by DK
Exponential growth and decay relate to many aspects of everyday life. For example, a radioactive isotope decays into another atomic form at an exponential rate, and that results in a half-life, where half the material takes the same amount of time to decay, irrespective of the starting quantity.
Sissa’s concept of wheat on a chessboard is an early example of how quickly numbers can increase with exponential growth.(Numbers from 1 million onward are approximate.) The wheat on this chessboard would total over 18 million trillion grains.
The second half of the chessboard
Recent thinkers have used the chessboard problem as a metaphor for the rate of change in technology over recent years. In 2001, computer scientist Ray Kurzweil wrote an influential essay describing the exponential growth in technology over previous years. He predicted that, like the wheat on the second half of the chessboard, the rate of technological development would rapidly grow out of control, following the model of doubling its previous growth with every leap forward.
Kurzweil argued that this rate of growth in technology would eventually lead to the singularity, which is defined in physics as a point at which a function takes an infinite value. When applied to technology, the singularity marks the point at which the cognitive ability of artificial intelligence will surpass that of humans.
See also: Zeno’s paradoxes of motion • Syllogistic logic • Logarithms • Euler’s number • Catalan’s conjecture
INTRODUCTION
Throughout the Middle Ages, the Catholic Church wielded considerable political power across Europe, and had a virtual monopoly of learning, but in the 1400s, its authority was being challenged. A new cultural movement, known as the Renaissance (“rebirth”), was inspired by renewed interest in the arts and philosophy of the Graeco-Roman Classical period.
The Renaissance thirst for discovery also accelerated a “Scientific Revolution”—classic texts of mathematics, philosophy, and science had become widely available, and inspired a new generation of thinkers. So too did the Protestant Reformation that challenged the hegemony of the Catholic Church in the 1500s.
Renaissance art also influenced mathematics. Luca Pacioli, an early Renaissance mathematician, investigated the mathematics of the golden ratio that was so important in Classical art, and the innovative use of perspective in painting inspired Girard Desargues to explore the mathematics behind it and develop the field of projective geometry. Practical considerations also prompted progress: commerce required more sophisticated means of accounting, and international trade drove advances in navigation, which demanded a deeper understanding of trigonometry.
Mathematical innovation
A major advance in the business of calculation came with the adoption of the Hindu-Arabic number system and an increase in the use of symbols to represent functions such as equals, multiplication, and division. Another significant development was the formalization of a number system of base-10, and Simon Stevin’s introduction of the decimal point in 1585.
To meet the era’s practical needs, mathematicians devised tables of relevant calculations, and John Napier developed a means of calculating with logarithms in the 1600s. The first mechanical aids to calculation were invented during this period, such as William Oughtred’s slide rule, and Gottfried Leibniz’s mechanical calculating device, which was a first step toward true computing devices.
Other mathematicians took a more theoretical path, inspired by the ideas in the newly available texts. In the 1500s, the solution of cubic and quartic equations occupied Italian mathematicians such as Gerolamo Cardano, while Marin Mersenne devised a method of finding prime numbers, and Rafael Bombelli laid down rules for using imaginary numbers. In the 1600s, the pace of mathematical discovery accelerated as never before, and several pioneering modern mathematicians emerged. Among these was philosopher, scientist, and mathematician René Descartes, whose methodical approach to problem-solving set the scene for the modern scientific era. His major contribution to mathematics was the invention of a system of coordinates to specify the position of a point in relation to axes, establishing the new field of analytic geometry, in which lines and shapes are described in terms of algebraic equations.
Another late-Renaissance mathematician who has become almost a household name is Pierre de Fermat, whose claim to fame rests largely on his enigmatic last theorem, which remained unsolved until 1994. Less well known are his contributions to the development of calculus, number theory, and analytic geometry. He and fellow mathematician Blaise Pascal corresponded about gambling and games of chance, laying the foundations for the field of probability.
The birth of calculus
One of the key mathematical concepts of the 1600s was developed independently by two scientific giants of the time, Gottfried Leibniz and Isaac Newton. Following on from the work of Gilles de Roberval in finding the area under a cycloid, Leibniz and Newton worked on the problems of calculation of such things as continuous change and acceleration, which had puzzled mathematicians ever since Zeno of Elea had presented his famous paradoxes of motion in ancient Greece. Their solution to the problem was the theorem of calculus, a set of rules for calculating using infinitesimals. For Newton, calculus was a practical tool for his work in physics and especially on the motion of planets, but Leibniz recognized its theoretical importance and refined the rules of differentiation and integration.
IN CONTEXT
KEY FIGURE
Luca Pacioli (1445–1517)
FIELD
Applied geometry
BEFORE
447–432 BCE Designed by the Greek sculptor Phidias, the Parthenon is later said to approximate the golden ratio.
c. 300 BCE Euclid makes the first known written reference to the golden ratio in his Elements.
1202 CE Fibonacci introduces his famous sequence.
AFTER
1619 Johannes Kepler proves that the numbers in the Fibonacci sequence approach the golden ratio.
1914 Mark Barr, an American mathematician, is credited with using the Greek letter phi (ϕ) for the golden ratio.
[The golden proportion] is a scale of proportions which makes the bad difficult [to produce] and the good easy.
Albert Einstein
The Renaissance was a time of intellectual creativity, in which disciplines such as art, philosophy, religion, science, and mathematics were considered to be much closer to each other than they are today. One area of interest was in the relationship between mathematics, proportion, and beauty. In 1509, Italian priest and mathematician Luca Pacioli wrote Divina Proportione (The Divine Proportion), which discussed the mathematical and geometric underpinnings of perspective in architecture and the visual arts. The book was illustrated by Pacioli’s friend and colleague Leonardo da Vinci, a leading artist and polymath of the Renaissance.
Since the Renaissance, the mathematical analysis of art by means of the “golden ratio,” “golden mean”—or, as Pacioli called it, the Divine Proportion—has come to symbolize geometrical perfection. The ratio can be found by dividing a straight line into two parts, so that the ratio of the longer length (a) to the smaller length (b) is the same as the ratio of the whole line (a + b) divided by the longer length (a). So: (a + b) ÷ a = a ÷ b. The value of this ratio is a mathematical constant denoted by the Greek letter ϕ (“phi”). The name ϕ comes from the ancient Greek sculptor Phidias (500–432 BCE), who is believed to have been one of the first to recognize the aesthetic possibilities of the golden ratio. He allegedly used the ratio in the design of the Parthenon in Athens.
Like π (3.1415…), ϕ is an irrational number (a number that cannot be expressed as a fraction) and can therefore be expanded to an infinite number of decimals in a nonrepeating random pattern. Its approximate value is 1.618. It is one of the wonders of mathematics that this seemingly unremarkable number should produce such aesthetically pleasing proportions in art, architecture, and nature.
Discovering phi
Some believe that proportions related to ϕ can be found
in ancient Greek architecture—and even earlier in ancient Egyptian culture, with the Great Pyramid built at Giza in c. 2560 BCE, which has a base to height ratio of 1.5717. Yet there is no evidence that ancient architects were conscious of this ideal ratio. Approximations to the golden ratio may have been the result of an unconscious tendency rather than any deliberate mathematical intention.
The Pythagoreans, a semi-mystical group of mathematicians and philosophers associated with Pythagoras of Samos (570–495 BCE) had the pentagram, or five-pointed star, as their symbol. Where one side of the pentagram crosses another, it divides each side into two parts, the ratio of which is ϕ. The Pythagoreans were convinced that the Universe was based on numbers; they also believed that all numbers could be described as the ratio of two integers. According to Pythagorean doctrine, any two lengths are both integer multiples of some fixed smaller length. In other words, their ratio is a rational number, so it can be expressed as the ratio of integers. Supposedly, when one of Pythagoras’s followers, Hippasus, discovered that this was not true, his fellow Pythagoreans drowned him in disgust.
LUCA PACIOLI
Luca Pacioli was born in 1445 in Tuscany. After moving to Rome in his youth, he received training from the artist–mathematician Piero della Francesca as well as the renowned architect Leon Battista Alberti, and gained knowledge of geometry, artistic perspective, and architecture. He became a teacher and traveled throughout Italy. He also took his vows as a Franciscan friar, combining monastic pursuits with teaching. In 1496, Pacioli moved to Milan to work as a payroll clerk. While there, he also gave mathematics tuition, one of his students being Leonardo da Vinci, who illustrated Pacioli’s Divina Proportione. Pacioli also devised a method of accounting that is still in use today. He died in 1517, in Sansepolcro, Tuscany.
Key works
1494 Summa de arithmetica, geometria, proportioni et proportionalita (Summary of arithmetic, geometry, proportions, and proportionality)
1509 Divina Proportione (The Divine Proportion)
Written records
The earliest written references to the golden ratio are found in the work of the Alexandrian mathematician Euclid, c. 300 BCE. Euclid’s Elements discussed the Platonic solids described earlier by Plato (such as the tetrahedron), and demonstrated the golden ratio (which Euclid called the “extreme and mean ratio”) in their proportions. Euclid showed how to construct the golden ratio using a ruler and compass.
The good, of course, is always beautiful, and the beautiful never lacks proportion.
Plato
Phi and Fibonacci
The golden ratio is also closely related to another well-known mathematical phenomenon— the set of numbers known as the Fibonacci sequence. It was introduced by Leonardo of Pisa, or Fibonacci, in his 1202 book Liber Abaci (The Book of Calculation). Subsequent numbers in the Fibonacci sequence are found by adding the previous two together: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89….
It took until 1619 for German mathematician and astronomer Johannes Kepler to show that the golden ratio is revealed if a number in the Fibonacci sequence is divided by the one that precedes it. The further along the sequence this calculation is attempted, the closer the answer is to ϕ. For example, 6,765 ÷ 4,181 = 1.61803. Both Fibonacci’s sequence and the golden ratio appear to exist widely in nature. For example, many species of flower have a Fibonacci number of petals, and the scales of a pine cone, viewed from below, are arranged in 8 clockwise spirals and 13 counterclockwise spirals.
Another golden ratio approximated in nature is the golden spiral, which gets wider by a factor of ϕ for every quarter turn it makes. The golden spiral can be drawn by splitting a golden rectangle (a rectangle with side lengths in the golden ratio) into successively smaller squares and golden rectangles, and inscribing quarter circles inside the squares. Natural spiral shapes, such as the nautilus shell, have a resemblance to the golden spiral, but do not strictly fit the proportions.
The golden spiral was first described by French philosopher, mathematician, and polymath René Descartes in 1638 and was studied by Swiss mathematician Jacob Bernoulli. It was classified as a type of “logarithmic spiral” by French mathematician Pierre Varignon because the spiral can be generated by a logarithmic curve.
Leonardo da Vinci supposedly used golden rectangles in his composition of The Last Supper (1494–98). Other Renaissance artists—such as Raphael and Michelangelo—also used the ratio.
Art and architecture
While the golden ratio can be found in music and poetry, it is more often associated with the art of the Renaissance in the 15th and 16th centuries. Da Vinci’s painting The Last Supper (1494–98) is said to incorporate the golden ratio. His famous drawing of the “Vitruvian Man”—a “perfectly proportioned” man inscribed in a circle and square—for Divina Proportione is also said to contain many instances of the golden ratio in the proportions of the ideal human body. In reality, the Vitruvian Man, which illustrated the theories of ancient Roman architect Vitruvius, does not quite align with golden proportions. Despite this, many people have subsequently attempted to relate the golden ratio to the notion of attractiveness in people (see box).
The problem with using the golden ratio to define human beauty is that if you’re looking hard enough for a pattern, you’ll almost certainly find one.
Hannah Fry
British mathematician
Against the golden ratio
In the 1800s, German psychologist Adolf Zeising argued that the perfect human body aligned with the golden ratio; it could be found by measuring the person’s total height and dividing this by the height from their feet to their navel. In 2015, Stanford mathematics professor Keith Devlin argued that the golden ratio is a “150-year scam.” He blamed Zeising’s work for the idea that the golden ratio has historically had a relationship to aesthetics. Devlin argues that Zeising’s ideas have led people to look back at historical art and architecture and retrospectively apply the golden ratio. Similarly, in 1992, American mathematician George Markowsky suggested that supposed discoveries of the golden ratio in the human body were a result of imprecise measurements.
A golden spiral can be inscribed within a golden rectangle. It is created by splitting the rectangle into squares and a smaller golden rectangle, then repeating the process in the smaller rectangle. If quarter circles are then inscribed in the squares, it creates a golden spiral.
Modern uses
Although ϕ’s historical use is debated, the golden ratio can still be traced in modern works, such as Salvador Dalí’s Sacrament of the Last Supper (1955), in which the shape of the painting itself is a golden rectangle. Beyond the arts, the golden ratio has also appeared in modern geometry, particularly in the work of British mathematician Roger Penrose, whose Fibonacci tiles incorporate the golden ratio in their structure. Standard aspect ratios for television and computer monitor screens, such as the 16:9 display, also come close to ϕ, as do modern bank cards, which are almost perfect golden rectangles.
The ratio of beauty
The mask created by Stephen Marquardt has been criticized for defining beauty based on white, Western models.
Studies indicate that facial symmetry plays a major role in determining a person’s perceived attractiveness. However, the proportions defined by the golden ratio appear to play an even greater role. People whose faces have proportions that approximate to the golden ratio (the ratio of the length of the head to its width, for instance) are often cited as being more attractive than those whose faces do not. Studies to date, however, are inconclusive and often contradictory; there is little scientific basis for believing that the golden ratio makes a face more attractive.
Stephen Marquardt, an American plastic surgeon, created a “mask” (see above) based on applying the golden ratio to the human face. The more closely a face aligns with the mask, the more beautiful it supposedly is. Some, however, see the mask—used as a template for plastic surgery—as an unethical, unfounded use of mathematics.
See
also: Pythagoras • Irrational numbers • The Platonic solids • Euclid’s Elements • Calculating pi • The Fibonacci sequence • Logarithms • The Penrose tile
IN CONTEXT
KEY FIGURES
Hudalrichus Regius (early 1500s), Marin Mersenne (1588–1648)
FIELD
Number theory
BEFORE
c. 300 BCE Euclid proves the fundamental theorem of arithmetic that every integer greater than 1 can be expressed as a product of primes in only one way.
c. 200 BCE Eratosthenes devises a method for calculating prime numbers.
AFTER
1750 Leonhard Euler confirms that the Mersenne number 231 − 1 is prime.
1876 French mathematician Édouard Lucas verifies that 2127 − 1 is a Mersenne prime.
2018 The largest known prime to date is found to be 282,589,933 − 1.
Prime numbers—numbers that can only be divided by themselves or 1—have fascinated scholars since the ancient Greeks of Pythagoras’s school first studied them, not least because they can be thought of as the building blocks of all natural numbers (positive integers). Until 1536, mathematicians believed that all prime numbers for n, when employed in the equation 2n - 1, would lead to another prime as the solution. However, in his Utriusque Arithmetices Epitome (Epitome of Both Arithmetics), published in 1536, a scholar known to us only as Hudalrichus Regius pointed out that 211 - 1 = 2,047. This is not a prime number, as 2,047 = 23 × 89.