Faith Lessons from Chaotic Fig Trees - Carlos E. Puente - UC Davis ...

THE KEY ROLE OF SCIENCE IN THE NEW EVANGELIZATION
2. Learn an urgent lesson
from a chaotic fig tree!
Carlos E. Puente
University of California, Davis
Ateneo Pontificio Regina Apostolorum
Rome, 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
X k+1 = αX k 1 − X k , α ∈ 0,4
... a population of “rabbits” as a function of time

e logistic map
X k+1 = αX k 1 − X k , α ∈ 0,4
... a population of “rabbits” as a function of time

e logistic map
X k+1 = αX k 1 − X k , α ∈ 0,4
... a population of “rabbits” as a function of time
α = 2.8

e logistic map
X k+1 = αX k 1 − X k , α ∈ 0,4
... a population of “rabbits” as a function of time
α = 2.8
different cases depending on the parameter α:

From the origin to bifurcations
Y = X
0 < α ≤ 1
X ∞ = 0

From the origin to bifurcations
Y = X
0 < α ≤ 1 1 < α ≤ 3
X ∞ = 0
X ∞ = α−1
α

From the origin to bifurcations
0 < α ≤ 1
X ∞ = 0
1 < α ≤ 3
X ∞ = α−1
α
Y = X
α = 3.2
period 2

From the origin to bifurcations
0 < α ≤ 1
X ∞ = 0
1 < α ≤ 3
X ∞ = α−1
α
Y = X
α = 3.2
period 2
α = 3.46
period 4

From the origin to bifurcations
0 < α ≤ 1
X ∞ = 0
1 < α ≤ 3
X ∞ = α−1
α
Y = X
α = 3.2
period 2
α = 3.46
period 4
… α ∞ ≈ 3.5699
period 2 n

Periodicities and chaos intertwined
(α > α ∞ ≈ 3.5699)
Y = X
α = 3.6
aperiodic
α = 4
aperiodic

Periodicities and chaos intertwined
(α > α ∞ ≈ 3.5699)
Y = X
α = 3.6
aperiodic
α = 3.74
period 5
α = 3.83
period 3
α = 4
aperiodic

e diagram of bifurcations
the tree contains periodic behavior, for any natural number

Dust galore and buds
after α ∞ the dusty strange attractors are common
in the white bands there are the “buds” of the tree

Intermediate bud of period 3
and similarly for all intermediate buds of period 9 and all others
… exquisite self-similarity ad infinitum

Thorns galore within the chaotic tree
the chaotic tree contains many multifractal thorns
they combine imbalances and holes, as in α ∞ :
they happen at ends of white bands of buds for all periods

Universality on the fig tree
(Feigenbaum, 1978)
bifurcations happen in an orderly fashion:
d n
⁄ d n+1 ⟶ F 1 = −2.50 …
openings
Δ n
⁄ Δ n+1 ⟶ F 2 = 4.669 …
durations

Universality on other chaotic trees
straight root, “tender branch,” branches, and dust (“fig leaves”)
f X = αα 1 − X 3
f X = αα 1 − X 3
relevant in biology, ecology, chemistry, physics, economics…
the dynamics of convection are reproduced if α is heat (Libchaber, 1979)

In the summit of chaos
all appears to wander in a compact strange attractor when α = 4:

In the summit of chaos
all appears to wander in a compact strange attractor when α = 4:
however, unstable extensions of the tree need to be excluded:

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 other points need to be excluded …
and as such is for every period, the 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:
points associated with the middle do paint an inverted square 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, Y = X, to reach the Origin
move away from nonlinearities (α > 1) to escape dust and thorns

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
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 disorder
trapped in empty dust,
jumps astir forever
enduring subtle thrust.
Crossing of the outset
leaving faithful root
looming tender offset
failing 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 reminding
at 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 point
that runs to the core.
On top of the fig tree
there is a clear light
that averts a fright.
In the midst of chaos
there is leaping game
discerning the way.
In the midst of chaos
there is a fine well
watering 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 help
that shelters from hell.

Other coincidences
the four states on the parable of the sower: fruit in the root
Y = 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 root
Y = 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 root
Y = 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
In the highest ugly heat
almost never a repeat
sad travel with no rest
ever close to the best
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 real driving seat
surely a fantastic feat
a way to erected root
faithful relief and fruit
également déplorable
périodicité interminable
obstination réprochable
oh hasard inévitable
sûrement misérable
oh détour incroyable
dynamique indésirable
étrange état exécrable
égoïsme détectable
oh fractal pitoyable
oh origine inégalable
oh paradis inalterable
oh l'amour véritable
oh poème formidable
miséricorde verifiable
oh le plus improbable
chaudement guérissable
oh purgation ineffable
oh victoire admirable
oh miracle vénérable