1. Alan
Turing
Fibonacci
Numbers
in
Nature
Melissa
Davis:
Alan
Turing
was
a
mathematical
genius.
He
speculated
that
there
was
a
relationship
between
math
and
nature
by
the
presence
of
Fibonacci
numbers
that
naturally
occur
in
plants.
Fibonacci
numbers
are
a
sequence
of
numbers,
where
you
can
add
one
of
the
numbers
with
the
number
to
the
right
of
it,
to
get
the
next
number.
For
example,
the
first
few
numbers
of
the
sequence
begin
as
follows:
0,
1,
1,
2,
3,
5,
8,
13,
21,
etc.
To
get
the
number
after
21,
simply
add
13
to
21,
which
gives
you
34.
These
numbers
may
correlate
to
the
number
of
petals,
leaves,
or
spirals
of
seeds
a
plant
has.
This
is
also
the
reason
why
four-­‐leaf
clovers
are
so
rare,
since
four
is
not
a
number
k
appears
in
the
Fibonacci
sequence.
Turing
specifically
looked
at
sunflowers
to
study
this
phenomenon.
Turing
examined
how
the
number
of
spirals
in
the
seed
patterns
of
sunflowers
typically
resulted
in
a
Fibonacci
sequence.
This
finding
was
significant,
as
it
provided
much
information
for
phyllotaxis,
the
study
of
the
way
plants
grow.
“The
appearance
of
patterns
in
the
phyllotaxis
-­‐
the
arrangement
of
leaves,
stems,
seeds
or
similar
-­‐
has
been
studied
by
many
well-­‐known
scientists,
including
Leonardo
Da
Vinci.”
–
BBC
News

2.
Here
is
a
demonstration
of
one
way
to
count
the
spirals
of
seeds.
The
total
number
of
rows
is
34,
which
is
a
Fibonacci
number.
Because
of
Alan
Turing’s
abbreviated
life
due
to
his
mistreatment
in
society,
Turing
was
never
able
to
confirm
his
findings.
However,
the
Manchester
Science
Festival,
the
Museum
of
Science
and
Industry,
and
the
University
of
Manchester
are
asking
for
help
from
the
public
to
confirm
Turing’s
work
on
Fibonacci
numbers
in
sunflowers.
The
project
entails
people
planting
and
growing
their
own
sunflowers,
and
then
counting
the
rows
of
seeds
as
Turing
did.
This
project
also
aims
to
honor
Turing
during
the
one-­‐hundred-­‐year
anniversary
of
his
death.
In
addition
to
sunflowers,
the
Fibonacci
sequence
is
found
in
the
number
of
petals
in
many
flowers.
For
example,
buttercups,
wild
roses,
and
larkspurs
have
five
petals;
delphiniums
and
coreopsis
have
eight
petals;
ragworts
and
marigolds
have
thirteen
petals;
and
daisies
can
have
eighty-­‐nine
petals.

3.
References:
Couder,
Yves.
"Sunflower."
Photo.
Flower
Patterns
and
Fibonacci
Numbers.
2002.
22
May
2012.
<http://www.popmath.org.uk/rpamaths/rpampages/sunflower.html>.
"The
Fibonacci
Series."
ThinkQuest.
Oracle
Foundation.
Web.
22
May
2012.
<http://library.thinkquest.org/27890/applications5.html>.
"Greater
Manchester
Sunflowers
to
Test
Alan
Turing
Theory."
BBC
News.
BBC,
22
Mar.
2012.
Web.
06
June
2012.
<http://www.bbc.co.uk/news/uk-­‐england-­‐manchester-­‐
17469241>.
GrrlScientist.
"Sunflowers
and
Fibonacci."
The
Guardian.
Guardian
News
and
Media,
29
Mar.
0016.
Web.
22
May
2012.
<http://www.guardian.co.uk/science/grrlscientist/2012/apr/16/1>.
"How
To
Count
the
Spirals."
Photo.
Museum
of
Mathematics.
22
May
2012.
<http://momath.org/home/fibonacci-­‐numbers-­‐of-­‐sunflower-­‐seed-­‐spirals>.
McKay,
Dennis.
"Larkspur."
Photo.
Drug
Discovery.
2009.
22
May
2012.
<http://digitalunion.osu.edu/r2/summer09/jaeger/MLA.html>.
“Sunflowers
and
Fibonacci
–
Numberphile.”
Video.
(2012).
Retrieved
May
22,
2012
from

4. http://www.youtube.com/watchfeature=player_embedded&v=DRjFV_DETKQ.
Wainwright,
Martin.
"Grow
a
Sunflower
to
Solve
Unfinished
Alan
Turing
Experiment."
The
Guardian.
Guardian
News
and
Media,
24
Nov.
0048.
Web.
22
May
2012.
<http://www.guardian.co.uk/uk/the-­‐northerner/2012/mar/26/alan-­‐turing-­‐sunf....>
Jing
(Sophie)
Xia:
However,
flowers
are
not
the
only
organisms
in
which
Fibonacci
numbers
are
present;
Fibonacci
numbers
are
also
found
in
pine
cones
and
plant
leaves.
Pine
cones
display
the
Fibonacci
Spirals
clearly.
The
best
way
to
examine
these
patterns
is
to
observe
pine
cones
from
the
base
where
the
stalk
connects
it
to
the
tree.
For
instance,
one
set
of
spirals
goes
in
one
uniform
direction
whereas
another
set
of
spirals
goes
in
the
opposite
direction
(see
images
below).
For
example,
in
one
direction,
there
are
8
whirls
whereas
in
the
other
direction,
there
are
13
whirls.
It
is
not
coincidence
that
both
8
and
13
are
Fibonacci
numbers.
Pine
cones
contain
evidence
of
Fibonacci
spirals
since
their
patterns
are
arranged
in
two
different
directions
of
spirals.

5.
In
addition,
many
plants
show
Fibonacci
numbers
in
the
arrangements
of
the
leaves
around
their
stems.
When
looking
down
on
a
plant,
one
can
notice
that
its
leaves
are
arranged
so
that
the
leaves
higher
up
on
the
stem
do
not
hide
leaves
below.
This
ensures
that
no
matter
where
the
leaves
are
located
on
a
stem,
they
are
able
to
receive
sunlight.
Fibonacci
numbers
are
evident
in
two
ways
in
terms
of
leaves
per
turn.
First,
they
occur
when
counting
the
number
of
times
they
go
around
the
stem.
Secondly,
it
occurs
when
counting
leaves
until
finding
a
leaf
directly
above
the
leaf
in
which
one
started
counting.
If
one
counts
in
the
opposite
direction,
there
is
a
different
number
of
turns
with
the
same
number
of
leaves.
The
number
of
turns
in
each
direction
and
the
number
of
leaves
met
are
three
consecutive
Fibonacci
numbers.
For
example,
one
must
rotate
three
turns
clockwise
to
meet
a
leaf
that
is
directly
above
the
first
leaf
counted.
On
the
way,
one
passes
by
five
leaves.
But
when
one
counts
counter-­‐clockwise,
they
only
turn
two
times.
Because
2,
3,
and
5
are
consecutive
Fibonacci
numbers,
this
example
demonstrates
the
existence
of
Fibonacci
numbers
in
plant
leaves.
References:
"Evolution."
How
Stuff
Works.
N.p.,
n.
d.
Web.
20
May.
2012.
<http://science.howstuffworks.com/environmental/life/evolution>.
"Fibonacci
Numbers
and
Nature."
Rabbits,
Cows
and
Bees
Family
Trees
.
N.p.,
n.
d.
Web.
20
May.
2012.
<http://www.maths.surrey.ac.uk/hosted-­‐sites/R.Knott/Fibonacci/fibnat.html>.
"Fibonacci
Numbers
and
the
Golden
Section."
N.p.,
n.
d.
Web.
20
May.
2012.
<http://britton.disted.camosun.bc.ca/fibslide/jbfibslide.htm>.
"Fibonacci
numbers
and
Golden
ratio."
Natural
occurrence
of
Fibonacci
numbers.
N.p.,
n.
d.
Web.
20
May.
2012.
<http://gwydir.demon.co.uk/jo/numbers/interest/golden.htm>.

6. Parveen,
Nikhat.
"Fibonacci
in
Nature."
N.p.,
n.
d.
Web.
20
May.
2012.
<http://jwilson.coe.uga.edu/emat6680/parveen/fib_nature.htm>.
Shiwei
Huang
Alan
Turing’s
interest
in
Fibonacci
series
was
inspired
by
zoologist
D’arcy
Wentworth
Thompson’s
book,
On
Growth
and
Form.
Thompson
wanted
to
explain
how
physical
and
mathematical
laws
could
explain
the
forms
and
patterns
of
living
things.
After
examining
the
patterns
of
fir
cones
and
sunflowers,
Thompson
observed
that
the
scales
of
a
fir
cone
and
the
florets
of
a
sunflower
are
grouped
in
the
numbers
in
Fibonacci
series.
However,
Thompson
claimed
that
the
appearance
of
Fibonacci
series
in
these
plants
were
purely
for
mathematical
reasons,
and
the
purpose
of
introducing
of
this
series
into
plants
throughout
the
course
of
natural
selection
was
not
worth
studying.
Alan
Turing
was
the
first
scientist
to
study
the
mechanisms
behind
the
development
of
pattern
in
living
organisms
using
computer
simulation.
When
Manchester
Electronic
Computer,
also
called
Ferranti
Mark
I,
was
installed
in
Manchester
University,
Turing
wrote
to
his
colleague,
“I
am
hoping
as
one
of
the
first
jobs
to
do
something
about
‘chemical
embryology.’
In
particular
I
think
one
can
account
for
the
appearance
of
Fibonacci
numbers
in
connection
with
fir
cones.”
Turing
formulated
his
reaction-­‐diffusion
model
from
the
observation
of
fir
cones
and
sunflowers.
He
believed
that
diffusing
chemicals
reacted
with
each
other
and
caused
the
development
of
forms
of
living
organisms.
He
postulated
that
the
reaction-­‐diffusion
model
could
be
applied
to
gastrulation
of
an
embryo,
which
is
the
rearranging
of
the
cells
in
an
embryo,
and
the
formation
of
leaf
pattern.
Modern
computer
has
simulated
Turing’s
reaction-­‐diffusion
mechanism,
and
it
has
successfully
produced
leopard-­‐like,
cheetah-­‐like,
and
giraffe-­‐like
stripes.
In
his
paper
“Chemical
basis
of
morphogenesis,”
he
called
these
interacting
chemicals
“morphogens.”
Unfortunately,
Turing’s
work
on
morphogenesis
was
considered
ahead
of
the
time,
and
he
died
and
left
a
large
number
of
research
materials
and
notes
that
could
not
be
understood
today.
Turing’s
reaction-­‐diffusion
system
(the
simplified
version):
∂c/∂t=f(c)+D∇2c
f(c)
represents
the
local
chemical
reaction
that
different
chemicals
are
reacted
and
formed.
D
is
the
diffusion
constant,
which
describes
the
flow
of
a
chemical
due
to
its
concentration
gradient
and
its
diffusion.
Simply
put,
the
equation
states
that
the
distribution
of
a
chemical
is
determined
by
the
chemical
reaction
that
generates
this
chemical
and
the
diffusion
of
this
chemical.
Under
certain
conditions,
two
or
more
chemicals
will
diffuse
and
react
with
each
other
in
the
embryo,
and
they
will
reach
a
stable
pattern
of
concentration.
For
instance,
if
there
was
a
ring
of
cells,
reaction-­‐diffusion
model
could
give
us
a
pattern
of
chemical
gradient
surrounding
the
ring
of
cells.
The
same
concentrations
would
occur
at
places
with
the
same
distances
to
each
other,
and
Turing
called
this
chemical
wave
if
it
was
stationary.
Conversely,
if
the
gradient
was
changing,
it
would
be
called
traveling
waves.
Turing
pointed
out
the
structure
of
the
embryo
could
break
this
pattern
and
cause
asymmetrical
chemical
waves.
Turing
believed
that
genes
catalyzed
the
production
of
morphogens
and
might
influence
the
rate
of
the
reaction
to
determine
the
pattern
in
animals.

7.
Program
sheet
written
by
Alan
Turing
during
his
study
of
fir
cone
patterning
’
Computer
output.
Turing
wrote
“How
did
this
happen?”
on
the
sheet.
Turing’s
numbering
on
the
sunflower
References:
Charvoin,
J.
and
Sadoc,
J-­‐F.
(2011)
“A
Phyllotactic
Approach
to
The
Structure
of
Collagen
Fibrils.”
<http://arxiv.org/pdf/1102.2359v2.pdf>

8. Copeland,
B.J.(2004)
“The
Essential
Turing,
Seminal
Writings
in
Computing,
Logic,
Philosophy,
Artificial
Intelligence,
and
Artificial
Life
plus
The
Secrets
of
Enigma.”
Oxford:
Clarendon
Press.
Engelhardt,
R.
(1994)
“Modeling
Pattern
Formation
in
Reaction-­‐Diffusion
System.”
<http://www.robinengelhardt.info/speciale/main.pdf>
Maini,
P.
K.
(2007)
“The
impact
of
Turing's
work
on
pattern
formation
in
biology.”
<people.maths.ox.ac.uk/maini/PKM%20publications/172.pdf>
Swinton,
J.
(2003)
“Watching
the
Daisies
Grow:
Turing
and
Fibonacci
Phyllotaxis.”
<user29459.vs.easily.co.uk/wp-­‐content/uploads/2011/05/swinton.pdf>
Thompson,
D.
W.
(1966)
“On
Growth
and
Form.”
Cambridge:
University
Press.
Turing,
A.
M.
(1952)
“The
Chemical
Basis
of
Morphogensis.”
<http://links.jstor.org/sici?sici=0080-­‐
4622%2819520814%29237%3A641%3C37%3ATCBOM%3E2.0.CO%3B2-­‐I
Jen-­‐Ling
Nieh:
Morphology,
at
the
most
basic
sense,
consists
of
two
aspects
–
shape
and
pattern.
The
changes
of
morphology
that
occur
during
biological
development
of
an
organism
are
called
“morphogenesis.”
Turing
proposed
that
both
shape
and
pattern
seem
to
be
set
up
in
embryos
by
the
same
mechanism,
a
pre-­‐pattern
of
chemical
changes
that
waits
for
the
appropriate
stage
of
development,
and
then
triggers
either
pigments,
to
create
pattern,
or
cellular
changes,
to
create
shape.
He
showed
that
this
kind
of
system
may
have
a
homogeneous
stationary
state
which
is
unstable
against
perturbations,
such
that
any
random
deviation
from
the
stationary
state
leads
through
diffusion
to
a
symmetry
break.
This
process
is
called
diffusion-­‐driven
instability.
Since
complex
spatial
patterns
are
commonly
found
in
nature,
for
example,
in
animal
skins
and
also
in
some
polymer
systems,
it
is
quite
natural
to
think
that
such
pattern
formations
could
be
caused
by
some
general
physicochemical
process.

9. Many
animals
develop
their
coat
patterns
in
stages.
Typically,
a
secondary
pattern
will
emerge
as
the
animal
transitions
to
adulthood.
The
following
examples
all
use
multiple
stages:
Turing's
mathematical
model
of
chemical
morphogenesis
helps
us
understand
why
tigers
and
zebras
have
stripes.
Turing's
Reaction-­‐Diffusion
model
from
1952
consists
on
a
set
of
equations
that
iteratively
simulate
the
distribution
of
a
chemical
agent
(activator)
modulated
by
the
presence
of
another
agent
called
inhibitor.
In
his
seminal
1952
paper,
Alan
Turing
predicted
that
diffusion
could
spontaneously
drive
an
initially
uniform
solution
of
reacting
chemicals
to
develop
stable
spatially
periodic
concentration
patterns.
It
is
believed
that
such
interactions
take
place
in
nature
to
form
patterns
that
can
be
found
in
mammals
and
fish,
and
the
first
model,
generating
spots.

10.
According
to
the
Reaction-­‐Diffusion
Model,
the
diffusion
of
an
activator
and
inhibitor
through
an
evolving
cellular
system
over
a
period
of
time,
the
concentration
gradients
dictating
cell
differentiation,
i.e.
a
zebra
skin
cell
can
be
black
or
white
according
to
the
concentration
of
a
white-­‐cell
activator
at
the
point
when
it
forms.
Biologists
would
call
these
activators
morphogens,
as
these
are
the
proteins
that
regulate
gene
expression.

11.
Currently,
pigmentation
patterns
in
animal
skins,
feathers
of
birds,
and
shells
of
snails
are
the
only
examples
in
which
we
can
detect
the
dynamic
nature
of
Turing
waves
as
a
time
course
of
the
pattern
change.
Especially,
the
two-­‐dimensional
(2-­‐D)
skin
pattern
of
fish
is
quite
convenient
to
study
because
their
waves
are
sometimes
alive
even
when
the
fish
has
grown
up
into
an
adult
(Shigeru
Kondo).
For
example,
when
a
striped
angelfish
(Pomacanthus
imperator)
grows,
the
branching
points
of
the
stripes
slide
horizontally
as
the
zip
opens
and
add
a
number
of
stripes;
eventually
the
spacing
between
the
stripes
remains
stable.
In
the
case
of
the
spotted
catfish
(Plecostoms),
both
division
of
the
spots
and
insertion
of
the
new
spots
occur
to
retain
the
density
and
size
of
the
spots.
Both
stripes
and
the
spots
are
the
most
typical
2-­‐D
patterns
generated
by
the
RD
mechanism,
and
the
time
course
of
the
pattern
change
possesses
the
characteristics
of
the
dynamics
of
RD
waves,
strongly
suggesting
that
the
RD
mechanism
underlies
the
process
of
pigment
pattern
formation
of
fish.
References:
www.scholarpedia.org/w/images/8/8d/TROPH.jpg">http://www.scholarpedia.org/w/im
ages/8/8d/TROPH.jpg
http://cgjennings.ca/toybox/turingmorph/texture1.png
http://cgjennings.ca/toybox/turingmorph/texture2.png
http://cgjennings.ca/toybox/turingmorph/texture3.jpg
www.urbagram.net/images/turing.jpg">http://www.urbagram.net/images/turing.jpg
www.urbagram.net/v1/revision/Morphogenesis?rev=1">http://www.urbagram.net/v1/r
evision/Morphogenesis?rev=1
http://27.media.tumblr.com/tumblr_lzeh1qUhRe1r3lyy3o1_500.jpg

12. Alexandra
Pourzia:
Alan
Turing
proposed,
based
purely
on
logical
reasoning,
that
pattern
formation
in
nature
involved
an
‘activating’
substance
and
an
‘inhibiting’
substance.
The
repetition
of
activator
and
inhibitor
could
create
patterns
such
as
stripes.1
Previously,
developmental
biologists
were
puzzled
by
pattern
formation
because
they
could
not
explain
it
using
the
linear
models
that
were
the
extent
of
their
knowledge
at
the
time.
Turing
proposed
a
nonlinear
model
by
introducing
diffusion
as
the
generator
of
instability
in
the
model,
instead
of
being
a
byproduct
of
the
model.
2
The
implications
of
Turing’s
mechanism
were
astounding:
he
predicted
the
mode
of
action
of
the
Hox
genes
in
Drosophila,
which
result
in
the
patterning
of
the
embryo’s
body
segments.
3
Hox
gene
patterning
by
body
segment
in
Drosophila
The
Hox
genes
induce
patterning
by
activating
transcription
of
their
unique
set
of
genes
while
repressing
others
not
related
to
their
segment.
They
in
turn
are
regulated
by
patterning
genes
(gap,
pair-­‐rule,
or
segment
polarity
genes),
which
follow
Turing’s
proposed
model
very
closely.
These
patterning
genes
are
induced
by
high
or
low
concentrations
of
maternal
proteins
in
the
embryo,
which
was
formed
from
the
maternal
egg
and
paternal
sperm.
For
example,
high
concentrations
of
maternal
protein
induce
the
expression
of
Bicoid
and
Hunchback,
while
inhibiting
Giant
and
Kruppel.
The
concentration
of
these
“morphogens”,
as
Turing
first
called
them,
lead
to
the
formation
of
a
pattern
–
segment
two
of
the
fly
embryo.3
Pair-­‐rule
genes:
expressed
between
certain
segments

13. References:
1. Hughes,
Virginia.
“Alan
Turing’s
60-­‐Year-­‐Old
Prediction
About
Patterns
in
Nature
Proved
True.“
Smithsonian.com.
The
Smithsonian
Institution,
21
Feb
2012.
Web.
20
May
2012.
<http://blogs.smithsonianmag.com/science/2012/02/alan-­‐turing-­‐
predicted-­‐natures-­‐stripes-­‐and-­‐patterns/>
2. Reinitz,
John.
“Pattern
formation.”
Nature.
Feb
2012.
3. “Hox
gene.”
Wikipedia.
Wikimedia
Foundation,
Inc,
n.d.
20
May
2012.
<http://en.wikipedia.org/wiki/Hox_gene>