Multi-Disciplinary Reading - Book Reviews

Fooled by randomness definitely first, followed by ‘The black swan’, then ‘Antifragile’ and ‘Skin in the game’ which is pretty much the chronological sequence in which they were written. Highly recommend ‘The bed of procrustes’ as well which can be read anytime. I have not read ‘Dynamic hedging’ and ‘Statistical consequences of fat-tails’ yet to comment on them.


THE POWER OF GEOGRAPHY: Ten Maps That Reveal the Future of Our World

by Tim Marshall, 2021


This book is a masterpiece on geopolitics. The central idea of this book is that the world is entering a new age of great-power rivalry in which numerous actors, even minor players, are increasingly playing an important role.

The big eye-opener for me was the complex politics that’s playing out in the middle east: Turkey has an acrimonious relationship with pretty much all countries in the region plus they have a love-hate relationship with Russia; Iran is actively working on building a corridor of influence all the way westward to Egypt; with decreasing importance of oil Saudi Arabia is trying to make a major pivot – they have befriended Israel to buy iron dome technology and they are investing heavily in technology. Middle East’s politics is complex where your enemy’s enemy might be your enemy as well.

Below are my notes –


*Australia’s size and location are both a strength and weakness. They protect it from invasion but also held back its political development.

*As a continent it experiences extreme diversity in its climate and topography, from deserts to tropical forests to snow-capped mountains. But the majority of it is taken up by what is known as the Outback, covering about 70 per cent of Australia, much of it uninhabitable.

*Almost 50 per cent of the people live in just three cities – Sydney, Melbourne and Brisbane.

*Its wool, lamb, beef, wheat and wine industries remain world leaders, it holds a quarter of the world’s uranium reserves, the largest zinc and lead deposits, it is a major producer of tungsten and gold, has healthy deposits of silver and it is still a major producer of coal.

*The Malacca Strait is the shortest route from the Indian Ocean to the Pacific. It sees 80,000 vessels pass through it each year, carrying about a third of the world’s traded goods including 80 per cent of the oil heading for Northeast Asia

*China is by far its biggest trading partner, although levels of investment fluctuate sometimes in line with those of diplomatic warmth. China buys almost a third of Australia’s exported farm produce, including 18 per cent of its beef exports and half its barley. It is also a major market for Australia’s iron ore, gas, coal and gold.


*Iran is defined by two geographic features: its mountains, which form a ring of crust on most of its borders, and the mostly flat salt deserts of the interior, along which run lower-range hills roughly parallel to each other. The mountains make Iran a fortress.

*Almost all Iranians live in the mountains. Because they are difficult to traverse, populated mountain regions tend to develop distinct cultures. Ethnic groups cling to their identities and resist absorption, making it harder for the modern state to foster a sense of national unity.

*Lack of water is one of several factors which have held Iran back economically. About one-tenth of the land is cultivated, a mere third of which is irrigated.

*Iran holds the world’s fourth-largest reserves of oil and second-largest of gas

*Strait of Hormuz: one-fifth of global oil supplies pass through it, closure would mean a world of pain. It would also cause massive disruption to oil and gas shipments from Iraq, Kuwait, Saudi Arabia and the UAE, leading to a huge rise in energy prices and potentially a global recession.

*When Tehran feels under pressure, especially when its oil exports are threatened, it uses a variation of a warning issued in 2018: ‘We will make the enemy understand that either everyone can use the Strait of Hormuz or no one.’


*Saudi Arabia is the world’s largest country without a river and the interior is dominated by two vast deserts. It is the largest continuous area of sand in the world, covering a region bigger than France.

*The Saudis view Syria as an Iranian land bridge linking Tehran, via Baghdad and Damascus, to the Iranian-funded Shia Hezbollah militia in Beirut.

*Qatar and Saudi Arabia have been at odds since the mid 1990s, especially since Qatar set up Al Jazeera TV, which the Saudis say is hostile to them.

*The Saudis have long opposed the Brotherhood as it seeks to topple royal dynasties. In 2013 Riyadh supported the military coup in Egypt which deposed the elected Muslim Brotherhood leader Mohamed Morsi and replaced him with General Sisi.

*Saudi’s Vision 2030 accepts that the economy must be diversified, with the focus on technology and the service sectors. Budget projections for the next few years envisage a steady draining of foreign reserves and the sovereign wealth fund.

*The state pays for an exceptionally generous welfare system. Rapidly declining income from oil and gas means that this is unsustainable; but without welfare, and with high unemployment, unrest is all but certain.

*Saudi’s domestic oil consumption has been on an upward trend. The country burns about a quarter of the oil it produces, and that burns through a huge part of the government’s income. Petrol and electricity are supplied to the public at a fraction of even the lowest prices in most developed countries.

*Saudi Arabia is the sixth-largest consumer of oil in the world and air conditioners use 70 per cent of Saudi Arabia’s electricity.

*The kingdom has the largest desalination operation in the world, which successfully produces the majority of its domestic needs. The massive desalination plants require large amounts of electricity, which comes from oil.

*Saudi Arabia owns 5 per cent of Tesla, and has invested heavily in General Motors’ push towards electric cars.

*China has sold the kingdom intermediate-range ballistic missiles, its oil imports have grown rapidly in the past few years, and Saudi Arabia has signed one of the twelve 5G contracts Huawei has won in the region.


*Greece includes more than 6,000 islands. Nowhere in Greece is more than 100 kilometres from water.

*Under international maritime laws a country has 200 nautical miles of Exclusive Economic Zone (EEZ) from its coastline.

*The waters around islands such as Crete, Rhodes and Lesbos belong to Greece, meaning that most of the Aegean is Greek territory – a fact that Turkey does not accept.

*Erdoğan has appeared in an official photograph of a 2019 visit to Istanbul’s National Defence University standing in front of a map showing half of the Aegean as belonging to Turkey.

*The term ‘Thucydides Trap’, originally about the growth of Athenian power and the fear that this caused in Sparta, now refers to the rise of China and the emotions this evokes in the US

*Four-fifths of Greece is mountainous, characterized by jagged peaks and spectacular deep gorges.

*There is limited scope for large-scale farming is why only about 4 per cent of GDP comes from agriculture; Greece imports significantly more food than it exports.

*The two world wars cost Greece 9,500 men but extended its land by 70 per cent.

*Greece, like Italy, believes it is being asked to be Europe’s border police, but without EU funding. Both fear they will be hosting refugees in squalid camps for years to come.

*The discovery of potentially huge reserves of natural gas in the eastern Mediterranean has complicated what was already a potential source of conflict between Greece and Turkey. Gas fields have been found off Egypt, Israel, Cyprus and Greece. Russia is watching the whole scene nervously as its dominant position supplying natural gas to Europe comes under threat.


*Around 97 per cent of Turkey’s land is in Asia, and most of it consists of Anatolia.

*The Bessarabian Gap is the lowland between where the Carpathian Mountains finish and the Black Sea begins. If you hold the Gap, you control the southern east–west route.

*Ottoman’s defeat by the Habsburg Empire at the gates of Vienna in 1683 marked the beginning of a long but steady decline, leading to the collapse of their empire in 1923.

*The Turks do not accept the treaty which then left Greece in control of most of the islands off the Turkish coast, and the loss of Kurdish and Arab territory in Syria still rankles with some.

*Within Turkish military circles, supporters of the concept of ‘Mavi Vatan’ – the Blue Homeland – are usually sceptical of their country’s membership of NATO and believe it to be an American plot (helped by Greece) to prevent Turkey from rising to its rightful place in the world. The Blue Homeland idea encompasses a world view in which Turkey will dominate the three seas around it – the Black Sea, the Aegean and the eastern Mediterranean.

*It’s thought that the name Istanbul comes from Greek speakers referring to visits as eis ten polin – ‘into the city’ – which transmuted into Istanbul.

*By 2020 Turkey had fallen out with Syria, Egypt, Saudi Arabia, the UAE, Kuwait, Israel, Iran, Armenia, Greece, Cyprus and France and had irritated all of its NATO allies by buying the S-400 missile defence system from NATO’s great rival – Russia.

*More than 50 per cent of Turkey’s 85 million population live in the greater Istanbul area or along the narrow coastal plains of the Black Sea and Mediterranean

*Around 90 per cent of the Euphrates River and 45 per cent of the Tigris River originate in the Anatolian highlands. The Euphrates flows into Syria and Iraq, and runs almost parallel with the Tigris before they merge in southern Iraq. The fertile land between them gave birth to the name ‘Mesopotamia’, or ‘between two rivers’.

*It’s often said that the Kurds are the largest nation without a state.

*Seventy per cent of Turkish military equipment is now built domestically, and the country has become the world’s fourteenth-largest arms exporter, although it’s worth noting that orders from NATO allies are few and far between. Its big-ticket project is the TF-X, intended as a state-of-the-art fighter jet to replace the F-16 by 2030.

*Turkey is expanding its capacity and now builds tanks, armoured vehicles, infantry landing craft, drones, sniper rifles, submarines, frigates, and in 2020 launched its first light aircraft carrier, which is capable of transporting helicopter gunships and armed drones.


*The word Sahel derives from the Arabic for shore, or coast, which is how early travellers thought of the area, having made the voyage across the world’s largest dry desert.

*It also forms a 6,000-kilometre-long corridor across Africa, connecting the Red Sea to the Atlantic.

*For thousands of years, periods of extreme dry or wet weather have caused the vast spaces of the Sahara to expand and contract, and thus have shaped the Sahel and its peoples – where they live, what they do and how they behave.

*Much of the Sahel fell under French control at the infamous Berlin conference of 1884–5 (where Europeans carved up Africa.

*French have thousands of expats in the Sahel states, including Niger, home to the uranium mines which help fuel the French nuclear industry and keep the lights on in French houses.

*In the last four decades of the twentieth century lake Chad shrank by 90 per cent causing a huge loss of fish, jobs and income among the millions of people in Chad and neighbouring countries who rely on its waters.

*Across the Sahel region 30 million people face ‘food insecurity’, of whom, according to the UN’s Food and Agriculture Organization (FAO), about 10 million are at extreme risk of hunger.

*Africa has the fastest demographic rate in the world. Between now and 2050 the population of the continent is expected to double from about 1.2 billion to 2.4 billion.

*Niger is the world’s fourth-largest producer of uranium but the government is locked into an unequal relationship with the French state-owned company Areva.

*Beijing is expanding military footprint in the Sahel. In 2015 China passed a law allowing the overseas deployment of the People’s Liberation Army. China persuaded Burkina Faso to end its recognition of Taiwan. In 2017 it opened its first foreign naval base – at Djibouti.

*China has funded an electric railway line from Djibouti to Ethiopia, and Chinese companies are busy building rail links connecting ports in Guinea and Senegal to landlocked Mali. Several hundred thousand Chinese workers are said to be involved in African Belt and Road activities.


*Tourism accounts for almost 10 per cent of the country’s GDP, with close to 1 million people a year venturing into an epic landscape of high mountains, tropical forests, burning deserts, nine World Heritage sites

*Water defines Ethiopia’s geopolitical position and importance. Fresh water is its main strength, and saltwater one of its weaknesses (landlocked). It has twelve large lakes and nine major rivers, most of which supply its neighbours, giving Ethiopia enormous political leverage over them.

*After water, the most defining geographical element about Ethiopia is that a rift runs through it – the East African Rift system. The mountains and valleys it created have long divided the country, and its leaders have always struggled to build the bridges, both literal and symbolic, necessary to bring it together. Most of the coffee plantations, which are the biggest foreign-exchange earner, are situated there.

*Ethiopia is the leading military power in the wider Horn of Africa area.

*Ethiopia sits at the centre of one of the most troubled regions in the world. In this century Sudan, South Sudan, Somalia, Ethiopia and Eritrea have all experienced civil wars, while Kenya has been rocked by wide-scale ethnic clashes and has suffered numerous terrorist attacks by the Somali-based Al-Shabab group.

*Ethiopia was, famously, never colonized but, having built its own empire, it has similar problems within its borders. Ethiopia has nine major ethnic groups among its population. There are nine administrative areas and two self-governing cities, all based on ethnicity.

*It’s the most populous landlocked country in the world.

*Approximately 90 per cent of Ethiopia’s imports and exports travel by sea, and almost all cargo goes via the deep-water port of Djibouti.

*Turkey has been expanding its economic footprint in Ethiopia and is now the second-biggest investor in the country behind China.

*Biggest bone of contention between Ethiopia and Egypt is GERD (Grand Ethiopian Renaissance Dam). It is Africa’s largest hydroelectric power plant being built on Nile.

*For the Egyptians the building of the GERD is an existential matter – this is one of the clearest examples of a country being a prisoner of its geography.

*Egypt is mostly desert, and so 95 per cent of its 104-million-strong population live along the Nile’s banks and delta. Egypt has said it will use ‘all available means’ to defend its interests which has led many analysts to speculate that a major ‘water war’ was on the cards between Egypt and Ethiopia.


*Spain is a vast fortress. From the Mediterranean Sea and the Atlantic Ocean, narrow coastal plains quickly bump up against great walls of mountains and the entire central region is a plateau with its own high ranges and deep valleys. Spain one of the most mountainous countries of Europe.

*Spain’s mountainous terrain and size have always hampered trade links and strong political control, and ensured that the different regions retain strong cultural and linguistic identities. Such are the complexities and passions of these differences that the Spanish national anthem does not have lyrics as no one can agree on what they should be.

*Even at the height of Spain’s powers, its internal geography limited its wealth creation and political unity.

*Arabic in particular left its mark on Spain: more Spanish words are taken from Arabic than any other language apart from Latin. The very name Gibraltar comes from Tariq ibn Ziyad: the rock became known as ‘Jabal Tariq’ (‘Tariq mountain’).

*General Franco imposed an economic system known as autarky – self-sufficiency, state control of prices, and limited trade with other countries. It had a devastating effect. The 1940s became known as ‘Los Años de Hambre’ – the years of hunger.

*Basque Country spans two states, but many Basques consider it still to be one nation called ‘Euskal Herria’. Its language, Euskara, predates the Indo-European tongues of the rest of Europe and is unrelated to any of them. For example, ‘I live in Bilbao’ translates as ‘Ni Bilbon bizi naiz’ and is constructed as ‘I Bilbao in to live am.’

*Catalonia is the wealthiest region in the country, a fact which has played a role in the recent upheavals. Catalonia became wealthy through its textile industry, but now has a diversified economy including heavy industry and tourism.

*Spain is one of Europe’s leaders in renewable energy, especially solar and wind.


*Everett Dolman created a maxim which echoes Halford Mackinder’s famous 1904 ‘Heartland’ geopolitical theory about control of the world, ‘Who rules East Europe commands the Heartland.’ Dolman’s version is: ‘Who controls low Earth orbit controls near-Earth space. Who controls near-Earth space dominates Terra. Who dominates Terra determines the destiny of humankind.’

*For the next few decades, the most important space exploration is Earth Space, particularly Low Earth Orbit (LEO). This is where our communication – and increasingly our military – satellites are being placed.

*More energy is required to get from the Earth’s surface to the Moon than from low Earth orbit to Mars.

*There are five ‘libration’ points near the Earth. These are places where the gravitational effects of the Earth and Moon cancel each other out, allowing objects stationed there to remain in position without having to use fuel.

*Military concept of ‘full-spectrum dominance’ now includes space, from low orbit to the Moon, and eventually beyond.

*Russia, China, the USA, India and Israel have developed ‘satellite-killer’ systems – specialist space weapons that destroy satellites.

*There are currently 3,000 dead satellites and 34,000 pieces of space junk at least 10 cm in size, and many smaller, orbiting the planet.

*Astronaut William Anders had taken the awe-inspiring ‘Earthrise’ photograph showing the surface of the Moon with the Earth in the background. It may be the most famous photograph ever taken and is credited with massive influence on the environmental movement.


Billion Dollar Whale, Wright & Hope, 2018 - This could be the script of a hollywood thriller, as it has political intrigue, international financial fraud involving swiss and wall street banks (Goldman), sovereign wealth funds from 1MDB, Mubadala and ADIA, hollywood glitz and glamour from high profile figures like Leonardo DiCaprio and Scorsese being involved (Wolf of Wall St. was financed by funds siphoned from 1MDB) and of course the mastermind behind it all who most of us haven’t even heard of in Jho Low.

1MDB scandal was a fraud of serious proportions involving $12 billion which was siphoned off from the sovereign fund of Malaysia, explicitly setup for this purpose by Jho Low in which the then president Najib Nazak (now cooling his heels in prison) and family were closely involved. It is amazing how much you can fake it and make it in high-profile finance so some of these deals these so called VCs and wall st. banks are making isn’t surprising. It is perhaps easier to con people not dealing with their own money.

The book details the modus operandi and the people involved in very good detail. If at all you have ever wondered how funds are moved offshore to tax havens and routed back, or how networking works in high-society, or in general how political funding works in developing countries where sovereign funds siphoned are re-routed back to finance election campaigns as was done here to get Najib Razak re-elected.

The story goes all the way to the white house and to royal families in Abu Dhabi and Saudi, to Chinese govt.’s protection for Jho Low once the scandal came to light. It takes different level of guts to do something like this where even the Prime Minister of a country and his wife are in prison but the man behind it all still roams free and still parties in his yacht with celebrities. Financial thrillers are the best kind of books to get into the habit of reading, so if you are contemplating starting somewhere, start here. 8/10


Alex in Numberland, Alex Bellos, 2010 - This book is a love letter to math and it explores math through the history of math and its progress. It also pays its due to Indian contribution to math, be it Vedic texts and their obsession with numbers large and small, Brahmagupta’s Zero or Madhava’s calculus and Ramanujan’s extraordinary intuition with numbers. For most of our evolution we had no numbers and when we did, we couldn’t count more than 5 and just used “more” or “a lot” for any number above 4 or 5.

I have been thinking philosophically about our obsession with numbers and how it feels satisfying while also being vacuous (like our CAGR / XIRR math). It is pointless to measure most of these. If you are doing well, it stands out so much making a compute unnecessary other than for intellectualising. I see some of these now as a conflict between the intuition (internal, single-player, infinite) and the intellect (external, multi-player, finite)

My notes -

A head for Numbers

  • Ethnomathematics - study of how culture/religion influenced mathematics

  • Munduruku (tribe in the Amazon) language has no tenses, plurals and no words for numbers beyond five. Working system of words and symbols for numbers is only ~10000 yrs old at most. Counting anything, be it fruit or children was considered ludicrous by most tribes

  • Staying in a forest for long, one forgets sense of numbers, time and space

  • We are taught numbers are evenly spaced (linear). Our perception of numbers though is logarithmic. We perceive less spaces between larger numbers (space between 10-20 is not same as space between 80-90 - we think in percentage increase 100% vs 12% intuitively!)

  • We perceive ratios more intuitively than we do counting. Which tree has more fruit or which tribe has more people is answered without “counting”

  • A tree 100m away from us and another a 100m behind that tree is not perceived the same (the far one appears shorter). We are however taught they are same (linear) removing this intuition

  • Our perception of time as well is logarithmic, hence time passes faster, the older we get (have oft wondered this), yesterday feels a lot longer than whole of last week

  • We use numbers mainly with quantity (cardinality or counts) and order (ordinality)

  • Ancient Indian numbers used -, =, ≡ and + for 1, 2, 3 and 4 (Very close to the numbers we use today)

The Counter Culture

  • Shepherds use 4 pebbles in their pocket to represent 80 sheep (base 20). Bases simplify how we count and throughout history, we have used 5,10 and 20 as bases. Without sensible bases, numbers become unmanageable

  • Base 12 (duodecimal arithmetik) is considered more natural and versatile - 12 is divisible by 2, 3, 4 and 6 while 10 only by 2 and 5

  • Sumerian cunieform has symbols for 1,10, 60 and 3600 (mix of base 10 and 60). Babylonians used the sexagesimal (base 60) system from Sumerians and made advances in astronomy and time - hence we use 60 seconds to the minute and 3600 to the hour (Attempt was made to decimalise time in 17th century - each day having 10 hours, with 100 minutes each with 100 seconds - 10k secs in a day but it failed)

  • Leibniz fell in love with binary and felt the 1 and 0 represented being and nothingness. I Ching also uses 64 symbols (Fu Hsi) that’s close to modern binary

  • Abacus was invented to count but was more useful with arithmetic. People used to using Abacus can perform math much faster by visually processing arithmetic (soroban) than using pen and paper which uses natural language that’s more tedious


  • Square of a number n is the sum of the first n odd numbers (4^2 = 1 + 3 + 5 + 7 = 16). Intuition in squares made of pebbles visual

  • Pythagorean brotherhood was a health camp, brotherhood and an ashram (math religious cult). Pythagoras discovered that length of vibrating string when halved increased the pitch by an Octave. Finding order in everyday things was their religious awakening

  • Pythagorean brotherhood is the model for several occult secret societies, including freemasonry

  • Right-angled triangle with sides with ratio of 3:4:5 was known as Egyptian triangle

  • Euclid’s Elements was a magnus opus of pedantry and rigour :slight_smile: Nothing assumed, except few basic axioms and everything followed logically from them

  • Since The Elements, logical reasoning has been the standard for all human enquiry

  • Platonic solids are perfectly symmetrical and there’s only 5 of them (tetrahedron, cube, octahedron, icosahedron, dodecahedron)

  • 4 ways to define centre of a triangle - orthocentre, circumcentre, centroid, midcircle (lookup definitions). Euler proved that these always lay on the same Euler line

  • Tessellate - cover a plane so that no region is uncovered (with repeating regular polygons like square, triangle or hexagon). Tesselation thus produced were periodic (repeating)

  • Non-periodic tesselations which for a long term were believed to be impossible can be produced by penrose’s dart and kite (fascinating)

  • 3-D versions of penrose’s tiles is present in quasicrystals. Again, crystals were assumed possible only from platonic solids (Mosaics in Iran, Iraq and Turkey had penrose patterns 5 centuries before it was discovered in the west)

  • Hinduism used geometry to illustrate the divine - Sri Yantra, made with 5 triangles pointing down and 4 pointing up (structure described in a long poem) is hard to construct

  • Origami is now the cutting edge of maths. Protein folding, arterial stents, robotics, solar panels and satellites have applications for origami

Something about Nothing (Zero)

  • Lalitavistara sutra expresses numbers higher than a koti (crore). 100 koti is ayuta, 100 ayuta is kankara and so on until tallakshana which stands for 10^53 (Entire universe measured in meters and then squared would be around 10^53)

  • There were several systems above like dhvajagravati and dhvajagranishamani that could count all the way up to 10^421 (Every atom in the universe 10^80, multipled by planck time 10^43 parts to the second would be 10^140 unique configurations to the universe since its beginning - still smaller than 10^421)

  • The ancient sanskrit texts also have numbers going all the way down to size of a carbon atom. Buddha is said to have been an expert in numbers large and small - a metaphysical obsession perhaps in groping towards the infinite

  • In contrast, Greek culture did not have the Indian hunger for numbers and had their largest number as myriad or M which stood for 10,000

  • MMCDLI is larger than DCLXXXVIII which goes against common sense. Neither Romans, Greeks or Jews had a symbol for zero

  • Vedas contain numbers from Dasa (10), Sata (100) all the way up to Parardha which stands for a trillion. Indian astronomy was way ahead because its astronomers and astrologers had the vocabulary for these large numbers

  • Brahmagupta showed in 7th century how shunya behaved towards other number siblings. (Fascinating stuff based on fortunes, debts and shunya)

  • The less maths was tied to actual things, the more powerful it became

  • Modern decimal system from Indian system with 10 numerals, place value and all-singing, all-dancing zero was brought to Europe Fibonacci in 12th century (Liber Abaci) through the Islamic world

  • Long division and long multiplication were the technological novelty of the 13th century

  • Zero cannot be annihilated or destroyed. It means nothing, it means eternity. The conceptual leap happened in a culture that accepted the void as the essence of the universe

Life of Pi (Geometry)

  • Many mathematicians are poor at arithmetic. Ability to calculate rapidly has no correlation with mathematical insight or creativity

  • Ancient civilisations realised that the ratio of circumference to the diameter of a circle was always constant (pi)

  • Archimedes loved to grapple with problems in the real world unlike Euclid who liked to deal in abstractions

  • Though the origin of calculus is debated between Leibniz and Newton. A version of it was invented in the 14th century by Indian mathematician Madhava

  • 35 digits of pi is sufficient to calculate circumference of earth to the centimeter. 1947, we had 808 digits to pi. 1949, ENIAC took 70 hrs to calculate it to 2037 digits. The numbers obeyed no obvious pattern. Today pi is known till 31.4 trillion digits

  • When people use “lowest common denominator” to mean something basic and unsophisticated, they actually mean “highest common factor” (LCD is usually big while HCF is small)

  • Numbers that cannot be expressed as fractions were “irrational”. Hippasus who proved their existence was drowned at sea by the Pythagorean brotherhood

  • Pi was present in places that did not involve geometry, from pendulum swings, to distribution of deaths in population to probabilities in coin tosses

  • Ramanujan formula calculated pi with remarkable speed and was a industrial strength pi making machine (fascinating formula! how the hell did he come up with it?)

  • Digits in pi are pre-determined but still mimic randomness very well

  • Reuleaux triangle (Similar concept to non-circular wheels - height from floor to top remains constant) and Watt’s drill can drill square holes

The X-Factor (Algebra)

  • Descartes’s La Geometrie introduces algebraic notation used today. He used alphabets from the end for unknowns (x, y and z). The printer ran out of y and z as french used them extensively and so chose “x” instead for unknowns

  • Ease in stating a problem has no correlation with ease in solving it (Eg. Fermat’s theorem)

  • Logarithms turn the complicated process of multiplication into the simpler process of addition (Division was subtraction, square roots, division by 2 and cube roots by 3)

  • Richter scale uses logarithms, so an earthquake of scale 7 is 10x higher in amplitude than one with 6

  • Peter Roget invented thesaurus due to his OCD of making lists to deal with his mental illness. He also invented log-log scale that helped with calculation fo fraction of power like 4^3.5

  • Descartes integrated algebra and geometry with his Catesian co-ordinate system (thus was born a way to visualise abstract notions)

Playtime (Puzzles)

  • Rhind papyrus contained the numbers 7, 49, 343, 2401, 16807 - probably first known occurence of geometric progression. Illustrates the counter-intuitive growth when a number is multiplied by itself a few times

  • Sudoku was invented by Kaji in 1980s but it took off only in the late 90s when it was published in newspapers (Led to 700% rise in pencil sales in UK and top 6/50 books were sudoku related in 2005)

  • Puzzles have always given rise to and grown mathematics. A bridge crossing puzzle from Russia led to the invention of Graph theory by Euler

  • Tangrams, Rubik’s cube, sudoku and fifteen puzzle - 4 internationally crazed math puzzles. Rubik’s cube also involved some clever engineering

Secrets of Succession (Combinatorics)

  • Every even number above 2 can be expressed as sum of two primes (Goldbach’s conjecture). Its as yet unproved though no even number has been found to disagree with it. Primes are scattered unpredictably on the number line

  • Persistence of a number - Number of steps it takes to get to single digit when the digits are multipled with each other Eg. 88 → 8 x 8 = 64 → 6 x 4 = 24 → 2 x 4 = 8 (Persistence is 3). Persistence of large numbers is surprisingly small - because they invariably have a zero somewhere, collapsing under their own weight

  • Recaman sequence generates numbers that look like a garden sprinkler when plotted (Music generated with these sequences sound chilling like a horror movie soundtrack)

  • Prime that can be written as 2^n - 1 is a Mersienne prime

  • GIMPS - Great Internet Mersienne Prime Search. There’s a linear relationship between number of digits in prime and computing power when plotted in log plot

  • The concepts of continuity and discreteness are not completely reconcilable (Zeno’s paradox for eg.)

Gold Finger (Fibs and Golden ratios)

  • Golden ratio - Ratio of A+B : B is same as ratio of A : B. This led to Greek fascination with phi (1.618) and their discovery of it in the Pentagram and subsequent worship

  • For most flowers number of petals is a fibonacci number (Even when its not, like a 4-leaved clover, the avg. is)

  • Golden ratio is approximated by the ratio of consecutive fibonacci numbers (1, 2, 1.5, 1.667, 1.6, 1.625, 1.615, 1.619…)

  • Nautilus shells and several spiral galaxies are logarithmic spirals - self-similar and never growing out of shape as they grow bigger

  • Phi is omnipresent. Peregrine falcons descend on prey in logarithmic spirals. Plants arrange leaves to the golden ratio (of angles) so each leaf gets sunlight

  • Cig. packs, credit cards and books often have length and breadth in golden ratio as we find it most appealing (Even the original iPod)

Chance is a fine thing (Probability)

  • Slot machines in casinos make $25 billion post payouts (2.5x total movie tickets sold)

  • “The book on games of chance” by Cardano was so ahead of its time that it was only published 100 yrs after his death

  • Randomness was not seen as randomness but as an expression of divine will (Caeser’s time)

  • Invention of probability was the root cause of the decline of superstition and religion

  • Pascal’s Wager - If there’s the slightest chance that God exists, a non-believer has nothing to lose by believing in him

  • Every bet in Roulette has a negative expected value. Craps on the other hand is the best deal (Expectation-wise)

  • Core concept of building gaming machines is expected value and the law of large numbers (IGT builds these)

  • IGT’s slot machines can be cherry dribblers or high-volatility - small prizes more often or large prizes rarely (payout rate remains same but plays with emotions differently). The sole objective is to keep the user playing because longer he plays, more he loses

  • Insurance is a negative expectation bet often taken to protect something someone cant afford to lose. Actuarial tables and how slot machines are designed aren’t very different

  • Insuring against losing a non-catastrophic amount of money is pointless (Don’t buy insurance against household electronics for eg.)

  • The belief that jackpot is “due” (because it hasn’t been in awhile) is gambler’s fallacy. Slot machines feed on it. True randomness has no memory of what came before. Truly random iPod shuffle felt less random, so Apple had to tweak it to be less random to make it appear more random :slight_smile:

  • Even the most miserly slot machines have a payback percentage of 85%, lotteries have 50% or less

  • Gambler’s ruin, is the eventual return to zero in a random walk on an iterative bets with long run negative expectations

  • Maximising wealth requires minimising the risk of losing it all. With small edges and judicious money management, huge returns can be achieved

  • Thorp’s “Beat the Dealer” was the first ever “quant” book, followed by his “Beat the market” for financial securities

Situation Normal (Statistics)

  • The ability to describe the world in quantitative rather than qualitative terms changes our relationship with our surroundings

  • Poincare noticed the distribution of sizes/weight of bread and saw the bell curve. Even before Poincare, Quetelet noticed it in frequency of murders in population

  • Maxwell and Boltzmann’s kinetic theory of gases heavily relied on Quetelet’s statistical thinking to explain pressure of gases

  • Reading Pascal’s triangle diagonally gives a fibonacci series. Ancient Indian texts also had versions of Pascal’s triangle

  • The bell curve is ubiquitous because we look for it actively and we often choose what serves our interests

  • Regression and correlation were big breakthroughs in modern scientific thought

The End of the Line (Non-Euclidean Geometry)

  • Hyperbolic geometry or non-Euclidian geometry came about in 19th century on assuming Euclid’s carefully laid out rules as False

  • “The Elements” was the bible of mathematics, so assuming fifth postulate (parallel postulate) to be false was untenable and yet thats what gave rise to a new form of mathematics and subsequently a new understanding of the world

  • Riemann’s lecture of 1954 dealt with positive (spherical) and negative curvature (hyperbolic) of space and integrated Euclidean and non-Euclidean geometry

  • Parallel lines in hyperbolic space get further and further apart from each other

  • Hyperbolic surfaces maximise area while minimising volume (Pringles chips). Some plants and corals expose large hyperbolic surface for nutrition

  • One geometry cannot be more True than the other, just merely more convenient - Poincare

  • Cantor figured out that infinity could come in various sizes (countable infinities and Hilbert hotel)

  • After Riemann and Cantor, mathematics umbilical cord with our experience of reality was cut-off

I thought this would be a daunting read but it was similar in structure to ‘The Joy of X’ and I enjoyed it as much as I enjoyed Strogatz’s book. I haven’t even captured a tiny portion of mathematical, anthropological and historical knowledge contained in this book. Read this even if you hated math in school. I was contemplating taking up serious math few months back and this book only made that resovle stronger. 11/10