
by
dropping
600
needles!)


    P(l | s) = lcosθ2πs0

    ∫ dθ = 2lπs

    •  I
cheated
and
used
a
computer:
•  l
=
40
•  s
=
50
•  100,000
needles
•  50,794
needles
cross
a
line
•  π
≈
3.15


    •  That’s
not
bad
for
a
virtual
pack
of
needles
and
a
virtual
pad
of
lined
paper.


    14


  • State
of
play
1946
•  By
the
early
20th
century
there
were
a
number
of
different
ways
to
calculate
π
to


    arbitrary
degrees
of
precision.


    •  It
wasn’t
uncommon
for
people
to
get
a
whole
series
of
digits
wrong!


    Year
 Source
 Method
 Decimal
places


    2000
BC
 Rhind
papyrus
 (16/9)2
 1


    250
BC
 Archimedes
 96‐gon
exhausRon
 3


    263
 Liu
Hui
 192‐gon
exhausRon
 5


    480
 Tsu
Ch’ung‐Chi

 355/113
 7


    1400
 Madhava
 Gregory’s
series
 11


    1430
 Al’Kashi
 3x228‐gon
exhausRon
 14


    1695
 Van
Ceulen
 262‐gon
exhausRon
 35


    1706
 Machin
 arctan
 100


    1946
 Fergurson
 arctan
 620


    15


  • Precision
wars
•  Al’Kashi
wanted
to
calculate
the
circumference
of


    the
universe
precise
to
the
width
of
a
hair.

–  He
thought
that
the
universe
was
about
600,000
Rmes


    wider
than
the
Earth.
–  Using
his
805,306,368‐gon
he
calculated
14
digits
of
π.



    That’s
about
right,
if
a
hair
is
1mm
wide.


    •  Changing
our
sense
of
scale
to
something
more
meaningful:
–  The
width
of
the
universe
if
about
30
billion
light
years


    (≈3x1026m.)


    –  The
width
of
a
Hydrogen
atom
is
≈1x10‐10m.
–  It
turns
out
that
40
digits
of
π
is
more
than
enough
for


    nearly
all
calculaRons.

(Add
another
25
digits
if
you
want
to
use
the
Planck
scale.)


    •  Anything
else
is
just
showing
off.

And
yet
the
calculaRons
conRnue
to
this
day.


    16


  • •  What
use
are
the
remaining
digits
of
π?
–  It’s
a
great
way
to
benchmark
a
new
supercomputer.

The
digits
of
π
give
an
endless
supply
of


    calculaRons
for
the
mulR‐petaflop
machines.
–  The
digits
of
π
provide
a
good
source
of
random
numbers
(True
randomness
is
a
constant


    annoyance
in
the
world
of
compuRng
science
and
staRsRcs.)
–  Some
staRsRcians
run
tests
on
the
digits
to
see
if
the
digits
really
are
random
and
whether
π
really


    is
a
normal
number.

So
far
π
has
passed
all
tests.
–  People
are
sRll
amazed
by
this
number
for
some
reason.


    •  Current
record
is
1,241,100,000,000
(1.24
trillion)
digits.


    The
age
of
the
computers


    •  The
numbers
of
decimal
places
obtained
since
1947
span
9
orders
of
magnitude!


    •  (I
needed
two
graphs
to
fit
all
the
points
in.)


    •  You
can
see
Moore’s
law
at
work.


    17


  • The
age
of
the
computers
•  There
are
quite
a
few
commonly
used
algorithms
for
calculaRng
π
to
very
large


    numbers
of
decimal
places.


    •  One
of
the
fastest
to
converge
is
the
Gauss‐Legendre
algorithm:


    an+1 = an + bn( ) /2

    b0 =1/ 2

    c0 =1/4

    d0 =1

    a0 =1

    bn+1 = anbn

    cn+1 = cn − dn an − bn( )2 /4

    dn+1 = 2dn

    Itera9on
 π
≈
 Decimals


    0
 2.9142135623730949
 0


    1
 3.1405792505221686
 2


    2
 3.1415926462135428
 7


    3
 3.141592653589794
 11
€

    π ≈an + bn( )

    2

    4cn

    Ler:
Yasumasa
Kanada
with
the
Hitachi
SR‐8000
supercomputer,
that
calculated
15,000,000,000
digits
of
π.
 18


  • The
age
of
the
computers
•  In
1995,
Bailey,
Borwein
and
Plouffe
discovered
a
fascinaRng
formula:


    •  This
can
be
used
to
determine
any
given
hexidecimal
digit
(base
16
digit)
if
π,
without
needing
to
calculate
any
of
the
rest.


    π =4

    8n +1−

    28n + 4

    −1

    8n + 5−

    18n + 6

    116

    n

    n= 0

    
Ler:
π
in
binary. 
 
Right:
22/7
in
binary


    •  ConverRng
a
hexidecimal
number
to
a
binary
number
is
trivial.

This
method
was
used
to
approximate
π
to
40,000,000,000,000
bits.


    •  There
is
now
a
search
for
an
algorithm
for
finding
digits
in
base
10.


    19


  • Fun
facts
about
π
•  Since
we
have
all
these
digits
we
may
as
well
do


    something
fun
with
them:
–  Take
combinaRons
of
3
digits,
divide
by
37
and
take
the


    remainder.

Assign
1=A,
2=B,
3=C
and
so
on.
–  This
gives
the
π
code.
–  Using
the
first
1,000,000
digits,
“PI”
appears
299
Rmes.



    “SASS”
never
appears
and
neither
does
“AIDAN”.
For
some
reason
“E”
appears
9,208
Rmes.

Why
should
e
appear
more
Rmes
than
π?
Hmm…


    •  Plenty
of
people
have
searched
for
paLerns:
–  The
three
digits
around
digit
315
are
3‐1‐5,
and
the
three


    digits
around
digit
360
are
3‐6‐0.
–  The
famous
“Feynman
point”
consists
of
six
nines
from
digit


    762,
and
Feynman
used
to
joke
that
π
is
equal
to
3.14159…999999…
etc


    •  But
that’s
just
the
start…


    20


  • Indiana
#246
•  In
1897
Edwin
Goodwin
proposed
a
Bill
to
the
Indiana
Legislature


    to
declare
the
value
of
π
to
be
3.2
(or
4,
the
mathemaRcs
behind
it
was
ambiguous).


    •  He
put
forward
some
rather
incredible
claims:
–  He
could
trisect
the
angle
with
just
a
pair
of
compasses
and
straight
edge
–  He
doubled
the
cube
(made
3√2
raRonal)
–  He
squared
the
circle
(made
π
algebraic)


    •  All
these
claims
had
already
been
shown
to
be
impossible.
•  Even
so,
the
bill
was
sent
(via
the
CommiLee
on
Swamp
Lands)
to


    the
CommiLee
on
EducaRon.


    •  The
bill
passed
the
first
reading
with
unanimous
consent.
•  Prof
Waldo,
who
was
passing
through
Indianapolis
at
the
Rme,


    read
the
bill.


    •  He
convinced
the
Legislature
to
postpone
the
second
reading
indefinitely.


    •  This
seems
to
be
the
only
aLempt
in
history
to
decrease
our
knowledge
of
π.
 21


  • Mnemonics
•  If
you’re
having
trouble
remembering
the
first
few
digits
of
π
there
are
plenty
of


    mnemonic
phrases
to
help
you.

The
number
of
leLers
in
each
word
represents
a
single
digit:


    How I wish I could recollect pi easily today!How I want a drink, alcoholic of course, after the heavy lectures involving quantum mechanics!

    •  Or
if
you’re
a
fan
of
sonnets
iambic
pentameter:
 Now I defy a tenet gallantly

    Of circle canon law: these integersImporting circles' quotients are, we see,

    Unwieldy long series of cockle bursPut all together, get no clarity;

    Mnemonics shan't describeth so reformedCreating, with a grammercy plainly,A sonnet liberated yet conformed.

    Strangely, the queer'st rules I manipulateBeing followéd, do facilitate

    Whimsical musings from geometric bard.This poesy, unabashed as it's distressed,

    Evolvéd coherent - a simple test,Discov'ring poetry no numerals jarred. 22


  • π
everywhere
•  π
seems
to
appear
in
all
over
the
place.

The
most
obvious
example
is
Euler’s


    relaRon.

(Which
is
preLy
weird
when
you
think
about
it.)


    •  We’ve
already
seen
it
in
some
infinite
series:


    •  And
staRsRcs:


    •  And
error
funcRons:


    •  And
integral
calculus:



    •  And
if
you
divide
the
length
of
a
river
by
the
distance
between
its
source
and
mouth,
and
the
river
is
sufficiently
old
and
long,
you
get
π.

And
nobody
knows
why.


    •  And…


    e−x2

    −∞

    ∫ dx = π

    Γ(1/2) = π

    π2

    =21×23×43×45×65…

    1− x 2−∞

    ∫ dx = π /2

    23


  • π
and
primes
•  If
you
take
two
integers
at
random,
what
is
the
probability
that
they
share
no


    common
divisor?

(Sound
impossible?)


    •  The
probability
that
an
integer
is
not
divisible
by
a
prime
q
is
1‐1/q.
•  So
the
probability
that
two
integers
share
no
divisors
is:


    •  Consider
mulRplying
the
following
series:


    P = 1− 1q2

    primes∏

    112

    +122

    +132

    +142

    +152

    +162

    +172

    +182

    +192

    × 1−

    122

    =112

    +132

    +152

    +172

    +192

    112

    +132

    +152

    +172

    +192

    × 1−

    132

    =112

    +152

    +172

    …24


  • π
and
primes
•  This
process
conRnues
unRl
every
term
is
removed,
except
1.
•  So
we
get
the
idenRty:


    •  Using
the
idenRty
we
derived
earlier:


    •  gives
the
probability
that
no
two
randomly
selected
integers
share
a
divisor
as:


    •  (And
if
you
don’t
find
that
interesRng
then
the
chances
are
I
just
wasted
40
minutes
of
your
Rme…)


    1n2n=1

    1−

    1q2

    primes∏

    =1

    1n2n=1

    ∑ = π2

    6

    P = 6π 2

    = 0.61…

    25


  • Conclusion
•  We
take
π
for
granted.

Today
geung
its
value
to
9
or
10
decimal
places
is


    extremely
easy.


    •  The
true
story
behind
the
search
for
π
reveals
that
geung
just
a
few
decimal
places
is
laborious
task.


    •  MathemaRcians
have
spent
centuries
geung
to
know
π
beLer
–  the
more
we
know
the
more
we
realize
π
finds
its
way
into
every
part
of
mathemaRcs,
oren
in


    strange
places.
–  Despite
know
more
unique
decimal
places
of
π
than
any
other
number
there
are
sRll
lots
of


    unanswered
quesRon.


    •  Learning
about
π
can
be
fun,
fascinaRng
and
informaRve.


    26


  • 3.14159265358979323846264338327950288419716939937510582097494459208998628034825342117067982148086513282306647093844609550582231725359408128481117450284102701938521105559644622948954930381964428810975665933446128475648233786783165271201909145648566923460104543266482133936072602491412737

    Backup


    27


  • DerivaRon
of
Machin’s
formula
•  The
equaRon
for
the
tangent
of
a
sum
of
angles
is:


    •  Applying
this
recursively
gives:


    •  And
then:


    •  Make
some
subsRtuRons:


    tan(A + B) = tanA + tanB1− tanA tanB

    tan(4A) = 4 tanA 1− tan2 A

    1− 6tan2 A + tan4 A

    tan(4A − B) = 4 tanA(1− tan2 A) − tanB(1− 6tan2 A + tan4 A)

    1− 6tan2 A + tan4 A + 4 tanA tanB(1− tan2 A)

    4A − B = arctan(1/ x)

    A = arctan(1/(x + n))

    B = arctan(y)

    28


  • DerivaRon
of
Machin’s
formula
•  From
the
previous
slide:


    •  Rearranging
gives:


    •  Then
let
x=1,
n=4,
which
gives
y=1/239:


    tan(4A − B) = 4 tanA(1− tan2 A) − tanB(1− 6tan2 A + tan4 A)

    1− 6tan2 A + tan4 A + 4 tanA tanB(1− tan2 A)

    4A − B = arctan(1/ x)

    A = arctan(1/(x + n))

    B = arctan(y)

    y =4x (x + n)3 − (x + n)( ) − (x + n)2 − 6 x + n( )2 +1( )4 (x + n)3 − x + n( )( ) + x (x + n)2 − 6 x + n( )2 +1( )

    arctan(1/ x) = 4arctan(1/(x + n)) − arctan(y)

    π4

    = 4arctan 15

    − arctan

    1239

    29


  • π
as
an
integer?


    •  Most
socieRes
realize
quite
quickly
that
π
is
not
an
integer.
•  However,
if
you
take
a
look
at
the
bible:


    –  ‘And he made a molten sea, ten cubits from the one brim to the other: it was round all about, and his height was five cubits: and a line of thirty cubits did compass it about.’ (I Kings 7, 23)

    •  So
π
=
3?

Oops!
•  This
approximaRon
actually
comes
from
the
Great
Temple
of


    Solomon
and
dates
back
to
around
1000BC.


    30


  • π
 0

    0

    1

    x 4 (1− x)4

    1+ x 2= x 6 − 4x 5 + 5x 4 − 4x 2 + 4( ) − 41+ x 2

    I = x 6 − 4x 5 + 5x 4 − 4x 2 + 4( )0

    1

    ∫ dx − 41+ x 2

    0

    1

    =227− 4arctan x[ ]0

    1

    =227−π

    31


  • Bibliography
•  There
are
loads
of
books
out
there
that
look
into
π
and
its
properRes.

Here
are
a


    few
that
are
currently
siung
on
my
shelf:
–  I’d
recommend
the
Penguin
DicRonary
of
Curious
and
InteresRng
Numbers
(David
Wells)
for
a
quick
overview
of
π
and
lots
of
easy


    recreaRonal
mathemaRcs.


    –  Finding
the
probability
of
a
shared
divisor
comes
from
Excursions
in
Number
Theory,
another
easy
book
that
leads
onto
harder
things.


    –  For
the
gory
details
of
Archimedes’
method
of
exhausRon
(as
well
as
loads
on
planetary
moRons)
check
out
100
Great
Problems
of
Elementary
MathemaRcs.

(They
use
the
original
derivaRons‐
it’s
difficult
stuff!)


    •  There’s
also
a
great
deal
online,
in
the
obvious
places.

The
usual
suspects
link
to
lots
of
other
sites:
–  Wiki:


    •  hLp://en.wikipedia.org/wiki/%CE%A0
•  hLp://en.wikipedia.org/wiki/Numerical_approximaRons_of_%CF%80
•  hLp://en.wikipedia.org/wiki/Category:Pi_algorithms


    –  Mathworld
Wolfram:
•  hLp://mathworld.wolfram.com/PiFormulas.html


    –  Machin
derivaRon:
•  hLp://milan.milanovic.org/math/english/pi/machin.html




    –  Chronology
of
π
approximaRons:
•  hLp://www‐groups.dcs.st‐and.ac.uk/~history/HistTopics/Pi_chronology.html



    32


  • Images
•  If
I
don’t
credit
an
image/plot
I’ve
made
it
myself.


    –  Slide
3:
hLp://plus.maths.org

–  Slide
4:
hLp://archimedes.galilei.com/archimedes_body.htm
–  Slide
7:
hLp://jeff560.tripod.com/stamps.html
–  Slide
8:
hLp://en.wikipedia.org/wiki/John_Wallis

,
hLp://en.wikipedia.org/wiki/Leonhard_Euler
,


    hLp://en.wikipedia.org/wiki/Leibniz


    –  Slide
12:
hLp://www.alcorngallery.com/Portraits/Portraits_display.php?i=24

–  Slide
16:
hLp://www.danstopicals.com/laRtude1.htm
,
hLp://www.nasa.gov/
–  Slide
18:
hLp://www.educ.fc.ul.pt/icm/icm2001/icm34/kanada.htm
–  Slide
19:
hLp://mathworld.wolfram.com/PiDigits.html
–  Slide
21:
hLp://www.civicheraldry.com/page/5735
,
hLp://www.agecon.purdue.edu/

–  Slide
26:
hLp://areason2write.wordpress.com/


    33