Burning Down the House

by Nicholas Mee on May 21, 2018

In 1998 two students, Larry Page and Sergey Brin, studying for PhDs in computer science at Stanford University, California founded a new tech company called Google. The company was based on an algorithm they had invented and incorporated into a new search engine—the Google browser. Less than two decades later Google is valued at well over $100 billion.

So what exactly is an algorithm and how can it be so incredibly valuable? Page and Brin’s algorithm was for ranking web pages; boring and tedious, perhaps, but incredibly useful if you want to sell advertising space on the internet. We tend to think of algorithms as repetitive procedures carried out step by step by computers until the end point of the calculation is reached. This is certainly true. Algorithms are simple-minded and especially suited to implementation in mechanical calculations, so computer programs are built around mathematical algorithms. But humans were employing algorithms long before computers were invented. At school we learn the ‘three Rs’, reading, writing and a handy set of algorithms known as arithmetic.

An Astrolabe for Little Lewis

Chaucer as a pilgrim from the Ellesmere manuscript of the Canterbury Tales.

Chaucer gives us a vibrant and authentic record of life in the late 14th century. In 1391 Chaucer wrote a manual known as A Treatise on the Astrolabe for ten-year old Lewis who he addresses as ‘Lyte Lowys my sone’. Chaucer refers in passing to the augrim numbers on the astrolabe. These strange sounding numbers are none other than the every day numerals that we are introduced to in our early days at school. If we are being specific today we refer to them as Hindu-Arabic numerals to distinguish them from Roman numerals and other number systems. Roman numerals remained in common use throughout Europe for at least a century after Chaucer’s time. Today, they have little more than a decorative purpose, but they are still sometimes used in dates and inscriptions. Chaucer’s word augrim is actually a variant of the modern word algorithm. We can see why he used this term by trying some arithmetic using Roman numerals. What is V times XX or X times LXXVIII?

Our notation is so ingrained in the way we think that we seldom reflect on just how well it is suited to its task, but good notation is a critically important part of mathematics. In modern numerals the two sums look much simpler: 5 x 20 = 100 and 10 x 78 = 780. The characters ‘LXXVIII’ and ‘78’ represent exactly the same number. Taken at face value they have exactly the same meaning and have indistinguishable mathematical content. But, on the other hand, they exist in completely different contexts and the change in notation makes all the difference in the world. In Roman numerals the answers to the sums are C and DCCLXXX. Multiplication of Roman numerals is an arcane and mysterious art. Without converting into our familiar ‘augrim’ numerals and back again we would struggle to reach the correct answers. It is indisputable that the Hindu-Arabic numeral system is much better suited to performing arithmetical operations and this is why Chaucer knew them as augrim numbers.

On the Art of Calculation with Hindu Numerals

Stamp issued by the Soviet Union to commemorate Al-Khwarizmi.

The word algorithm (or augrim) derives from al-Khwarizmi, the surname of the great 9th century Persian mathematician Muhammad ibn Musa al-Khwarizmi whose books were translated into Latin in the 12th century. Latin manuscripts are often referred to by their opening two words. The Latin translation of al-Khwarizmi’s book on the art of Hindu arithmetic was known as Dixit algorizmi, which simply means ‘and so said al-Khwarizmi’.

Prior to the adoption of Hindu-Arabic numerals the only practical way to perform arithmetic was to use a counting board. A series of lines were inscribed on the counting boards to indicate units, tens, hundreds and so on, and stones were positioned on the lines to represent numbers and then manipulated as they would be on an abacus. These stones or pebbles were known in Latin as calculi, and this is the origin of our words calculation and calculus. (It is also the origin of the chemical name calcium and it is why it is not only mathematicians who suffer from a build up of calculus on their teeth.) By Chaucer’s time these stones had become known as augrim stones. Tokens known as jetons, similar in purpose to gambling chips, were made for use on the counting boards. For several centuries there was a rivalry between those who preferred their accountancy performed on counting boards and those who preferred to perform their arithmetic on paper.

Gregor Reisch: Madame Arithmetica, 1503.

Given the clear advantages of Hindu-Arabic numerals, it is surprising how much time passed before they were universally adopted. This woodcut of Madame Arithmetica by Gregor Reisch illustrates the two methods of accountancy. Modern numerals were regarded with suspicion for many years. For instance, in 1299 the city of Florence issued a decree banning their use. This was partly because they could be altered, so a 0 might become a 6 or a 9 by the addition of a flourish. Even two centuries later in the late 15th century the Mayor of Frankfurt ordered his officials not to use them.

 

The Exchequer

In the UK, the Ministry of Finance is known as the Treasury or the Exchequer and the Head of the Treasury is the Chancellor of the Exchequer. The Exchequer is named after the counting table that was used to perform calculations for taxes and goods in medieval times.

The Exchequer of Ireland (Facsimiles of Irish Manuscripts, volume III, plate xxxvii.)

The Dialogus de Scaccario (Dialogue concerning the Exchequer) written in the late 12th century by Richard FitzNeal describes the operation of the Treasury. Calculations were performed on a large 10 foot by 5 foot table with a raised lip around its edges four fingers in height so that no counters would fall off. On the table was placed a black cloth overlaid with a chequer-pattern of green squares each about the size of a hand, with columns representing pounds, shillings and pence. The table resembled a chess board, known in French as échiquier, hence the name Exchequer for the government office of the Treasury. The Exchequer then came to refer to the twice yearly meetings at Easter and Michaelmas when government financial business was transacted.

With a bit of luck the modern Exchequer will be able to employ some sophisticated algorithms to calculate the tax liabilities of Google and then enforce the payment of a fair proportion of their profits to the UK Treasury.

Bonfire of the Tally Sticks

During the Middle Ages debts were recorded on tally sticks. The tally sticks were notched and then split lengthways, so the two parties could keep matching halves of the record. The example shown here records a debt owed to the rural dean of Preston Candover in Hampshire of a tithe of 20d each on 32 sheep, amounting to a total sum of £2 13s 4d.

Tally sticks were used by the Exchequer to record tax receipts for seven centuries from the time of King Henry I in around 1100AD until they were finally abolished in 1828. Six years later an order was issued to burn the vast quantities of tally sticks that had accumulated over the centuries. The stove in which the blaze was lit set the chimney on fire, leading rapidly to a conflagration that grew until most of the buildings of the Houses of Parliament were destroyed. The blaze over the Thames was captured by the greatest painter of the age John Turner.

The Burning of the Houses of Parliament by J.M.W Turner.

The Houses of Parliament were rebuilt in a grand gothic style designed by architects Charles Barry and Augustus Pugin. They were not completed for several decades; the final touches being added in 1870, some years after the deaths of both leading architects.

 

Further Information

For more about numbers and Google see my recent post: The Googleplex.

 

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The Chamber of Secrets

by Nicholas Mee on April 29, 2018

By the mid-1930s, just five fundamental particles were known. This concise collection of building blocks revealed the true nature of matter and light. Three types of particle: electrons, protons and neutrons, form the wide array of atoms known to chemistry, and the whole electromagnetic spectrum including light is composed of photons. The fifth particle is the positron, the anti-particle of the electron, predicted by Paul Dirac and discovered by Carl Anderson in cosmic rays.

Inside the CMS detector during maintenance. The outer polygonal rings are the muon chambers that track muons created within the LHC.

Who Ordered That?

The world of particle physics seemed neat and tidy with everything in its place. Then, in 1936, Carl Anderson and Seth Neddermeyer announced the discovery of another new particle in cosmic rays, a particle now known as the muon. The whole physics community was taken aback at the unexpected arrival of a particle with no obvious role in the grand scheme of things. Nuclear physicist Isador Rabi captured their surprise in his memorable reaction: Who ordered that?!

Our understanding of particle physics has come a long way since the 1930s. The Large Hadron Collider (LHC) blasts high energy protons together and a whole variety of particles are produced in these collisions. Most such particles are composed of quarks and antiquarks and interact via the strong force. These particles are known as hadrons. There are just a handful of particles, known as leptons, that do not feel the strong force.

The fundamental particles fit neatly into the standard model table of particles. There are three generations of matter particles plus their antiparticles. Each generation contains two types of quark and two leptons. The first generation consists of the up and down quarks, the electron and the electron neutrino, as shown in the first column below.

The particles of the Standard Model. The three generations of fundamental matter particles (fermions) are shown on the left. The force exchange particles (bosons) are shown on the right.

Muons behave just like very heavy electrons, with around 207 times as much mass. Like electrons, they feel the electromagnetic and weak forces, but not the strong force. Unlike the electron, the muon is unstable, decaying into an electron and two neutrinos with a lifetime of around 2 microseconds. The muon is the second generation equivalent of the electron as shown in the standard model table of particles above.

The tauon is the third generation equivalent of the electron. Discovered in 1975, it has almost 3500 times the mass of an electron. It is highly unstable because of its greater mass and decays in less than a trillionth of a second. The charged leptons, that is the electron, the muon, the tauon, and their antiparticles, interact via both the electromagnetic and weak forces. There are also three types of uncharged lepton and they only interact via the weak force. They are known as neutrinos.

The Dragon Of Smoke Escaping From Mount Fuji by Katsushika Hokusai.

Muons undergo numerous interactions when passing through matter, but lose very little energy in each collision. Being so much more massive than electrons, they just brush the electrons to one side as they pass by. Muons are also deflected much less than electrons by the electromagnetic fields within a solid material, so they generate much smaller electromagnetic ripples and lose much less energy in this way as well.

Tauons are more massive than muons and would pass even further through matter, but they decay so rapidly they do not have enough time to travel a significant distance. The upshot is that muons are the most penetrating of the particles produced in the LHC apart from the will-o-the-wisp neutrinos that head off into deep space and disappear without a trace. The outermost parts of the two main detectors at the LHC, known as ATLAS and CMS (Compact Muon Solenoid), are dedicated to tracking the muons created in the proton-proton collisions within the machine.

Enter the Dragon

Muons are continually created in our atmosphere as high energy cosmic rays from distant regions of the galaxy, mainly protons, collide with atomic nuclei in the atmosphere. Every second dozens of these muons pass through our bodies. It is estimated that about 150 muons pass through every square metre of the Earth’s surface every second. These ultra high energy muons can penetrate up to a kilometre of solid ground. They travel further through less dense materials such as air and this has given rise to an important practical application.

Japanese physicists have adapted sensitive muon detectors designed for particle physics experiments to monitor the innards of active volcanoes in a technique known as muon transmission imaging or muography. Detectors are positioned around the volcano and the flux of muons from different directions is measured. There is a greater transmission of muons through low density material such as a cavity within the volcano, so a 3D image of the interior of the volcano can be generated much as an X-ray is used in medicine. By imaging the volcano over a period of time it is possible to see the magma chamber filling with molten lava which offers the potential to save lives by evacuating the area prior to an eruption.

Muograph of Mount Iwo-dake on Satsuma-Iwojima Island. Credit: Hiroyuki Tanaka.

 

The Riddle of the Sphinx

Muons are now being used to investigate the mysteries of the ancient world. The Pharaoh Khufu ruled Egypt over 4500 years ago. Khufu is remembered for building the Great Pyramid on the plateau of Giza close to present day Cairo. Khufu’s monumental tomb is believed to have been constructed in just twenty years. Within the pyramid are three chambers whose names are modern inventions with no historical significance. They are the King’s Chamber, the Queen’s Chamber and an unfinished chamber cut into the bedrock, as shown in the diagram below. The pyramid was looted in antiquity and all that remains inside is the base of a sarcophagus in the King’s Chamber. But the Great Pyramid is vast and there has long been speculation that other hidden chambers may exist deep within its structure.

The muographers have taken up the challenge of revealing secret rooms hidden within the Great Pyramid. Last year a team of physicists led by Kunihiro Morishima of Nagoya University, Japan announced the discovery of a large cavity about 30 metres long above the Grand Gallery within the pyramid. They made this remarkable discovery by placing muon detectors in the Queen’s chamber and using the imaging methods developed for monitoring volcanoes. To confirm the reality of the discovery three different techniques for detecting muons were used, so the physicists are confident that their results are correct.

Perhaps, this is the secret chamber where the royal treasures of pharaoh Khufu were hidden 4500 years ago and they have rested there undisturbed ever since. As yet, no-one knows. The team is currently designing a mini flying robot with the aim of investigating further.

 

Further Information

There is more information about how the standard model was developed in my book Higgs Force: Cosmic Symmetry Shattered.

 

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The Cosmic Mystery Tour

April 6, 2018

My new book The Cosmic Mystery Tour will be published by Oxford University Press later this year. It is a short, easy read offering a lightning trip around some of the greatest mysteries of our universe with lots of attractive illustrations. The trip is interwoven with brief tales of the colourful characters who created modern science, […]

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Hawking Crosses the Event Horizon

March 22, 2018

Stephen Hawking died on 14 March 2018. As a student in the 1960s he was diagnosed with motor neurone disease and given just two years to live. Confounding these predictions, he lived to become the world’s most famous scientist since Einstein. His incredible determination to succeed in the face of any obstacle and his drive to […]

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Hawking Radiation

March 19, 2018

What is the connection between a steam engine and a collapsed star? Not much, you might think. There is, however, a very deep and subtle connection that is still not completely understood. Brewing Up New Theories of Physics James Prescott Joule, the son of a wealthy Manchester brewer, was taught physics by John Dalton, famous […]

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The Pale Blue Dot

March 7, 2018

Voyager I was launched by NASA in September 1977, on course for the outer solar system and beyond. Carl Sagan realised the mission was an opportunity to highlight the immensity of the cosmos and acquire a new perspective on our place within it. After some persuasion, NASA agreed and in 1990 Voyager’s cameras were directed […]

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All for One and One for All

February 10, 2018

In days of old, when knights were bold, it was essential that a knight should bear an elegant mathematical symbol on his coat of arms. Well, perhaps not, but at least the Borromeo family used a design that is well known to mathematicians. The coat of arms of the Borromeo family of merchants and bankers […]

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The Wheel of Fortune

November 7, 2017

We never stray far from devices that chop up our days into hours, minutes and seconds. We are now all synchronized and no-one is out of step with the rest of the world. It is difficult to imagine how different life must have been when days came and went and the passage of time was […]

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Golden Spacequakes

October 24, 2017

Long ago in the year 132 AD the Imperial Astronomer Zhang Heng designed an earthquake detector. The History of the Later Han Dynasty reports that his ingenious invention would alert the Chinese emperor to catastrophic seismic events in distant regions of the empire. Zhang Heng’s seismoscope is described as a bronze vessel two metres in diameter with […]

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The Path to Immortality

October 11, 2017

It is 1697, we enter a dimly lit tavern in one of the less inviting districts of London. Huddled in the shadows we see a man in a loose cloak sitting expectantly with his accomplice at a small table. He has long grey hair, a sharp nose and a determined look in his piercing eyes. […]

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