India: The Land Of Human Computers?

With a recent warts-and-all Bollywood biopic of Shakuntala Devi, is India the land of “human computers” – i.e. people who can do incredible mathematical feats without the aid of a calculator?

By: Ringo Bones

For much of the 20^th Century, India has become a go-to country for those in the search of people who can do amazing mathematical feats without the aid of a pocket calculator or even a slide rule. From the number theories of Srinivasa Ramanujan to scores of others who can recite the value of pi to several decimal places that necessitate the use of an electronic device much more advanced than a battery-operated electronic pocket calculator, India seems to be the go to place to find them.

Recently, Indian math wizard Shakuntala Devi, often described as a “human computer”, became the subject of a new film that premiers on the online streaming giant Amazon Prime Video on Friday, July 31, 2020. Born in November 4, 1929 in Bengaluru, India, and in her interviews, Shakuntala Devi said she was “doing mathematical calculations from the age of 3 in my head” and that her father, a circus artist, discovered her felicity with numbers while playing cards with her when he discovered that she was beating him not by cheating – but by memorizing the cards.

At the age of 6, Shakuntala Devi first displayed her extraordinary mathematical skills in a public performance in the city of Mysore in Karnataka the southern state where she was born. She taught herself reading and writing and for decades travelled around the world doing impossibly complex mental calculations before audiences in universities and theaters and in radio and television studios.

In 1950, when Shakuntala Devi participated in a BBC television show, her answer to a problem differed from the host’s. That was because, as she pointed out, there was a flaw in the question. She was proved right when experts re-examined the numbers. In 1977 in the American city of Dallas, she beat Univac, one of the fastest supercomputers ever built during that time. And for her 1982 Guinness Book of World Records recognition s the fastest human computer, she multiplied two 13-digit numbers, randomly picked by a computer, in front of an audience of 1,000 at the Imperial College of Science and Technology in London. It took her 28 seconds, including the time to recite the 26-digit answer. For much of her professional life, she strove to simplify mathematics for students before passing away back in April 21, 2013 in the Bangalore Hospital, Bengaluru, India.

Johann Daniel Titius: Original Author Of Bode’s Law?

Even though he’s not a well known household name like Newton, did the astronomer and mathematician Johann Daniel Titius the original author of Bode’s Law?

By: Ringo Bones

It has since been rechristened as the Titius-Bode Law and in his honor, an asteroid – 1998 Titius -  and a crater on the Moon was named after him, the 18^th Century German mathematician and astronomer Johann Daniel Titius never became a well-known household name like the Englishman Isaac Newton. But nonetheless, Titius did make some important contributions to mathematics, physics, astronomy and biology during his lifetime.

Johann Daniel Titius (1729 – 1796) was born on January 2, 1729 in Konitz Royal Prussia – a fiefdom of the Crown of Poland – to Jakob Tietz, a merchant and council member from Konitz, and Maria Dorothea, née Hanow. His original name was Johann Tietz, but as was customary in the 18^th Century, when he became a university professor, he Latinized his surname to Titius. Teitz attended school in Danzig (Gdansk) and studied at the University of Leipzig (1749-1752). He died in Wittenberg, Electorate of Saxony on December 16, 1796.

Titius proposed his law of planetary distances in an unsigned interpolation in his German translation of the Swiss philosopher Charles Bonnet’s Contemplation de la nature (“Contemplation of Nature”). Titius fixed the scale by assigning 100 to the distance of the planet Saturn from the Sun. On this scale, planet Mercury’s distance from the Sun is approximately 4. Titius therefore proposed that the sequence of planetary distances (starting from Mercury and moving outward) has the form:  4,4 + 3,4 + 6,4 + 12,4 + 24,4 + 48,4 + 96,…

There was an empty place at distance 28, or 4 + 24 (between the planets Mars and Jupiter), which Bode asserted, the Founder of the Universe surely has not left unoccupied. Titius’ sequence stopped with the planet Saturn, the most distant planet then known. His law was reprinted, without his credit, by Johann Elert Bode in the second edition of his Deutliche Anleitung zur Kenntniss des gestirnten Himmels (Clear Guide to Knowledge of the Starry Heaven) in 1772. In later editions, Bode did credit Titius, but this mostly escaped notice and during the 19^th Century the law was usually associated with Bode’s name.

Titius published a number of works on other areas in physics, such as a set of conditions and rules for performing experiments and he was particularly focused in thermometry. In 1765, he presented a survey of thermometry up to that date. He wrote about the metallic thermometer constructed by Hans Loeser. In his treatises on both theoretical and experimental physics, he incorporated the findings of other scientists, such as the descriptions of experiments written by Georg Wolfgang Kraft in 1738.

As a confirmed polymath, Titius was also active in biology, particularly in classification of organisms and minerals. His biological work was influenced by Carolus Linnaeus. Lehrbegriff der Naturgeschichte Zum ersten Unterrichte, his most extensive publication in biology, was on the systematic classification of plants, animals and minerals, as well as the elemental substances: ether, fire, air, water and earth. The standard author abbreviation Titius is used to indicate Johann Daniel Titius as the author when citing a botanical name.

The Mysterious Mathematics Behind Bode’s Law: The Most Puzzling Law Of Science?

Often cited as the most productive – and most puzzling – scientific law at the same time, are there any “mathematical” mysteries behind Bode’s Law?

By: Ringo Bones

This rather “curious” scientific law was named after an 18^th Century German astronomer and mathematician named Johann Elert Bode, but contrary to popular belief, it was actually discovered by Johann Daniel Titius – a German mathematician – back in 1766. However, the empirical relation that gives the approximate distances of the planets from the Sun did not attract attention to the 18^th Century astronomical community until it was publicized by Johann Elert Bode – whose name has since then associated with it – back in 1772.

To the uninitiated, Johann Elert Bode (1747-1826) was an 18^th Century era German astronomer and mathematician who popularized an empirical law that was later named after him, which gives the approximate distances of the planets from the Sun. Bode was also famous for naming the planet Uranus that ended the confusion in the astronomical community at the start of the 19^th Century when the British astronomer William Herschel desired to name the then newly discovered planet as Georgium Sidius after King George III of England.

After examining the work of fellow German mathematician, Johann Daniel Titius, Bode noted that the distances of the various planets from the Sun fell into a curious mathematical sequence. Bode then published a paper which arbitrarily assigned numbers to the planets: 0, 3, 6, 12, 24, 48, 96, and 192. Thus the planet Mercury was numbered 0, planet Venus 3, planet Earth 6, planet Mars 12, and so on, each number being double the last one. When 4 was added to each of these numbers and the result is divided by 10, figures emerged which almost exactly equaled the planets’ distances from the Sun, measured in astronomical units. By the way, an astronomical unit is a unit of distance between the planet Earth and the Sun – which is around 93-million miles or 150-million kilometers.

The only trouble with the law was that back in the time when Bode published it in 1772, there were no planets found at positions 24 or 192. But astronomers searching in position 24 located the asteroids – around the start of the Nineteenth Century – i.e. the discovery of asteroid Ceres in 1801. The planet Uranus, which was discovered back in 1781, occurs at position 192 and conforms almost exactly to Bode’s calculations. Only the outermost planets – Neptune and the dwarf planet Pluto – failed to obey Bode’s Law. Although many attempts have been made to derive a physical explanation for the law, none has completely succeeded.  Today, many astronomers dismiss Bode’s Law as a coincidence and that Bode’s Law is not a rule governing planetary systems. Yet it remains one of the most mysterious statements of natural law formulated by man with the help of mathematics.

Did A 13 Year Old Girl’s Mathematical Skills Help Design The Spitfire’s Weapons System?

It would also have been much of a dream job for boys within her age but did a 13 year old girl helped design the weapons system of the iconic Supermarine Spitfire?

By: Ringo Bones

Now, 80 years after the start of the Battle of Britain on July 10, 1940, the RAF has finally recognized the role of an unseemly inventor and mathematical genius. In 1934, Hazel Hill, a teenage girl from north London, carried out the calculations that proved the new generation of fighter planes – i.e. Spitfires and Hurricanes – should carry eight fifty caliber machine guns, instead of just four. In a documentary researched by her granddaughter, Felicity Baker, a journalist, Hazel Hill’s contribution that allowed the Spitfire to dominate the Battle of Britain and denied the Nazi’s their British conquest finally got the recognition it deserve. Yet – until now – the compelling story of the schoolgirl who helped to win a war has been sadly untold. Hazel Hill’s only recognition was in a memoir written by her father’s superior officer in the UK Air Ministry.

Fortunately for Hazel Hill and her dad, the historic mathematical collaboration happened way before Number 10 declared that the Supermarine Spitfire’s design details were part of the UK’s Official Secrets Act or she could certainly have been denied access to it. In the summer of 1934, Hazel Hill, a 13 year old girl from north London, was approached by her father, Captain Fred Hill, a scientific officer in the UK Air Ministry who was trying to make the case for the new generation of fighter planes. Despite her youth, Captain Hill drew upon his daughter’s mathematical intellect and discussed plans with her as to how it could be possible to arm Spitfires with eight 50 caliber machine guns, as opposed to the four which had been originally suggested. Along with her father, she worked through the night on complex calculations that would shape the future of fighter planes like the Spitfire and the Hurricane. The work was done by lamplight over a kitchen table in north London. Night after night throughout the early months of 1934, Captain Fred Hill and his 13 year old daughter burned the midnight oil plotting graphs and laboring over complex algorithms.

When they got access to the new “calculating machines” of the time – which to our eyes today, resemble very rudimentary vacuum tube based computers – father and daughter worked long into the night analyzing the data that was previously obtained at their kitchen table. Their complicated calculations showed conclusively that each Spitfire needed to be capable of firing 1,000 rounds a minute – per gun. They also calculated the exact distance the Spitfire – whose top speed was about 360 mph – had to be from the enemy to hit them, just 755 feet.  The biggest thing was the huge increase in speed of the new fighters, which was way beyond anything people had experienced before – says mathematician Niall MacKay, the current head of the Department of Mathematics at the University of New York.

  It was tiring, unrewarding work but they both sensed how vital it would prove to be. And their instincts would before long be ratified by history because their intricate calculations would go on to help the RAF secure victory in the Battle of Britain – a triumph that many historians now believe changed the course of World War II. Bent together over their graphs, father and daughter concluded that the new generation of aircraft being built by the UK government to prepare for future war should be armed not with four powerful machine guns but eight – an idea was seen as deeply radical, even improbable at the time. Yet only then, the Hills had come to believe, would a new generation of Spitfires and Hurricanes have sufficient firepower to bring down enemy aircraft. A scientific officer in the UK Air Ministry, Captain Hill managed to convince his superior officers of the importance of his and Hazel’s findings – and six years later, in 1940, their calculations were put to the test in the skies above Britain as the RAF fought Adolf Hitler’s much feared Luftwaffe in a four month battle that has been described as the most important military campaign ever fought. The Battle of Britain is often referred to as the first major military battle which was fought entirely by air forces. Who knew that Reginald Joseph Mitchell’s iconic design could still be improved by a 13 year old girl from north London?

Can Mathematical Modeling Be Used To Stop The Spread of COVID 19?

Can mathematical modeling help us in keeping the individual spread – or basic reproduction number - of COVID 19 to less than 1?

By: Ringo Bones

Since COVID 19 transmission started in late January 2020, the use of mathematical modeling has been at the forefront of shaping the decisions around different non-pharmaceutical interventions to confine the spread of the virus. Mathematical modeling can be used to understand how a virus spreads within a population. The essence of mathematical modeling lies in writing down a set of mathematical equations that mimic reality. These are then solved for certain values of the parameters within the equations.

 The solutions of the mathematical model can be refined when we use information that we already know about the virus spread, for example, available data on reported number of infections, the reported number of hospitalizations or the confirmed number of deaths due to the infection. This process of model refinement – or calibration – can be done a number of times until the solutions of the mathematical equations agree with what we already know about the virus spread. The calibrated model can then be used to tell us more about the future behavior of the virus spread.

One outcome of mathematical models is the predicted epidemic curve representing the number of infections caused by the virus over time. Using different parameters in the model, which may illustrate different interventions, or calibrating the model against different data, can change the predicted epidemic curve.

Mathematical modeling is a powerful tool for understanding transmission of COVID 19 and exploring different scenarios. But, instead of focusing on which model is correct, we should accept that “one model can’t answer it all” and that we need more models that answer complementary separate questions that can piece together the jigsaw and halt the COVID 19 spread.

Farewell Katherine Johnson

Could the United States have won the so-called space race against the then Soviet Union without the help of NASA's African-American mathematician Katherine Johnson?

By Ringo Bones

Fortunately, she got her due credit while still alive given that her most important mathematical works were done during Jim Crow era America. As of February 24, 2020, former NASA mathematician Katherine Johnson, also known as Katherine Goble passed away in Newport News, Virginia. Born in August 25, 1918 in White Sulphur Springs, West Virginia, USA became well known as America’s NASA mathematician whose calculations of orbital mechanics during her employment at NASA were critical to the success of the first and subsequent manned spaceflights.

Katherine Johnson was better known to the generation born after the Apollo moon missions as the NASA African-American mathematician portrayed by Taraji P. Henson in the 2016 movie Hidden Figures about a group of trailblazing African American women mathematicians employed by NASA during the start of America’s Civil Rights movement at the start of the 1960s. Although Katherine Johnson’s mathematical work began earlier in the National Advisory Committee for Aeronautics / NACA – the predecessor of NASA – back in 1953. Before being made famous by the movie Hidden Figures in 2016, Katherine Johnson was awarded with the Presidential Medal of freedom – America’s highest civilian honor – by President Barack Obama in 2015.

During the early days of programmable digital computers – whose active components of which were still largely made with subminiature vacuum tubes first manufactured during 1947 – astronauts were not exactly keen on putting their lives in the care of these early electronic calculating machines, which were prone to hiccups and blackouts according to NASA. So pioneering astronaut John Glenn asked the NASA engineers to “get the girl” – referring to Katherine Johnson to run the computer equations by hand for improved reliability. Johnson and her team of African American women mathematicians did vital work for NASA that eventually made the United States won the space race by successfully landing the first men on the moon and  taking them back safely to earth before President John F. Kennedy’s end of the 1960s deadline.