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.

Jeffrey Epstein Was A Mathematics Professor?

While his tenure at the esteemed Manhattan prep school was only a brief one, historically speaking, Jeffrey Epstein is not the only mathematics professor with an “iffy” sexuality by today’s standards?

By: Ringo Bones

When it comes to mathematics professors who had dabbled in “paedophilia”, it seems that only the most scholarly can attest that there are already two of them – i.e. Charles Lutwidge Dodgson, also known as Lewis Carroll and the disgraced billionaire financier who had recently allegedly committed suicide in prison named Jeffrey Epstein. But is there any truth to the “alleged paedophilia” to both math professors?

Even though US President Donald Trump seems to have got off Scott-free when it comes to his “paedophile adventures” with Jeffrey Epstein, it was Prince Andrew who got a grilling by public opinion after an ill-advised interview at the BBC Panorama program. But does the “mathematical profession” really attract some “perverts”?

Dalton – the esteemed Manhattan prep school  where Jeffrey Epstein became a mathematics professor back in the 1970s has long been known for its rigorous academics, repeatedly ranking among the United States’ best private schools while drawing the sons and daughters of New York’s titans of finance, media and art. And students who are enrolled in Epstein’s class vividly remembered the then mathematics professor dressing in furs with open chest revealing chest hairs and blingy gold jewelry. Many say that the only reason Epstein got the job is that a number of New York’s upper crust acquired millions via Epstein’s financial advised backed by his mathematical acumen – although Epstein eventually quit after getting richer off the New York Stock Exchange.

Even though Victorian era mathematician Charles Dodgson – aka Lewis Carroll – who wrote Alice’s Adventures In Wonderland had an extensive collection of photos of naked girls aged 8 to 11. Though Charles Dodgson signed his real name to only his “serious” mathematical works, mathematicians for decades have been intrigued by the rich skein of symbolic logic that is woven into fantasies like in Alice’s Adventures In Wonderland and Through The Looking Glass.

The Mathematical Merits of Jeopardy’s 69 Ban?

I thought it was the premise of the upcoming third Bill and Ted movie, but is there any so-called “mathematical merits” of Jeopardy’s 69 ban?

By: Ringo Bones

I have no idea when it started, but it still surprises me that middle-school kids still giggle whenever the number 69 is uttered in an unguarded moment. But during the last week of April 2019, a new ruling on the iconic TV game-show Jeopardy has been divulged preventing contestants from betting $69 on the Final Jeopardy stage of the game citing the awkward sexual nature of the number.

It is not only the number 69 that got the axe on Jeopardy – it also includes the so-called “Number of the Beast” – i.e. 666 as in $666 Final Jeopardy bets. The betting ban also includes numbers of Neo-Nazism significance, like the number 88, the number 14, and the number 1488 – odd since before the Obama Administration era episodes of Glenn Beck’s show on Fox News, the only “Nazi significance” I know of the number 88 was the 88 millimeter shell used by Nazi era Germany. Outside of joining Richard Butler’s Aryan Nation, it was probably Glenn Beck who made Neo-Nazi numerology more or less common knowledge. Fortunately, a $420 Final Jeopardy bet is still valid.

Ernest William Brown: More Mathematician Than Astronomer?

Despite being more well-known for his astronomical work in lunar theory, Is Ernest William Brown more of a mathematician than an astronomer?

By Ringo Bones 

Born in November 29, 1 in Kingston upon Hull, UK, Earnest William Brown FRS was an English mathematician and astronomer, who spent the majority of his career working in the United States and thus became a naturalized American citizen in 1923. In 1907, he was appointed Professor of Mathematics at Yale University. 

His life’s work was the study of the Moon’s motion (lunar theory) and the compilation of extremely accurate lunar tables. Brown also studied the motion of the planets and calculated the orbits of Trojan asteroids. During the height of his professorship at Yale, Brown was also an active member of the American Mathematical Society as its president from 1915 to 1916. 

Since 1923, the Lunar Tables of Ernest William Brown have reduced the Moon’s complicated motion to a numerical theory that yielded serviceable tables, which only proves that his mathematical skills are way better than his mathematical skills. Brown’s Tables were adopted by nearly all of the national ephemerides in 1923 for their calculations of the Moon’s position and continued to be used with some modification until 1983. With the advent of programmable digital computers, Brown’s original trigonometric expressions, given in the introduction to his 1919 tables (and from which the tables had been compiled), began to be used for direct computation instead of the tables themselves. This also gained some improvement precision, since the tables had embodied some minor approximations, in a trade-off between accuracy and the amount of labor needed for computations in those days of manual calculation. 

By the middle of 20^th Century, the difference between Universal and Ephemeris Time had been recognized and evaluated and the troublesome empirical terms were removed. Further adjustments to Brown’s theory were made, arising from improved observational values of the fundamental astronomical constants used in the theory and from reworking Brown’s original analytical expansions to gain more precise versions of the coefficients used in the theory. Eventually in 1984, Brown’s work was replaced by results gained from more modern observational data – including data from lunar laser ranging - and altogether new computational methods for calculating the Moon’s ephemeris. 

A heavy smoker, Brown suffered from bronchial trouble for much of his life. He was afflicted by ill-health during most of the six years of his retirement and died in New Haven, Connecticut in 1938.