Time of Mathematics

Ishango Bone
The Ishango Bone is possibly the oldest mathematical artefact still in existence: it was discovered in 1950, in the Democratic Republic of Congo in central Africa, and is named after the region where it was found. It is dates back to the Upper Paleolithic period of human history, and is approximately 20,000 years old.

The bone is 10 cm long and contains a series of notches, which many scientists believe were used for counting. The grouping of the notches might even suggest some more advanced mathematical understanding, like decimal numbers or prime numbers.

Mesopotamian Accounting Tokens
In ancient Mesopotamia, almost 10,000 years ago, scribes and merchants started using small, three-dimensional clay objects as counters, to represent certain quantities, units or goods. Thousands of these were found on archaeological sites across the Middle East, like these from Tepe Gawra in Iraq (from around 4000 BCE):

The cone, sphere and flat disc were used to represent small, medium and large measures of grain. The tetrahedron probably measured the amount of work done in one day.

These two tablets from Susa in Iran were created around 3200 BCE and used a more advanced technique: the counters were pressed into the clay while it was still soft, to create a record:

Again, the triangular and circular impressions represent smaller and larger measures of grain. The patterns across the rest of the tablet were the official seals of the scribes.

Mesopotamian Tablets
This is the oldest known clay tablet with mathematican computations – it was created around 2700 BCE in Sumer, one of the earliest civilisations that flourished in the Middle East.

It shows a multiplication table in cuneiform, which may have been used by student scribes to learn mathematics.

Sumerian Multiplication Table
This tablet shows a multiplication table that was created around 2600 BCE in the Sumerian city of Shuruppak. It is one of the oldest mathematical tablets we have ever discovered.

The table has three columns. The dots in the first two columns represent distances ranging from around 6 meters to 3 kilometres. The third column contains the product of the first two, which is the area of a rectangle with the given dimensions.

Sumer was a region of ancient Mesopotamia in the Middle East. They invented Cuneiform as one of the earliest writing systems, by pressing small, wedge-shaped markers into clay tablets like this one. They also developed the base-60 number system.

Babylonian Tablet (Plimpton 322)
This Babylonian clay tablet, called Plimpton 322, was created around 1750 BCE in Sumeria, during the reign of Hammurabi the Great.

While more than 1000 years older than Pythagoras, the rows and columns on this table contain Pythagorean triples: integer solutions for the equation a2+b2=c2. For example, (3, 4, 5) is a Pythagorean triple because 32+42=52.

The exact purpose of the tablet has been debated by archeologists. Some think that it was a “teachers aid”, designed to help generate right-angled triangles. Others think it may be a very early trigonometry table.

YBC 7289
This circular tablet from the Yale Babylonian Collection, called YBC 7289, was created around 1800 – 1600 BCE in ancient Babylon. It shows the geometric diagram of a square with its diagonals.

The cuneiform numerals indicate that one side of the square is 30 units long, and show how to find the length of the diagonal: 302+302≈42 units.

The tablet shows that Babylonian scribes knew Pythagoras’ theorem, more than 1000 years before Pythagoras was even born. They were also able to calculate square roots and had an estimate for 2 accurate to 6 decimal places. It is the highest computational accuracy ever seen in the ancient world!

While this simple tablet may have just been a practice exercise by a novice scribe, its mathematical and historical importance is enormous.

Babylonian Area Tablets
These two clay tablets from the Yale Babylonian Collection were created between 1800 and 1600 BCE, and contain exercises by student scribes, to calculate the area of different geometric shapes.

Tablet YBC 7290 shows how to calculate the area of a trapezium, by multiplying the average of the bases and the average of the sides.

Tablet YBC 11120 shows how to calculate the area of a circle, using the approximation π=3.

Rhind Papyrus
The Rhind Papyrus is one of the most famous mathematical documents from ancient Egypt. It was written around 1550 BCE by a scribe called Ahmose, who is maybe the earliest contributor to maths in history, whose name we still know today.

The papyrus is around 2 meters long and contains 84 problems about multiplication, division, fractions, and geometry. It was probably used as a kind of “textbook” by other scribes.

One of the most notable sections is a 2n table. This shows how you can write rational numbers of the form 2n, where n is an odd number, as sums of unit fractions.

The papyrus is named after Scottish antiquarian Alexander Henry Rhind, who purchased it in Luxor, Egypt. Today, most of its remains are located at the British Museum in London.

Tomb of Menna
Menna was a chief scribe in ancient Egypt, and in charge of measuring the size of fields for farming, inspected crop yields, reporting to the Pharaoh’s central field administration, and calculating taxes.

The wall paintings in his tomb show the different measuring and calculating techniques used more than 3,000 years ago. For example, in the first row, you can see how long distances were measured using ropes with knots at regular intervals.

The tomb was built around 1420 BCE in the Valley of the Kings.

Thales of Miletus
Thales of Miletus (c. 624 – 546 BCE) was a Greek mathematician and philosopher.

Thales is often recognised as the first scientist in Western civilisation: rather than using religion or mythology, he tried to explain natural phenomena using a scientific approach. He is also the first individual in history that has a mathematical discovery named after him: Thales’ theorem.

Pythagoras of Samos
Pythagoras of Samos (c. 570 – 495 BCE) was a Greek philosopher and mathematician. He is best known for proving Pythagoras’ Theorem, but made many other mathematical and scientific discoveries.

Pythagoras tried to explain music in a mathematical way, and discovered that two tones sound “nice” together (consonant) if the ratio of their frequencies is a simple fraction.

He also founded a school in Italy where he and his students worshipped mathematics almost like a religion, while following a number of bizarre rules – but the school was eventually burned down by their adversaries.

Zeno of Elea
Zeno of Elea (c. 495 – 430 BCE) was a Greek philosopher who his known for his famous paradoxes, which have fascinated mathematicians for centuries.

One example is the paradox of motion: imagine that you want to run a 100 meter race. You first have to run half the distance (50 meters). But before doing that, you have to runn a quarter of the distance (25 meters). Before running a quarter, you have to run 18th,116th, and so on. This is an infinite number of tasks, which means that you’ll never arrive!

Democritus
The Greek mathematician Democritus (c. 460 – 370 BCE), may be the first person to speculate that all matter was made up of tiny atoms and is considered the “father of modern science”. He also made many discoveries in geometry, including the formula for the volume of prisms and cones.

Plato
Plato (c. 425 – 347 BCE) was a philosopher in ancient Greece, and – together with his teacher Socrates and his student Aristotle – laid the very foundation of Western philosophy and science.

Plato founded the Academy of Athens, the first higher learning institution in the Western world. His many writings on philosophy and theology, science and mathematics, politics and justice, make him one of the most influential thinkers of all time.

Eudoxus of Cnidus
Eudoxus of Cnidus (Εὔδοξος ὁ Κνίδιος, c. 390 – 337 BCE) was an ancient Greek astronomer and mathematician. Among his most enduring contributions to astronomy are his planetary models.

History remembers him as the first to write mathematical explanation of the planets. He developed the method of exhaustion in mathematics, which laid the foundation for integral calculus. Eudoxus traveled to several places around the Mediterranean to study. He studied under Plato in Athens, Greece and under Egyptian priests in Heliopolis, Egypt. He later returned to Athens to teach in Plato's Academy during the time Aristotle was a student.

Aristotle
Aristotle (Ἀριστοτέλης, c. 384 – 322 BCE) was a philosopher in Ancient Greece. Together with his teacher Plato, he is considered the “Father of Western Philosophy”. He was also the private tutor of Alexander the Great.

Aristotle wrote about science, mathematics, philosophy, poetry, music, politics, rhetoric, linguistics, and many other subjects. His work was highly influential during the Middle Ages and into the Renaissance, and his views on ethics and other philosophical questions are still being discussed today.

Aristotle is also the first known person to formally study logic, including its applications in science and mathematics.

Euclid of Alexandria
Euclid of Alexandria (Εὐκλείδης, around 300 BCE) was a Greek mathematician and is often called the father of geometry. He book The Elements first introduced Euclidean geometry, defines its five axioms, and contains many important proofs in geometry and number theory – including that there are infinitely many prime numbers. It is one of the most influential books ever published, and was used as textbook in mathematics until the 19th century.

Euclid taught mathematics in Alexandria, but not much else is known about his life.

Euclid’s Elements
Around 300 BCE, Euclid of Alexandria wrote The Elements, collection of 13 books that contained mathematical definitions, postulates, theorems and proofs, and covering topics like geometry and number theory.

It is one of the most famous books ever written, and one of the most influential works in the history of mathematics. Copies were used as textbooks for thousands of years and studied all around the world, with thousands of new editions published

No original copies of the Elements still exist today. This small papyrus fragment dates back to around 100 AD, and may be a part of the oldest existing copy of Euclid’s work.

It is part of the Oxyrhynchus papyri, which were found in 1897 in an ancient rubbish dump in Egypt. The diagram shows the 5th proposition in book 2 of the Elements, a geometric version of the identity x+yx−y=x2−y2.

Pingala
Pingala (पिङ्गल) was an ancient Indian poet and mathematician who lived around 300 BCE, but very little is known about his life. He wrote the Chandaḥśāstra, where he analysed Sanskrit poetry mathematically. It also contained the first known explanations of binary numbers, Fibonacci numbers and Pascal’s triangle.

Bamboo Multiplication Table
Here you can see a set of 21 Bamboo Strip that were created around 2300 years ago in China. When arrenged correctly, they form a multiplication table in base 10, written in ancient Chinese calligraphy.

While earlier civilisations like the Babylonians created multiplications tables in base 60, this is by far the oldest known decimal multiplication table – and it looks very similar to what we still use today.

Archimedes
Archimedes (c. 287 – 212 BCE) was an ancient Greek scientist and engineer, and one of the greatest mathematicians of all time. He discovered many concepts of calculus and worked in geometry, analysis and mechanics.

While taking a bath, Archimedes discovered a way to determine the volume of irregular objects using the amount of water they displaced when submerged. He was so excited by this discovery that he ran out on the street, still undressed, yelling “Eureka!” (Greek for “I have found it!”).

As an engineer, he built ingenious defence machines during the siege of his home city Syracuse in Sicily. After two years, the Romans finally managed to enter, and Archimedes was killed. His last words were “Do not disturb my circles” – which he was studying at the time.

Archimedes Palimpsest
A palimpsest is a scroll or parchment from which the text has been washed or scraped off so that it can be reused. This method was common in the Middle Ages – even for documents by brilliant scientists and mathematicians.

Archimedes of Syracuse lived in the 3rd Century BCE and was one of the greatest mathematicians in history. A Greek copy of some of his work, created around 1000 CE in Byzantium, was later overwritten by Christian monks in Palestine. More recently, forgers added pictures to increase the value of the documents.

In 1998, scientists started studying the Archimedes Palimpsest, and used X-rays, ultraviolet and infrared light to uncover the hidden original text.

Eratosthenes of Cyrene
Eratosthenes of Cyrene (c. 276 – 195 BCE) was a Greek mathematician, geographer, astronomer, historian, and poet. He spent much of his life in Egypt, as head of the library of Alexandria.

Among many other achievements, Eratosthenes calculated the circumference of the Earth, measured the tilt of the Earth’s axis of rotation, estimated the distance to the sun, and created some of the first maps of the world.

He also invented the “Sieve of Eratosthenes”, an efficient way to calculate prime numbers.

Apollonius of Perga
Apollonius of Perga (c. 200 BCE) was a Greek mathematician and astronomer best known for his work on the four conic sections.

Suàn shù shū
The Suàn shù shū (筭數書), which means Book on Numbers and Computation, is one of the oldest mathematical manuscripts from China. It was written around 200 BCE and consists of 200 strips of bamboo.

There are 69 problems, each with a solution, covering topics like arithmetic, fractions, integer factorisation, geometric sequences, inverse proportions, unit conversion, and error handling. Geometry problems show how to find the area of circles and rectangles, as well as the volume of three-dimensional solids, while assuming that π=3.

Hipparchus of Nicaea
Hipparchus of Nicaea (Ἵππαρχος, c. 190 – 120 BCE) was a Greek astronomer and mathematicians, and one of the greatest astronomers of antiquity.

Hipparchus made detailed observations of the night sky and created the first comprehensive star catalog in the western world. He is considered the father of trigonometry: he constructed trigonometric tables and used these to reliably predict solar eclipses. He also invented the astrolabe and solved different problems in spherical trigonometry.

Heron of Alexandria
Heron of Alexandria ( Ἥρων ὁ Ἀλεξανδρεύς, c. 10 – 70 CE) was a Greek mathematician and engineer. He lived in the city of Alexandria in Egypt, and is one of the greatest “experimenter” of antiquity.

His inventions include windmills, pantograph, as well as a radial steam turbine called aeolipile or Hero’s engine. Hero’s formula allows you to calculate the area of any triangle, using just the length of its three sides.

Nicomachus of Gerasa
Nicomachus of Gerasa (c. 60 – 120) was an ancient Greek mathematician who also spent much time thinking about the mystical properties of numbers. His book Introduction to Arithmetic contains the first mention of perfect numbers.

Claudius Ptolemy
Claudius Ptolemy (c. 100 – 170 CE) was a Greco-Roman mathematician, astronomer, geographer and astrologer. He is best remembered for the Ptolemaic or Geocentric model of our universe – that Earth is in the centre and all planets and the sun revolve around this.

While we know today this model is incorrect, Ptolemy’s scientific impact is indisputable. He developed trigonometric tables with many practical applications, which remained the most accurate for many centuries. He also created detailed maps of the Earth, and wrote about music theory and optics.

Diophantus
Diophantus was a Hellenistic mathematician who lived in Alexandria. Most of his works are about solving polynomial equations with several unknowns. These are now called Diophantine equation and remain an important area of research today.

It was while reading one of Diophantus’ books, many centuries later, that Pierre de Fermat proposed one of these equations had no solution. This became known as “Fermat’s Last Theorem”, and was only solved in 1994.

Liu Hui
The mathematician and writer Liu Hui (c. 225 – 295 CE) lived during the Three Kingdoms period of China. He might be the first mathematician to understand and use negative numbers, while writing a commentary with solutions for The Nine Chapters on the Mathematical Art, a famous Chinese book about mathematics.

Hypatia
Hypatia (c. 360 – 415 CE) was a prominent astronomer and mathematician in ancient Alexandria. She was also the first female mathematician whose life and work are reasonably well recorded. She edited or wrote commentaries on many of the scientific books of her time, and constructed astrolabes and hydrometers.

She was renowned during her life as a great teacher, and she advised Orestes, the Roman prefect of Alexandria. Orestes’ feud with Cyril, the bishop of Alexandria, led to Hypatia being murdered by a mob of Christians.

Zu Chongzhi
Zu Chongzhi (祖沖之, 429 – 500 CE) was Chinese astronomer, mathematician, writer, politician and inventor.

He calculated Pi accurately to 7 decimal places – a record which was not surpassed until 800 years later. To do this, he approximated a circle with a 24,576-sided polygon.

Zu also discovered the formula 4 over 3 πr cubed for the volume of a sphere. His precise astronomical observations allowed him to create a new, more accurate calendar and to predict solar eclipses. He also calculated that Jupiter takes almost 12 years to orbit the sun.

Aryabhata
Aryabhata (आर्यभट, 476 – 550) was one of the first mathematicians and astronomers in the golden age of Indian mathematics. He defined trigonometric functions, solved simultaneous quadratic equations, found approximations for π, and realised that π is irrational.

Guass
This code is a pure Julia implementation of the primal-dual predictor corrector found in cvxopt, written with an emphasis on robustness, brevity, portability and numerical stability.

Examples:
LASSO, Trend Filtering, Group-LASSO, Support Vector Machines (primal, dual, kernel), Support Vector Regression, Quantile Regression, Robust Regression, Non-Negative Least Squares, Convex.jl integration, JuMP Integration

Euclid of Alexandria
Euclid, sometimes called Euclid of Alexandria to distinguish him from Euclides of Megara, was a Greek mathematician, often referred to as the "father of geometry". He was active in Alexandria during the reign of Ptolemy I (323–283 BC). His Elements is one of the most influential works in the history of mathematics, serving as the main textbook for teaching mathematics (especially geometry) from the time of its publication until the late 19th or early 20th century. In the Elements, Euclid deduced the principles of what is now called Euclidean geometry from a small set of axioms. Euclid also wrote works on perspective, conic sections, spherical geometry, number theory, and rigor.

Examples
LASSO, Logistic Regression, Maximum-Volume inscribed Ellipsoid, Non-Negative Matrix factorization, Dictionary Learning, Tensor Factorization, Deep Learning.

Word-Building
There's more to the life of a machine learning model, I am sad to report, than optimizing for training accuracy. And when you can't have everything at once, the best thing you can do is compromise. But what better compromise than the optimal one?

Paper

Efficent Evaluation of Scaled Proximal Operators
Here I discuss discussion the linear algebraic tricks used to solve proximal subproblems efficiently. Be forewarned, the number of times I used the Woodbury Matrix Identity is nearly criminal. This project led to the development of QSip.

Paper

Tail Bounds for Stochastic Gradient Descent
My first paper in grad school. Here I investigated at the role concentration plays in randomized algorithms for optimization, in particular stochastic gradient descent.

Paper

4 Years of Advice Animals
I saw a beautiful spindle diagram in a natural history museum in montreal (it looked a little like this or this) and I wondered if I could do something similar for memes on the internet. Perhaps memes obeyed similar forces, living and dying in the microclimates of our imagination. A long shot, perhaps. But philosophical pretentions aside, I enjoy at least the visual analogy. Here it is - a D3 visualization of Jason Braughmerger's meticulously mined reddit dataset showing all posts in /r/AdviceAnimal to Dec 2014 with over 50 up-votes. Enjoy!

Site
Making of (coming soon)