Euclid with acompass( a detailfrom “Theschool ofAthens”)
A short story........ Joannes Stobaeus (Ιωάννης Στοβαίος ) 5 century AD, was the compiler of a valuable series of extracts from Greek authors.He has written:One day a young man visited Euclid and asked him toteach him Geometry. When he learnt the first theorem, heturned to the famous mathematician and asked him “andnow which is the profit of what I learnt?”Euclid called a servant and said to him :” Give to this youngman 3 δεκάρες (cents), so he will have a profit from whathe was taught”.
Use of mathematicsMany times during every school year, studentsask me why they have to learn mathematics.I tell them to take a paper and a pencil andwrite down in which cases they usemathematics during the day.The students comments are:
Use of mathematics When● we buy a car● when Irene follows a recipe to make a cake● when Vasilis goes on foot to his school in the morning following the shortcut,● when we decorate our home or our classroom we are using math principles. People have been using these same principles for thousands of years, across countries and continents.
2000 BCBabylonians used Algebra and Geometry toconstruct buildings. Also they used them inAstronomy. They created a type of calendarand they were able to predict solar and lunareclipse.Reconstruction of the ziqquratof Babylon
1500 BC -Egypt Nile!! the longest river in the world. Every year the flood brought disaster and problems in the fields.● Simple geometry fields borders● Simple metric system to study the Niles flow and flood every year.
Geometric periodThe period of the ancient Greek historybetween the 11th and 8th century BC is knownas Geometric period, and took its name fromthe geometric shapes on the ceramics.
Mathematics and ArtMathematics and Art have a long historicalcorrelation.Many artists have been inspired bymathematics from the ancient years until today.
Golden ratio and ArchitectureThe Golden Ratio appeared in ancient architecture.The examples are many, such as the Great Pyramid in Giza, Egypt which is considered as one of theSeven World Wonders of the ancient world, and the Greek Parthenon . It was also used by the Renaissance artists, who used the Golden Ratio in the design of Notre Dame, etc.
Pyramids and φ (φ from Φειδίας-ancient Greek sculptor - 5th century BC)
ParthenonParthenon is a temple dedicated to the Greekgoddess Athena, built in the 5th century BC. Inits facade and on the floor the golden rectangleis applied (a rectangle whose length to widthratio is the golden ratio)
Ancient Theater of Epidaurus 340-330 BC a/b=(a+b)/a = φ=1.62 b a
Golden ratioThe astronomer Johannes Kepler (1571-1630) wrote, "Geometry has two greattreasures: one is the theorem of Pythagoras,the other one is the division of a segment intomean and extreme ratios, which is, the GoldenMean. The first theorem may be compared toa measure of gold, the second one to aprecious jewel."
Maths and PaintingIt is suggested that mathematics and paintingare so closely related and have so manysimilarities that it is reasonable to considerthem simply as two different but complementaryways of visualizing the world.Mathematics have influenced many artmovements, such as Cubism, Constructivism,Linear perspective, Surrealism.
CubismCubism was a 20th century avant-garde artmovement, pioneered by Pablo Picasso andGeorges Braque Pablo Picasso Pablo Picasso
ConstructivismConstructivism was anartistic and architecturalmovement originating inRussia from 1919 onwardwhich rejected the idea of"art for arts sake" in favourof a praxis directed towardssocial purpose.1915 Dance by Rodchenko
Linear perspectivePerspective (from Latin perspicere, to see through) in thegraphic arts, such as drawing, is an approximaterepresentation, on a flat surface (such as paper), of animage as it is seen by the eye. Pietro Peruginos usage of perspective in this fresco at the Sistine Chapel (1481–82) helped bring the Renaissance to Rome .
Melozzos usage of upward foreshortening in his frescoes at Loreto
Wassily Kandinsky- Abstract paintingHe began completely abstract painting. His forms evolvedfrom fluid and organic to geometric and, finally, topictographic
Leonardo Da Vinci (1452-1519)Leonardo incorporated many mathematicalconcepts into his artwork despite never havingreceived any formal mathematical training. Hehas often been described as the archetype ofthe Renaissance man.
Salvador Dalí (1904–1989) incorporatedmathematical themes in several of his laterworks. His 1954 painting Crucifixion (CorpusHypercubus) depicts a crucified figure upon thenet of an hypercube
Salvator DaliThe Sacrament of the Last Supper (1955)Christ and his disciples are pictured inside agiant dodecahedron.
M.C. Escher (1898-1972) is known for his mathematically inspired work.
Platonic solidsThe Platonic solids (tetrahedron, cube-hexahedron,octahedron,dodecahedron,icosahedron) and other polyhedra are a recurringtheme in Western art. Floor mosaic in St Marks Basilica, Venice by Paolo Uccello
Maths and Music"Music," wrote the great 17th-century Germanmathematician Gottfried Leibniz, "is thesensation of counting without being aware youwere counting."But there is more to this connection thancounting. The French baroque composerRameau declared in 1722:"I must confess that only with the aid ofmathematics did my ideas become clear."
The Pythagoreans of ancient Greece are thefirst researchers known to have investigatedthe expression of musical scales in terms ofnumerical ratios. Their central doctrine wasthat "all nature consists of harmonyarising out of numbers". Also ancientChinese, Egyptians and Mesopotamians areknown to have studied the mathematicalprinciples of sound.
It is believed that some composers wrotetheir music using the golden ratio and theFibonacci numbers to assist them.The golden ratio is apparent in theorganization of the sections in the musicof Debussys Image, "Reflections inWater", in which the sequence of keys ismarked out by the intervals 34, 21, 13,and 8 (a descending Fibonacci sequence),and the main climax sits at the φ position.
Maths and literature “Pure mathematics is, in its way the poetry of logical ideas”..... Albert EinsteinFrom the ancient years until today, many Authors use math in stories, plays orpoems in a variety of different ways.In producing their art writers may :1. Call on math to illuminate a theory (Aristophanes-the Birds, Dostoevsky-The Brothers Karamazov , Tolstoy – War and Peace ).2. Be inspired by mathematical themes to create a work of art based on thethemes (Denis Guedj -Le theoreme du perroquet, A. Doxiadis -Uncle Petros andGoldbachs Conjecture, T. Michaelides- Pythagorean crimes, Growney andUpson).Meanwhile, the last years a new kind of literature has appeared : Mathematicalfiction.3. Want to produce an educational work( Enzensberger);4. Want to write the imagined life of an intriguing mathematician ( Petsinis-The French Mathematician ).Topics such as logic, geometry, measurement, number theory, statistics,topology, set theory, and calculus arose naturally from the readings.
th 8 century BCHomer gives a poetic description of number 700and a geometric presentation of Sicily in chapterμ of Odyssey : «... και φτάσαν στο τριγωνικό νησί της Σικελίας/όπου του Ήλιου έβοσκαν τα όμορφα θρεφτάρια/γελάδια σε κοπάδια εφτά, αρνιά σε άλλα τόσα,/πενήντα το καθένα τους...»“...and they arrived at the triangular island ofSicily / where the Suns beautiful cattlesgrazed/the cows in seven herds, and the sheepin as many/ fifty in each of them...”
Prometheus (of Aeschylus ,460 BC) was chained to the rocks, punished by Zeus for daring to offer to the mortals the knowledge which was suitable only for Gods . Among the gifts that he gave to the people, there are goods destinated only for immortals: «...και μην αριθμόν, έξοχον σοφισμάτων, εξηύρον αυτοίς...» …..(«Μα και τον αριθμό, την πιο τρανή σοφία βρήκα για χάρη τους εγώ...»)“I also found out thenumber, the most greatwisdom , for them...”
Aristophanes- The BirdsIn the end of the 5th century BC , Aristophanes refers to the unsolvedproblem “circle squaring” , in his comedy “The Birds”.The astronomer Meton (Μέτωνας) is at the scene of the theater saying:«με το ορθό ραβδί αρχίζω να μετρώ= I start measuring with my right stickώστε να γίνει ο κύκλος τετράγωνος για χάρη σου˙= so that the circlebecomes a square, for your sakeκαι στο κέντρο του θα είναι η αγορά=and at its center there will be themarketστην οποία θα οδηγούν όλοι οι δρόμοι= to which all roads will lead συγκλίνοντας στο κέντρο, όπως σ’ ένα αστέρι,=converging in the center,the way it happens in a star,που ενώ είναι κυκλοτερές=which (the star) , although it is circular,στέλνει παντού ευθείες ακτίνες λαμπρές».= it sends straight bright rayseverywhere«Αλήθεια, ο άνθρωπος είναι Θαλής!»=Really, the man is Thales!
Dostoyevsky-The Brothers Karamazov During the 19th century, there was a period when a non- Euclidean geometry appeared throughout Europe Dostoyevsky used these new mathematical ideas to illuminate his philosophy. In a part of the “Brothers Karamazov”, Ivan explains to his younger brother, Alyosha, that while he believes in God, he has difficulty believing in the world that God created. “Even if the parallel lines converge and I actually witness it, I shall witness it and say they have converged, but all the same I shall not accept it”.
Tolstoy-War and PeaceIn War and Peace, Tolstoy provides anexample of the use of ratio and linear equationsto clarify how the disadvantaged (such as theRussians) can win battles against moreadvantaged (such as the French) if theyhave enough spirit and energy. Ratio andproportion can be used to relate the power ofan army to its spirit!
Mathematics in movieshttp://www.math.harvard.edu/~knill/mathmovies/Many directors and script writers have beeninspired by mathematics. There is more than700 movies containing mathematical dialoguesand images.
A lonely trip based on mathematics..... Back in 1977 , when the voyager spaceship started its lonely trip into the vast space-void, the NASA officials placed inside it, a golden record with recordered political messages, th spoken greetings in 55 languages, the 5 symphony of Beethoven and engraved mathematical symbols on it.
This diagram defines the location of our sunutilizing 14 pulsars of known directions from oursun. The binary code defines the frequency ofthe pulses.
The way of communication with unknown civilizations...
The greetings in European languages << Οίτινες ποτ`εστέ χαίρετε. Ειρηνικώς πρός φίλουςελαλήσαμεν φίλοι.>> ("Greetings to you, whoever you are. We come in peaceto those who are friends.") Herzliche Grüße an alle (in German) Salutări la toată lumea ( in Romanian) Sayın Türkçe bilen arkadaşlarımız, sabah şeriflerinizhayrolsun (in Turkish) Witajcie, istoty z zaświatów (in Polish)
Archimedes will be remembered when Aeschylus is forgotten, becauselanguages die and mathematical ideas do not. “Immortality“ may be a silly word, but probably a mathematician has the best chance of whatever it may mean....C.H.HARDY : A mathematicians Apology