What exactly does this page display?

It displays an interactive visual simulation (HTML5 Canvas) drawing a tree, its shadow, and a matching right-angled triangle. When the angle is set to 22.5° (π/8), the ratio of shadow to height (L/H) equals tan(22.5°) = √2 − 1 ≈ 0.41421356. The simulation allows adjusting the angle with the value updating instantly.

22.5° = π/8 is half of 45° (π/4). Using the half-angle formula tan(θ/2) = (1−cos θ)/sin θ at θ = 45°, we get tan(π/8) = (2−√2)/√2 = √2 − 1. Thus, 22.5° is not an arbitrary choice, but an exact mathematical result.

How does the simulation calculate L/H?

If H is the tree height and L is the shadow length, then L/H = tan(θ) where θ is the sun's angle from the vertical. In the simulation, H is relative (26% of the canvas height), and L = H × tan(θ). The value is displayed directly at the top of the screen and in the stat cards.

Is this simulation linked to empirical evidence?

The theoretical derivation here is the mathematical basis for the Field Measurement Protocol (tree3-en.html). A preliminary survey covered 30 Yemeni trees; the mango subsample (n=5) showed a mean (H−h)/H of 0.41297, with TOST p=0.0008. This is an exploratory finding on a small sample and requires independent verification before any generalization. The small sample limits the strength of the inference.

Tree of Unification — Visual Simulation | 22.5° Angle yields √2−1

An interactive simulation drawing a tree, its shadow, and a matching right-angled triangle. At 22.5° (π/8), the shadow-to-height ratio L/H equals tan(22.5°) = √2 − 1 ≈ 0.41421356. Use the "Sun Motion" button to change the angle, "Measure" to show H, L, and L/H on the canvas, and "Reset" to return to 22.5°.

The θ arc matches above and below — measured from the vertical, not the horizontal

This page presents the interactive visual simulation of the mathematical identity tan(22.5°) = √2 − 1 ≈ 0.41421356. The exploratory field study (n=5) reported an initial numerical similarity, further explored in the Measurement Protocol.

Angle θ (from vertical)22.5°

L / H (Shadow ÷ Height)0.41421356

tan(θ)0.41421356

√2 − 1 (Reference)0.41421356

v4.1: Calculation fixed in the proof section; the sun freezes at the current angle; the sun is clamped within the canvas.
Only at θ = 22.5°: L/H = √2 − 1 = 0.41421356

📐 Three levels of interpretation — do not conflate them

This page combines three distinct epistemic layers: an exact mathematical identity, a numerical visual simulation, and a pilot field study. Keeping them separate strengthens credibility and does not detract from the project's value.

Level 1 — Mathematical identity (exact)

The number √2−1 is not a choice; it is an exact consequence of the half-angle formula:

tan(22.5°) = tan(45° / 2)
= (1 − cos 45°) / sin 45°
= (1 − √2/2) / (√2/2)
= (2 − √2) / √2
= √2 − 1 ≈ 0.4142135624

This is a mathematical identity — valid regardless of any tree or shadow.

↓

Level 2 — Numerical simulation: Pythagoras applied

The following example is independent and illustrative (at ≈ 56.54°): it shows how Pythagoras links the sides of a right triangle. The numbers are not from the current drawing (θ=22.5°); they were chosen to illustrate the application.

Z = 192, C = 290.74

Z² + C² = 192² + 290.74²
= 36864 + 84530.74
= 121394.74
A = √121394.74 ≈ 348.42

tan⁻¹(290.74 / 192) ≈ 56.54°

In the default simulation (θ=22.5°), the ratio L/H becomes tan(22.5°) = √2−1 automatically — see it at the top of the page.

↓

Level 3 — Pilot field study

A preliminary measurement on a small sample of Yemeni mango trees gave a mean (H−h)/H close to the theoretical value:

n_mango = 5 (out of 30 trees)
mean (H−h)/H = 0.41297
TOST p = 0.000801 (preliminary)
90% CI = [0.40491, 0.42102]

Inference limits: the sample is small; the mean indicates a numerical similarity with the mathematical identity, not proof of a general natural law for trees. Results are exploratory and require independent verification before any generalization.

Conclusion: each level serves its purpose

Z' + C' + A' = 1

where Z' = Z/(Z+C+A), C' = C/(Z+C+A), and A' = A/(Z+C+A).
Z and C are the Pythagorean legs, A is the hypotenuse. This is a normalized representation, not a direct consequence of Pythagoras alone.

Level 1 gives the exact value — Level 2 shows it visually — Level 3 suggests a preliminary numerical similarity.

🌳 Tree of Unification · Tree Above · Triangle Below