“Like the crest of a peacock, like the gem on the head of a snake, so is mathematics at the head of all knowledge.” (Vendanga Iyotisa)

Mathematics has a long story of important human endeavour and is a result of intuitive reasoning combined with experiential learning and exceptional achievements. Although the initial steps in mathematics came about as a result of need by ancient civilisations, the contribution by outstanding personalities cannot be ignored. Among these, one finds three great mathematicians who lived in the 3rd century BC.

Euclid of Alexandria is arguably the most famous mathematician whose life is enshrined in acutest mystery, with very little known about him and not even his birthplace being certain. His Elements of Geometry is regarded as the most successful mathematical textbook ever written. Euclid did not emphasise the practicality of the subject or its usefulness, but was completely devoted to mathematics for its own sake, for the beauty of the logical arguments, the exact reasoning it entails and the completeness in its details. The terms ‘Euclidean Geometry’ and ‘Euclidean Algorithm’ are almost household concepts and feature in a multitude of environments including discrete mathematics, cryptography and geometry. Euclid’s proofs of the infinitude of primes, the irrationality of the square root of 2 and the sum of the interior angles of a triangle are only few of his great masterpieces.

A contemporary of Euclid was Archimedes of Syracuse, considered by many as the greatest creative mathematical genius of the ancient world. Away from the anecdote of shouting “Eureka” naked in the streets of Syracuse upon discovering a method for finding the volume of a golden crown, his innovative ideas gave mathematics a big boost and his work enlightened others to follow suit. He showed how to trisect an angle, construct a regular heptagon, and find the volume and surface area of a sphere, and obtained a value of π correct to three decimal places. A problem attributed to Archimedes is the so-called ‘Stomachion’, which gives a light and enjoyable dimension to deeper mathematical concepts (see Sound Bites). The aim of this puzzle is to rearrange the pieces to form interesting objects such as people, animals and other shapes.

Apollonius of Perga refined and consolidated some of the previous findings of Archimedes and Euclid, hence making them more complete, and added his own contributions, especially in geometry. His most renowned work is the Conics, organised into eight books (the last of which was lost), and containing 389 propositions. In this work, he defined the circular conic and the right circular cone, showed that the different conic sections can be produced from the intersection of a plane with a cone, and coined the terms ellipse, parabola and hyperbola which are still in use today. These conics are also used by architects in the buildings they project (see Photo of the Week).

The creativity and commitment to learning exhibited by these three mathematicians cannot but be cherished. The importance in the simplicity of their results tends to be overseen by many, but we must bear in mind that they laid the foundations of today’s mathematics and as such their findings will always remain relevant and highly applicable.

John Baptist Gauci is a senior lecturer at the Department of Mathematics within the Faculty of Science of the University of Malta.

Did you know?

• The base 10 number system is the most widely used system nowadays, but this was not the only system used throughout the history of mankind.

• It is believed that Mayans used a base 20 system, whereby they used their fingers and toes to facilitate counting larger numbers, while the Babylonians operated a base 60 system.

• The motivation for a base of 60 is subject to many interpretations, but probably the one which screams innovation at its best attributes the ability to count to 12 on one hand (by using the thumb to point to each finger bone of the remaining four fingers) and the other to keep track of the number of dozens so counted.

• The relatively large number of divisors of 60 (namely 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30 and 60) simplifies the mathematical operations of multiplication and division.

• A base of 60 is still in use today in some parts of Asia and is also utilised in measuring time across the world.

For more trivia, see: www.um.edu.mt/think

Sound bites

The intriguing affinity between mathematics and games has captivated the interest of generations and continues to disclose some of the intuition, imagination and creativity underlying a subject that is considered by some as dull and abstract.

• The ‘Stomachion’ (see the main article) is a puzzle similar to the more popular ‘Tangram’. Such simple dissection puzzles pose complex mathematical problems, such as ‘Which are all the different convex polygons that can be assembled with the seven tangram pieces?’ These puzzles are also used to enhance the teaching of mathematics with children of different ages by developing their creative thinking and problem-solving skills. Numerous academic papers have been published in recent years to illustrate how tangrams can be used in the teaching of mathematics and to assess their effectiveness as a teaching and learning tool.

• Another extremely popular puzzle is without any doubts Sudoku. In 2012, McGuire, Tugemann and Civario answered the sudoku minimum number of clues problem, namely “What is the minimum number of clues that can be given such that a sudoku puzzle has a unique completion?” There are over 49,000 sudoku puzzles that are determined uniquely by starting with 17 clues. The conjecture that 16 clues are not enough to determine a unique solution of a sudoku puzzle had been around for a number of years, and this was finally proven in 2012 through an exhaustive computer search. The method used is described in detail in the paper found here: https://arxiv.org/abs/1201.0749.

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