MORFOGENEZA
1. Introducere
PARTEA 1-a
Principii geometrice de umplere a spatiului
12/17/2023
Similarity
• Similarity is one of the "big ideas" in geometry. Note that two
things may be similar in colloquial English, but it is a much
stronger statement to say that they are mathematically
similar.
Here is a picture of a panther:
Which of these pictures is a good copy
of the original picture?
Explain how you made your decision.
If two triangles have all
three pairs of sides in
proportion, the triangles
must be similar.
Scaling Effects
Scaling of a 3D Object:
2
AL
Surface area
L
Volume
V L
Mass (weight)
m  V  L

Moment of inertia
I  mR  L
Torque (weight
acting
on a moment arm)
  R  mg  L
3

3
2
5
4
Mechanical Resonance
Cantilever beam

1
f0 
2
K
m
K  L1 ,
m  L3
1
f0  L
Scaling Effects
Decrease size of a bridge
Structure is designed to
carry load. Weight of the
Stress scales
objects passing the bridge
scales L3. Cross section of
3
bridge scales L2.
Stress scales L3/L2=L or
2
M/M2/3=M1/3
Ten times smaller bridge just
suffers one tenth the stress
of the larger bridge
Thus bridge could be built in
a more slender version and
with less material costs.
This explains why elephants have in proportion much thicker bones than a human being
L
L
L

Scaling Effects
A  L2  m

2
3
The largest egg from a living bird
belongs to the ostrich. It is over 2000
times larger than the smallest
egg produced by a hummingbird.
Ostrich eggs are about 180 mm long
and 140 mm wide and weigh 1.2 kg.
Hummingbird eggs are
13 mm long and 8 mm wide
and they weigh only half of a gram.
Scaling Effects
J. Swift
Scaling Effects
• Lilliputs estimated 12 times their own height, and
calculated his mass and volume (123)=1728
times their own
• To be perfect hosts they fed him 1728 portions
of food and wine, BUT the amount of calories
necessary for an organism scales not with its
volume, or mass,
• but with its surface through which the heat is
lost, i.e. L2, V2/3 or M2/3
• Measurements gave M3/4 dependence
Corelatie functionala
Conditii de minim
Conditii care implica complexitatea
Conditii care implica evolutia
Conditii care implica conservarea sistemului (a entitatii)
Conditii de stationaritate
Intr-o dinamica variabila
Proprietati generale ale Sistemelor Dinamice:
NEDISIPATIVE :
• sistem inchis
• guvernat de forte conservative
• evolueaza reversibil
Sistem dinamic disipativ
sistem deschis
evolutie ireversibila
traiectoriile modeleaza perioada TRANZITORIE
la limita:
vor converge catre un ATRACTOR
sau
vor diverge la infinit fata de un REPULSOR
BAZIN = multimea starilor initiale
pornind de la care sistemul tinde
catre un atractor
SEPARATOARE = punctele ce nu
apartin bazinului
Concluzii:
-Motive care conduc la fenomene de morfogeneza:
-Primul motiv este conservarea si invarianta sistemului
-Motive geometrice – de simetrie , de acces liber,
–Motive functionale – umplerea spatiului, conditii de
energie minima, conditii de effort, conditii de
adaptare (evolutia), conditii de stres,
conditii de optim sau conditii de fiabilitate
-Motive fizico – chimice la nivel microscopic, sau chiar
molecular, forte electrice sau van der waals, simetrii
la nivel molecular,
-