The geometry of space

The geometry of space takes many forms: in the symmetry of buildings, the design of bridges, and in the sculptures that decorate our towns and cities.
Several artists, notably the ‘master of optical illusion’ Maurits Escher, have incorporated into their designs certain ‘impossible features that cannot exist in three-dimensional space. These include an impossible cube inspired by his designs and an impossible triangle by the Swedish artist Oscar Reutersvärd.
An attractive sculpture in the form of a Möbius strip is Continuity by the Swiss architect Max Bill; carved from a single piece of granite, it weighs eighty tonnes. A Brazilian sculpture in the form of an enormous helix is Expansion, symbolising progress. The American sculptor Alexander Calder has produced many geometrically inspired mobiles, such as his 1959 Black Cascade.
Some buildings are based on a ‘ruled surface’, a curved surface built from closely packed straight lines; an example is the German pavilion constructed for the Expo 67 World Fair in Montreal. Also at Expo 67 was Buckminster Fuller’s American pavilion, a massive geodesic dome, constructed of pentagons and hexagons made up from triangles. The carbon molecules known as fullerenes or buckyballs are named after him.  
[Austria 1981; Brazil 1953; Germany 1997; Great Britain 2001; Sweden 1982; Switzerland 1974; USA 1998, 2004]

20th-century painting

In the late 19th and early 20th century, artists were fascinated by non-Euclidean geometry and the fourth dimension. This interest in mathematics provoked them to look at the world in a new way and record their observations in paintings and sculpture.
Robert Delaunay (1885–1941) rejoiced in the visual impact of colour. In Rhythme: Joie de vivre (1931) the circles vividly represent the haloes around glowing electric street lights. Delaunay and his contemporaries were aware of the new geometries and the writings of Poincaré. Interpreting this work, Delaunay remarked ‘All is halo’, adding that he had never seen a straight line in his life.
 In Broadway Boogie-Woogie (1943) Piet Mondriaan (1872–1944) created spatial ambiguity by means of coloured lines in a rectangular arrangement. For years he shared other artists’ enthusiasm for portraying the fourth dimension, and in his earlier paintings of white and coloured rectangles separated by straight black lines, the white space represents the fourth dimension.
Victor Vasarely (1908–1997) was much influenced by Mondriaan’s works and studied colour theory, perception and illusion. In Vega-chess, he created this illusion by repeating the same motif over and over again; because of its 4-fold rotational symmetry, this painting can be hung with any side at the top. His 1975 design Tridimensional design is similarly based on symmetry – in this case, 6-fold rotational symmetry.
Bridget Riley (b. 1931) has always made great play of geometry in her paintings. Several are black-and-white geometrical designs, others are coloured displays of parallelograms and other tilings, and some consist of coloured parallel lines, as in this millennium stamp entitled World of Music.
[France 1976, 1977; Great Britain 1999; Hungary 1979; Liberia 1997]

Mathematics in nature

Mathematics occurs throughout nature – from honeycombs and ammonites to the geometry of crystals and snowflakes. There are three regular types of tiling pattern (or tessellation) for the plane: those formed by equilateral triangles, squares and regular hexagons. The hexagonal pattern was discussed by the Greek mathematician Pappus (AD 300) and appears in nature in the form of a bees’ honeycomb.
The delicate structure of a snowflake has six-fold rotational symmetry – rotation by 60° leaves the pattern unchanged – and no two snowflakes are exactly the same. Their hexagonal form was recognised by the Chinese in the second century BC and was later investigated by Kepler, Descartes and others.
The Fibonacci sequence occurs throughout nature, and is related to the logarithmic spiral, found on snail shells and ammonites.
As liquids crystallise they assume the form of polyhedral of various types: fluorite crystals appear as octahedra (eight triangular faces), while lead and zinc sulphide crystals appear as cuboctahedra and truncated tetrahedra.
[Bulgaria 1970; Germany 1968; Great Britain 1989; Hungary 1969; Luxembourg 1973; Switzerland 1961]

Bletchley Park codebreakers

In 1932, led by Marian Rejewski, Polish codebreakers managed to break the codes used by the Germans in their Enigma machines; these contributions awere commemorated on a Polish Enigma stamp.
With the outbreak of the Second World War the Enigma codes had become much more complex and many lives were lost through an inability to break them. It was some time before a team of codebreakers at Bletchley Park in England led by Alan Turing managed to do so. This was followed by the successful breaking by Bill Tutte and others of the Tunny code used by Hitler and other German War leaders to send their messages. To aid them in this, the first computer COLOSSUS was developed in great secrecy at Bletchley Park. It has been claimed that the achievements of Turing and his fellow codebreakers shortened the War by about two years.
[Great Britain 2012; Poland 1983; St Helena 2005]

Pioneers of computing

The central figure of 19th-century computing was Charles Babbage (1791–1871), who pioneered the modern computer age with his ‘difference engine’ and ‘analytical engine’. The difference engine, a digital machine conceived in the 1820s, was a complex arrangement of gears and levers designed to mechanise the calculation of mathematical tables and print the results. Never built during his lifetime, a full-scale version was constructed from his detailed drawings in 1991.
Babbage’s analytical engine can be regarded as the forerunner of the modern programmable computer. Designed to be run by steam power, it contained a store (or memory) and was to be programmed by means of punched cards with holes in specific locations to convey information. Joseph Marie Jacquard (1752–1834) had used such an idea in his ‘Jacquard loom’ to mechanise the weaving of complicated patterns.
In England Alan Turing introduced the idea of a stored-program computer in which the data and instructions are held in an internal store until needed, and helped in the design and construction of COLOSSUS, the first electronic digital machine. Shortly after, the electrical engineers John Mauchly (1907–1980) and J. Presper Eckert (1919–1995), professors at the Moore School in the University of Pennsylvania, designed the ENIAC (Electronic Numerical Integrator And Computer) machine. ENIAC, The Hungarian-born mathematician John von Neumann (1903–1957) contributed to the development of ENIAC and pioneered its successor EDVAC.
The invention of the world wide web by Tim Berners-Lee in the early 1990s led to the information superhighway, whereby all types of information from around the world became easily accessible.  
[France 1934; Great Britain 1991, 2010; Hungary 1992; Malagasy Republic 1990; Marshall Islands 2000; St Vincent 2000]

The New World

The founders of American independence included several highly learned people who encouraged the study of mathematics and science in the late 18th century. Benjamin Franklin (1706–90) invented the Franklin stove, bifocal spectacles, the odometer and lightning rod. He also carried out experiments in electricity, such as his celebrated one on lightning conduction in which he flew a kite in a thunderstorm. Although never claiming to be a mathematician, he was fascinated by magic squares and constructed a remarkable 16 × 16 square in which the numbers in any row, column or 4 × 4 sub-square, have the same value.
Thomas Jefferson (1743–1826), the third president of the United States, extolled the virtues of science and wrote of the importance of calculation (extracting roots, solving quadratic equations and using logarithms). Interested in classical architecture he designed his home, Monticello, and the central rotunda of the University of Virginia. While ambassador in Paris he became enthused by the metric system being proposed in France and strongly advocated decimalising the American coinage, but it was not until 1866 that the United States Congress passed a law legalising the use of metric measurements. Benjamin Banneker (1731–1826) was a self-taught mathematician and astronomer. When 22 years old he designed and built an accurate striking clock, although he had never seen one previously. In later life he constructed accurate astronomical tables. In 1791 he compiled the first of several almanacs, as ‘the creation of a free man of the African race’, and sent it to Jefferson with a plea to end slavery. Banneker was appointed by George Washington, himself a noted surveyor, to help with the surveying and layout of the new capital city.
[Great Britain 1976; Micronesia 1993; USA 1956, 1979, 1980]

Longitude

To determine latitude north or south of the equator a mariner simply measured the angle between the sun or pole star and the horizon. However, to determine longitude east or west of home he had to compare local time with the same time at home: each hour’s difference corresponds to 15° longitude, or about 1000 miles at the equator. Thus, accurate clocks were required that could be used on board a rolling ship, unaffected by changes in temperature and involving no hanging weights. Many lives were lost through inaccurate longitude readings, and in 1714 the British Parliament set up the ‘Board of Longitude’ which offered a £20,000 prize for a reliable timepiece.
In the absence of accurate chronometers, astronomical methods had been sought by Galileo, Huygens, Newton, Halley and others, but it was the chronometers of John Harrison (1693–1776) that won the day. Over a period of forty years he constructed five timekeepers of increasing complexity. The first was used on a voyage to Lisbon in 1735, during which it erred by only a few seconds. Eventually, after much prevarication by the Board of Longitude, the prize was awarded to Harrison for his H5 chronometer.
[Ascension 1971, 1979; Great Britain 1993]

Halley’s comet

Edmond Halley (1656–1742) is primarily remembered for the comet whose return he predicted and which is named after him. While still an undergraduate at Oxford University, Halley sailed to St Helena to explore the skies of the southern hemisphere and prepare the first accurate catalogue of the stars in the southern sky. On his return he was elected to the Royal Society of London where he conversed with other scientists and carried out his researches into the solar system, comets and geophysics.
In the 1680s Halley persuaded Isaac Newton to develop his ideas on gravitation and publish them in the Principia Mathematica; Halley himself paid for the publication. In 1704 he became Savilian Professor of Geometry in Oxford, where he prepared a definitive edition of Apollonius’s Conics. In 1720, he became Astronomer Royal and thereafter spent most of his time at the Greenwich observatory.
The earliest recorded appearances of Halley’s comet was in 240 BC when it was reported by Chinese astronomers. Thereafter it has reappeared every 75 to 80 years. It made a spectacular return in 1066, being depicted in the Bayeux tapestry of the Norman conquest, and the comet’s appearance in 1301 inspired the Italian painter Giotto to include it in his painting Adoration of the Magi.
 Halley himself observed the comet in 1682, and realised that it was the same one that had been seen in 1531 and 1607. In 1705 he predicted its return in late 1758 or early 1759, and its appearance on Christmas Day 1758, several years after Halley’s death, did much to vindicate Newton’s theory of gravitation. It was on the basis of this prediction that the comet came to be named ‘Halley’s comet’.
[Ascension 1986; Great Britain 1986; Montserrat 1986; St Helena 1977, 1986]

Newton’s gravitation

The story of Isaac Newton and the apple is well known. Seeing an apple fall to the ground, he realised that the gravitational force that pulled it to earth was the same as that which kept the moon orbiting around the earth and the earth orbiting around the sun. This planetary motion is governed by a universal law of gravitation, the inverse-square law: the force of attraction between two objects varies as the product of their masses and inversely as the square of the distance between them – so if the distance is tripled, the force decreases by a factor of 9.
In his Principia Mathematica (1687), possibly the greatest scientific work of all time, Newton used this law to explain Kepler’s laws of elliptical planetary motion and account for cometary orbits, the variation of tides, and the flattening of the earth at the poles due to the earth’s rotation.
[Great Britain 1987; Grenada 1987; Monaco 1987; Nicaragua 1971; North Korea 1993]

Reforming the calendar

Before the time of the Romans many different calendars were in use: the Egyptians used a 365-day solar-based calendar, while the Greek, Chinese and Jewish lunar-based calendars consisted of 354 days with extra days added at intervals. The early Roman year had just 304 days; in 700 BC this was extended to 355 days, with the addition of two new months, Januarius and Februarius.
In 45 BC Julius Caesar introduced his ‘Julian calendar’. This had 365¼ days, the fraction being taken care of by adding an extra ‘leap day’ every four years. The beginning of the year was moved to January and the lengths of the months (other than February) alternated between 30 and 31 days. Later writers determined the length of the solar year with increasing accuracy. In particular, Omar Khayyam and Ulugh Beg independently measured it as about 365 days, 5 hours and 49 minutes – just a few seconds out. The Julian year was thus 11 minutes too long, and by 1582 the calendar had drifted by ten days with respect to the seasons.
 In that year, with the aid of the mathematician Christopher Clavius, Pope Gregory XIII issued an Edict of Reform, removing the extra days and correcting the over-length year by omitting three leap days every 400 years, so that 2000 was a leap year but 1900 was not. The Gregorian calendar was quickly adopted by the Catholic World and other countries eventually followed suit: Germany and Denmark in 1700, Britain and the American colonies in 1752, Russia in 1917, and China in 1949.
Meanwhile, the line from which time is measured (0° longitude) was located at the Royal Observatory in Greenwich in 1884, giving rise to an international date line near Tonga. In 1972 atomic time replaced earth time as the official standard, and the year was officially measured as 290,091,200,500,000,000 oscillations of atomic caesium.
[Germany 1982; Great Britain 1975, 1984; Italy 1945; Tonga 1984; Vatican 1982, 2012]

Fibonacci (Leonardo of Pisa)

Hindu–Arabic methods of calculation were also used by Fibonacci (Leonardo of Pisa) in his Liber Abbaci [Book of calculation] of 1202. This celebrated work contained many problems in arithmetic and algebra, including the celebrated problem of the rabbits that leads to the ‘Fibonacci sequence’ 1, 1, 2, 3, 5, 8, 13, …, in which each successive term is the sum of the previous pair.
The ratios 1/1, 2/1, 3/2, 5/3, … of successive terms of the Fibonacci sequence tend to a limit, often called the ‘golden ratio’ and equal to ½(1 + sqrt 5) = 1.618… . This number appears throughout mathematics: it is the ratio of a diagonal and a side of a regular pentagon, and a rectangle with sides in this ratio is often considered to have the most pleasing shape. The removal of a square from such a golden rectangle leaves another one; this process is shown on the Swiss stamp, which also features the closely related logarithmic spiral, found on nautilus shells.
The Fibonacci sequence occurs throughout nature. The spiral arrangements of scales on a pine cone tend to exhibit 8 right-hand and 13 left-hand spirals, while much larger Fibonacci numbers (34, 55, etc.) appear in the spiral arrangements of seeds in the head of a sunflower.

[Dominica 1999; Great Britain 1996; Israel 1961; Macau 2007; Switzerland 1987]

Chess

The game of chess probably originated in the 6th century in India, where it developed from a game called ‘chaturanga’. It quickly spread to Persia, where it was known as ‘shatranj’, and from there to the Arabic world. During the 8th and 9th centuries the Moors took the game to Spain, and thence to the rest of Europe where it became widely established by the 11th century.
The oldest and most celebrated European work on chess is the beautifully illustrated Book of Chess, Dice and Boards, commissioned in 1283 by King Alfonso the Wise of Castile and León; a diagram from this book appears on the stamp from Yemen. Other chess books followed – notably, William Caxton’s The Game and Playe of Chesse, which appeared in 1476. An Italian book of the late 15th century illustrates a royal chess party that took place in Florence in 1493.
 [Djibouti 1980; Great Britain 1976; Yemen 1967]

Alexandria

Around 300 BC, with the rise to power of Ptolemy I and the military successes of Alexander the Great, mathematical activity moved to the Egyptian part of the Greek empire. In Alexandria Ptolemy founded a university that became the intellectual centre for Greek scholarship for over 800 years, and also started its famous library which eventually held hundreds of thousands of manuscripts. The celebrated Pharos lighthouse at Alexandria was one of the seven wonders of the ancient world.
[Great Britain 2003; Greece 1955; Hungary 1980]

Ancient mathematics

Geometrical arrangements of stones have been found in many places. Celebrated examples include the circular pattern of megaliths at Stonehenge in England and the linear arrangements at Carnac in Brittany. Although their exact purpose is unknown, it is likely that their construction had religious significance and was designed to demonstrate astronomical events such as sunrise on midsummer’s day.
[France 1965; Great Britain 1990; Gambia 1997]

Mathematics: a philatelic history

There are many hundreds of postage stamps relating to mathematics, ranging from the earliest forms of counting to the modern computer age. Here you will meet many of the mathematicians who contributed to this story – influential figures such as Pythagoras, Archimedes, Newton and Einstein – and will learn about those areas, such as navigation, astronomy and art, whose study aided this development.
This website is aimed at anyone interested in mathematics and its applications. Although parts of it assume some knowledge of school or college level mathematics, much of it will be of interest to readers without this background. In particular, I hope that it will also attract a philatelic audience. This is not a history of mathematics in the conventional sense of the word. Several important mathematicians and topics are omitted, due to the absence of suitable stamps featuring them, while others may have assumed undue prominence because of the abundance of attractive images. Where appropriate I have let the stamps dictate the story.
Postage stamps are an attractive vehicle for presenting mathematics and its development to general audiences. For some years I have presented an illustrated lecture entitled Stamping through Mathematics to school and college groups and to mathematical clubs and societies. Since 1984 I have also contributed a regular ‘Stamp Corner’ to The Mathematical Intelligencer, and in 2001 I wrote a book entitled Stamping through Mathematics, published by Springer, New York. We are grateful to the postal authorities whose stamps we have featured here. Anyone who feels that their rights have been infringed is invited to contact us and we will correct the situation as soon as possible.  
[Czech Republic 2000; Germany 1977; Great Britain 1987; Greece 1955; Portugal 1981]