A geometric and numerical frame, reproducible in the browser: proof of the 22.5° angle, an interactive simulation, a bar chart, the preliminary measurement table, and a TOST equivalence test computed in-page with automated verification of the t distribution.
The frame begins with an exact trigonometric result, then builds an interactive simulation on it, then presents preliminary field measurements that are examined exploratorily against a pre-specified equivalence margin.
How can an exact geometric relationship (tan 22.5° = √2−1) be linked to measurable field observations, and what is actually inferred from a numerical proximity between a crown/height ratio in a small sample and the theoretical value?
An exact trigonometric proof of tan(22.5°) = √2−1, the shadow ratio relation S/H = tan(θ) within the geometric model, an interactive simulation, preliminary measurements of five mango trees, and a TOST equivalence test reproducible within the page with visible reference tests.
An exact mathematical proof for the trigonometric result. A geometric model with specified conditions (horizontal ground, straight light ray) for the shadow relation. Exploratory field data (n=5) showing numerical proximity, not proof. This frame makes no proven claim of a connection to the Riemann hypothesis.
If the angle θ is measured from the vertical, then the sun's elevation angle above the horizon is α = 90° − θ. For a solar ray making elevation angle α with a horizontal ground, a right triangle yields that the ratio of the horizontal shadow to the vertical height is:
At θ = 22.5°, α = 67.5°, and therefore:
Let x = tan(22.5°). Since tan(45°) = 1, the double-angle formula gives:
Rearranging gives x² + 2x − 1 = 0. Solving the quadratic:
Since 22.5° is an acute angle, x is positive, therefore:
Change the object height and angle θ to see the theoretical shadow length under the geometric model. θ is measured from the vertical, not the horizon. Angle ranges are complementary: θ ∈ [5°,60°], α = 90°−θ ∈ [30°,85°].
Note: changing the sun elevation α updates θ to its complement; changing θ directly updates α to 90°−θ. Bounds are limited to the complementary ranges above.
Five mango trees in Shu'ub, Sana'a; recorded measurement date: October 14, 2026. H is total height, h is measured lower height, C = H−h is the vertical difference. The studied ratio is R = (H−h)/H.
| Tree | H (m) | h (m) | C = H−h (m) | R = C/H | Difference from √2−1 |
|---|---|---|---|---|---|
| 1 | 7.80 | 4.50 | 3.30 | 0.42308 | +0.00886 |
| 2 | 8.50 | 5.00 | 3.50 | 0.41176 | −0.00245 |
| 3 | 7.20 | 4.20 | 3.00 | 0.41667 | +0.00245 |
| 4 | 9.00 | 5.40 | 3.60 | 0.40000 | −0.01421 |
| 5 | 7.50 | 4.40 | 3.10 | 0.41333 | −0.00088 |
Corrections applied: row 5 sign (was +0.00088, now −0.00088) and row 1 rounding (was +0.00887, now +0.00886). All values are computed from the displayed H and h.
A successful TOST comparing mean R to √2−1 does not prove the shadow law S/H = tan(θ), because the two are different quantities. The honest phrasing is: "mean crown/height ratio in this sample is numerically close to √2−1 within the specified margin", without claiming causality or generalizing to mango trees at large.
The two-one-sided equivalence test at α = 0.05 corresponds to a 90% confidence interval for the mean lying entirely inside [μtarget−Δ, μtarget+Δ]. The margin Δ must be specified before data collection; changing it after seeing the result invalidates the test's confirmatory meaning.
| Quantity | Value |
|---|
p₁ is the one-sided test of the lower bound (H₀: μ ≤ μtarget−Δ); p₂ is the one-sided test of the upper bound (H₀: μ ≥ μtarget+Δ); p_TOST = max(p₁, p₂). Equivalence at α=0.05 is concluded when p_TOST < 0.05. Note that n=5 is small, statistical power is limited, and the conclusion depends on the chosen margin.
These tests run automatically on page load; results are visible below. Reference values are taken from standard t distribution tables.
| t | df | Expected | Actual | |Error| | Result |
|---|
Each bar represents R for one tree; the dashed line is the theoretical value √2−1 ≈ 0.41421. The chart compares values but does not establish that the similarity is causal.
| Claim | Status | Limits of inference |
|---|---|---|
| tan(22.5°) = √2−1 | Proven mathematically | Exact trigonometric result |
| S/H = tan(θ) | Correct within the geometric model | Requires horizontal ground and consistent angle/shadow measurements |
| Mean R for trees is close to √2−1 | Numerical description of a small sample | Five trees at one site; insufficient for generalization |
| Mean equivalence within Δ (value computed above) | Test conditional on chosen margin | Margin is practical/conventional and must be justified before data collection; power is limited at n=5 |
| General link to the Riemann hypothesis | Not proven | Numerical similarity does not create a proof or a mathematical connection |
The displayed 90% CI is the t-based interval for the sample mean (n=5, df=4). The TOST in Section 3 is computed from the same five data points and reports the margin, degrees of freedom, p₁, p₂, and p_TOST. Passing the test does not amount to proving exact equality or a general natural law. The reference tests of the t distribution (Section 3) show whether the displayed p-values can be trusted before relying on them.
frame2-en.html and open in a modern browser. Calculations, plots, and CSV export work offline. The "Download PDF" button requires loading html2pdf.js from CDN on first use, or hosting it locally.The file begins with #-comment metadata identifying the frame, author, DOI, collection date and site, units, and a summary of descriptive statistics, followed by the column header row:
tree, H_m, h_m, C_m, R, target_sqrt2_minus_1, difference_from_target
Columns tree, H_m, h_m are the measured data; C_m, R, difference_from_target are derived; target_sqrt2_minus_1 is a theoretical constant. The file does not contain additional raw measurements or a record of instruments and uncertainty, which must be added before use in a new analysis.
After proving in Section 1 that \( \tan(22.5^\circ) = \sqrt{2} - 1 \) in the 2D plane (The Tree and the Shadow), we now transition to 3D geometry. What happens if we move from a right-angled triangle to a cube?
In a cube, the angle between the space diagonal (passing through the center) and any of its connecting edges is called the "Magic Angle". It is famous in Nuclear Magnetic Resonance (NMR) because it cancels out specific dipolar interactions in a magnetic field. Its value is:
It can also be expressed using the tangent function (tan):
What is the relationship between this angle and 22.5°?
In the Tree and Shadow (plane): \( \tan(22.5^\circ) = \sqrt{2} - 1 \).
In the Cube (space): \( \tan(54.7356^\circ) = \sqrt{2} \).
This gives us an exact mathematical identity:
Both depend on the square root of 2, but in different dimensions! This proves that the constant \( \sqrt{2} \) plays a pivotal role in both planar and spatial geometry.
Plane (2D)
Tree & Shadow
Space (3D)
Cube Diagonal