Threesology Research Journal
The Remarkable Trinities

(The Study of Threes)
http://threesology.org


Vladimir Igorevich Arnold (alternative spelling Arnol'd) 12 June 1937 – 3 June 2010). He was a Soviet and Russian mathematician. While he is best known for the Kolmogorov–Arnold–Moser theorem regarding the stability of integrable systems, he made important contributions in several areas including dynamical systems theory, catastrophe theory, topology, algebraic geometry, classical mechanics and singularity theory, including posing the ADE classification problem, since his first main result—the partial solution of Hilbert's thirteenth problem in 1957 at the age of 19.


Information and picture source:


Wikipedia: Vladimir Arnold
VArnold (11K)
Les trinités remarquables


Thu, 6 Mar 2014 15:01:12 -0700

Cette page est un recueil de ce que Vladimir Arnol'd appelle des trinités mystérieuses en mathématiques.




(1)
Corps K
R C H
(2)
Sphère unité de K
S0 S1 S3
(3)
Espace projectif KP1
S1 S2 S4
(4)
Classes caractéristiques
Whitney Chern Pontrjagin
(5)
Polytopes réguliers simpliciaux,
Nombre de faces
Homologie IP
Tetraèdre
(4,6,4,1)
(1,0)
Octaèdre
(6,12,8,1)
(1,2)
Icosaèdre
(12,30,20,1)
(1,8)
(6)
Polytopes réguliers simples,
Nombre de faces
Homologie IP
Tetraèdre
(4,6,4,1)
(1,0)
Cube
(8,12,6,1)
(1,4)
Dodécaèdre
(20,30,12,1)
(1,16)
(7)
Groupes de symétrie dans SO(3,R)
Cardinal
Alt4
12
Sym4
24
Alt5
60
(8)
Groupes binaires dans SU(2,C)
Cardinal
Chambres
Dim. des représentations irréductibles
Somme
A3
24
2(1+3+3+5)
(1,1,1,2,2,2,3)
12
B3
48
2(1+5+7+11)
(1,1,2,2,3,3,4,2)
18
H3
120
2(1+11+19+29)
(1,2,3,4,5,6,2,3,4)
30
(9)
Posets des racines positives
A3
Racines pour A3
B3
Racines pour B3
H3
Racines pour H3
(10)
Polytopes dans R4
Nombre de faces
Homologie IP
24-cellule
(24,96,96,24,1)
(1,19,10)
96-cellule
(48,240,288,96,1)
(1,43,?)
600-cellule
(120,720,1200,600,1)
(1,115,250)
(11)
Polytopes duaux
Nombre de faces
Homologie IP
24-cellule
(24,96,96,24,1)
(1,19,10)
48-cellule
(96,288,240,48,1)
(1,91,?)
120-cellule
(600,1200,720,120,1)
(1,595,250)
(12)
Groupes de Coxeter
Nombre de Coxeter
A3 4
D4 6
G2 6
D4 6
E6 12
Diagramme E6
Poset E6
B3 6
F4 12
F4 12
E6 12
E7 18
Diagramme E7
Poset E7
H3 10
H4 30
E8 30
E8 30
E8 30
Diagramme E8
Poset E8
(13)
Groupes de réflexions
Degrés des invariants
Groupe de symétrie
D4
(2,4,4,6)
Sym3
F4
(2,6,8,12)
Sym2
H4
(2,12,20,30)
Sym1
(14)
Associaèdres généralisés
Groupe de symétrie
Exposants
Grassmanniennes
D4
Sym3
1,3,3,5
Gr(3,6)
E6
Sym2
1,4,5,7,8,11
Gr(3,7)
E8
Sym1
1,7,11,13,17,19,23,29
Gr(3,8)
(15)
Diagramme de Dynkin affine
Coefficients de ? ?
Groupe de symétrie
Ê6
(1,1,1,2,2,2,3)
Sym3
Ê7
(1,1,2,2,3,3,4,2)
Sym2
Ê8
(1,2,3,4,5,6,2,3,4)
Sym1
(16)
Triangles euclidiens
Groupe de symétrie
(p/3,p/3,p/3)
A2 affine
Sym3
triangle A2
(p/2,p/4,p/4)
C2 affine
Sym2
triangle B2
(p/2,p/3,p/6)
G2 affine
Sym1
triangle G2
(17)
Singularités
Nombre de Milnor
x3+y3+z3
8
x2+y4+z4
9
x2+y3+z6
10
(18)
Groupes de réflexions
Dim. des représentations irréductibles
Somme
G2
(2,1,3)
6
F4
(2,4,3,2,1)
12
E8
(1,2,3,4,5,6,2,3,4)
30
(19)
Groupes de réflexions complexes
Cardinal
Degrés des invariants
Dim. des représentations irréductibles
A2(3)=G4
24
(4,6)
(1,1,1,2,2,2,3)
A2(4)=G8
96
(8,12)
2x(1,1,2,2,3,3,4,2)
A2(5)=G16
600
(20,30)
5x(1,2,3,4,5,6,2,3,4)
(20)
Groupes sporadiques
Groupe de Fischer F24 Bébé Monstre B Monstre M
(21)
Groupes de réflexions complexes
G4
G5
G7
G6
G8
G10
G11
G9
G16
G18
G19
G17
(22)
Systèmes de racines elliptiques
Poids & nombre de Coxeter (Saito)
Écriture comme produit
E6(1,1)
1,1,1 & 3
A2×D4
E7(1,1)
1,1,2 & 4
A3×A3
E8(1,1)
1,2,3 & 6
A2×A5
(23)
Groupes de Coxeter hyperboliques
Forme du diagramme
E6h
(3,3,4)
E7h
(2,4,5)
E8h
(2,3,7)
(24)
Repliements non standards
Nombre de Coxeter
A4 ? H2
5
pliage de A4 vers H2
D6 ? H3
10
pliage de D6 vers H3
E8 ? H4
30
pliage de E8 vers H4
(25)
Singularités unimodales exceptionnelles
Nombres de Gabrielov et Dolgachev
Fonction
Décomposition
U12
4,4,4
x3+y3+z4
A3×D4
1,3,4,4 & 12
W12
2,5,5
x2+y4+z5
A3×A4
1,4,5,10 & 20
E12
2,3,7
x2+y3+z7
A2×A6
1,6,14,21 & 42
(26)
Singularités unimodales exceptionnelles
Nombres de Gabrielov et Dolgachev
E14 & Q10
3,3,4 & 2,3,9
E13 & Z11
2,4,5 & 2,3,8
E12 & E12
2,3,7 & 2,3,7
(27)
Espaces projectifs à poids
Miroir des singularités elliptiques
(3,3,3) (2,4,4) (2,3,6)
(28)
Suites d'entiers
Séries génératrices algébriques
(12n)!n! / (6n)!(4n)!(3n)! (18n)!n! / (9n)!(6n)!(4n)! (30n)!n! / (15n)!(10n)!(6n)!
(29)
Exposants particuliers
(1,5,7,11) (1,5,7,11,13,17) (1,7,11,13,17,19,23,29)
(30)
Suites d'entiers
Coefficients multinomiaux
Fonction hypergéometrique 2F1
(3n)! / (n)!(n)!(n)!
2F1(1/3,2/3;1;27 x)
(4n)! / (2n)!(n)!(n)!
2F1(1/4,3/4;1;64 x)
(6n)! / (3n)!(2n)!(n)!
2F1(1/6,5/6;1;432 x)
(31)
Nombre de parties exceptionnelles
et nombre de modules basculants
E6 : 5844
418=11×38
E7 : 61866
2431=17×143
E8 : 808005
17342=29×598
(32)
Posets de type affine-minuscule
f-vecteur
Antichaines
ordre
f-vecteur
exposants
E6 : 20
(20,30,12,1)=B3
66
12
(66,120,65,10)
(1,5,7,11)=F4
E7 : 32
(32,48,18,1)=H3
119
18
(119,224,126,20)
(1,7,11,17)
E8 : 56
(56,84,30,1)
232
30
(232,448,259,42)
(1,11,19,29)=H4
(33)
Triangles (surfaces de translation)
Kenyon & Smillie, Puchta
(3 p/12, 4 p/12, 5 p/12) (4 p/18, 6 p/18, 8 p/18) (6 p/30, 10 p/30, 14 p/30)

Référence: Vladimir Arnol'd


Polymathematics : is mathematics a single science or a set of arts?

dans Mathematics: Frontiers and Perspectives.


Vladimir Arnold's Remarkable Trinities



Here's another trinity from Arnold, secured from an abstract entitled Mathematical Kinds, or Being Kind to Mathematics written by:
David Corfield
Dept. of Philosophy
University of York
dc23@york.ac.uk


All mathematics is divided into three parts:


  1. Cryptography (paid for by CIA, KGB and the like).
  2. Hydrodynamics (supported by manufacturers of atomic submarines).
  3. Celestial mechanics (financed by military and by other institutions dealing with missiles, such as NASA).

H.O.B. note: my inclination is to perceive all of mathematics as an art; with its varieties of formulaic expression like the many forms of artistic expression encountered in various genres such as theatre, dance, musical composition, sculpting (ice, stone, wood, etc.), etc., etc.. This does not in any way imply that a mathematically portrayed perspective is tantamount to a greater clarity nor value of truth, whereas in fact a math equation can very often confuse perceptions more attuned to visualizing reality in a different way. Most people do not see nor portray perceptions within the narrowness that enumeration can sometimes convey with a demeanor very often perceived by most people as an alien visitor from some other-worldly domain speaking a language of gibberish. While some arrogantly claim this is because most people are mathematically disinclined (in other words, stupid), they overlook the descriptive word "language".


Mathematics may be similar to so-called normal language development in that there is a critical period for developing a "full spectrum" usage of mathematics as a second language. Many a student verbalizes ("talks through") efforts directed towards solving a math problem, whether or not the verbalisation is a silent or audible level. If anyone can indeed acquire the usage of two or more languages given the right environment, the lack of learning "higher" mathematics by most people may indicate public instruction of mathematics needs to be sorely revamped. While some people do indeed learn another language after the "critical development period", perhaps due to some retention of a plasticity in their brain physiology, most people don't. Most mathematics "language" environments are not total language developing environments, they are piece-meal varieties. In terms of language, piece- meal varieties of exposure may lead a person to acquire the usage and/or understanding of a few words or phrases, but there is no reading, writing nor speaking fluency.


With respect to Vladimir Arnol'd's question "Is mathematics a single science or a set of arts?", denoting mathematics as a science can obscure the perception that it can be compared with aligning a given artistic expression with a determined set of form and function, such as musical notation, using the same colors, the same sculpting tools, the same warm-up routine to practice the same dance steps, etc. In short, it is a repetition of thought which may represent a cognitive limit. Expressing oneself outside the lines of a defined form and function is the act of adding a variable to a standardized formula which creates another genre of expression.


While some art form types (for example, landscape painting) are routinely titled the same, individual characteristics of the artist (or mathematician) involve personalized techniques, but has more to do with perspective and not the landscape itself. While some artists attempt to render a composition with an intended specificity of truth value, others think the value is subjective and thus permits them a usage of color, medium and subject matter as they sit fit. While some view mathematics as dry, colorless and uninteresting beyond a functional application, others view it as a myriad ensemble of beautiful pictures. If we do not recognize something as an art nor a science, nor give it a new name denoting a "combinatorial" thereof, we should thus look elsewhere for an explanation.


For example, both art and science, regardless of which subject is brought to mind, may be little more than expressions of our present day inability to apprehend a greater realization beyond that which anyone of us may attempt to describe or define. In other words, they are like the dam of a beaver, the mound of a termite colony, the nest of a bird, the rummaging of rodents, the chasing of prey, the graffiti on cave walls, migrations of birds and butterflies, etc... While humanity is patting itself on the back for all its claimed assumptions of being highly gifted, innovative and intelligent, such back patting may reflect just another form of primate grooming... like picking bugs out of fur in order to snack on. Bon appétit!




From page 270, Hitory of Mathematics, originally published in 1912, with a facsimile edition in 2001. ISBN # 1-4007-0053-9. Author W.W. Rouse Bell.


...He (Descartes) was accustomed to date the first ideas of his new philosophy and of his analytical geometry from three dreams which he experienced on the night of Novermber 10, 1619, at Neuberg, when campaigning (as a soldier) on the Danube, and he regarded this as the critical day of his life, and one which dtermined his whole future...




Friday, March 14, 2014
Updated and Reposted: Saturday, January 17, 2015 7:46 AM


Your Questions, Comments or Additional Information are welcomed:
Herb O. Buckland
herbobuckland@hotmail.com