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H.S.M. Coxeter, Projective Geometry 2nd ed.“6. Still working in the Euclidean plane, draw a line segment OC, take G twothirds of the way along it, and E two-fifths of the way from G to C. (For instance, make the distance in centimeters OG = 10, GE = 2, EC = 3.)If the segment OC represents a stretched string, tuned to the note C, the samestring stopped at E or G will play the other notes of the major triad. By drawinga suitable triangle, verify experimentally that H(OE,CG). (Such phenomena explain our use of the word harmonic.)”
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Cremona, Elements of Projective Geometry, Oxford University Press, 1913 Matthews, Projective Geometry, Longmans, Green and co., 1914 Veblen and Young, Projective Geometry Volume 2, 1918 Young, Projective Geometry, Carus Mathematical Monographs, MAA, 1930 Baer, Linear Algebra and Projective Geometry, Academic Press, 1952 Seidenberg, Lectures in Projective Geometry, D Van Nostrand and Co, 1962 Pedoe, Introduction to Projective Geometry, Macmillan, 1963 Fishback, Projective and Euclidean Geometry 2ed, John Wiley & Sons, 1969 Bennett, Affine and Projective Geometry, John Wiley & Sons, 1995 Kadison and Kromann, Projective Geometry and Modern Algebra, Birkhauser, 1996Beutelspacher and Rosenbaum, Projective Geometry: From Foundations to Applications, Cambridge University Press, 1998 Casse, Projective Geometry, an Introduction, Oxford University Press, 2006
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Linnaeus Wayland Dowling, Projective Geometry, McGraw-Hill Book Co, Inc., 1917 (Forgotten Books)Notes.- The idea of four harmonic points, or harmonic division, was known to theearly Greek geometers, but who first invented it is not definitely known. Apolloniusof Perga (247 BC) mentions it is his book on conic sections.The harmonic property of a complete quadrangle is contained in the Collections ofPappus (300AD). It was made the foundation for Von Staudt’s Geometric der Lage,1847.Three cords consisting of the same substance and having the same size and tension,and whose lengths are in harmonic progression, will vibrate in harmony when struckin unison. The name harmonic is probably due to that fact.
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Harmonic sets in mathematics Drawing in perspectiveLewis Goupy, Brook Taylor. 1720 Harmonics in music
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Harmonic sets defined by ratios Ceva’s Theorem Menelaeus’ Theorem
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Harmonic sets defined by ratios Harmonic Set: H(AC,BD)
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Harmonic sets defined by ratios Harmonic Set: H(AC,BD)
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Harmonic set from a circle and its tangent Harmonic Set: H(AC,BD)
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Harmonic set from two circles and their tangents Harmonic Set: H(AC,BD)
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Harmonic set on Euler’s line A = circumcenter B = centroid C = 9-point circle center D = orthocenter Harmonic Set: H(AC,BD)
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Harmonic sets in projective geometry… What is projective geometry? A non-Euclidean geometry which developed out of the mathematics of perspective drawing. A study of geometric properties that are invariant under projections.Abraham Bosse, 1665 A real projective plane is an Leon Battista Alberti, 1435 extension of the real Euclidean plane, extended by strategically adding points and a line at infinity.
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Euclidean Geometry Projective GeometryConstructions with compass and Constructions with just a straightedgestraightedgeParallel Postulate: Given a line and a There are no parallel lines: Any twopoint not on the line, there is just one lines are incident with a unique point.line through the point parallel to the lineA study of properties invariant under A study of properties invariant underrigid motions, like length, angle, area projections, like…
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Definition: A projectivity is a composition of perspectivities.
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Theorem : A projectivity preserves harmonic sets.
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Definition: A projective collineationis a composition of perspectivecollineations.
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Collineations in perspective drawingConsider translation from the red square to the blue square inperspective.
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Collineations in perspective drawingTranslation in perspective is a perspective collineation.
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Collineations in perspective drawingReflection in perspective is a perspective collineation. English Boy Using Reflection in Mirror in Foyer of Grand Hotel to Fix His Tie Photographic Print by Alfred Eisenstaedt
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Collineations in perspective drawing180˚ rotation in perspective is a perspective collineation.
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Collineations in perspective drawing45˚ rotation in perspective is NOT a perspective collineation.
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Collineations in perspective drawing45˚ rotation in perspective is a projective collineation, thecomposition of two reflections.
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Harmonic sets and perspective drawingthen H(BD, AC) ?
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Harmonic sets and perspective drawingthen H(BD, AC) ?
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Harmonic sets, the harmonic sequenceand the harmonic mean of two numbers 1/3 1/2 1 1/3 1/2 1 1/4 1/3 1/2 2/4 2/3 2/2 1/5 1/4 1/3 3/5 3/4 3/3 1/6 1/5 1/4 4/6 4/5 4/4 1/(n+2) 1/(n+1) 1/n n/(n+2) n/(n+1) n/n
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Harmonic sets and its relationship with harmonics in musicDefinition: The frequency of a sound wave is the number ofcycles per second, measured in Hertz.Definition: The pitch of a sound is the perception of frequency.
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Harmonic sets and its relationship with… harmonics in musicDefinition: A harmonic of a sound wave is an integer multipleof the fundamental frequency of the sound wave.
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Harmonic sets and its relationship with… harmonics in musicDefinition: Overtones are frequencies higher than thefundamental frequency.Many musical instruments arecreated to have harmonicovertones.The human voice can createovertones.
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Harmonic sets, the harmonic sequence, the harmonic mean of two numbers, and harmonics in music 1 55/55 1/2 55/110 1/3 55/165 1 110/110 2/3 110/165 2/4 110/220 1 165/165 3/4 165/220 3/5 165/275 1 220/220 4/5 220/275 4/6 220/330 1 275/275 5/6 275/330 5/7 275/385 n/(n) … n/(n+1) … n/(n+2) …
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Harmonic sets and its relationship with harmonics in music Just intervals1 1/2 1/3 1:1/3 = 3:1 1:1/2 = 2:1 1:2:3 2 octaves 1 octave1 2/3 2/4 1:1/2 = 2:1 1:2/3 = 3:2 2:3:4 1 octave Perfect fifth1 3/4 3/5 1:3/5 = 5:3 1:3/4 = 4:3 3:4:5 Major sixth Perfect fourth1 4/5 4/6 1:2/3 = 3:2 1:4/5 = 5:4 4:5:6 Perfect fifth Major third1 5/6 5/7 1:5/7 = 7:5 1:5/6 = 6:5 5:6:7 Subminor fifth Minor third1 7/8 6/8 1:3:4 = 4:3 1:7/8 = 8:7 6:7:8 Perfect fourth Supermajor second1 8/9 7/9 1:7/9 = 9:7 1:8/9 = 9:8 7:8:9 Supermajor third Major second
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Harmonic sets and an android app by Stephen Brown
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