2. Learn an urgent lesson from a chaotic fig tree!

Carlos E. PuenteUniversity of California, Davis

THE KEY ROLE OF SCIENCE IN THE NEW EVANGELIZATION

Ateneo Pontificio Regina ApostolorumRome, May 2013

Thesis

We humans, with a soul,

may learn from recent advancements

regarding natural complexity

in order to find the peace of Christ

and share the good news in a novel way…

e fig tree and Holy Scripture

Adam and Eve covered with fig leaves (Gn 3:7)

a symbol of Israel together with the vine (Hos

2:14)

King Hezekiah healed by prophet Isaiah (Is

38:21)

e fig tree and Holy Scripture

Adam and Eve covered with fig leaves (Gn 3:7)

a symbol of Israel together with the vine (Hos

2:14)

King Hezekiah healed by prophet Isaiah (Is

38:21)

Nathanael is seen under the fig tree (Jn 1:47–51)

the parable of the barren fig tree (Lk 13:6–9)

Jesus curses a fruitless fig tree (Mt 21:18–22, Mk

11:12–23)

the lesson from a fig tree (Mk 13:28–31, Lk

21:29–33)

e logistic map

... a population of “rabbits” as a function of time

𝑋𝑘+1=𝛼 𝑋𝑘 (1−𝑋𝑘 ) ,𝛼∈ [0,4 ]

e logistic map

... a population of “rabbits” as a function of time

𝑋𝑘+1=𝛼 𝑋𝑘 (1−𝑋𝑘 ) ,𝛼∈ [0,4 ]

e logistic map

... a population of “rabbits” as a function of time

𝑋𝑘+1=𝛼 𝑋𝑘 (1−𝑋𝑘 ) ,𝛼∈ [0,4 ]

𝛼=2.8

e logistic map

... a population of “rabbits” as a function of time

𝑋𝑘+1=𝛼 𝑋𝑘 (1−𝑋𝑘 ) ,𝛼∈ [0,4 ]

𝛼=2.8

different cases depending on the parameter :

The logistic dynamics

𝑋∞=0

0<𝛼≤1

𝑌=𝑋

The logistic dynamics

𝑌=𝑋

𝑋∞=0

0<𝛼≤1 1<𝛼≤3

The logistic dynamics

𝑋∞=0

period 2

0<𝛼≤1

𝑌=𝑋

1<𝛼≤3

The logistic dynamics

1<𝛼≤3

period 4

𝑌=𝑋

𝑋∞=0

period 2

0<𝛼≤1

The logistic dynamics

1<𝛼≤3

period 4

𝑋∞=0

period 2

0<𝛼≤1

period

aperiodic

aperiodic

𝑌=𝑋(𝛼>𝛼∞≈3.5699)

The logistic dynamics

𝑌=𝑋

period 5aperiodic

period 3

(𝛼>𝛼∞≈3.5699)

aperiodic

The logistic dynamics

e diagram of bifurcations

e diagram of bifurcations

the tree contains periodic behavior, for any natural

number

e diagram of bifurcations

e diagram of bifurcations

after the dusty strange attractors are common

e diagram of bifurcations

after the dusty strange attractors are common

in the white bands there are the “buds” of the tree

Intermediate bud of period 3

Intermediate bud of period 3

… exquisite self-similarity ad infinitum

Thorns galore

the chaotic tree contains many

multifractal thorns

they combine imbalances and holes, as

in :

Thorns galore

the chaotic tree contains many

multifractal thorns

they combine imbalances and holes, as

in :

they happen at ends of white bands of buds for all

periods

Universality on the fig tree

bifurcations happen in an orderly fashion:

(Feigenbaum, 1978)

0…openings

…durations

Universality on other chaotic trees

straight root, “tender branch,” branches, and dust

(“fig leaves”)

𝑓 ( 𝑋 )=𝛼 𝑋 (1−𝑋 )3𝑓 ( 𝑋 )=𝛼 𝑋 (1− 𝑋 3 )

Universality on other chaotic trees

straight root, “tender branch,” branches, and dust

(“fig leaves”)

𝑓 ( 𝑋 )=𝛼 𝑋 (1−𝑋 )3𝑓 ( 𝑋 )=𝛼 𝑋 (1− 𝑋 3 )

relevant in biology, ecology, chemistry, physics, economics…

Universality on other chaotic trees

straight root, “tender branch,” branches, and dust

(“fig leaves”)

𝑓 ( 𝑋 )=𝛼 𝑋 (1−𝑋 )3𝑓 ( 𝑋 )=𝛼 𝑋 (1− 𝑋 3 )

relevant in biology, ecology, chemistry, physics, economics…

the dynamics of convection are reproduced if is heat

(Libchaber, 1979)

In the summit of chaos

all appears to wander in a compact strange

attractor when :

In the summit of chaos

however, unstable extensions of the tree need to be excluded:

all appears to wander in a compact strange

attractor when :

Oscillations on the summit

pre-images of eventual repetitions every three generations

are not part of the attractor:

Oscillations on the summit

pre-images of eventual repetitions every three generations

are not part of the attractor:

similar trees for every period need to be excluded …

Oscillations on the summit

pre-images of eventual repetitions every three generations

are not part of the attractor:

similar trees for every period need to be excluded …

the strange attractor is therefore dust

An escape to the origin

there are also few points that improbably return to the root

pre-images of zero rest despite high heat:

An escape to the origin

there are also few points that improbably return to the root

pre-images of zero rest despite high heat:

points associated with the middle paint an inverted

square root!

An escape to the origin

there are also few points that improbably return to the root

pre-images of zero rest despite high heat:

points associated with the middle paint an inverted

square root!

the ideas remind us of the concept of purgatory

Plain common sense

it is much better not to miss the point!

in order to avoid chaos and its infernal turbulence:

Plain common sense

it is much better not to miss the point!

in order to avoid chaos and its infernal turbulence:

come down the tree lowering your heat to find the root

Plain common sense

it is much better not to miss the point!

in order to avoid chaos and its infernal turbulence:

come down the tree lowering your heat to find the root

abandon yourself to the threshold, , to reach the Origin

Plain common sense

it is much better not to miss the point!

in order to avoid chaos and its infernal turbulence:

come down the tree lowering your heat to find the root

abandon yourself to the threshold, , to reach the Origin

move away from nonlinearities () to escape thorns and

dust

A scientific prophetic tree?

A scientific prophetic tree?

an allegorical fig tree without fruit, consistently cursed

A scientific prophetic tree?

an allegorical fig tree without fruit, consistently cursed

this tree and other trees have tender branches and

many buds

A scientific prophetic tree?

an allegorical fig tree without fruit, consistently cursed

this tree and other trees have tender branches and

many buds

the ax is at the root of the trees, Nathanael is at the

root

we may curse the fig tree as He rebuked the wind

Our options

order

simplicity

serenity

peace

disorder

complexity

turbulence

chaos

Our options

order

simplicity

serenity

peace

decreasing

God’s way

below Y = X

abandonment

disorder

complexity

turbulence

chaos

increasing

our way

above Y = X

selfishness

Our options

order

simplicity

serenity

peace

decreasing

God’s way

below Y = X

abandonment

obedience

blessing

resting

Heaven

disorder

complexity

turbulence

chaos

increasing

our way

above Y = X

selfishness

rebelliousness

curse

wandering

hell

Feigenbaum’s parabel

Foliage of disordertrapped in empty

dust,jumps astir forever

enduring subtle thrust.

Crossing of the outset

leaving faithful rootlooming tender

offsetfailing to yield fruit

Cascade of bifurcations,

increasing heat within,

inescapable succession

of branches bent by wind.

Sprouting of dynamics

attracted to the strange,

o infinity remindingat the origin: the

flame.

Could it be, oh my friends

that science offers a rhyme?

For a rotten tree foretells

the very advent of time.

Could it be, oh how plain,

that nature extends a call?

For old parable proclaims

the crux in growing small.

In the midst of chaos

there is a small gate

leading to fine rest.

In the midst of chaos

there are loyal paths

inviting to a dance.

On top of the fig tree

there is a key pointthat runs to the

core.

On top of the fig tree

there is a clear lightthat averts a fright.

In the midst of chaos

there is leaping game

discerning the way.

In the midst of chaos

there is a fine wellwatering the brain.

On top of the fig tree

there is a clean frame

that cancels the blame.

On top of the fig tree

there is mighty helpthat shelters from

hell.

Other coincidences

the four states on the parable of the sower: fruit in the rootY = X is “our shelter from the wind”

Other coincidences

the four states on the parable of the sower: fruit in the rootY = X is “our shelter from the wind”

“abandon yourself, pick up your + and follow me”

“whoever loses his life for my sake will save it”

“if a grain of wheat dies, it produces much fruit”

Other coincidences

the four states on the parable of the sower: fruit in the rootY = X is “our shelter from the wind”

“abandon yourself, pick up your + and follow me”

“whoever loses his life for my sake will save it”

“if a grain of wheat dies, it produces much fruit”

in the summit of chaos: wheat vs. chaff, sheep vs. goats

“though 1,000 fall at your side, 10,000 at your right…”

“can you bury the haughty in the dust together?”

Other coincidences

the four states on the parable of the sower: fruit in the rootY = X is “our shelter from the wind”

“abandon yourself, pick up your + and follow me”

“whoever loses his life for my sake will save it”

“if a grain of wheat dies, it produces much fruit”

in the summit of chaos: wheat vs. chaff, sheep vs. goats

“though 1,000 fall at your side, 10,000 at your right…”

“can you bury the haughty in the dust together?”

“people do not pick figs from thornbushes”

the power of sainthood is visualized geometrically:

the escape point from chaos is the Ω point

Le plus improbable

In the most hellish beat

a silly and vain defeat

also a spectacle disjoint

an epsilon from the point

chaudement guérissable

oh purgation ineffable

oh victoire admirableoh miracle vénérable

In the symbolic tree

thorns, dust in a spree

but in humble zero set

line cancels all debt

In the unbent choice

always a clear voice

an appointed up lift

deterministic a gift

In the highest ugly heat

almost never a repeat

sad travel with no rest

ever close to the best

sûrement misérableoh détour incroyabledynamique indésirable

étrange état exécrable

également déplorablepériodicité

interminableobstination réprochableoh hasard inévitable

In the real driving seat

surely a fantastic feat

a way to erected root

faithful relief and fruit

égoïsme détectableoh fractal pitoyableoh origine inégalableoh paradis inalterable

oh l'amour véritableoh poème formidablemiséricorde verifiableoh le plus

improbable

Voice by Sandrine Matiasek

Animations by Mahesh L. Maskey