THE H A F F PROPORTIONAL DIVIDER (= Reduktions-Zirkel in German) Model 195aE ******************************************************************************** PURPOSE: Engineers have used this tool to transfer proportions from one drawing to a new drawing in bigger or smaller scale. Open-air painters used it to trans- fer distances in the landscape or art painters took proportions of the beautiful model to sketch it on paper - as base of the artwork in mind. USAGE: With closed legs loosen the TOP SCREW. With the OPPOSITE SCREW move the axle up or down to the SCALE-MARK of your choice. Tighten the TOP-SCREW. Open the LEGS, sample the original distance with ONE SIDE - the OPPOSITE SIDE will be in the desired SCALE-PROPORTION. There are 2 SCALE-MARKS: o In »LINES« the sampled DISTANCES are set in proportion. ( 1:1 ; 4:3 ; 3:2 ; 5:3 ; 2 ; 2.5 ; 3 ; 4 ; 5 ; ... 9 ; 10 ) o In »CIRCLES« when its D I A M E T E R IS TAKEN AS THE SAMPLE, the OPPOSITE SIDE will cut the circumference in equal sections: ( 4 ; GS ; 5 ; 6 ; 7 ; 8 ; 9 ; 10 ; ... 16 ; ... 20 ) E.g.: »4« will make 4 Segments ( = 90° Sectors = Corners of a Square ), »5« will make 5 Segments ( = 72° Sectors = Corners of a Pentagon), ... out of the full circle ... The M A R K »GS« = "Goldener Schnitt" = "Golden Cut" = "THE GOLDEN RATIO" -------------------------------------------------------------------------------- Separates a LINE AB in C to the well known proportion: |===============================|==================| A C B AB/AC = AC/CB = 2/( SQRT(5) - 1 ) = 1.618033988749... = PHI = Φ (*) It is attributed as "divine", "beautiful", "harmonious", "aesthetics", ... in ART & ARCHITECTURE and has some UNIQUE PROPERTIES in MATHEMATICS too ... Φ^{2} = Φ + 1 1/Φ = Φ - 1 Φ = 1 + 1/Φ Φ = 1+1/(1+1/(1+1/(1+1/(1+1/ ...))...) How this (*) FORMULA came out of ??? ******************************************************************************** [1st] A GEOMETRICAL CONSTRUCTION ... IF you want to find the »GS«-POINT (C) in a given LINE-AB =!= 2, THAN ... o Draw a rectangular 1/2 LENGTH-OF-LINE-AB =!= 1 in POINT-B to POINT-X o Draw the HYPOTENUSE of this RECTANGLE from POINT-A to POINT-X o With PYTHAGORAS's RULE the lenth of the HYPOTENUSE is SQRT(5) o Put the COMPASS in X: (a) Mark the LENGHT = 1 on the HYPOTENUSE o Put the COMPASS in A: (b) Transfer THIS POINT to LINE-AB as POINT-C o POINT-C is »GS«, the "Golden Cut" of LINE-AB The GEOMETRICAL CONSTRUCTION showed the relation ... AC SQRT(5) - 1 SQRT(5) - 1 ---- = --------------------- = ------------- = 1.618 = Φ CB 2 - ( SQRT(5) - 1 ) 3 - SQRT(5) ===== AB 2 ---- = --------------------- = 1.618 = Φ AC SQRT(5) - 1 ===== ******************************************************************************** [2nd] AN ANALYTICAL SOLUTION ... From (*) AB / AC = AC / CB we get AC^{2} = AB * CB and let^{ } AB =!= 2 = AC + CB than CB = 2 - AC now substituted AC^{2} = 2 * ( 2 - AC ) follows AC^{2} + 2*AC - 4 = 0 Using the well known formula to solve a squared equations ... x^{2} + px + q = 0 ===>> x_{1,2} = - p/2 +/- SQRT( (p/2)^{2} - q ) AC = - 1 +/- SQRT( 1 + 4 ) = SQRT(5) - 1 { only + is real } ... we get AB / AC = 2 / ( SQRT(5) - 1 ) = 1.618 = Φ qed. ******************************************************************************** A wonderful BOOK about "Proportional Harmonies in Nature, Art & Architecture"... ISBN 978-1-59030-259-0 György Doczi » THE POWER OF LIMITS « (2005) Shambhala, Boston & London eg. FIBONACCI-Serie: 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, ... (a) Each number is the sum of the two previous ones (b) Any number in this serie divided by the following one is ~0.618... and any number divided by the previous one is ~1.618... = Φ ******************************************************************************** REMARK: As a "Rule-Of-Thumb" 60% / 40% is a good guess of the »GS«-POINT (C) PART TWO = AN ENGINEERING APPROACH = BOWEN Model 770 ... Back to the ENGINEERING TOOLS Main Page impressum: ******************************************************************************** © C.HAMANN http://public.BHT-Berlin.de/hamann 02/02/23 |