GEOMETRY AND ARITHMETIC IN DIFFERENT CULTURES CONDUCTED BY DEPARTMENT OF MATHEMATICS UNIVERSITY OF MORATUWA MS SHANIKA FERDINANDIS MR. KEVIN RAJAMOHAN History and Philosophy of Mathematics MA0010

FOR YOUR ATTENTION NOTE 1 : STUDENTS ARE REQUIRED TO BE PRESENT FOR ALL THE LECTURES. NOTE 2 : IT IS STRONGLY ADVISED FOR THE STUDENTS TO GO THROUGH THE KEYWORDS, TO BECOME FAMILIAR WITH THE TERMS AND THEIR MEANINGS NOTE 3 : REFER THE DICTIONARY WHEN YOU COME ACROSS A NEW TERM. 9/3/2015 Department of Mathematics,UOM

9/3/2015 Department of Mathematics,UOM Geometry A perspective from different cultures

Beauty of Geometry 9/3/2015 Department of Mathematics,UOM

What is Geometry? Geometry is the branch of mathematics that is concerned with points, lines, surfaces and solids, and their relation to each other. E.g.: One branch of Geometry is concerned with tessellation, which means covering a surface by repeated use of a single shape. A good application of this concept is the division of coverage areas as hexagonal shapes in the telecommunication industry. In modern times, geometric concepts have been generalized to a high level of abstraction and complexity, and have been subjected to the methods of calculus and abstract algebra, so that many modern branches of the field are barely recognizable as the descendants of early geometry. 9/3/2015 Department of Mathematics,UOM

Tessellation patterns 9/3/2015 Department of Mathematics,UOM

Hexagonal tessellation : used in wireless mobile communication 9/3/2015 Department of Mathematics,UOM

History of geometry 9/3/2015 Department of Mathematics,UOM

Babylonian Geometry 9/3/2015 Department of Mathematics,UOM

History of Geometry Babylonian Geometry (3000 B.C.) 1. We have evidence ( from clay tablets) for the usage of square roots, cube roots in geometric problems. (Even 1500 years before Pythagoras) 2. Exercise 1:Calculate the diagonal d of a rectangle gate of height h and width w. (good approximation for h>w) 9/3/2015 Department of Mathematics,UOM

Babylonian geometry cont.. 3. Discovered some computations of area and volume. They were all empirically discovered. 4. Area of a circle was given by c 2 /12 where c is the circumference of the circle( here π =3). 5. The accuracy of the formulae were subjective. 6. Geometry was not a separate mathematical field in this period. It was always connected with practical problems such as surveying, construction, astronomy. 9/3/2015 Department of Mathematics,UOM

Egyptian geometry 9/3/2015 Department of Mathematics,UOM

Egyptian geometry 1. Rhind papyrus and Moscow papyrus are the chief information sources of this era. 2. Egyptians also used geometry as a practical tool to solve problems. (E.g. : Defining the field shapes around the river Niles,combination of astronomy and geometry to construct temples) 3. Area of a circle was given by (8d/9) 2 where d is the diameter of the circle.( here π =3.1605). 4. Some crude approximations. ab c d 9/3/2015 Department of Mathematics,UOM

Egyptian geometry cont.. 5. Some excellent approximations : Volume of a frustum of a square based pyramid. Exercise 2. 1) Draw a truncated pyramid of height h with a square base where a and b are the side length of the top and the bottom of the frustum. 2)Show that the volume V of the frustum is given by 6. They expressed problems verbally ("story problems") and solved them without proof. 9/3/2015 Department of Mathematics,UOM

A modern version of the “ Story problem” "If you are told: A truncated pyramid of 6 for the vertical height, by 4 on the base by 2 on the top. You are to square this 4, result 16. You are to double 4, result 8. You are to square 2, result 4. You are to add the 16, the 8, and the 4, result 28. You are to take one third of 6, result 2. You are to take 28 twice, result 56. See, it is 56”. You will find it right 9/3/2015 Department of Mathematics,UOM

Greek geometry 9/3/2015 Department of Mathematics,UOM

Classical period ( B.C.) Growth of geometry in this period is attributed to different educational institutes. Major contributions in this period were Euclid’s Elements and Apollonius’ conic sections. Ionian School 1. Thales ( BC) leaded the school. 2. Calculated heights of pyramid by comparing its shadow with the shadow cast by a stick of known height. 9/3/2015 Department of Mathematics,UOM

Classical period cont.. The Pythagoreans 1. Pythagoras was the founder. 2. The concept that geometrical figures are abstractions and are clearly distinct from real physical objects or pictures was developed (E.g. : For Egyptians a line was merely a rope or an edge of a field but wasn’t an abstraction) 3. Discovery of Pythagorean triples :integers which could be the sides of a right triangle. When m is odd m,(m 2 - 1)/2,(m 2 +1)/2 are such a triple. 4. They were not able to express non- whole number ratios. 9/3/2015 Department of Mathematics,UOM

Classical period cont.. 5. Such ratios were identified as incommensurable ratios. E.g.. such findings challenged the Pythagorean doctrine that All phenomena in the universe can be reduced to whole numbers or their ratios. 6. In modern mathematics we call incommensurable ratios as irrational numbers. 7. Pythagoreans identified numbers with geometry. 8. Pythagorean theorem, Theorems on triangles, parallel lines,polygons,circles are credited to the Pythagoreans. 9/3/2015 Department of Mathematics,UOM

Classical period cont.. Eleatic School 1. The relation of the discrete to the continuous was heavily debated. 2. Notions such as length, area and volume are continuous were brought to the surface. 3. Zeno’s 4 paradoxes ( refer in detail) 4. Volume of cone, pyramid was discovered. 9/3/2015 Department of Mathematics,UOM

Classical period cont.. Sophist school 1. Many mathematical results obtained, were due to the three famous construction problems dealt by this school. 2. They are, a) Constructing a square,equal in area to a given circle (Exercise: Find an approximate way to do this!!) b) Constructing the side of a cube whose volume is double that of a cube of a given edge. c) Trisecting any angle ( is it possible for any angle??) 9/3/2015 Department of Mathematics,UOM

Classical period cont.. Further contributions were made by Platonic, Eudoxus’ Aristotle schools. Direct method and indirect method of proof, Dealing irrational numbers using geometry, Notion of magnitude and separation between number and geometry. Definition of an Axiom and Postulate, Definition of a point, line, curve and surface are some of the results of their research. 9/3/2015 Department of Mathematics,UOM

Classical period cont.. Axiom :A general truth for all science. Postulate: Acceptable principal for a particular science. Point: is indivisible and has a position. Curve : is generated by a moving point. Surface: is generated by a moving curve. FACT : The Academy of Plato had the motto “Let none unversed in geometry enter here".  How many of you would have got entrance to the Academy of Plato???? 9/3/2015 Department of Mathematics,UOM

Euclid’s contribution to Classical geometry 1. Euclid’s work known as the Elements ( A collection of 13 books) has influenced the course of mathematics as no other book has. 2. He comes up with 5 axioms & 5 postulates ( This will be covered in the next lecture), from which he develops and proves different prepositions. E.g. Preposition 47: Pythagorean Theorem. Preposition 29: A straight line falling on parallel straight lines makes alternate interior angles equal to one another, corresponding angles equal and the interior angles on the same side equal to two right angles. 9/3/2015 Department of Mathematics,UOM

Euclid’s contribution to classical geometry cont.. 3. He introduced Geometrical algebra: Where algebraic problems are solved using geometry (Though algebra has still not been explored in the period) Some concepts in geometrical algebra I. Numbers are represented by Line segments. a II. Product of two numbers becomes an area of a rectangle, with sides whose lengths are the numbers. b a (area= a x b) III. Addition of two numbers is translated into extending one line length by the length of the other. a + b = ( a+b) 9/3/2015 Department of Mathematics,UOM

Euclid’s contribution to classical geometry cont.. E.g. will be considered as “If a straight line is cut at random, the square on the whole is equal to the squares on the segments and twice the rectangle contained by the segments” Exercise 3: Answer : Here ab could be considered as rectangle and x 2 as a square. Therefore we would need to construct as square equal in area to a given triangle and find its length. 9/3/2015 Department of Mathematics,UOM

Euclid’s contribution to classical geometry cont.. Hint : If the diameter of a semicircle of length (a+b) is divided into two segments of length a and b find the length of the perpendicular line drawn from the common point of the segment s to the semi circle. Exercises : 4. Euclid also introduced solid geometry concepts. 5. Another Classical geometer is Apollonius who worked on conic sections, which resulted in the finding of ellipse, parabola and hyperbola. 9/3/2015 Department of Mathematics,UOM

Alexandrian period (300 B.C. – 600 A.D.) Key features 1. Geometry of this era gave a base for the development of Trigonometry. 2. Active involvement in the work of Mechanics 3. Approximation of π. 4. Students are expected to explore more about geometry of this period!!!! 9/3/2015 Department of Mathematics,UOM

Alexandrian geometry cont… Archimedes worked in finding area and volume of many basic figures and objects (cylinder, cone etc..)by the method of exhaustion. Was heavily involved in mechanics (Archimedes’ principle ) New curves which were of less interest to the classical mathematicians, were discovered. We shall plot some of them using Matlab. 1. Spiral : 2. Conchoid : 3. Cissoid: (Try to plot them analytically ) 9/3/2015 Department of Mathematics,UOM

9/3/2015 Department of Mathematics,UOM (a=5)

9/3/2015 Department of Mathematics,UOM (a=100,b=3)

Alexandrian geometry cont… Heron who was heavily influenced by Egyptian mathematics found the formula for the area of a triangle where s is the perimeter and a,b,c are the sides of the triangle. The growth of Spherical trigonometry during this period was due to the involvement in astronomy. 9/3/2015 Department of Mathematics,UOM

Geometry in other cultures Hindu Geometry (A.D ) Square,circle and semi circle shapes were heavily used in their altar construction. This civilization developed a system of uniform weights and measures that used the decimal system, a surprisingly advanced brick technology which utilized ratios, streets laid out in perfect right angles, and a number of geometrical shapes and designs, including cuboids, barrels, cones, cylinders, and drawings of concentric and intersecting circles and triangles. Arabic geometry Heavily influenced by Euclid, Archimedes and Heron. Geometry was used in Astronomy ( tangent and cotangent ratios) The earliest thought of non-Euclidean geometry is attributed to Islamic mathematicians. 9/3/2015 Department of Mathematics,UOM

9/3/2015 Department of Mathematics,UOM Arithmetic A perspective from different cultures

What is Arithmetic?? 9/3/2015 Department of Mathematics,UOM In common usage, the word refers to a branch of mathematics which records elementary properties of certain operations on numbers (Addition, multiplication etc..) Professional mathematicians sometimes use the term (higher) arithmetic when referring to number theory.

History of arithmetic 9/3/2015 Department of Mathematics,UOM

Egyptian arithmetic 9/3/2015 Department of Mathematics,UOM

Egyptian arithmetic 9/3/2015 Department of Mathematics,UOM Hieroglyphic number symbols made the number system. Arithmetic was essentially additive (Multiplication and division were reduced to additive processes) E.g. 19/8 was performed as follows (The goal was to find fractions which have unity numerators) / / =19 Therefore 2 +1/4 +1/8 = 19/8

Egyptian arithmetic cont… 9/3/2015 Department of Mathematics,UOM Even multiplication was done using “doubling procedure”.( Now known as Russian Peasant method of multiplication) E.g.: 11 X Egyptians performed arithmetic steps to solve problems, they did not provide justification for the steps. Therefore = 143

Babylonian arithmetic 9/3/2015 Department of Mathematics,UOM “Akkadian system” was the Babylonian Arithmetic. Number system was base-60 (sexagecimal) and positional notation. (Can you think of some 60-base measurements, we still perform??) Addition,subtraction, multiplication and division of whole numbers were performed. They had a table with approximate values for reciprocals. They devised methods to solve quadratic formulae but they neglected the negative roots.

Classical Greek arithmetic 9/3/2015 Department of Mathematics,UOM Till the 6 th century Greek mathematicians did not distinguish numbers from Geometrical dots. Pythagoreans depicted numbers as dots. E.g.: If the pebbles could be arranged as triangles those were called triangular numbers.

Classical Greek arithmetic cont… 9/3/2015 Department of Mathematics,UOM Exercise :Prove that the summation of two consecutive triangular numbers produces a square number using Pythagorean arithmetic. They have also worked on polygonal numbers. Exercise: Show that the n th pentagonal and hexagonal numbers are Classical Greeks rejected the notion of irrational numbers.(They limited such numbers to geometry) Some famous proofs were made during this time. E.g. “Prime numbers are infinite” ( MA1010)

Proof of irrationality: by reductio ad absurdum Some Greeks accepted the fact of incommensurable ratios and proved them. An example is given below 9/3/2015 Department of Mathematics,UOM

Alexandrian Greek arithmetic 9/3/2015 Department of Mathematics,UOM Their number system. They formed a method to find root of a number A. Method:

9/3/2015 Department of Mathematics,UOM Heron found square & cube roots by purely arithmetic procedures.( Without the aid of geometrical algebra) Numbers were no longer visualized as line segments. Students are expected to read on the Hindu and Arabic influences in arithmetic. The Symbolism of the numbers we use today (1,2..) and the discovery of zero are some key features of these periods.

The End 9/3/2015 Department of Mathematics,UOM