Determine the Geometric Version of the Golden Mean (Ratio or Proportion)

Euclid's "Elements", Book II Proposition 11, gives a method for determining the geometric Golden Mean (or Golden Ratio or Golden Proportion). Via Microsoft Excel, an {X,Y} chart is designed and explained how it arrives at one of the roots of the Golden Mean(-1), 0.618033988749895, and its square. The Golden Mean is noticeable in Nature, in phyllotaxy, nautilus shells, pine cones, pineapples, sunflowers, etc. It has been used by artists for centuries and perhaps by ancient architects as well. It has been observed by astronomers in galactic images.

Steps

The Tutorial

  2. Let AB be the given straight line; thus it is required to cut AB so that the rectangle contained by the whole and one of the segments is equal to the square on the remaining segment.
  3. Let the square ABDC be described on AB [see Proposition I. 46[; let AC be bisected at the point E, and let BE be joined; let CA be drawn through to F, and let EF be made equal to BE (as by radius); let the square FH be described on AF, and let GH be drawn through to K.
  4. Say that AB has been cut at H so as to make the rectangle contained by AB, BH equal to the square on AH.
  5. For, since the straight line AC has been bisected at E, and FA is added to it, the rectangle contained by CF, FA together with the square on AE is equal to the square on EF. [By Proposition II.6]
    • But EF is equal to EB; therefore the rectangle CF, FA together with the square on AE is equal to the square on EB.
  6. But the squares on BA, AE are equal to the square on EB, for the angle at A is right [by Proposition I.47 - Pythagorean Theorem]; therefore the rectangle CE, FA together with the square on AE is equal to the squares on BA, AE.
  7. Let the square on AE be subtracted from each; therefore the rectangle CF, FA which remains is equal to the square on AH.
  8. Now the rectangle CF, FA is FK, for AF is equal to FG; and the square on AB to AD; therefore FK is equal to AD.
  9. Let AK be subtracted from each; therefore FH which remains is equal to HD.
  10. And HD is the rectangle AB, BH for AB is equal to the square on HA.
  11. Therefore the given straight line AB has been cut at H so as to make the rectangle contained by AB, BH equal to the square on HA.
  12. Q.E.F. - quod erat faciendum, "that which was to have been done". The Greek loses something in the translation; it means "precisely what was required to be done".

Explanatory Charts, Diagrams, Photos

  1. Review the chart of the Golden Mean from Euclid's Elements. Book II, Proposition 11. Double-click on the Excel diagram to enlarge it.

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

  • Please see the References Section for Propositions referred to above.
  • I opine that the key insight here is realizing that the diagonal extended from the midpoint of the square made from the original line keeps things in proportion, as when another square is made by extending that radial length from that midpoint -- it is the far side of that square which cuts the original line in precisely the correct proportion.

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References

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