How good you were in Maths !! - Tables !!!

 

How good were in you Maths !!  ~ remember that when we were young, we were taught Tables !  - upto 16 – in the class, students in chorus 1 x 8 = 8, 2x 8 = 16, 24,32,40,48,56,64,72,80,88,96 ………. 

This morning SYMA conducted a Medical camp at Chennai Middle School, Ramnagar, Triplicane – the infrastructure, class rooms, cleanliness all impressed me beyond a point ! [that will be subject matter of another post] – this about a blackboard containing tables ..  do children glued to mobiles and calculators still learn Tables ?!?!?

Perhaps  there isn't a single person who invented them. They developed gradually in ancient civilizations.  Some references put that the  earliest known multiplication tables come from ancient Mesopotamia. Babylonians wrote mathematical tables on clay tablets, using their base-60 (sexagesimal) system.   Egyptians had sophisticated methods for multiplication, but generally used doubling rather than memorising a complete set of tables. For example, to multiply by 13, they could successively double and add the required quantities.

                Indian mathematicians developed and used multiplication tables extensively. The decimal place-value system and zero, which emerged in India, made arithmetic and multiplication much easier. By the early medieval period, Indian mathematical texts contained systematic multiplication methods and tables. Chinese mathematics: China also developed multiplication tables very early. The famous “Nine-Nine Table” (九九表) was traditionally memorised as a sequence of products from 1×1 through 9×9.  

Nobody can really be called the inventor. Multiplication itself is thousands of years old, and tables evolved as a practical shortcut for repeated multiplication. The  “2 to 20 tables” tradition familiar in Indian schools is much more extensive than the standard 1–10 tables commonly taught in many Western countries. Aryabhata, Brahmagupta and Bhaskaracharya did not invent the school multiplication tables as such. They developed and codified arithmetic methods that made systematic calculation possible..

Before Aryabhata — multiplication was repeated addition -  5 × 4 = 5 + 5 + 5 + 5 = 20.  Indian mathematicians explicitly regarded multiplication as a form of repeated addition. The terminology was quite sophisticated: the number being multiplied and the multiplier had specific Sanskrit names, and the result had its own terminology.  Calculations were sometimes performed on a board covered with dust, hence the term dhūlikarma (“dust work”).  

Aryabhata, in his Āryabhaṭīya, gave a systematic treatment of mathematics (gaṇita) alongside astronomy. His importance to the later tradition is not that he created “2 times table, 3 times table…” but that he codified mathematical procedures in a highly systematic form. His work became the foundation for generations of Indian commentators and mathematicians. An important point is that these texts were written in compact Sanskrit verse, partly because mathematical rules were intended to be memorised. So there was already a strong tradition of learning numerical rules by heart—something that eventually fits very naturally with the school practice of reciting multiplication tables.

About a century later came Brahmagupta. His Brāhmasphuṭasiddhānta, written in 628 CE, contains an extensive treatment of arithmetic, including multiplication, division, fractions, negative numbers and zero.  Consider: 347 × 26. Today we instinctively understand that:

6 × 347 = 2082

20 × 347 = 6940

therefore 2082 + 6940 = 9022.

That depends on understanding place value — units, tens, hundreds — and the decimal numeral system. Indian mathematicians developed increasingly sophisticated arithmetic using this system. Historical descriptions of Indian multiplication methods show procedures remarkably close in principle to the long multiplication taught in schools today.   

Much later in the 12th century,  Bhaskaracharya’s famous Līlāvatī, part of the Siddhānta Śiromaṇi, contains extensive arithmetic and specifically discusses methods of multiplication and squaring. Bhaskara II gives methods of multiplication that involve arranging numbers in particular positions and multiplying their individual digits—methods that are recognisably related to the algorithms used later in traditional arithmetic teaching. And Līlāvatī is particularly interesting because it doesn't merely give rules. It presents problems for students to solve, including arithmetic progressions, interest, fractions, geometry and other numerical problems.

The modern Indian school tradition of Maths tables  is not something we can attribute to Aryabhata or Brahmagupta individually. Rather, it grew out of the broader Indian tradition of: number → place value → repeated addition → multiplication rules → memorised computational rules → school arithmetic. The fact that Indian mathematical texts were composed in concise, memorable verses also encouraged oral memorisation. Over centuries, this developed into the familiar practice of children reciting tables aloud.

The traditional Indian method of multiplication was so influential that medieval Europeans sometimes referred to Indian arithmetic as “modus Indorum” — the method of the Indians.  So when an Indian child today confidently rattles off “13 ones are 13, 13 twos are 26…”, there is a very long mathematical ancestry behind that apparently simple school exercise.

Interesting ! – back to Basics ! – back to schools !!

 

Regards – S Sampathkumar

4.10.2026

Kuthiraikkaran vilakku - some Science - some Maths !!

 

Sure, you have been to many temples – observed people pouring Oil in lamps hanging from high ceiling … one often wonders, how they hang in good balance – Symmetry, distribution of mass, and some more Scientific reasons come to mind.  

Starting with calculating the weight - there are few things more frustrating than weighing an item twice only to receive two radically different results. One seeks a  precise measurement, and one cannot  trust  digital scale too.

 

Here is a photo of lamp (background edited and colour enriched)  and some Science behind the surprising design – the parts are not symmetric – and it defies imagination on how equilibrium and the mass distribution are designed – how artisans were so Scientific  -  “Kuthiraikkaran Vilakku” (Horseman’s lamp) seen at Padmanabhapuram Palace. 

A lamp hanging on chains stays up because the chain is in tension, and the lamp’s  weight is pulled downward by gravity while the chain pulls upward with an equal and opposite force. When those forces balance, the lamp is in static equilibrium and remains suspended without accelerating.

The important idea is the center of mass. The lamp hangs stably when the vertical line through its center of mass falls below the suspension point, so gravity does not create a turning effect that would make it tip over.

 

If the lamp is shaped like the one in this  photo, the decorative body and the tray are to be precisely arranged  so the whole assembly balances around the hanging point. 

In hanging lamps, the chain is flexible, so each link carries tension along its own direction, and the chain naturally settles into the shape needed to support the load. The hanging shape of a chain is called a catenary (the curve formed by a flexible chain or cable hanging freely under its own weight between two fixed points. Its name derives from catena, the Latin word for "chain"), and its geometry helps distribute force smoothly from the lamp to the ceiling. This is why chains are often used instead of rigid hooks when the design needs some swing or slight adjustment.

 

Catenary pic – Wikipedia By Geek3 - Own work, CC BY-SA 3.0, https://commons.wikimedia.org/w/index.php?curid=7728690

 

When the hanging lamp is nudged, it may sway like a pendulum, but if the center of mass is low enough and the suspension is properly aligned, gravity tends to pull it back toward the resting position.   

The lamp hangs in balance because its center of mass is arranged so the vertical line of gravity falls through the support point, and the turning effects on either side cancel out. In physics terms, it is in static equilibrium: the net force is zero and the net torque is zero. 

The heavy decorative horse and the lamp tray are not just ornamentation; they are positioned to shift the combined center of mass to a stable point below the chain attachment. When the center of mass hangs directly under the suspension point, gravity creates no twisting tendency, so the lamp does not tip.  When you observe the photo,   the chain is attached above the horse figure, and the lamp bowl is offset but still balanced by the mass distribution of the full hanging assembly. The support point, the horse body, and the tray together form a system where clockwise and counterclockwise torques balance each other. 

It is beautifully positioned as though it is a seesaw that has been carefully weighted so both sides match. Even if the parts look uneven, the object can remain steady as long as the overall center of mass is below the hanging point.  It takes care of both Static equilibrium as also dynamic stability – meaning it is firm at rest and does not fall even when disturbed and is moving.  Only a  well-designed hanging object can be in static equilibrium and also be dynamically stable, because gravity pulls its center of mass back under the suspension point. 

“Kuthiraikkaran vilakku” at  Padmanabhapuram Palace is a  horse-shaped hanging lamp, hailed for its balance and physics.   In this horseman-style lamp, the decorative horse is not just ornamentation; it is part of the mass distribution that helps position the center of gravity correctly. Even though the bowl and horse look visually uneven, the total weight is arranged so the system hangs in static equilibrium.   The lamp’s geometry works like a carefully weighted hanging system.   The horse body acts like a counterweight and also moves the mass distribution away from the bowl. The tray-like lamp part adds weight low down, which helps keep the overall center of gravity beneath the chain attachment, making the system stable. 

Imagine drawing a vertical line straight down from the chain hook. If that line passes through the combined center of mass of the horse, hook, and tray, the lamp stays balanced. If it were to move outside that support region, the lamp would rotate until balance is restored. So the geometry is really a weighted balance problem: the shape is designed so that the center of mass stays under the support point, and that is why the lamp can hang neatly without toppling. 

Padmanabhapuram Palace is a magnificent 16th-century wooden palace located in Thuckalay, Kanyakumari district, Tamil Nadu. Notably, while the palace is physically situated in Tamil Nadu, it is owned, controlled, and maintained by the Government of Kerala as it served as the ancient capital of the erstwhile Kingdom of Travancore. It is widely celebrated as one of Asia's largest and finest surviving examples of traditional wooden architecture

 
Interesting !
 
Regards – S Sampathkumar
24.5.2026

A goniometer – what ? பாகைமானி !!

 

A goniometer – what ?   பாகைமானி என்றால் என்னவென்று தெரியுமா ! 

We can measure Angles in Degrees.  There are 360 degrees in one Full Rotation (one complete circle around). Perhaps the first knowledge prize that we got in our school days was a Geometry box !  it was a set of basic equipment required for regular use in Schools in doing  diagram and graphs. The box usually consisted  of a compass, a 15cm ruler, a divider, 2 set squares and a protractor.  Protractors usually have two sets of numbers going in opposite directions. The outer scale starts from 0° and goes up to 180° in a clockwise direction, and the inner scale has the same range in the counterclockwise direction.  

பாகை என்பது கோணத்தை அளப்பதற்குரிய ஒரு அலகு ஆகும். இது 60 கலைக்குச் சமனானது ஆகும். இது ° என்னும் குறியீட்டினால் குறிக்கப்படுவது வழக்கம். 60° என எழுதும்போது அது 60 பாகை என்பதைக் குறிக்கும். ஒரு தளத்தில் அதிலுள்ள ஒரு புள்ளியை முழுவதுமாகக் சுற்றி அமையும் கோணம் 360 பாகை (360°) ஆகும்.  

வடிவவியலில், பாகைமானி அல்லது கோணமானி (Protractor) கோணங்களை அளவிடும், பொதுவாக ஊடுபுகக்கூடிய பேர்பெக்சாலான, ஒரு சதுர, வட்ட அல்லது அரை வட்டக் கருவியாகும்.  

A goniometer is an instrument that either measures an angle or allows an object to be rotated to a precise angular position. The term goniometry derives from two Greek words, γωνία (gōnía) 'angle' and μέτρον (métron) 'measure'.  Better known as the  protractor, it  is a commonly used type in the fields of mechanics, engineering, and geometry.   The first known description of a goniometer, based on the astrolabe, was by Gemma Frisius in 1538.

A protractor,  a measuring instrument, is  typically made of transparent plastic or glass, for measuring angles. Some protractors are simple half-discs or full circles. More advanced protractors, such as the bevel protractor, have one or two swinging arms, which can be used to help measure the angle.  Most protractors measure angles in degrees (°). Radian-scale protractors measure angles in radians. Most protractors are divided into 180 equal parts. Some precision protractors further divide degrees into arcminutes. A protractor divided in centiturns is normally called a "percentage protractor".

 

Interesting !

 
With regards – S Sampathkumar
28.3.2026

Gauss Law .. .. and Nature's beau stripes !!

 

Remember a Danush / Sivakarthikeyan movie (where they attend a tuition centre) – the master would ask about ‘Gauss law’ – and that provoked to read something. 

Geometry (from Ancient Greek (geōmetría) 'land measurement')   is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures. Geometry, along with arithmetic, one of the oldest branches of mathematics. A mathematician who works in the field of geometry is called a geometer. Until the 19th century, geometry was almost exclusively devoted to Euclidean geometry, which includes the notions of point, line, plane, distance, angle, surface, and curve, as fundamental concepts. 

Geometry is all about lines -  lines in geometry can be classified into several types, primarily categorized as straight or curved. Straight lines can be further defined as horizontal, vertical, parallel, intersecting, or perpendicular.

 

Here is God’s creation with lines … .. an equine called Zebra.  Zebras  are African equines with distinctive black-and-white striped coats. There are three living species: Grévy's zebra (Equus grevyi), the plains zebra (E. quagga), and the mountain zebra (E. zebra). Zebras share the genus Equus with horses and asses, the three groups being the only living members of the family Equidae. Zebra stripes come in different patterns, unique to each individual. Several theories have been proposed for the function of these patterns, with most evidence supporting them as a deterrent for biting flies.   

Johann Carl Friedrich Gauss [1777 –1855] was a German mathematician, astronomer, geodesist, and physicist, who contributed to many fields in mathematics and science. He was director of the Göttingen Observatory and professor of astronomy from 1807 until his death in 1855. While studying at the University of Göttingen, he propounded several mathematical theorems. As an independent scholar, he wrote the masterpieces Disquisitiones Arithmeticae and Theoria motus corporum coelestium. Gauss produced the second and third complete proofs of the fundamental theorem of algebra. In number theory, he made numerous contributions, such as the composition law, the law of quadratic reciprocity and the Fermat polygonal number theorem. He also contributed to the theory of binary and ternary quadratic forms, the construction of the heptadecagon, and the theory of hypergeometric series. Due to Gauss' extensive and fundamental contributions to science and mathematics, more than 100 mathematical and scientific concepts are named after him. 

Gauss's Theorema Egregium (Latin for "Remarkable Theorem") is a major result of differential geometry, proved by Carl Friedrich Gauss in 1827, that concerns the curvature of surfaces. The theorem says that Gaussian curvature can be determined entirely by measuring angles, distances and their rates on a surface, without reference to the particular manner in which the surface is embedded in the ambient 3-dimensional Euclidean space. In other words, the Gaussian curvature of a surface does not change if one bends the surface without stretching it. Thus the Gaussian curvature is an intrinsic invariant of a surface.

 
Interesting !!  ~  here is a Zebra pictured at Bannerghatta zoological park.
 
Regards – S Sampathkumar
2.6.2025

Thinking ! Philosophy ! ~ Thales of Miletus

 

When one is calm and has time at his disposal, he tends to think, think logically said a great Philosopher (Anon.!)

 

In geometry, a famous  theorem states that if A, B, and C are distinct points on a circle where the line AC is a diameter, the angle ∠ ABC is a right angle. This theorem is a special case of the inscribed angle theorem and is mentioned and proved as part of the 31st proposition in the third book of Euclid's Elements.  (pic credit : Wikipedia) 

How much do you think !?! some great philosophers have left their indelible impression on the generations, making them think.    Philosophy (from Greek  philosophia, literally "love of wisdom") is the study of general and fundamental questions about existence, knowledge, values, reason, mind, and language. Such questions are often posed as problems to be studied or resolved. Philosophical methods include questioning, critical discussion, rational argument, and systematic presentation. Classic philosophical questions include: Is it possible to know anything and to prove it?  What is most real? Philosophers also pose more practical and concrete questions.     Historically, "philosophy" encompassed any body of knowledge 

Macedonia, a small kingdom in northern Greece, established a growing empire from 359 B.C. to 323 B.C. through the reign of several kings. It also gave to the World many great philosophers.   These brilliant scholars began to use reason and logic to try and unravel the workings of the cosmos. They also explored the intricacies of human morality. In ancient Greece, philosophers contemplated and theorized about many different ideas such as human nature, ethics, and moral dilemmas. Ancient Greek philosophers can be categorized into three groups: the Pre-Socratics, the Socratics, and the Post-Socratics. 

Pre-Socratic philosophers mostly investigated natural phenomena. They believed that humans originated from a single substance, which could be water, air, or an unlimited substance called “apeiron.” One well-known philosopher from this group was Pythagoras, the mathematician who created the Pythagorean Theorem.

The Socratic philosophers in ancient Greece were Socrates, Plato, and Aristotle. Plato  studied ethics, virtue, justice, and other ideas relating to human behavior. Following in Socrates’ footsteps, he became a teacher and inspired the work of the next great Greek philosopher, Aristotle.  The Post-Socratic philosophers established four schools of philosophy: Cynicism, Skepticism, Epicureanism, and Stoicism.

 

The earliest among them was ‘Thales of Miletus’ who  lived in Ancient Greece. He was the first scientist in history. Thales looked for patterns in nature to explain the way the world worked rather than believing everything happened only because one of the Greek gods commanded it. He replaced superstitions with science. He was the first person to use deductive logic to find new results in geometry and, through requiring proof of theorems, took mathematics to a new, higher level. When pronouncing his name, we say thail-eez, emphasizing the first syllable.

There may have been other scientists before Thales, but if there were, we do not know their names. Thales of Miletus (623  – c. 548  BC)   was one of the Seven Sages, founding figures of Ancient Greece. The first philosophers followed him in explaining all of nature as based on the existence of a single ultimate substance. Thales theorized that this single substance was water. Thales thought the Earth floated on water.  

In mathematics, Thales is the namesake of Thales's theorem (read para 2 again!), and the intercept theorem can also be known as Thales's theorem. Thales was said to have calculated the heights of the pyramids and the distance of ships from the shore. In science, Thales was an astronomer who reportedly predicted the weather and a solar eclipse. The discovery of the position of the constellation Ursa Major is also attributed to Thales, as well as the timings of the solstices and equinoxes. He was also an engineer, known for having diverted the Halys River. Thales of Miletus is credited with the discovery of 5 geometric theorems:

•        A circle is bisected by its diameter

•        Angles opposite two equal sides of a triangle are equal

•        Opposite angles of two intersecting angles are equal

•        An angle inscribed in a semi-circle is a right-angled triangle

•        We can determine a triangle if we have the length of its base and two angles at the base are given 

 

Just in case you wonder, what all these is about ! –  the  Chimpanzee in this pic taken by me appears to be thinking ! – and the images of Thales of Miletus portray him sporting a flowing white beard.

 

தாடிகள் எல்லாம் தாகோர் அல்ல ! எனினும் தாடி உடையோரைக்  கண்டால், அவர்கள் அறிவார்ந்தவர்கள் என நினைக்கும் குழுமத்தில் அடியேனும் ஒருவன் !

 
Interesting !
 
With regards – S Sampathkumar
26.8.2024

Series and Sequences ! - Jasminum Sambac - அடுக்கு மல்லிகை

How good were you in Mathematics in schools !  :   a²+b² = (a+b)² - 2ab  !! 

 

Maths was tougher ! – not that other subjects were easy !!   In Arithmetics, - Sequence and series are considered basics. An itemized collection of elements in which repetitions of any sort are allowed is known as a sequence, whereas a series is the sum of all elements. An arithmetic progression is one of the common examples of sequence and series.

 

திருக்கோயில்களுக்குச் சென்று இறைவனை வழிபடுகிறோம். நமக்கு அனைத்து நல்லனவற்றையும்  அளிக்கும் இறைவனுக்கு நமது நன்றியை அன்பை வெளிப்படுத்துவதற்கு அபிஷேக வழிபாடு மலர் அர்ச்சனை ஆகியவற்றை மனம் குளிரச் செய்து வழிபடுகிறோம்.  இறைவனுடைய அருளைப் பெறுவதற்குத் புஷ்பங்களும், திரவியங்களும்  எளிமையான வழியாகும்.    மல்லிகை,  முல்லை,  செண்பகம், தாமரை,மகிழம், ரோஜா,  அல்லி, விருச்சி,  செங்காந்தள்;   போன்ற நறுமலர்கள் திருக்கோவில்களில் எம்பெருமானுக்கு சாற்றப்பெற்று வருகின்றன.  அந்த காலத்தில் நந்தவனத்தில் இருந்து கொய்த அன்றலர்ந்த   பூக்களைப்  திருக்கோவிலில் உள்ள  மண்டபத்தில் வைத்து தொடுத்தனர். இதற்கென கல்லால் தனி மேடை அமைக்கப்பட்டிருக்கும்.  இதற்கு "பூப்பலகை' என  பெயர்.   எம்பெருமான் கழலிணைகள் சிறப்பானவை.  இறைவனுடைய திருவடிகளை  "மலரடி' என்று போற்றுகிறோம். 

In our time there used to be Competitive exams like the ones conducted by BSRB – cracking them was considerably challenging and rewarding too.  One of them was test reasoning which included :  A numerical series is given in which a number is wrongly placed. You are asked to identify that particular wrong number. A numerical series is given in which a specific number is missing. You are required to find out that missing number. A complete numerical series is followed by an incomplete numerical series. You need to solve that incomplete numerical series in the same pattern. 

An arithmetic progression or arithmetic sequence (AP) is a sequence of numbers such that the difference from any succeeding term to its preceding term remains constant throughout the sequence. The constant difference is called common difference of that arithmetic progression. For instance, the sequence 5,9,13,17,21,25,29,33 .. .. .. is an arithmetic progression with a common difference of 4. 

Acharya Pingala  was an ancient Indian poet and mathematician, the author of the Chandaḥśāstra (also called the Pingala-sutras), the earliest known treatise on Sanskrit prosody.    

After dwelling long, this is no post on Mathematics but on flowers – Jasmines are very fragrant – there is a variety called ‘layered jasmine’.  Jasminum sambac is an evergreen vine or shrub. The plant bears dark green glossy foliage and white flowers. Jasminum Sambac ‘Belle of India’ is an everblooming,  fragrant variety with long slender oval buds which open to pure white double blooms.  The scent from the flowers that blossom in the vine is heavenly. 

‘ஜாஸ்மினும் சம்பக்’ (Jasminum Sambac) எனும் தாவரவியல் பெயரால் அழைக்கப்படும் அடுக்கு மல்லிகை - இருவாட்சி மல்லிகை  எனவும் அழைக்கப்படுகின்றது.  இருவாட்சி மணம்  கொண்ட மலர் மட்டுமல்ல, ஒரு பறவையையும் குறிக்கும்.  சூழலியல் சுட்டிக்காட்டிகளில் மிக முக்கியமானது இருவாட்சி என தமிழிலும், Horn Bill என ஆங்கிலத்தில் அழைக்கப்படும் பறவை இனம்.   மழைக்காடுகளில் இருவாட்சி பறவை இல்லையென்றால், மழை இல்லை என எடுத்துக்கொள்ளலாம்.  மலைப் பகுதிகளில் மரங்கள் வெட்டப்படுவதால், இருவாட்சி பறவைகளின் எண்ணிக்கை நாளுக்கு நாள் குறைந்து வருவதாக பறவை ஆர்வலர்கள் வேதனை தெரிவிக்கின்றனர். மிகவும் உயரமான மரங்களில் வசிக்கும் இருவாட்சி பறவைகள், மரங்கள் அழிக்கப்படுவதின் காரணமாக தன் இனத்தை பெருக்கிக்கொள்ள முடியாமல் தவிக்கின்றனவாம். .  

 

Here are couple of photos – Belle of India – adukku malligai - Jasminum Sambac – seen at my terrace garden.
 
With regards – S Sampathkumar
27.8.2023

  

Reflections !!

“Reflection”   -  the noun means :  the throwing back by a body or surface of light, heat, or sound without absorbing it. Eg.  "the reflection of light" 

When one tries to remove the vessel used for burning milk for making good coffee, one touches the vessel – the hot vessel burns hand – the immediate learning is – touching a vessel on a hot stove burns ! – the coffee may still taste good !!   Common  brain makes the connection almost simultaneously. There’s little need for reflection because the “hot vessel =burn” connection is one that  mind makes almost immediately. Similarly, driving two-wheeler in  bad weather and going way too fast might result in skidding, spinning off and losing control, giving undesirable results – there is instant lesson in that too.  Academic learning however is seldom that obvious. 

Reflections are one’s thoughts about a particular subject. After days of reflection some  write back. Reflection is a deeper form of learning that allows us to retain every aspect of any experience, be it personal or professional — why something took place, what the impact was, whether it should happen again — as opposed to just remembering that it happened. 

A famous quote reads :  “We don’t learn from experience. We learn from reflecting on experience.” Reformers have been talking about the importance of reflection in the learning process since ages.  Another thinker equates the learning process as “Input-Process-Output-Reflect” and notes that without reflection, it is very difficult to have genuine learning. If the goal is not merely coverage but actual learning, than reflection is no longer optional – it’s an essential piece to transition a classroom from “covering material” to being “focused on learning”. 

Reflective practice is, in its simplest form, thinking about or reflecting on what you do. It is closely linked to the concept of learning from experience, in that you think about what you did, and what happened, and decide from that what you would do differently next time. Thinking about what has happened is part of being human. However, the difference between casual ‘thinking’ and ‘reflective practice’ is that reflective practice requires a conscious effort to think about events, and develop insights into them. 

Before concluding, in Mathematics, a reflection is a mapping from a Euclidean space to itself that is an isometry with a hyperplane as a set of fixed points; this set is called the axis or plane  of reflection. For example the mirror image of the small Latin letter p for a reflection with respect to a vertical axis would look like q. Its image by reflection in a horizontal axis would look like b. A reflection is an involution: when applied twice in succession, every point returns to its original location, and every geometrical object is restored to its original state. 

In Science, Reflection is the change in direction of a wavefront at an interface between two different media so that the wavefront returns into the medium from which it originated. Common examples include the reflection of light, sound and water waves. The law of reflection says that for specular reflection (for example at a mirror) the angle at which the wave is incident on the surface equals the angle at which it is reflected. 

In acoustics, reflection causes echoes and is used in sonar. In geology, it is important in the study of seismic waves. Reflection is observed with surface waves in bodies of water. Reflection is observed with many types of electromagnetic wave, besides visible light. Reflection of VHF and higher frequencies is important for radio transmission and for radar.   

Most modern TVs have glossy screens, which act like a mirror for any light source in a room (from windows to lamps). Some TVs, including many older models, have more matte screens, which don't show the same mirror-like reflections. Ambient light still adversely affects matte screen TVs though. Instead of bouncing the light right back at you, a matte finish spreads that light energy across the whole screen. Reflections are lessened, but black level gets brighter, so they look more washed out overall.  Reflections on your TV can make it nearly impossible to watch. Luckily, there are ways to reduce irritating TV glare. 

A research team from the University of Maryland has developed an artificial intelligence-powered (AI) method to reconstruct complex scenes and objects in 3D using only the reflections in a person’s eye. “The reflective nature of the human eye is an underappreciated source of information about what the world around us looks like. By imaging the eyes of a moving person, we can collect multiple views of a scene outside the camera’s direct line-of-sight through the reflections in the eyes,” explain the researchers.

 

What you are seeing in the photo here is reflection of coconut trees in a small pond at Ananthazhwan avathara sthalam, Hale Kiranguru, near Mandya !!

With regards – S. Sampathkumar
18.7.2023

  

from humble Cabbage - to - complex Fibonacci numbers !!

How good were you in Mathematics in schools !  :   a²+b² = (a+b)² - 2ab  !!  - how much do you like Cabbage in your food ?  - the humble cabbage – encompasses several cultivars of Brassica oleracea, a leafy green, red (purple), or white (pale green) biennial plant grown as an annual vegetable crop for its dense-leaved heads.  The word “cabbage” comes from the French word caboche, which means “head.” That makes sense given the shape. And the metaphor exists in English, where we talk about a head of cabbage.

Some of its cousins are more interesting !! and tastier !!!  -  Cauliflower is one of several vegetables in the species Brassica oleracea in the genus Brassica. It is an annual plant that reproduces by seed. Typically, only the head is eaten – the edible white flesh is sometimes called "curd" (with a similar appearance to cheese curd). The cauliflower head is composed of a white inflorescence meristem. Cauliflower heads resemble those in broccoli, which differs in having flower buds as the edible portion.

Its much elitist cousin is - Broccoli an edible green plant in the cabbage family  whose large flowering head, stalk and small associated leaves are eaten as a vegetable. Broccoli is classified in the Italica cultivar group of the species Brassica oleracea. Broccoli has large flower heads, usually dark green, arranged in a tree-like structure branching out from a thick stalk which is usually light green. The mass of flower heads is surrounded by leaves. Broccoli resembles cauliflower, which is a different but closely related cultivar group of the same Brassica species.  It is eaten either raw or cooked and is used in soups too.

Acharya Pingala  was an ancient Indian poet and mathematician, the author of the Chandaḥśāstra (also called the Pingala-sutras), the earliest known treatise on Sanskrit prosody.   Fibonacci (1170 – 1240)  also known as Leonardo Bonacci, Leonardo of Pisa, or Leonardo Bigollo Pisano   was an Italian mathematician from the Republic of Pisa, considered to be "the most talented Western mathematician of the Middle Ages". The name he is commonly called, Fibonacci, was made up in 1838 by the Franco-Italian historian Guillaume Libri  and is short for filius Bonacci ('son of Bonacci').

Fibonacci popularized the Indo–Arabic numeral system in the Western world primarily through his composition in 1202 of Liber Abaci (Book of Calculation). He also introduced Europe to the sequence of Fibonacci numbers, which he used as an example in Liber Abaci.  In mathematics, the Fibonacci numbers, commonly denoted Fn , form a sequence, the Fibonacci sequence, in which each number is the sum of the two preceding ones. The sequence commonly starts from 0 and 1 - the first few values in the sequence are:

0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144.

The Fibonacci numbers were first described in Indian mathematics, as early as 200 BC in work by Pingala on enumerating possible patterns of Sanskrit poetry formed from syllables of two lengths. They are named after the Italian mathematician Leonardo of Pisa, later known as Fibonacci, who introduced the sequence to Western European mathematics in his 1202 book Liber Abaci.

Romanesco broccoli   is an edible flower bud of the species Brassica oleracea, which also includes regular broccoli and cauliflower.  It is chartreuse in color,  and has a form naturally approximating a fractal. Romanesco broccoli has a nutty flavor and a firmer texture than regular broccoli when cooked. 

The Romanesco superficially resembles a cauliflower, but it has a visually striking fractal form.  It  is chartreuse in color, with the form of a natural fractal. Nutritionally, romanesco is rich in vitamin C, vitamin K, dietary fiber, and carotenoids.  The inflorescence (the bud) is self-similar in character, with the branched meristems making up a logarithmic spiral, giving a form approximating a natural fractal; each bud is composed of a series of smaller buds, all arranged in yet another logarithmic spiral. This self-similar pattern continues at smaller levels. The pattern is only an approximate fractal since the pattern eventually terminates when the feature size becomes sufficiently small. The number of spirals on the head of Romanesco broccoli is a Fibonacci number.

Interesting !  - some photos of  Romanesco broccoli  taken from twitter. 

With regards – S. Sampathkumar
2.3.2023

 

  

Strike, algorithm and more - Money Monster

A  century ago – there was a strike by   the workers of Buckingham and Carnatic Mills in the city of Madras, against the managing company, Binny and Co. The strike, which lasted from June to October 1921, caused severe losses to the Madras economy. It also created a rift in the ruling Justice Party.

Half a century or so ago, industrial activities were at their peak at Ambattur – a blooming Industrial estate with mushrooming small scale units. Sadly, strikers broke the backbone of small upcoming entrepreneurs – by those orchestrated strikes by Left Unions, the worst sufferers were the workers themselves, as one after the other Units started closing.  Now the Estate is struggling more so with migrant workers not being available due to Covid 19. A couple of decades ago, in Banks and PSUs – Unions could threaten the management calling strikes too often – all that has changed ! totally.  One would remember witnessing – ‘agitations / demonstrations at lunch or at the start of the day itself – workers shouting – worker’s unity zindabad ! management murdabad !!’.

The right to strike in the Indian constitution set up is not absolute right but it flow from the fundamental right to form union. As every other fundamental right is subject to reasonable restrictions, the same is also the case to form trade unions to give a call to the workers to go on strike and the state can impose reasonable restrictions.

Section 2(gg)(q) in The Industrial Disputes Act, 1947 – defines :  " strike" means a cessation of work by a body of persons employed in any industry acting in combination or a concerted refusal, or a refusal under a common understanding, of any number of persons who are or have been so employed to continue to work or to accept employment; Whenever employees want to go on strike they have to follow the procedure provided by the Act otherwise there strike deemed to be an illegal strike. Section 22(1) of the  Industrial Dispute Act, 1947 put certain prohibitions on the right to strike.

An interesting movie purportedly about stock market – the  story unfolds the day after a major listed company, Ibis, has declared a “glitch" in its algorithms has led to the loss of $800 million. The chief executive officer, Walt Camby (Dominic West), due to appear as a guest on Money Monster, is said to be mid-air between Geneva and New York and therefore unable to attend the show.  May be of more interest to those investing in Shares ?

How much have you invested in shares ? – is that more than 50% of your portfolio – and how closely do you follow the market ?- daily, hourly or at a higher frequency ! – it is close follow-up and mental Maths one may think .. .. more wise people would use algorithms.  Instead of following the market, and buying (online or through a broker) intra-day – you could set rules.

I am an absolute novice in this :  but suppose, if you want to invest Rs.2 lakhs – and would like to keep your portfolio by buying 100 shares of a stock when its 100 day moving average goes above 200 day moving average !  and sell your owned stocks when their 100 day moving average goes below 200 day moving average ! (sounds pretty complex!) – you can have a program written which would constantly monitor the stock price and the moving average indicators and execute ‘sell / buy’ orders when the set conditions are met.  You need not monitor the market on hourly basis but this will happen based on the ‘algorithm’.  

The basis of any computer application and its functions is algorithm. With the advent of advanced technologies, almost every sector is basing its effectiveness on this piece of logical coding. Algorithms leverage user data, past patterns and a designated set of prespecified instructions to achieve the set goals. In the financial domain, the rise in demand for reducing transaction cost is fuelling the demand for use of algorithms in everything, from customer service to stock trading.  In Mathematics and Computer Science, an algorithm, is a finite sequence of well-defined instructions, typically used to solve a class of specific problems or to perform a computation.

Algorithmic trading (automated trading, black-box trading, or simply algo-trading) is the process of using computers programmed to follow a defined set of instructions for placing a trade in order to generate profits at a speed and frequency that is impossible for a human trader.  The algorithmic trading strategies follow defined sets of rules, and are based on timing, price, quantity or any mathematical model. Apart from profit opportunities for the trader, algorithmic-trading makes markets more liquid and makes trading more systematic by ruling out emotional human impacts on trading activities.

On a not so fine day, flamboyant television financial expert Lee Gates is in the midst of the latest edition of his show, Money Monster. Less than 24 hours earlier, IBIS Clear Capital's stock inexplicably cratered, apparently due to a glitch in a trading algorithm, costing investors $800 million. Lee planned to have IBIS CEO Walt Camby appear for an interview about the crash, but Camby unexpectedly left for a business trip to Geneva, Switzerland.  Midway through the show, a deliveryman wanders onto the set, pulls a gun and takes Lee hostage, forcing him to put on a vest laden with explosives. The man reveals that his name is Kyle Budwell, who invested $60,000—his entire life savings—in IBIS after Lee endorsed the company on the show. Kyle was wiped out along with the other investors. Unless he gets some answers, he will blow up Lee before killing himself. Once police are notified, they discover that the receiver to the bomb's vest is located over Lee's kidney. The only way to destroy the receiver—and with it, Kyle's leverage—is to shoot Lee and hope he survives.

With the help of longtime director Patty Fenn, Lee tries to calm Kyle and find Camby for him, though Kyle is not satisfied when both Lee and IBIS chief communications officer Diane Lester offer to compensate him for his financial loss. He also is not satisfied by Diane's insistence that the algorithm is to blame. Diane is not satisfied by her own explanation, either, and defies colleagues to contact a programmer who created the algorithm, Won Joon. Reached in Seoul, Joon insists that an algorithm could not take such a large, lopsided position unless someone meddled with it.

Lee appeals to his TV viewers for help, seeking to recoup the lost investment, but is dejected by their response. New York City police find Kyle's pregnant girlfriend Molly and allow her to talk to Kyle through a video feed. When she learns that he lost everything, she viciously berates him before the police cut the feed. Lee, seemingly taking pity on Kyle, agrees to help his captor discover what went wrong.

It turns out that Camby bribed a South African miners' union, planning to have IBIS make an $800 million investment in a platinum mine while the union was on strike. The strike lowered the mine's owners stock, allowing Camby to buy it at a low price. If Camby's plan had succeeded, IBIS would have generated a multibillion-dollar profit when work resumed at the mine and the stock of the mine's owner rose again. The gambit backfired when the union stayed on the picket line. Camby attempted to bribe the union leader, Moshe Mambo, in order to stop the strike, but Mambo refused and continued the strike, causing IBIS' stock to sink under the weight of its position in the flailing company.  In the aftermath, the SEC announces that IBIS will be put under investigation, while Camby is charged with violations of the Foreign Corrupt Practices Act.  .. that's movie Money Monster!

Some movies carry strong messages !  .. .. ..  some may think and relate this to the happening down South, in what was put up by media,  to be a people’s movement – a factory was sought to be closed. There were strikes, lockdown, police shoot out – who was right, on whose side was truth – whether strikes were manipulated ? whether there were political gains – Qs and more Qs – it is clear that it has impacted people in a big way, affecting their livelihood and the stoppage has denied wage earning of many directly and indirectly too !! 

Interesting and life offers so many learnings, if only we are ready to !! 

With regards – S. Sampathkumar
10^th Apr 2o22.