Squaring the Circle

  # Squaring the Circle
  *Hand-crafted wiki article — demonstrating manual cross-linked HTML wiki format*

  **Squaring the circle** is the ancient geometric problem of constructing a square with the same area as a given circle using only compass and straightedge.

  ## The Problem
  Given a circle of radius *r*, construct a square of side *s* such that:


  s² = πr²


  With conventional π (3.14159...), this is impossible because π is **transcendental** — it cannot be expressed as a finite combination of algebraic operations.

  ## The Golden Pi Solution
  Using [Golden Pi](true-value-of-pi.html) (4/√Φ), the circle can be squared through the Golden Ratio:


  π = 4 / √Φ = 4 / 1.27201965... = 3.14460551...


  The key insight is that Φ is an algebraic number (it satisfies Φ² = Φ + 1), so √Φ and therefore 4/√Φ are also algebraic — enabling exact geometric construction.

  ## Step-by-Step Construction
  <ol>
    - Construct a line segment of length Φ using the golden section division
    - Construct √Φ as the geometric mean of 1 and Φ
    - Divide 4 by √Φ (constructible as the fourth proportional)
    - Use this length to construct the square's side equal to the circle's area
  </ol>

  ## Historical Significance
  For over 2,000 years, squaring the circle was considered the ultimate geometric challenge. The [Pyramid of Giza](evidence/pyramid-of-giza/index.html) may have encoded the solution thousands of years ago.


    **See also:**
    [True Value of Pi](true-value-of-pi.html) ·
    [Golden Ratio (Φ)](/wiki/golden-ratio/phi/index.html) ·
    [Kepler's Right Triangle](/wiki/golden-pi/kepler-triangle/index.html) ·
    [Pyramid of Giza](/wiki/evidence/pyramid-of-giza/index.html)