# How to Show Phi Is Quadratic

Two Methods:The tutorialHelpful Guidance

Euclid, in his book, "Elements", gave us the proportion for Phi as a is to b as b is to a+b, i.e. a/b=b/(a+b). This article will step through the process of obtaining the quadratic roots of this relation, i.e. that {Phi1, Phi2} = (±1 ± sqrt(5)) / 2, the Golden Ratio for both b=1 and b=-1. Phi is a very important number in Science and Nature, especially in Nature's example of phyllotaxis (whereby leaves, etc. grow out in a spiral at an offsetting angle of 137.5º [=360º * Phi^2], or in other Fibonacci Sequence patterns, in order to get as much sunlight as possible for each leaf/branch/fruit, etc).

## Steps

### Method 1 The tutorial

- 1
**Open a new workbook in Excel and do (list in text) the following algebraic steps (reset column width to accommodate):**- Do a/b = b/(a+b)
- Do a(a+b) = b^2 ... where "b^2" denotes b squared.
- Do a^2 + ab - b^2 = 0
- Let a = x, b=b, c=b^2
- Do x^2 + bx - c = 0
- Let a=1
- Let ax^2 + bx - c = 0; this is a standard quadratic form.
- Then let the quadratic roots be given by ...
- {x1, x2} = (-b ± sqrt(b^2 - 4ac)/ 2a
- Let c=-b^2 and let a=1
- Let {x1, x2} = (-b ± sqrt(b^2 - 4*1*(-b^2))/ (2*1)
- Let {x1, x2} = (-b ± sqrt(5b^2)/ 2
- Let {x1, x2} = {Phi1, Phi2} and let b = 1
- Let {Phi1, Phi2} = (-1 ± sqrt(5*(1^2)))/2
- Let {Phi1, Phi2} = (-1 ± sqrt(5))/2; Done!

- 2
**Compute the results of the formulas:**- Let Phi1 = (-1 + sqrt(5)) / 2 = 0.618033988749895, which is the artistic ratio for Phi.
- Let Phi2 = (-1 - sqrt(5)) / 2 = -1.61803398874989

- 3
**Please note that had we let b=-1, the results would have been:**- Let Phi1 = (--1 + sqrt(5)) / 2 = 1.618033988749895, which = Phi, formally in Math.
- Let Phi2 = (--1 - sqrt(5)) / 2 = -0.618033988749895

### Method 2 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 Category:Microsoft Excel Imagery, Category:Mathematics, Category:Spreadsheets or Category: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.

## Article Info

Categories: Mathematics | Algebra