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Transcript
10.10.2016 The Building Bloks of Geometry In the history of architecture geometric rules based on the ideas of proportions and symmetries formed fixed tools for architectural design Sacred geometry St. Vitus Cathedral 1 10.10.2016 Golden ratio (divine proportion) ab a a b 1 5 1.618 2 Architecture was profoundly influenced by the golden ratio. The Egyptian Pyramids, Stonehenge, the Acropolis, all gothic cathedrals,… these architectural wonders and many more are said to have been designed and built using the golden ratio. Euclid: Elements, book II Proposition 11 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. 2 10.10.2016 Here are some definitions you will need to remember. • Point – names an exact location on a plane. A point is that which has no part. Euclid: Elements, book I • Line – a collection of points forming a straight path that extends infinitely in opposite directions. A line is breadthless length. A straight line is a line which lies evenly with the points on itself. Euclid: Elements, book I • Plane – a perfectly flat surface that extends forever in all directions. • Segment – part of a line between two endpoints. • Angle – formed by 2 rays with a common endpoint called a vertex. Pleural of vertex is vertices. Right triangle Parallelogram 3 10.10.2016 Congruent - figures that have the same size and shape. Segments that have the same length are congruent. Angles that have the same measure are congruent. The symbol for congruence is , which is read “ congruent to”. Types of angles Acute angle - any angle which measures less than 90° Right angle - any angle which measures exactly 90° Obtuse angle - any angle which measures >90°, but <180° Straight angle - any angle which measures exactly 180° 4 10.10.2016 Vertical angles – 2 angles formed by intersecting lines. They are always equal in measure. They are across from one another. Angle 1 , 3 are vertical angles. Angle 2 , 4 are vertical angles. Angle 1 and angle 2 are not vertical. Supplementary angles – 2 angles whose measures add up to 180°. Supplementary angles can be placed so that they form a straight line. Angle 1 and angle 2 are supplementary. The line passing through points A, B, and C is a straight line. True or False • If 2 distinct lines do not intersect, then they are parallel. • If 2 lines are parallel, then a single plane contains them. • If 2 lines intersect, then a single plane contains them. • If a line is perpendicular to a plane, then it is perpendicular to all lines in that plane. 5 10.10.2016 Solids vertex edge face cylinder pyramid cone prism cube tetrahedron sphere octahedron 6 10.10.2016 Angle between line and plane Dihedral angle (Двугранный угол) -angle between two intersecting planes Angle between two planes (α, β, green) in a third plane (pink) which cuts the line of intersection at right angles. 7 10.10.2016 Euclid: Elements, book I Proposition 1: To construct an equilateral triangle on a given finite straight line. Euclid: Elements, book I Proposition 46 To describe a square on a given straight line. Let AB be the given straight line. It is required to describe a square on the straight line AB. Construction: Draw AC at right angles to the straight line AB from the point A on it. Make AD equal to AB. Draw DE through the point D parallel to AB, and draw BE through the point B parallel to AD 8 10.10.2016 Euclid: Elements, book II Proposition 14 To construct a square equal to a given rectilinear figure. Let A be the given rectilinear figure Construct the rectangular parallelogram BCDE equal to the rectilinear figure A (Book I, prop 45). To construct a square equal to a given paralellogram. Geometric mean theorem Geometric mean of the two segments equals the altitude. 9 10.10.2016 http://geotest.geometry.cz/2016/?lang=EN 10