The trig functions that appear in CET's prediction formulas aren't arbitrary — they're there because the stella octangula projects onto the sky the same way a point on a unit circle projects onto its axes. This page shows the geometry and the live prediction together.
A unit circle has radius 1. Pick any point on its edge: it sits exactly 1 unit from the center. That point has two coordinates — cos θ, how far it is horizontally, and sin θ, how far it is vertically. The angle θ is measured from the positive horizontal axis.
cos and sin are shadow projections. They answer: if you shone a light straight down at this point, how long is the shadow on the floor? Straight across — how long is the shadow on the wall? Neither shadow can exceed 1, because the radius is 1. And both are always related by the identity cos²θ + sin²θ = 1.
At 45°, cos and sin are equal — both √2/2 ≈ 0.707 — and their sum is √2 ≈ 1.414. Drag the slider toward 0° and cos grows toward 1 while sin shrinks toward 0. Toward 90° and they trade places. The sum cos θ + sin θ peaks at √2 when θ = 45° and equals 1 when θ = 0° or 90°.
That sum — cos θ + sin θ — turns out to be exactly what appears in the Eta Carinae prediction. Not by accident.
A stellar explosion driven by the bilateral crossing has a definite structure: a bipolar axis, an equatorial plane, and secondary features at angles determined by the crossing geometry. What we observe depends on where we're standing relative to that axis.
Call the angle between the bipolar axis and our line of sight i — the inclination. If i = 0°, we're looking straight down the barrel: we'd see a ring. If i = 90°, we're in the equatorial plane: we'd see the full bipolar structure edge-on. Most observations fall somewhere between.
At inclination i, the bipolar axis projects two ways at once. cos i tells you how much of the axis points directly toward you — its depth projection. sin i tells you how much of it is spread across the sky plane — its width projection. Both components are always present simultaneously.
This is what makes the bilateral crossing different from a single axis: it has two directions that both project. The observer sits at one point; the crossing structure has two relevant faces. The trig captures which face each observer sees, and in what proportion.
The secondary condensation features — the rings and lobes outside the main bipolar axis — sit at an angle θ from the bipolar axis. That angle is set by the crossing geometry acting on both the depth and width projections simultaneously. The formula that results is:
θ = arctan(1 ÷ (cos i + sin i))
The 1 in the numerator comes from the crossing geometry itself — it's normalized. The denominator combines both projections. The arctan converts the ratio back into an angle.
Eta Carinae is a massive stellar system about 7,500 light-years away. Its observed inclination — the angle of its bipolar axis to our line of sight — is approximately 41°. The CET prediction for secondary condensation features at that inclination is:
Drag the inclination slider across the full range. Notice how little θ moves. At 41° (Eta Carinae's observed inclination) the prediction gives 35.3°. The observed secondary condensation angle in Eta Carinae is approximately 35° to 36°, measured from the bipolar axis. The agreement is exact within measurement uncertainty, derived with no free parameters.
This is what zero free parameters means in practice. The formula θ = arctan(1/(cos i + sin i)) takes one measured input — the inclination — and returns the secondary angle. No tuning. No fitted constant. The geometry does the work.
Drag the inclination slider from 1° to 89° and θ stays within a narrow band — roughly 25° to 45°. It never escapes that range regardless of the inclination. This isn't a coincidence; it's what the sum cos i + sin i does.
The sum cos i + sin i is always between 1 (at i = 0° or 90°) and √2 ≈ 1.414 (at i = 45°). That means 1/(cos i + sin i) is always between 1/√2 and 1. So arctan of that ratio is always between arctan(1/√2) ≈ 35.26° and arctan(1) = 45°.
The prediction is geometrically constrained to a 10°-wide window: 35.26° to 45°. Any observed secondary condensation angle in that range is consistent with the bilateral crossing structure. Eta Carinae at 35°–36° sits right at the geometric floor — the crossing geometry's most stable expression.
No other geometric framework predicts this window from first principles. The bilateral crossing produces it necessarily, as a consequence of how cos i + sin i behaves.
"The trig isn't applied to the geometry. The geometry is the trig."
Sin and cos appear in the Eta Carinae formula because projection is what sin and cos do. The bilateral crossing has two faces that project simultaneously onto every observer's sky plane. The formula that combines those projections is the formula for the sum of two shadows from the same unit radius. Arctan converts the ratio back to the angle we can measure.
The result — a secondary condensation angle of 35.3° at 41° inclination — is confirmed in the literature. It follows from three elements: the crossing geometry, the inclination angle, and the three trig functions that describe projection. No other inputs. Nothing tuned. The Stellar Predictions page puts Eta Carinae alongside the other three confirmed cases. The Geometric Physics page covers what zero free parameters means as a scientific standard.
No results — try a different term.