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~ 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: 

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 
S^{0}  S^{1}  S^{3} 
^{(3)} Espace projectif KP^{1} 
S^{1}  S^{2}  S^{4} 
^{(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 
Alt_{4} 12 
Sym_{4} 24 
Alt_{5} 60 
^{(8)} Groupes binaires dans SU(2,C) Cardinal Chambres Dim. des représentations irréductibles Somme 
A_{3} 24 2(1+3+3+5) (1,1,1,2,2,2,3) 12 
B_{3} 48 2(1+5+7+11) (1,1,2,2,3,3,4,2) 18 
H_{3} 120 2(1+11+19+29) (1,2,3,4,5,6,2,3,4) 30 
^{(9)} Posets des racines positives 
A_{3} 
B_{3} 
H_{3} 
^{(10)} Polytopes dans R^{4} Nombre de faces Homologie IP 
24cellule (24,96,96,24,1) (1,19,10) 
96cellule (48,240,288,96,1) (1,43,?) 
600cellule (120,720,1200,600,1) (1,115,250) 
^{(11)} Polytopes duaux Nombre de faces Homologie IP 
24cellule (24,96,96,24,1) (1,19,10) 
48cellule (96,288,240,48,1) (1,91,?) 
120cellule (600,1200,720,120,1) (1,595,250) 
^{(12)} Groupes de Coxeter Nombre de Coxeter 
A_{3} 4 D_{4} 6 G_{2} 6 D_{4} 6 E_{6} 12 Diagramme E6 Poset E6 
B_{3} 6 F_{4} 12 F_{4} 12 E_{6} 12 E_{7} 18 Diagramme E7 Poset E7 
H_{3} 10 H_{4} 30 E_{8} 30 E_{8} 30 E_{8} 30 Diagramme E8 Poset E8 
^{(13)} Groupes de réflexions Degrés des invariants Groupe de symétrie 
D_{4} (2,4,4,6) Sym_{3} 
F_{4} (2,6,8,12) Sym_{2} 
H_{4} (2,12,20,30) Sym_{1} 
^{(14)} Associaèdres généralisés Groupe de symétrie Exposants Grassmanniennes 
D_{4} Sym_{3} 1,3,3,5 Gr(3,6) 
E_{6} Sym_{2} 1,4,5,7,8,11 Gr(3,7) 
E_{8} Sym_{1} 1,7,11,13,17,19,23,29 Gr(3,8) 
^{(15)} Diagramme de Dynkin affine Coefficients de ? ? Groupe de symétrie 
Ê_{6} (1,1,1,2,2,2,3) Sym_{3} 
Ê_{7} (1,1,2,2,3,3,4,2) Sym_{2} 
Ê_{8} (1,2,3,4,5,6,2,3,4) Sym_{1} 
^{(16)} Triangles euclidiens Groupe de symétrie 
(p/3,p/3,p/3) A_{2} affine Sym_{3} 
(p/2,p/4,p/4) C_{2} affine Sym_{2} 
(p/2,p/3,p/6) G_{2} affine Sym_{1} 
^{(17)} Singularités Nombre de Milnor 
x^{3}+y^{3}+z^{3} 8 
x^{2}+y^{4}+z^{4} 9 
x^{2}+y^{3}+z^{6} 10 
^{(18)} Groupes de réflexions Dim. des représentations irréductibles Somme 
G_{2} (2,1,3) 6 
F_{4} (2,4,3,2,1) 12 
E_{8} (1,2,3,4,5,6,2,3,4) 30 
^{(19)} Groupes de réflexions complexes Cardinal Degrés des invariants Dim. des représentations irréductibles 
A_{2}^{(3)}=G_{4} 24 (4,6) (1,1,1,2,2,2,3) 
A_{2}^{(4)}=G_{8} 96 (8,12) 2x(1,1,2,2,3,3,4,2) 
A_{2}^{(5)}=G_{16} 600 (20,30) 5x(1,2,3,4,5,6,2,3,4) 
^{(20)} Groupes sporadiques 
Groupe de Fischer F_{24}  Bébé Monstre B  Monstre M 
^{(21)} Groupes de réflexions complexes 
G_{4} G_{5} G_{7} G_{6} 
G_{8} G_{10} G_{11} G_{9} 
G_{16} G_{18} G_{19} G_{17} 
^{(22)} Systèmes de racines elliptiques Poids & nombre de Coxeter (Saito) Écriture comme produit 
E_{6}^{(1,1)} 1,1,1 & 3 A_{2}×D_{4} 
E_{7}^{(1,1)} 1,1,2 & 4 A_{3}×A_{3} 
E_{8}^{(1,1)} 1,2,3 & 6 A_{2}×A_{5} 
^{(23)} Groupes de Coxeter hyperboliques Forme du diagramme 
E_{6}^{h} (3,3,4) 
E_{7}^{h} (2,4,5) 
E_{8}^{h} (2,3,7) 
^{(24)} Repliements non standards Nombre de Coxeter 
A_{4} ? H_{2} 5 
D_{6} ? H_{3} 10 
E_{8} ? H_{4} 30 
^{(25)} Singularités unimodales exceptionnelles Nombres de Gabrielov et Dolgachev Fonction Décomposition 
U_{12}
4,4,4 x^{3}+y^{3}+z^{4} A_{3}×D_{4} 1,3,4,4 & 12 
W_{12}
2,5,5 x^{2}+y^{4}+z^{5} A_{3}×A_{4} 1,4,5,10 & 20 
E_{12}
2,3,7 x^{2}+y^{3}+z^{7} A_{2}×A_{6} 1,6,14,21 & 42 
^{(26)} Singularités unimodales exceptionnelles Nombres de Gabrielov et Dolgachev 
E_{14} & Q_{10} 3,3,4 & 2,3,9 
E_{13} & Z_{11} 2,4,5 & 2,3,8 
E_{12} & E_{12} 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 _{2}F_{1} 
(3n)! /
(n)!(n)!(n)! _{2}F_{1}(1/3,2/3;1;27 x) 
(4n)! /
(2n)!(n)!(n)! _{2}F_{1}(1/4,3/4;1;64 x) 
(6n)! /
(3n)!(2n)!(n)! _{2}F_{1}(1/6,5/6;1;432 x) 
^{(31)} Nombre de parties exceptionnelles et nombre de modules basculants 
E_{6} : 5844 418=11×38 
E_{7} : 61866 2431=17×143 
E_{8} : 808005 17342=29×598 
^{(32)} Posets de type affineminuscule fvecteur Antichaines ordre fvecteur exposants 
E_{6} :
20 (20,30,12,1)=B_{3} 66 12 (66,120,65,10) (1,5,7,11)=F_{4} 
E_{7} :
32 (32,48,18,1)=H_{3} 119 18 (119,224,126,20) (1,7,11,17) 
E_{8} :
56 (56,84,30,1) 232 30 (232,448,259,42) (1,11,19,29)=H_{4} 
^{(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
dans Mathematics: Frontiers and Perspectives.
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:

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 otherworldly 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 socalled 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 piecemeal 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 warmup 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, History of Mathematics, originally published in 1912, with a facsimile edition in 2001. ISBN # 1400700539. 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 November 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 determined his whole future...
Friday, March 14, 2014
Updated and Reposted: Saturday, January 17, 2015... 7:46 AM
Reupdated: Saturday, September 8th, 2018... 2:10 PM
Herb O. Buckland
herbobuckland@hotmail.com