The kūrma
The Śrī Yantra on a spherical cap — the tortoise-backed form.
Some texts describe the yantra not as flat but as raised on a dome, curved like a tortoise’s shell. That form has been used, for the better part of forty years, to explain why the plane figure is hard to draw: the argument runs that the true figure was spherical all along and the plane version is a degraded copy.
This platform takes the kūrma seriously as an object and rejects it as an explanation. Both halves of that matter, and they are separable.
Figure
View
Projection
Measured
| cap radius | 20.000° |
| rise / radius | 0.176327 |
| sphere radius / flat radius | 2.9238 |
| concurrency | 3.97e-13 |
| concentricity | 1.59e-13 |
| base line bows off its chord | 2.814% |
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What the sphere actually buys
The plane figure has five parameters. Imposing concurrency and concentricity leaves three free — an infinite family, which is why no two published figures agree. Moving to a sphere adds the cap radius as a sixth parameter, and the natural hope is that the extra freedom pins the figure down.
It does not. The sphere buys exactly one parameter, and one is not enough: the spherical problem is under-determined in the same way and to the same degree as the plane one. The kūrma is a real form with real geometry. It is not a solution to the plane figure’s indeterminacy, and no amount of curvature makes it one.
The correction
Rao’s 1998 paper — the only complete spherical formulation in existence, and badly under-credited — reports that the fit to the traditional numbers improves as the cap flattens toward a plane. Checked against the same data, it does not. The misfit has a genuine minimum near r ≈ 20°, not at the limit.
| cap radius | rise / radius | RMS misfit | against the noise floor |
|---|---|---|---|
| 0.5° | 0.004 | 2.2960% | indistinguishable |
| 10° | 0.087 | 2.2836% | indistinguishable |
| 20° | 0.176 | 2.2682% | indistinguishable |
| 30° | 0.268 | 2.3234% | indistinguishable |
| 45° | 0.414 | 2.8472% | ruled out |
| 60° | 0.577 | 4.3973% | ruled out |
| 75° | 0.767 | 7.5066% | ruled out |
Every point is a full solve: at that cap radius, the spherical figure with concurrency and concentricity exact that best fits all eighteen traditional height-and-width statements. The improvement from flat to 20° is 0.0279 percentage points — about 1.0% of the tradition’s own internal inconsistency. The minimum is real and it is also negligible.
The honest qualification: that minimum sits inside the tradition’s own noise floor of 2.675% of R. So it is a correction to a published claim, and not evidence that the tradition intended a 20° dome. A minimum you cannot distinguish from the measurement error around it is a fact about the arithmetic, not about the intention.
What the curve does establish, decisively, is a negative. Past about 45° the misfit clears the noise floor and keeps climbing — 4.40% at 60°, 7.51% at 75°, nearly three times the noise. Whatever the tradition was describing, it was not a deep dome.Kulaichev’s own illustrations run to 82°.
The proportions, at r = 20°
| rise divided by radius | 0.176327 |
| sphere radius per unit of flat radius | 2.9238 |
| cap radius | 20.000° |
A dome rising about a sixth of its own radius. Shallow — which is why the bowing of the base lines is measurable rather than obvious, and why a kūrma photographs so much like a plane plate.
The full argument is Part VI of the book: what the kūrma is, Rao’s formulation, what the sphere buys, testing the hypothesis, and the accurate kūrma.