# Concerning Pi, again

For Pi Day this year (2018), I will provide some results related to this interesting mathematical constant. These results are mostly inspired or taken from answers I gave at quora.com about  $\pi$ or about similar mathematical topics.

The millionth decimal digit of π is 1 (verified with Mathematica).

The 10 millionth decimal digit of π is found to be 7, and the 100 millionth decimal digit of π is 2.

The billionth decimal digit of $\pi$ (in base 10) is 9 (verified with Mathematica).

The 2 billionth decimal digit of π is found to be 0 (this result takes a longer time to compute with Mathematica).

Here are some (repeated) number sequences or numeric strings found among the first 2 billion decimal digits of π.

The numeric string 777777777 appears at the 24, 658, 601 st decimal digit of π :

9919408245759718530477777777724846769425931046864

The numeric string 111111111 appears at the 812, 432, 526 th decimal digit of π :

2891450444990691713511111111100399876875718824885

Here are also two numeric strings from 1 to 9 in increasing order and decreasing order:

The numeric string 123456789 appears at the 523, 551, 502 nd decimal digit of π:

7260489917323889207212345678922486448188070486710

The numeric string 987654321 appears at the 719, 473, 323 rd decimal digit of π :

5221398663241526793698765432160793011913242320510

The numeric strings above can be calculated or found with the help of the following link or web page:

Irrational Numbers Search Engine

The numerical value of $\pi^{\pi}$ to 1000 decimal digits is equal to:

36.46215960720791177099082602269212366636550840222881873870933592293407436888169990462007987570677485436814688343670070542736699139359264431565675267180230917777595737242260530320050233549595161382594571885422222305402433199779769167302876444780028452117394296018175249159350019492001619423210110480018557258718860782819839215304503453543238476218257664861595609057280314341958390400811991506636066295817900302292747422204210046403709493285441101884797707466358510710362803891181156618083260884536505255311095948029552909133361385823497120761861157606574436205295895657736468959837126404885207348833917602169536002174958035720670509318770633086434935593202425189496008880555048213388792769304015639745480898380866639428337794250284522113741878602793251848366623602214652151453276098545038540482041645516919082097210265379423765817354600472953993840487842116366153365057093066926223775915023204727672609958737278566633593210689698807508602353552267490321670730929373240451651980750157959708689483469068

Two expressions involving π and infinite sums:

Representation of π in continued fraction form:

The sum of π and e, the base of natural logarithms, is equal to:

$\pi + e=\displaystyle \sum _{k=0}^{\infty } \frac{(3 k)^2+1}{(3 k)!}+2 i \ln \left(\frac{1-i}{1+i}\right)i$

The letter i  represents the imaginary unit of complex numbers.

Another expression involving π, e, and an infinite product:

$\pi = \displaystyle 2 e \prod _{k=1}^{\infty } \left(\frac{2}{k}+1\right)^{(-1)^{k+1} k}$

And here is an identity relating the Golden Ratio, π, e, and the imaginary unit i:

$\displaystyle \varphi=e^{i\pi/5}+e^{-i\pi/5}=\frac{1+\sqrt{5}}{2}$

The value of π can be deduced from the identity above:

$\displaystyle\pi =5 i \ln \left(\frac{1}{2} \left(\varphi -\sqrt{\varphi ^2-4}\right)\right)$

# Concerning the relationship between science and philosophy

This post consists of the  elements of an answer I wrote at quora.com ; the question there was: “Is philosophy the top of all kinds of sciences?”

I think it would be convenient to distinguish between the general term “science”, referring to the state or fact of knowing, or to knowledge acquired by study and learning, and the modern meaning of “science”, mostly referring to mathematics and to the exact sciences using the rules of the scientific method (astronomy, physics…).

Philosophy and science were not separate in Antiquity.

In the original sense, philosophy meant the love, study, or pursuit of wisdom, or the knowledge of things and their causes, theoretical as well as practical.

Pythagoras was a mathematician, and at the same time it is said that he was the first one to call himself a philosopher, or “lover of wisdom”.

Plato was a philosopher who recommended the knowledge and the study of geometry. In The Republic, Plato thought that the best ruler was the king-philosopher.

Aristotle studied nature and wrote works about physics, biology, logic, etc, from a philosophical point of view.

According to the OED:

“In the Middle Ages, ‘the seven (liberal) sciences’ was often used synonymously with ‘the seven liberal arts’, for the   group of studies comprised by the Trivium (Grammar, Logic, Rhetoric) and the Quadrivium (Arithmetic, Music, Geometry, Astronomy).”

The expression Natural philosophy was frequently used for centuries :

“Natural philosophy or philosophy of nature (from Latin philosophia naturalis) was the philosophical study of nature and the physical universe that was dominant before the development of modern science. It is considered to be the precursor of natural science.

From the ancient world, starting with Aristotle, to the 19th century, the term “natural philosophy” was the common term used to describe the practice of studying nature. It was in the 19th century that the concept of “science” received its modern shape with new titles emerging such as “biology” and “biologist”, “physics” and “physicist” among other technical fields and titles; institutions and communities were founded, and unprecedented applications to and interactions with other aspects of society and culture occurred. Isaac Newton‘s book Philosophiae Naturalis Principia Mathematica (1687), whose title translates to “Mathematical Principles of Natural Philosophy”, reflects the then-current use of the words “natural philosophy”, akin to “systematic study of nature”. Even in the 19th century, a treatise by Lord Kelvin and Peter Guthrie Tait, which helped define much of modern physics, was titled Treatise on Natural Philosophy (1867).

In the last few centuries, alchemy separated from chemistry, astrology separated from astronomy, and there was also a certain separation between philosophy on one side, and mathematics and the exact sciences on the other side.

Mathematics became progressively the most prominent and the essential scientific discipline, it is acknowledged as the language of science and of the physical world.

Philosophy is nowadays often regarded as a reflection, view or study of the general principles of a particular branch of knowledge, or activity. There is a philosophy of science, philosophy of mathematics, philosophy of education ,etc.

Some theories or views related to epistemology (which is concerned with the general theory and the study of knowledge) and philosophy, such as rationalism, empiricism, and positivism, share a number of principles with the scientific approach to events and phenomena.

(Source of the image above: Wikimedia Commons)

A scientist or a physicist can also be a philosopher. Important thinkers can be philosophers and create philosophical systems, but modern philosophers must take into account the advances, discoveries and theories in modern science. A historical example would be Immanuel Kant elaborating his philosophical system and philosophical ideas at the end of the eighteenth century in light of and in relation to the exact sciences known at that time, especially Euclidean geometry and Newtonian classical physics and mechanics.

# About the importance or relevance of black swan events in History

Here is a slightly modified reproduction of an answer I wrote at quora.com concerning the black swan theory and black swan events in History.

The Black Swan theory or notion can be sometimes useful but it cannot be reliable or subject to generalization for History.

Explaining (or not being able to explain ) historical events by often relying on the concept of Black Swan can be counterproductive and prevents researchers and thinkers from analyzing objectively and impartially human history in order to find patterns , regularities , and reasonable interpretations in accordance with the principles of the scientific method.

I’ll give an example:
The British had a guy called Oliver Cromwell.Then the French had a guy called Napoleon Bonaparte . Then the Germans had a guy called Hitler. So one could discern a pattern here in the historical succession of great powers.

Another example:
There are many important scientists and scholars but a few have an importance or greatness of the biggest caliber.
The British had a very important scientist called Isaac Newton , then the French had a scientist called Pierre Simon Laplace , then the Germans had a scientist called Albert Einstein ( Einstein was born in Germany , then he lived in Switzerland , then he came back to Germany , then he went to the United States.But in his most productive years he was in the sphere of influence of the German language and of the German culture).

An interesting field for studying history is Cliodynamics, but I think the methods of cliodynamics can also be ameliorated and surpassed.

Certain events might be unexpected for a group of people , but they could be expected and/or predictable for another group of people.

A famous event such as the French Revolution was unexpected and could have been viewed as a Black Swan by the French nobles and aristocrats and by the supporters of the Ancien Régime , but it was expected , anticipated and brought about by the ‘bourgoisie’ , the peasants and the common people in France.The English Civil War or the English Revolution can be thought of as an antecedent to the French Revolution.
For the last two centuries the French Revolution and its causes have been explained , interpreted and reinterpreted in many ways by many people and thinkers.

Some events may seem surprising or unexpected , but the big picture and the general structure of historical events can be analyzed , found and interpreted.

As an additional example , I think there is no need to refer to black swans in relation to the 2016 presidential elections in the USA . Things and events to come can be viewed objectively by analyzing past historical events the right way , impartially and coherently , and by making the right and correct connections .

I also think specific patterns , regularities , and connections between important events can be found in human history for (definite) periods extending over centuries and even millennia.

# A poem I wrote years ago

I was fifteen- soon to be sixteen- years old ; I had been reading (important) books about science, physics, philosophy , and other similar topics,  and all those ideas in my head intermingled and inspired me to write a poem involving particle physics and particle collisions and combining elements of science and philosophy .

I wrote the poem in French , using the French alexandrine poetic meter of twelve syllables, but I didn’t follow the poetic rules very closely.

I will provide the final version of the poem here , with a line by line English translation. Different people have different tastes and opinions , I hope it will be liked .

The hydrogen-1 atom mentioned in the title of the poem is also called “protium” , but this last word is not much used in French. Protium is the most common hydrogen isotope, having one proton ( and one electron) and no neutrons.

A proton is supposed to be talking or telling the story in the poem . I think I was a little inspired by the poem ” Le Bateau ivre ” by Arthur Rimbaud .

Here it is :

Bombardement d’atomes par un proton d’hydrogène 1H
Bombardment of atoms by a proton of protium 1H

Synchrotrons , canons à électrons, cyclotrons
Synchrotrons, electron guns, cyclotrons

Soyez prêts, particules, deutons, neutrons, hélions
Be prepared, particles, deutons/deuterons, neutrons, helions

En attendant que les hommes préparent les canons
Until men prepare the guns

Le moment est arrivé, l’appareil frappe
The moment has come, the apparatus strikes

Dans son coeur vidé moi, le proton j’attrape
In its emptied heart I , the proton take

Le coup et je vais croiser les atomes en grappe
The blow and I go meet the atoms in clusters

Je fuis dans l’espace et le temps calculables
I flee in computable space and time

Ma vitesse est vertigineuse, incroyable
My speed is vertiginous, incredible

Non pas celle de la lumière, infranchissable
Not that of light, insurmountable

C’est le lieu de la relativité impie
It is the place of impious Relativity

Masses, longueurs, lois de la physique varient
Masses, lengths, physical laws vary

Ma trajectoire déterminée sera suivie
My particular/determined path will be followed

Par d’autres microcosmes malheureux
By other unfortunate microcosms

Le trajet est terminé, le choc a eu lieu
The journey is over, the shock/collision occurred

Je donne la vie à de nouveaux corps heureux
I give life to new happy/fortunate bodies

Quanta de matière utilisés pour la paix
Quanta of matter used for peace

Dans le monde de la science un pas est fait
In the world of science a step/discovery has been made

L’humanité en marche en connaît les bienfaits
Humanity in motion/advancing knows the benefits (of this discovery)

# Some notes about the possibility of a mathematical theory of History

I will present some general remarks and some personal opinions and findings (with a constant concern for accuracy and objectivity) about the attempts at mathematizing History , historical events , processes and phenomena.

My many readings (see my various book pages and the books I have read) and my analysis of History made me realize that important historical events and phenomena are (highly) periodic , and that exact correspondences (or similarities or “homologies” , one of these terms could be chosen, used  and defined) can be found between historical events separated by definite periods of time .

The history of humanity can be considered as the result of the interactions between the lives and actions of human beings  moving and acting in time. All humans have a role to play in the unfolding of historical events , but great men (and women) and great thinkers/scientists/reformers constitute the main group of humans who change and drive historical phenomena and happenings across cultures , nations and empires.

Evolution and progress take place in human history , such as technological/scientific progress, the increase in the global human population over millennia and the increase in the surface of political entities, from city-states to nation-states and to bigger entities , etc, but there are also general principles and definite periodicities and regularities in world history. Among these regularities are the stages or phases of gradual growth and decline through which most great powers and empires pass as they rise/fall and go up and down in time. Certain essential periodicities or cycles in history are accompanied by a change or transmutation in values (“moral” , behavioral , sexual, etc).
Relevant concepts can be defined such as the notion of bi-millenary (exact) correspondence and of bi-millenary periodical return of historical events. I will explain and clarify these concepts more when I have time in future writings.

Some philosophical and religious theories talk about eternal return/recurrence and cyclical time (the notion of eternal return constantly permeates the philosophy of Nietzsche), but these notions are not defined in a precise or scientific way.

I think that Mathematics and the scientific method , i.e observing phenomena and collecting data, creating hypotheses with the adequate mathematical model, experimental/empirical verification of these hypotheses and building  coherent theories, can be used in and applied to human history, provided that this is done in the proper and correct way. Historical chronology plays an important role, and the chronology of events before the Common or Christian era ought to be revised and corrected.

Another prerequisite for the impartial and objective study of history is to abandon preconceived ideas and to have a global perspective of human history , avoiding euro-centrism , afro-centrism and all kinds of ethnocentrisms , and avoiding to get stuck in certain habits such as classifying people and cultures into Western and non-Western. One should take into account the fact that geopolitical groupings and alliances change with the passing of time , centuries and decades.

A new discipline called Cliodynamics was created in the last decade . It is an area of research using mathematical , quantitative approaches and modelings to explain historical processes and societies. The practitioners of this discipline have made some interesting observations about historical events and have tried to formulate mathematically backed theories to interpret historical facts , however I think they have not found or discovered the right , convenient , correct and/or precise way to mathematize History and historical phenomena .

If the right mathematical theory of human history can be elaborated, using the scientific method and testing hypotheses in history could be equivalent to and lead to the (precise) prediction of important events taking place in the future of humankind.

# Books about physics, astrophysics and astronomy regarded as important classics

This post is mostly inspired (with some additions and modifications) from and answer I wrote at quora.com .

I will try to give a list of famous , influential books or classic books having a significant historical importance in the fields of physics , astrophysics and astronomy . It’s a somewhat extensive list but it’s not exhaustive.

Starting with Antiquity :

Then advancing to more recent times:

Below is a page from the Astronomia Nova (in 1609) showing the three models of planetary motion known in the seventeenth century (free image from Wikipedia) :

• Recherches sur la théorie des quanta (Researches on the quantum theory) , and The Current Interpretation of Wave Mechanics: A Critical Study , by Louis de Broglie .
• Collected papers , The interpretation of Quantum Mechanics , and Statistical Thermodynamics , by Erwin Schrödinger .
• The Physical Principles of the Quantum Theory , by Werner Heisenberg .
• Books and papers by Paul Dirac , such as and Lectures on Quantum Field Theory .
• Space, Time and Gravitation: An Outline of the General Relativity TheoryThe Internal Constitution of Stars , and The Nature of the Physical World , by Arthur Eddington .
• Problems of Cosmology and Stellar Dynamics , An Introduction to the Kinetic Theory of Gases , and  The Growth of Physical Science , by James Hopwood Jeans .
• The Theory of Sound , by John William Strutt, 3rd Baron Rayleigh .
• Problems of Atomic Dynamics , Atomic Physics , Principles of Optics , Experiment and Theory in Physics , and A General Kinetic Theory of Liquids , by Max Born .
• Books and papers by David Bohm , such as Quantum Theory , Causality and Chance in Modern Physics , The Undivided Universe.

Some more recent well known , insightful and/or widely used books would include :

• The Large Scale Structure of Space-Time , by Stephen Hawking and George F. R. Ellis .
• Speakable and Unspeakable in Quantum Mechanics , by John Stewart Bell .
• Classical-Mechanics , by Herbert Goldstein .
• Classical Electrodynamics , by J.D. Jackson .
• Galactic Dynamics , by Binney and Tremaine .
• The Quantum Theory of Motion: an account of the de Broglie-Bohm Causal Interpretation of Quantum Mechanics , by Peter Holland .
• Photons and Atoms: Introduction to Quantum Electrodynamics , by Claude Cohen-Tannoudji , Gilbert Grynberg and Jacques Dupont-Roc .
• Introduction to Elementary Particles , by D.J. Griffiths .
• Condensed Matter Field Theory , by Alexander Altland .
• The Standard Model and Beyond , by Paul Langacker .
• The Road to Reality , by Roger Penrose .
• Not Even Wrong: The Failure of String Theory and the Search for Unity in Physical Law , by Peter Woit .
• The Trouble With Physics: The Rise of String Theory, The Fall of a Science, and What Comes Next , by Lee smolin .
• Farewell to Reality: How Modern Physics Has Betrayed the Search for Scientific Truth , by Jim Baggott .

https://en.wikipedia.org/wiki/Hi…

https://en.wikipedia.org/wiki/Hi…

Astronomy in the medieval Islamic world

Indian astronomy

Chinese astronomy

# Regarding gravitational waves

Gravitational waves have become a popular topic recently , and this post presents material I have written for an answer at quora.com (with a few modifications).

Gravitational waves are changes in curvature similar to ripples in space-time . They are an indirect result of the special theory theory of relativity , and were explicitly proposed by Einstein in 1916 in the framework of his theory of general relativity . He showed that the acceleration of mass generates gravitational fields which are time-dependent and are capable of transporting energy (as gravitational radiation ) from their source at the speed of light .
Gravitational waves are quadrupolar in nature , whereas electromagnetic waves are dipolar . Oscillating multipole moments of the mass distribution of a system produce gravitational radiation .
Many attempts have been made to detect gravitational waves , but no direct evidence of their existence has been observed until their recent detection in February 2016 .

The Einstein field equations describe the interactions between space-time curvature and mass , as Maxwell’s equations describe and specify the relationship between electric charge and electromagnetic fields .
The field equations have a solution represented by a weak oscillating perturbation to the curvature of space-time , and this solution is a gravitational wave .
These waves can be regarded as an oscillating perturbation to a flat Minkowski space-time metric , or also as a tidal force oscillating between free test masses , or as a strain oscillating in space-time .
More explicitly , one can show that a wave equation represents the solutions in free space for the metric perturbations of a nearly flat space-time , with waves propagating at the speed of light ( this is a weak gravitational field approximation) .
One can take a coordinate system where the metric has components :
$g_ {\mu\nu} = \eta_ {\mu\nu} + h_ {\mu\nu}$
where
$\eta_ {\mu\nu}$  is the Minkowski metric in special relativity , and

After some calculations the  solution to Einstein’s equations in free space can be written as :

where

So the metric perturbations propagate in free space as waves at the speed of light .
A primary example of a source of gravitational waves is a pair of neutron stars , or two black holes , or one of each type of these astrophysical objects .
Observing supernova explosions or the orbital motion of binary pulsars may possibly give and indirect proof of the existence of gravitational waves .
The image below represents gravitational waves generated by two neutron stars orbiting each other (image source : File:Wavy.gif ) :

Ways of detecting gravitational waves include resonant mass detectors , free mass detectors , detectors in space , cosmic background measurements , and monitoring pulsar signals .
External disturbances and the effects of thermal noise in the detecting system should be avoided , the possible interaction between detectors and gravitational waves being very weak .
In 1974 Russell Hulse and Joseph Taylor discovered and observed the orbital period of a binary pulsar . They confirmed that the orbit was accelerating at the rate predicted by the emission of gravitational waves according to the theory of general relativity .
The LIGO (Laser Interferometer Gravitational-Wave Observatory) detectors
are used to attempt to observe directly cosmic gravitational waves . They can detect extremely small strains (of the order of  one part in 10²¹ ).
In the quantum theory of gravity , a quantum field whose excitations are gravitons represents the gravitational field .
Gravitons may be regarded as the normal modes of oscillation of a (gravitational) gauge field , produced by a mass current of accelerating masses .
Some (online) links and resources :
Gravitational wave
McGraw-Hill Encyclopedia of Science and Technology , 10th Edition .

To make this answer complete  , it should be noted that the expression gravity waves is also used to refer to waves studied in oceanography , meteorology and fluid dynamics .
Used in this sense , a gravity wave is a liquid surface layer wave controlled by gravity and not by surface tension .
The surface tension of water becomes unimportant at wavelengths greater than a few centimeters . On the ocean surface or interfaces , all significant waves are gravity waves .
In meteorology , gravity waves are transverse atmospheric waves where the restoring force is caused by the effect of gravity on density and pressure fluctuations .
See for example the Wikipedia article Gravity wave .
The expressions gravity waves and gravitational waves are sometimes used interchangeably for both meanings (i.e. for waves related to general relativity and waves related to fluid dynamics) , so this might cause some confusion.

As an update to the information above , something new took place in the history of the detection of gravitational waves on 11 February 2016 .
For the first time, scientists have observed ripples in the fabric of spacetime called gravitational waves, arriving at Earth from a cataclysmic event in the distant universe. This confirms a major prediction of Albert Einstein’s 1915 general theory of relativity and opens an unprecedented new window to the cosmos.[…]
The gravitational waves were detected on Sept. 14, 2015 at 5:51 a.m. EDT (09:51 UTC) by both of the twin Laser Interferometer Gravitational-wave Observatory (LIGO) detectors, located in Livingston, Louisiana, and Hanford, Washington.[…]
Based on the observed signals, LIGO scientists estimate that the black holes for this event were about 29 and 36 times the mass of the sun, and the event took place 1.3 billion years ago. About three times the mass of the sun was converted into gravitational waves in a fraction of a second—with a peak power output about 50 times that of the whole visible universe. […]
The discovery was made possible by the enhanced capabilities of Advanced LIGO, a major upgrade that increases the sensitivity of the instruments compared to the first generation LIGO detectors, enabling a large increase in the volume of the universe probed—and the discovery of gravitational waves during its first observation run.
As and additional note , it is generally preferable to have other precise experiments confirming the detection and presence of gravitational waves.

# Books about mathematics regarded as noteworthy classics

This post is taken (with some modifications from an answer I gave at quora.com  .

Classic books or classics may refer to great and historically important books , or to books widely popular , read and used ,or both . I will try to mention both types of books.
I will cite first a number of books which I think are of primary importance in the history of mathematics , and therefore are generally regarded as classics . This is not a exhaustive list .
Moving a few centuries later to modern times :
Some recent modern and well known math books that may be regarded as classics :

# The Lagrangian of the Standard Model of particle physics

I will present some notes and explanations related to the Standard Model of particle physics and its Lagrangian . The text in this post is inspired from two answers I gave at quora.com .

The Standard model and its Lagrangian form a vast topic . I will attempt to give relevant and accurate information about it.

The story of the Standard Model started in the 1960s with the elaboration of the theory of quarks and leptons  , and continued for about five decades until the discovery of the Higgs boson in 2012.
For a timeline of the history of the Standard Model see the Modern Particle Theory timeline .
The formulation of the Lagrangian of the Standard Model with its different terms and parts mirrored the theoretical and experimental advances associated with particle physics and with the Standard Model.

The Lagrangian function or Lagrangian formalism is an important tool used to depict many physical systems and used in Quantum Field Theory . It has the action principle at its basis .

In simple cases the Lagrangian essentially expresses the difference between the kinetic energy and the potential energy of a system .

The Standard Model of particle physics describes and explains the interactions between the essential components and the fundamental particles of matter , under the effect of the four fundamental forces: the electromagnetic force , the gravitational force , the strong nuclear force , and the weak (nuclear) force.
However , the Standard model is mainly a theory about three fundamental interactions , it does not fully include or explain gravitation .
The Standard model (or SM) is  a gauge theory representing fundamental interactions as changes in a Lagrangian function of quantum fields.  It depicts spinless , spin-(1/2) and spin-1 fields interacting with one another in a way governed by the Lagrangian which is unchanged by Lorentz transformations.

The Lagrangian density or simply Lagrangian of the Standard Model contains kinetic terms , coupling and interaction terms (electroweak and quantum chromodynamics sectors) related to the gauge symmetries of the force carriers (i.e. of the elementary and fundamental particles which carry the four fundamental interactions) , mass terms , and the Higgs mechanism term .

Explicitly , the parts forming the entire Lagrangian generally consist of :
Free fields : massive vector bosons , photons , and leptons.
Fermion fields describing matter.
The Lepton-boson interaction.
Third-order and fourth order interactions of vector bosons.
The Higgs section.

Leptons are the elementary particles not taking part in strong interactions.
All leptons are fermions. They include the electron , muon , and tauon , and the electron neutrino , muon neutrino , and tauon neutrino.
All leptons are color singlets , and all quarks are color triplets.

In the Standard model , the Higgs mechanism provides an explanation for the generation of the masses of the gauge bosons via electroweak symmetry breaking.

Different reference works , books , e-books or textbooks use different or slightly different notations and symbols to describe or designate the entities and terms within the Lagrangian of the SM .

Below is a detailed image of the Lagrangian of the Standard Model  (Source: http://einstein-schrodinger.com/Standard_Model.pdf ).
However I have rearranged it and modified it with the help of Photoshop to make it look more presentable and more readable.

The Lagrangian function in the Standard Model , as in other gauge theories , is a function of the field variables and of their derivatives.

$G_ {\mu \nu}$ is the gauge field strength of the strong SU(3) gauge field.
Gluons are the eight spin-one particles associated with SU(3).
A particle which couples to the gluons and transforms under SU(3) is called ‘colored’ or ‘carrying color’.
Gluons and quarks are confined in hadrons.

$W_ {\mu \nu}$ is the gauge field strength of the weak isospin SU(2) gauge field .

The field strength tensor $W_ {\mu \nu}$ is given by :

where $g_2$ is the electroweak coupling constant , a dimensionless parameter.

The charged $W^+$ and $W^-$ bosons and the neutral Z boson represent the quanta of the weak interaction fields between fermions , they were discovered in 1983 .

$B_ {\mu \nu}$ is the gauge field strength of the weak hypercharge U(1) gauge field.
The field strength tensor $B_ {\mu \nu}$  is given by :

$B_ {\mu \nu} = \frac {\partial B_ {\nu}} {\partial\mu} - \frac {\partial B_ {\mu}} {\partial\nu}$

In the Standard model , electrons and the other fermions are depicted by spinor fields .
The group U(1) is the set of one-dimensional unitary complex matrices .
U(1) represents the symmetry of a circle unchanged by rotations in a plane.

SU(2) is called ‘the special unitary group of rank two’. It is a non commutative group related to SO(3) , the sphere symmetry in 3 dimensions.
SU(2) is the set of two-dimensional complex unitary matrices with unit determinant.

SU(3) , the special unitary group of rank three , is used in quantum chromodynamics (QCD) .
SU(3) is the set of three-dimensional complex unitary matrices with determinant equal to 1 .
The natural representation of SU(3) is that of 3×3 matrices acting on complex 3D vectors.
The generators of the group SU(3) are eight 3×3 , linearly independent , Hermitian , traceless matrices called the Gell-Mann matrices . These generators can be created from Pauli spin matrices (which are used with the group SU(2) ) .

The SM Lagrangian displays invariance under SU(3) gauge transformations for strong interactions , and under SU(2)xU(1) gauge transformations for electroweak interactions.

The electromagnetic group is not directly the U(1) weak hypercharge group component of the standard model gauge group. The electric charge is not one of the basic charges carried by particles under the unitary product group SU(3)xSU(2)xU(1) , it is a derived quantity.

All the masses vanish in the absence of the Lagrangian term related to the Higgs , due to the invariance of SU(3)xSU(2)xU(1) .

In some texts the gauged symmetry group of the SM is written with subscripts such as:
$\text {SU} _c (3)\times\text {SU} _L (2)\times U_Y (1)$
In the notation above , the subscript ‘c’ denotes color.
The subscript ‘L’ denotes left-handed fermions.
The subscript ‘Y ‘  distinguishes the group related to the quantum number  of weak hypercharge , expressed by the letter Y , from the group associated with ordinary electric charge, expressed by Q .
$U_ {\text {em}} (1)$ denotes the electromagnetic group.

The Higgs field in the Standard model is a complex scalar doublet. It is generally represented by :

In the image of the SM Lagrangian above , the Higgs field has the form

The field h(x) is real .

In the SM Lagrangian image above , $\phi _ 0$  is equal to v .

As an additional note , the  equation of the Lagrangian  is usually made of a definite number of terms and Lagrangians.
In order to make such an equation look less like a big behemoth and make it more compact ,  it would be simpler to view it or write it first as the sum of Lagrangians :

$\mathcal{L}=\mathcal{L}_1+\mathcal{L}_2+\text{...}+\mathcal{L}_n$

or equivalently :

$\mathcal {L} = \sum _ {i = 1}^n\mathcal {L} _i$

Then each Lagrangian in the equation could be expanded and explained.

Standard Model

The Standard Model of Particle Physics

Standard Model (mathematical formulation)

http://arxiv.org/pdf/hep-ph/0304186v1.pdf

Gauge Theory of Weak Interactions: Walter Greiner, Berndt MÃ¼ller: 9783540878421: Amazon.com: Books

The Structure and Interpretation of the Standard Model, Volume 2 (Philosophy and Foundations of Physics): Gordon McCabe: 9780444531124: Amazon.com: Books

The Standard Model: A Primer: Cliff Burgess, Guy Moore: 9781107404267: Amazon.com: Books

An Introduction to the Standard Model of Particle Physics: W. N. Cottingham, D. A. Greenwood: 9780521852494: Amazon.com: Books

The Theory of Almost Everything: The Standard Model, the Unsung Triumph of Modern Physics: Robert Oerter: 9780452287860: Amazon.com: Books

Here is also a link to one of the  important papers in the history of the Standard Model written in 1967 by Weinberg and entitled ‘A Model of Leptons’ :
http://physics.princeton.edu/~mcdonald/examples/EP/weinberg_prl_19_1264_67.pdf

# Some algorithmic texture generation with Mathematica

The LineIntegralConvolutionPlot[] function in Mathematica is defined by the Wolfram Mathematica Documentation Center as :

LineIntegralConvolutionPlot[{vx,vy},{x,xmin,xmax},{y,ymin,ymax}]
generates a line integral convolution plot of white noise with the vector field {vx,vy}.

LineIntegralConvolutionPlot[] can also generate the plot of an image convolved with a vector field .

I will give some plots of line integral convolutions of a number of vector fields . These plots have often visually appealing forms.

Here is the first plot :

The Mathematica code for the image above is :

Below are some more line integral convolution plots using various “ColorFunction” options .The frame has been removed from these plots :

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And below is another line integral convolution plot with the external frame modified with Photoshop . Looks like a nice piece of art …

The Mathematica code for the image above is :

A last example of a line integral convolution plot with a large size :

Mathematica code for the image above ;