Φ The True Value Of Pi Π

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Harry Lear's Measuring Pi Squaring Phi

May 12, 2026

One of the most remarkable figures in the golden π movement is Harry Lear — a man who took the debate out of pure mathematics and into the physical world. Rather than arguing over infinite series or continued fractions, he walked into his workshop with a CNC machine and a 1,000 mm sheet of poster board, and cut a circle. His website Measuring Pi Squaring Phi is a monument to the belief that if math and measurement disagree, it's the math that needs correcting.

Who Is Harry Lear?

Harry Lear is an independent researcher based in the United States. His background is in engineering and physical measurement, not academic mathematics — and he considers this an advantage. "Become an engineer AND a mathematician," he wrote on his site, urging university math departments to physically cut a circle instead of trusting inherited formulas.

His central thesis is simple: if you physically measure the circumference of a circle with a precisely known diameter, you get a value for π that converges on 4/√φ = 3.144605511…, not the conventional 3.141592654… Out to at least four decimal places, the physical measurement disagrees with Archimedes. And from that disagreement, Lear built an entire geometric framework.

"As soon as you physically measure the Pi circumference of a one-unit diameter circle, you will throw away all your current math equations for Pi and see that they are in error at the 3rd decimal point."
— Harry Lear, Measuring Pi Squaring Phi

The CNC Measurement — Proof 7(a)

Lear's most concrete proof is Proof 7(a): a direct physical measurement of π using a 1,000.0 mm diameter circle. Using a beam compass and an NT Model CL-100P adjustable rotary circle cutter (a manual tool with a carbon tool steel blade — later described as a $60 alternative to a $100,000 CNC machine), he cut a circle of precisely 1,000.0 mm diameter on a 40×60 inch poster board.

The steps were exacting: a flat, level drawing table; calibration to a Starrett engineering tape measure in millimeters; drawing the circle with a 500.0 mm radius via beam compass; then cutting the circumference and measuring it. The result consistently pointed to π = 3.1446…, not 3.1416…

The Seven Geometric Proofs

Beyond physical measurement, Lear developed a series of seven geometric proofs (chronologically numbered, with Proof 7 split into 7a and 7b). Each proof centers on the same foundational relationship: the Kepler Golden Ratio Right Triangle — a right triangle whose sides are in the ratio 1 : √φ : φ.

All seven proofs share a common structure: a circle is drawn whose radius or diameter derives from the sides of a Kepler Triangle, and around it a square is constructed. When the circle's circumference is equated to the square's perimeter (the act of "squaring the circle"), the only value of π that satisfies the equality is 4/√φ.

• Proof 1: Four blue circles (circumference = 2 units each) whose diameters sum to the diameter of a large yellow circle. Squaring the yellow circle's circumference to the square's perimeter yields π = 4/√φ.
• Proof 2: Same structure but circumferences redefined in terms of π². Equating P = C again reduces to π = 4/√φ. An addendum shows that substituting conventional π = 3.14159… causes P ≠ C by ~0.038.
• Proof 3–5: Progressive refinements building on Kepler Triangle geometry and the squaring operation from different starting givens.
• Proof 6: A generalized proof testing any proposed value of π against the condition "circumference = perimeter" for Kepler Triangle-constructed circles and squares. Only π = 4/√φ passes, carried to 40 decimal places. Lear explicitly notes this is NOT circular reasoning — the conditional (C = P) is the test, not the assumption.
• Proof 7(a): Physical CNC/rotary cutter measurement of a 1,000 mm diameter circle.
• Proof 7(b): Simplified math proof using the same physically measured circle, presented step-by-step for even a high school algebra student.

Critique of Archimedes' Polygon Method

Lear reserved some of his sharpest criticism for the Archimedean polygon method — the 2,300-year-old technique of inscribing and circumscribing polygons around a circle to approximate π. Archimedes used 96-sided polygons; later mathematicians pushed to 10,000, then millions of sides.

Lear's objection is both simple and profound: the tangent and arctangent of a curved side is not a valid mathematical operation. When you inscribe a polygon inside a circle, the polygon's sides are chords — straight lines that cut the circle. But Archimedes' method, and all subsequent refinements using calculus, treat the arc itself as a triangle side when computing trigonometric functions. Lear wrote:

"If it's not a straight side, it is not a right triangle and the notion of tan and arctan go out the window. Even using the calculus, it is a gross error to assume that you can compute the tan or arctan of a curved side, since then the curved triangle is not truly a triangle."

In Lear's view, this tiny error — treating arcs as if they were straight — accumulates across the hundreds or thousands of polygon iterations used in modern π calculations, producing the ~0.003 unit gap between 3.14159… and the true value 3.14460…

Five More Universal Constants (π2–π6)

After squaring the circumference of a circle to the perimeter of a square — the "squaring the circle" operation — Lear discovered that the relationship between π and φ generates a family of universal constants, not just one. He classified what we now call golden π as π1, and named five additional constants π2 through π6:

These constants form a complete mathematical family linking π, φ, and the geometry of squared circles across dimensions — from linear measures (π1, π2, π3) through area (π4), surface area (π5), and volume (π6). That such a coherent family emerges from the single ratio 4/√φ is, in Lear's view, powerful evidence that it is the correct value.

Why This Matters

Harry Lear's work sits at the intersection of physical measurement and geometric proof — two pillars of science that should never disagree. For 2,300 years, Archimedes' polygon method has been the accepted basis for π, refined by ever-finer approximations but never fundamentally challenged. Lear's contribution is to challenge it on two fronts simultaneously: the physical front (cut a circle and measure it) and the geometric front (discover the special squaring hidden in Kepler's Triangle).

His five additional constants further suggest that golden π is not an isolated correction but part of a unified dimensional framework — a system where π and φ are not merely connected but are different expressions of the same underlying geometric truth.

References

• Measuring Pi Squaring Phi — Harry Lear's main website
• Geometric Proofs of Pi — All seven proofs with diagrams and video walkthroughs
• Five More Universal Constants — π2 through π6 spreadsheet and explanation
• Harry Lear — "A very simplified Proof 7 (a & b) for the Proof of Pi"

π = 4/√φ = 3.144605511029693144…