Math 103 Lecture #12 class notes page 1 Math 203 Lecture 12 notes: Note: this lecture is very “visual,” i.e., lots of pictures illustrating tessellation concepts Tessellation – a complete covering of a plane by one or more figures in a repeating pattern, with no overlapping. There are only 3 regular polygons that can tessellate a plane without being combined with other polygons: 1) equilateral triangle 2)square 3) regular hexagon. Once a basic design has been constructed, it may be extended through 3 different transformations: translation (slide), rotation (turn), or reflection (flip). FLIP: A figure has line symmetry if there is a line of reflection such that each point in the figure can be reflected into the figure. Line symmetry is sometimes referred to as mirror symmetry, axial, symmetry, bilateral symmetry, or reflective symmetry. The line of symmetry may be vertical, horizontal, or slanted. A figure may have one or more lines of symmetry, or none. Regular polygons with n sides have n lines of symmetry. Any line through the center of a circle is a line of symmetry for a circle. The circle is the most symmetric of all plane figures. The musical counterpart of horizontal reflection (across the y-axis) is called retrogression, while vertical reflection (across the x-axis) is called inversion. SLIDE: Translation (or slide translation) is one in which a figure moves without changing size or shape. It can be thought of as a motion that moves the figure from one location in a plane to another. The distance that the figure moves, or slides, is called the magnitude. A sophisticated application of translation to music, involves the movement of an exact sequence of notes to another location on the scale, called transposition. TURN: The operation of rotation of a figure in a place involves turning the figure about a given point in a place, called the center of rotation. The center of rotation may or may not be a point in the figure itself. The properties of size and shape remain invariant under the operation of rotation. A figure has point symmetry if a half-turn makes the figure coincide with itself. The point is called the point of symmetry. Point symmetry is a special case of rotational symmetry. A figure has rotational symmetry if it can be rotated through less than 360° about a point to coincide with itself. A figure has n-fold symmetry if it coincides with itself when rotated through an angle of 360•/n about its point of symmetry. For example, a square has fourfold symmetry. Point symmetry can be achieved by reflecting the figure across both axes in turn, in any order. Thus the musical counterpart is called retrograde inversion. Creating an Escher-type tessellation: Using a square: Alter opposite side by slide translation Alter adjacent sides by rotation at opposite vertices Using an equilateral triangle: Alter two adjacent sides by rotation at the vertex between the sides Alter the 3rd side by rotation about the midpoint Using a regular hexagon: Alter opposite sides by slide translations Alter pairs of adjacent sides by rotation about the vertex between the sides Use a template to cover the paper (plane). Add artwork and color. Math 103 Lecture #12 class notes page 2 Base units in SI (Metric System) length – meter (m) mass – kilogram (kg) temperature – degrees Celsius (°C) or Kelvin time – second (s) luminous intensity – candella (cd) amount of substance – mole (mol) electrical current – ampere (A) A related unit (not a base unit): volume – liter (l or L) “metric sense” means being able to select the best unit of measure for a situation and being able to estimate the measurement. Examples: A newborn baby might weigh 3 kg An average adult might weigh 60 kg A bedroom might measure 3x4 m A paperclip is about 1 cm wide and weighs 1 g Room temp is about 22 °C Your shoe size might be 26 cm A football field is about 100 m A bath might use 250 L of water SI prefixes (as of 1991): 24 –1 10 yotta Y 10 deci d 21 –2 10 zetta Z 10 centi c 18 –3 10 exa E 10 milli m 15 –6 10 peta P 10 micro µ 12 –9 10 tera T 10 nano n 9 –12 10 giga G 10 pico p 6 –15 10 mega M 10 femto f 3 –18 10 kilo k 10 atto a 2 –21 10 hecto h 10 zepto z 1 –24 10 deka da 10 yocto y Rules for using metric prefixes: 2. Prefix symbols are printed in roman type (upright) without spacing between the prefix symbol and the unit symbol. 3. The grouping formed by the prefix symbol attached to the unit symbol constitutes a new inseparable symbol which can be raised to a positive or negative power. 4. Compound prefixes are not to be used. 5. A prefix should never be used alone. Math 103 Lecture #12 class notes page 3 To convert between one metric unit and another, move the decimal point the correct number of places in the direction of the desired unit. Ex: 240 cm = m 0.08 kg = µg 0.8 L = ml 31 cm3 = ml 3500 mm = cm 1200 cc = L 0.03 hm = dm 1 cubic decimeter (a cube 10 cm x 10 cm x 10 cm) = 1 liter 3 and 10 x 10 x 10 cm = 1000 cubic cm (cc or cm ) 3 and 1/1000 of a liter is a milliliter, then 1 cc = 1 cm = 1 ml = Metric temperature: “30 is hot, 20 is nice. 10 wear a coat, 0 is ice.” Body temp: 37°C A nice feature of the metric system is the relationship between length, volume, and mass. 1 liter of water at 4° C has a mass of 1 kg 1 cc of water at 4° C has a mass of 1 g www.nist.gov/metric Metric Program (email) [email protected] National Institute of Standards and Technology U.S. Department of Commerce The English (or Customary) System of Measurement has 85 base units. Some units date back to about 6000 B.C. and often were based on parts of the body. Each country introduced its own standards and there was often no relationship between them. cubit span palm hand yard fathom foot rod inch pace furlong mile acre knot Some units are defined simply to measure particular objects. fortnight carat square bolt Catty cord Firkin skein Which is heavier, an ounce of gold or an ounce of feathers? Troy: 480 grains = 1 ounce Avoirdupois: 436 grains = 1 ounce Which is heavier, a pound of gold or a pound of feathers? Troy: 12 ounces = 1 pound (5760 grains) Avoirdupois: 16 ounces = 1 pound (6968 grains) Exact vs. Approximate Measurement Exact: √2 Approximate: 1.414 Exact: 4/3 Approximate: 1.3 Exact: π Approximate: 3.1415 or 22/7 Exact: 0.666… Approximate: 1.67 Precision: the last place value with a non-zero-digit. .