Dynamic Symmetry MATHEMATICS APPENDIX

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Dynamic Symmetry MATHEMATICS Appendix

By Karen Halbert


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    Summary Basic Dynamic Symmetry Panel Measurements






    Formulas for other panels:

    The formula for the Top Intercept: Baroque Diagonal Reciprocal is:

    width – (height * height)/width

    The formula for the Top Intercept: Sinister Diagonal Reciprocal is:

    (height*height)/width

    Check that these two intercepts add up to the width. Also note, for example that for the 6x8 panel the proportion of the implied vertical rectangle is 4.5 x 6 the same as 6 x 8. Note that the intercepts switch positions on the top edge for the last three (wider) panel sizes; ie, the Baroque Reciprocal is moved to the right of the Sinister Reciprocal intercept 

     


    Construction of Golden Rectangle Inside a Square 


    1. Draw a line from the lower left corner to the midpoint of the right side. From the Pythagorean Theorem the length will be the square root of (1 + one half squared ) or 1 + 1⁄4 = square root of 5/4 or 1⁄2 square root 5, denoted \/5/2.

    2. With a compass draw an arc from the lower right corner with radius 1⁄2 up to the line from 1). Mark a point on the line.

    3. Draw another arc from the lower left corner up to the left edge with radius as indicated \/5/2 – 1⁄2, the part left over after subtracting 1⁄2 from the line to the right edge’s midpoint.

    4. This number is the reciprocal of the golden mean: 

      a. Golden mean=1⁄2+\/5/2
      b. 1/golden mean=1/(1/2+\/5/2)=(1/2-\/5/2)/(1/2+\/5/2)((1/2-\/5/2)=(1/2- \/5/2)/(1/4-5/4) = \/5/2 -1/2 since 1⁄4-5/4=-1

    5. r:1 is the same proportion as1:1/r (multiply top and bottom by r) 


    Harmonic Armature Proportions
    with Junctions at 1/3, 1⁄4, 1/5 and 1/6 divisions

    B x A, height B and width A Intercepts on the edges at midpoints 


    Use the (x,y) coordinate system with x-axis extending to the right from the origin and the y-axis extending up from the origin, for these proofs. Hence (0,B) represents the point on the main y- axis (vertical) B units above the origin, for example. The point (A,B) represents the point on the x-y plane A units to the right of the origin and B units above the origin.

    Recall from high school geometry or algebra that the equation of a line is:
    y = mx + b, where m is the slope of the line, and b is the y-intercept    .

    • The slope of the line, m, is defined to be the change in y (verOcal distance) divided by the change in x (horizontal distance).

    • The y-intercept is that point, (x, y) were y is 0.

    • Note that the x-axis is defined as y = 0, while the y-axis is defined by the line, x=0.

    Theorem: The point, a3, divides the canvas (rectangle) in thirds (horizontally): 

    Proof
    a3 is intersection of two lines.:

    1) Y=B/(A/2)*x

    2) Y = ((B/2)/A)*x + B/2

    Set the y coordinates equal to each other and solve for x: B/(A/2) * x = ((B/2)/A)*x + B/2
    2B/A * x = B/2A * x + B/2
    (2B/A – B/2A)*x = B/2

    (3/2)*(B/A)*x = B/2
    3/A*x = 1 or x = A/3
    Therefore, a3 is 1/3 of A, the distance from left edge to right edge.

    Theorem: The point, a5, divides the canvas (rectangle) in fifths (horizontally): 

    Proof
    a5 is intersection of two lines:

    1) Y = ((B/2)/A)*x + B/2

    2) Y=-B/(A/2)*x+B

    Set the y coordinates equal to each other and solve for x: ((B/2)/A)*x + B/2 = -B/(A/2) *x + B
    (B/2A)*x +(2B/A)*x = B/2
    ((1/2A) + 2/A)*x = 1⁄2

    (1/A) * (1/2 + 2)*x = 1⁄2
    (1/A) * (5/2)*x = 1/2
    (5/2)*x = A*1/2; multiplying both sides by A
    5x = A or x = A/5; the x-coordinate of the point     a5 is 1/5 of A. 

    Therefore, a5 is 1/5 the distance from left edge to right edge.

    Use the same process to find a4, a6 and other intersections in the Harmonic Armature. It can be shown that all the intersections are at rational (fractional) divisions of the rectangle (panel).

    The intersection of the main Baroque and Sinister Diagonals is in the center. Show this using the two equations indicated.

    a4 is 1⁄4 of the width by inspection – but the interested student may show that it divides the canvas in fourths using the same method as for a3 and a5.

    Where would the 1/6 mark be? 
    Note there are other proofs. 

      Corollary: The Root 2 Rectangle is divided into thirds by the verticals through the intersections of the main diagonals and their reciprocals.



      Root 2 Dynamic Symmetry Armature

      Proof: In a Root 2 rectangle, the reciprocals to the main diagonals are the same as the lines in the Harmonic Armature from the corners to the midpoints of the top and bottom edges. (Recall that the square root of 2 is approximately 1.414.... Hence a 10x14 or 20 x 28 panel is close to a root 2 rectangle. 1: Root 2 seems to be popular proportion historically. Not also that the Root 2 rectangle can be divided into two (vertical) small Root 2 Rectangles. 

      Self-Similar Rectangle Theorem

      Assumption: the slope of a line perpendicular to a given line is the negative reciprocal.

      Theorem: the reciprocals to the main diagonals mark off vertical lines that form new (vertical) rectangles proportional to the original rectangle. 

      Proof: Mark off a rectangle as indicated using the (x,y) coordinate system. 



      Now mark off a vertical line from the intersection of the reciprocal at the bottom edge at (w1,0).

      Substitute (w1,0) into the reciprocal line equation to obtain


      QED.



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