THE PROBLEM OF DIVISION INTO EXTREME AND MEAN

THE PROBLEM OF DIVISION INTO ‘EXTREME AND MEAN RATIO’ Elina Andritsou Ioannis Alonistiotis φ Model Lyceum Evageliki School of Smyrna 4 Lesvou street, Nea Smirni, Athens, Greece

“ Geometry has two great treasures. One is theorem of Pythagoras, the other is the division of a line into extreme and mean ratio (Golden section). The first we may compare to a measure of gold, the second we may name a precious jewel. ” Johannes Kepler

THE PROBLEM The division of a line AB into two segments using a point C so that, the large segment AC and the small segment CB with the whole of the line AB form equal ratios

THE ORIGIN The origin is not historically verified. Some historians attribute it to the Pythagoreans and connect it to the study of the equation x 2+ax= a 2 as shown in geometric language in the Book II of Euclid's Elements , or the discovery of asymmetry in Ancient Greece, and others connect it with the construction of the pentagon from Theaietitos.

WHY WE SELECTED THIS PROBLEM; Plays an important role in the evolution of maths Is absolutely connected with the problem of the golden section and the number φ Solves geometrically second degree equations Has applications in geometry and architecture Shows differences in the construction of the division and its use

THE PROBLEM OF DIVISION INTO “EXTREME AND MEAN RATIO” IN THE ELEMENTS OF EUCLID’S Proposition II. 11 : 1 st construction Propositions IV. 10 and IV. 11 : construct regular pentagons Definition VI. 3: division into “extreme and mean ratio” Proposition VI. 30 : 2 nd construction Proposition ΧΙΙΙ. 16 and ΧΙΙΙ. 17 : constructs the icosahedron and dodecahedron

PROPOSITION II. 11 (GOLDEN RATIO) To cut a given straight line so that the rectangle contained by the whole and one of the segments equals the square on the remaining segment. that was before ratios were defined, and there an equivalent condition was stated in terms of rectangles, namely, that the square on AH equal the rectangle AB by BΗ. That construction was later used in Book IV in order to construct regular pentagons.

, PROPOSITION II. 11 : PROOF

Proposition II. 11 : II. 6 I. 47 Therefore, the AH is proportionate means of AB and HB.

DEFINITION VI. 3 Straight line is said to have been cut in extreme and mean ratio when, as the whole line is to the greater segment, so is the greater to the less. The line AB is cut in extreme and mean ratio at C since The definition specifies the point that divides the straight line in extreme and mean ratio.

PROPOSITION VI. 30 : PROOF With side section a we construct a square ABCD (I. 46) AKZE be similar to the parallelogram ABCD (VI 29) After (ABCD) = (DEZH) deducting the rectangle ADHK which is common , then equity will occur (AKZE) =(BCHK).

PROPOSITION VI. 30 : PROOF Rectangles AKZE and BCHK is equal-area and equiangular, sides containing equal angles are inversely proportional (VI. 14) , ie valid: since AB> AK will be AK> KB.

PROPOSITION ΧΙΙΙ. 16 AND ΧΙΙΙ. 17 Constracts the icosahedron and dodecahedron • Proposition 16 • To construct an icosahedron and comprehend it in a sphere, like the aforesaid figures; and to prove that the square on the side of the icosahedron is the irrational straight line called minor. Proposition 17 To construct a dodecahedron and comprehend it in a sphere, like the aforesaid figures; and to prove that the square on the side of the dodecahedron is the irrational straight line called apotome

GEOMETRIC SOLVING OF THE 2 ND DEGREE EQUATION - NUMBER Φ

PROPOSITION IV. 10 AND IV. 11 the side and the diagonal of the regular pentagon is also a product of self-similar of the Golden Section. The golden ratio appears in a number of geometric elements of regular pentagon.

THE PROBLEM OF DIVISION IN MODERN TEXTBOOKS OF GEOMETRY x x a φ. ggb

We can see that the ratio BC / CA is φ.

GOLDEN RATIO IN HUMAN BEINGS Α C B

THE PROBLEM OF DIVISION AFTERE UCLID Hypsicles of Alexandria (2 nd BC): "Supplement" or Book XIV of the Elements Hero of Alexandria: appears in the determination of the surface of the pentagon and the dodecagon Pappus of Alexandria Synagogue: appears on the construction of the icosahedron and dodecahedron and comparison theorems of the volumes Arab period: considered related projects

EUROPEAN TRADITION Fibonacci (about 1180 -1250), Fibonacci with the ‘problem of the rabbits’ is creating the following sequence: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, . . . that is called sequence of Fibonacci and is formed according to the rule: u 0 = 1, u 1 = 1, un = un-1 + un-2 The limit of the ratio between every next and last term of this sequence is the value of φ.

15 TH -16 TH CENTURY the division in extreme and mean ratio is revitalized relatively to its applications in geometry and architecture. Leonardo da Vinci introduced the term "balance". Luka Pacioli (1509) published "The divine proportion", which although is specially dedicated to the problem of division in extreme and mean ratio.

CONCLUSIONS The ancient Greeks saw more like a constant ratio with application to construct regular polygons (in books 6 and 13) and had not connected it with geometric solving 2 nd degree equation. René Descartes The construction was based on equations and similar areas of rectangles and squares as the product of two line segments is surface, not straight section.

BIBLIOGRAPHY v. School book v. Dipl_kontogeorgis v. Dipl_vazoura v. Euclid volume I, IV v. Bonia v. Gold_number v. Gourlia v. Herz-Fischler v. Wikipedia v. Acharilaou v. Fysikidrasi v. Eisatopon