Lecture 9 - Symmetry (1) PDF

MATH, SYMMETRY, AND OUR WORLD Math 10 Based on the slides by J. Balmaceda (IM, UP Diliman) and N. Sheik (Lahore University of Management Sciences) INTRODUCTION Highly significant developments in math are almost always those in which ideas and techniques from one set of mathematical sources make a...

MATH, SYMMETRY, AND OUR WORLD Math 10 Based on the slides by J. Balmaceda (IM, UP Diliman) and N. Sheik (Lahore University of Management Sciences) INTRODUCTION Highly significant developments in math are almost always those in which ideas and techniques from one set of mathematical sources make a huge impact on another set of mathematical sources. These abstract ideas often find unanticipated applications even outside mathematics. SYMMETRY Symmetry is a mathematical operation, or transformation Occurs in the sciences, arts, architecture, and in nature. In art – often used as an aesthetic element – a kind of balance in which the corresponding parts are not necessarily alike but only similar. – Generally is a balance between various parts of an object. SYMMETRY PROPERTIES “DICTATE THE VERY EXISTENCE” OF ALL PHYSICAL FORCES – STEVEN WEINBERG In Physics, Noether's symmetry theorem states that each symmetry of a system leads to a physically conserved quantity. symmetry under translation corresponds to momentum conservation, symmetry under rotation to angular momentum conservation, symmetry in time to energy conservation, etc. EVARISTE GALOIS, 1811-1832 Galois developed the branch of abstract algebra called Group Theory, which is the mathematical study of symmetry. Group Theory is now a fundamental subject in mathematics. All math majors enrol in two semesters of Group Theory. Galois discovered symmetry in the roots of polynomial equations – that eventually led him to prove that the general 5th degree (quintic) polynomial, and higher degree polynomials, could not be solved by some “formula” This is unlike the case of the quadratic, cubic, or quartic polynomials where formulas had been earlier discovered the quadratic formula was known 4,000 years ago by ancient Babylonians; in the 16th century, formulas for solving 3rd and 4th degree polynomials were found by Italian mathematicians. BELOW ARE EXAMPLES OF SYMMETRIC SHAPES How different are these two shapes?  THE REGULAR TRIANGLE HAS 6 SYMMETRIES. 3 Rotations Rotation by 120° Rotation by 240° Rotation by 360° (or 0°) 3 Reflections These are the three reflections about the perpendicular On the other hand, the regular hexagon will have bisectors of the three sides 12 symmetries. The group of symmetries of a regular n-gon is called a DIHEDRAL GROUP. MULTIPLICATION TABLE OF THE “GROUP D6 (SYMMETRIES IN A TRIANGLE) ” * I R R240 F1 F2 F3 120 1 I I R120 R240 F1 F2 F3 R120 R120 R240 I F3 F1 F2 R240 R240 I R120 F2 F3 F1 F1 F1 F2 F3 I R120 R240 2 F2 F2 F3 F1 R240 I R120 3 F3 F3 F1 F2 R120 R240 I Elements of the group of symmetries: 3 rotations (counter-clockwise) : I = R0 , R120 , R240 ; 3 reflections : F1, F2, F3 - flips about line thru vertex 1, 2, 3, respectively Symmetry on the plane Symmetry in space (3D) (2-dimensions) SYMMETRIES IN THE EUCLIDEAN SPACE ARE EXPRESSED BY ISOMETRIES An isometry of the plane is a mapping that preserves distance, and therefore shape.  Recall: The four plane (euclidean) isometries. 1. Translation 2. Reflection 3. Rotation 4. Glide Reflection SYMMETRIC PATTERNS A plane pattern has a symmetry if there is an isometry of the plane that leaves the pattern unchanged. There are three types of symmetric patterns on the plane. Finite patterns (Rosettes) Frieze patterns Wallpaper patterns Penrose Tilings FINITE PATTERNS (ROSETTES) Patterns formed by periodic rotations or by some dihedral symmetry group.  FRIEZE PATTERNS: 1-DIMENSIONAL PATTERNS Frieze patterns are patterns that have translational symmetry in (only) one direction. We imagine that they go on to infinity in both directions or can wrap around THE 7 DISTINCT FRIEZE PATTERNS Translation only Glide reflection only Translation and horizontal line reflection (glide reflection follows) Translation and vertical line reflection Translation and 180⁰ rotation Translation, 180⁰ rotation, vertical line reflection and glide reflection Translation, 180⁰ rotation and vertical/horizontal reflection EXERCISE Which pattern on the left is equivalent to the pattern of footprints above? How about for the pattern below? Answer: footprints – 2 , triangular pattern - 6  WALLPAPER PATTERNS: 2D PATTERNS oWallpaper patterns are repeating patterns (in two directions) on a surface/plane, and are also known as tessellations or tilings. oThe name comes from the word tessella, the small square tile used in ancient Roman mosaics. oThe different types of wallpaper patterns depend on the symmetries (isometries) they contain. Tiles in Alhambra, Spain THE 17 TYPES OF WALLPAPER PATTERNS The study of wallpaper patterns is part of (2-dimensional) geometric crystallography. Some Philippine researchers study weaving patterns of local mats and textiles. WEAVING PATTERNS BY INDIGENOUS PHILIPPINE GROUPS DIFFERENT MOTIFS DIFFERENT RATTAN WEAVES THE ALHAMBRA – A CELEBRATION OF SYMMETRY All 17 wallpaper patterns have been found on the walls and grounds of the Alhambra.  TESSELLATIONS Other spaces, such as hyperbolic planes and the sphere, can also be tesselated.  Day and Night, 1938 Encounter, 1944 M.C. ESCHER Dutch artist that is Fascinated with paradox and “impossible” figures Reptiles, 1943 CRYSTALLOGRAPHIC RESTRICTION One important property of wallpaper patterns is that the patterns can only admit rotations by 180°, 120°, 90° or 60°. This fact is known as the crystallographic restriction theorem and is true even for 3-dimensional crystals. In particular, crystals do not admit rotations by 72° (5-fold rotations). PENROSE TILINGS AND QUASICRYSTALS Penrose tilings have 5-fold rotations! Sir Roger Penrose  PENROSE TILINGS ARE DIFFERENT FROM WALLPAPER PATTERNS. These are aperiodic tilings (not periodic or regular) More than one tile or pattern is repeated to fill the plane. Penrose tilings are the symmetrical patterns underlying the structure of quasicrystals. QUASICRYSTALS CONTAIN 5-FOLD ROTATIONS - (IN VIOLATION OF THE CRYSTALLOGRAPHIC CONDITION). Quasicrystals are now known to be naturally occurring. QUASICRYSTALS WERE DISCOVERED BY DAN SHECHTMAN OF ISRAEL IN 1982. HE WAS RIDICULED BY MANY SCIENTISTS AND WAS ASKED TO LEAVE HIS RESEARCH GROUP WHEN HE ANNOUNCED HIS ‘DISCOVERY’. He received the Nobel Prize for Chemistry in 2011 for the discovery of quasicrystals. FRACTALS ARE PATTERNS THAT EXHIBIT SCALING SYMMETRY; Koch snowflake Sierpiński triangle Mandelbrot set FRACTALS IN NATURE FRACTALS IN ARTS FRACTALS IN ARCHITECTURE In Hindu temples such as the Virupaksha temple at Hampi, the parts and the whole have the same character. The Eiffel Tower uses a repetition of a triangle that generates a shape known amongst fractal geometrists as the Sierpinski Triangle, resulting in a sturdy and cost-effective design. If, instead of its spidery construction, the tower had been designed as a solid pyramid, it would have consumed a large amount of iron, without much added strength.  FRACTALS IN ARCHITECTURE Fractal pattern in Ba-ila (South Zambia) settlement. Fractal generation of Ba-ila simulation. - First iteration is similar to a single house - Second to a family ring - Third to village as whole SYMMETRY IN THE ARTS AND HUMANITIES SUMERIAN SYMMETRY PATTERNS IN ISLAMIC ART Isfahan, Iran, 15th c. Fez, Morocco, 1325 J. Balmaceda, Institute of Mathematics, UP Diliman  There are many intriguing symmetries in the works of Leonardo Da Vinci. The Vitruvian Man Leonardo Da Vinci: the Renaissance Man Michelangelo’s Holy Family Raphael’s Crucifixion SYMMETRY IN LANGUAGE AND LITERATURE Poetry (literary forms like haiku, tanka, dalit, tanaga), rhyme, palindromes – A man, a plan, a canal – Panama Alice in Wonderland, by Lewis Carrol BUILDINGS IN CLASSICAL GREECE AND ROME WERE HIGHLY SYMMETRICAL AND CAREFULLY PROPORTIONED. The Parthenon, ancient Greece The Colosseum, ancient Rome CRYSTAL STRUCTURE IN CHEMISTRY C60  SYMMETRY IN MEDICINE Symmetry is an important feature of human anatomy and the absence of symmetry in medical images may indicate the presence of pathology. Sample images of MRI: A) Normal brain MRI; B) MRI with acute stroke; C) Alzheimer disease; D) MRI with the tumor disease SYMMETRY IN LIFE FORMS BILATERAL SYMMETRY MEN WITH HIGH LEVELS OF FACIAL SYMMETRY ARE LESS PRONE TO MENTAL DECLINE BETWEEN 79 AND 83. - JOURNAL OF EVOLUTION AND HUMAN BEHAVIOUR, 2009  SOME FINDINGS OF SCIENTIFIC RESEARCH There is research supporting the claim that bilateral symmetry is an important indicator of freedom from disease, and worthiness for mating. A higher degree of symmetry likely indicates a better coping system for environmental factors and is a preferred trait during mate selection in a variety of species.  “The mathematical sciences exhibit order, symmetry, and limitation; and these are the greatest forms of the beautiful.” Aristotle, 384-322 BC