Determine Numeric Golden Mean from Geometric Version

This article demonstrates that the geometric version of the Golden Mean is a:b as b:(a+b) and the numeric version of the Golden Mean is 1:1.61803399 as 1.61803399:(1+1.61803399) or -.61803399/-1 as -1/(-.61803399-1), where 1.61803399 and -.61803399 are the roots which sum to 1, after figuring out the quadratic equation and roots. This article will demonstrate how to convert from the geometric version to the numeric version.

Steps

The Tutorial



  1. Use b/x = x/(b+x) instead of a:b as b:(a+b).
  2. Cross-multiply to get b^2 +bx = x^2, or 0 = x^2 - bx - b^2. This latter form fits the quadratic form of ax^2 + bx + c, in that a = 1, x = x, b = -b and c = -b^2.
  3. Convert to the standard formula for the roots of a quadratic equation is {x1, x2} = (-b ± sqrt(b^2 - 4ac)) / 2a.
  4. Substitute in our terms in the standard formula and get {x1, x2} = (-(-b) ± sqrt((-b)^2 - 4*1*-b^2)) / 2*1.
    • Collecting and simplifying, we have {x1, x2} = (b ± sqrt(b^2 + 4b^2)) / 2.
    • Further collecting and simplifying, we have {x1, x2} = (b ± sqrt(5b^2)) / 2.
  5. Let now b = 1 and we have {x1, x2} = (1 ± sqrt(5)) / 2 and the answers computed from this are root x1 =  1.61803399 and root x2 = -.61803399. Adding the squares of these numbers results in 3. The added squares are always 3 times the square of b. At any rate, we have successfully determined the numeric Golden Mean from the geometric Golden Mean.
    • If b = -1, root x1 = .61803399 and root x2 = -1.61803399 and we have again determined the Golden Mean numerically from its geometric roots.

Explanatory Charts, Diagrams, Photos

  1. Double-click on this image twice to blow it up to where it is legible. It explains how to convert the Golden Mean from its Geometric Version to its Numeric Version, which is the point of this article.

Helpful Guidance

  1. Make use of helper articles when proceeding through this tutorial:
    • See the article How to Create a Spirallic Spin Particle Path or Necklace Form or Spherical Border for a list of articles related to Excel, Geometric and/or Trigonometric Art, Charting/Diagramming and Algebraic Formulation.
    • For more art charts and graphs, you might also want to click on Microsoft Excel Imagery, Mathematics, Spreadsheets or Graphics to view many Excel worksheets and charts where Trigonometry, Geometry and Calculus have been turned into Art, or simply click on the category as appears in the upper right white portion of this page, or at the bottom left of the page.

Tips

  • If we'd let b = π, then x1 = 5.08203692 and x2 = -1.941611039, the sum of which is π. The sum of the squares of x1 and x2 = 29.6088132, which is 3 times the square of π (which is 9.8696044). Adding 29.6088132, + 9.8696044 = 39.4784176, which is 4*(π^2). This is of interest because 4πr^2 is the surface area of a sphere, and we'd be setting r=1, then multiplying r by 4π^2. Seems like it could be a relative density function? Not sure!
  • If we'd let b = 2π, then x1 = 10.1660738 and x2 = -3.883222077, the sum of which is 2π. The sum of the squares of x1 and x2 = 118.4352528, which is 3 times the square of 2π (which = 39.478176). Adding 118.4352528 + 39.478176 = 157.9136704, which is 16*(π^2).

Related Articles

References

  • Excel workbook was used, "sqrt3*sqrt3=cube,GM,tippingPoint" in arriving at specific numbers.
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