Nine triangles · 43 cells · 49 conditions
Almost every Śrī Yantra in circulation — printed, cast or tattooed — is subtly wrong. Where three lines should cross at one point, they miss. Here the whole figure is solved from its defining conditions in 60-digit arithmetic, and the worst error anywhere in it is —.
Bhū-prastāra · the flat figure
Nine isosceles triangles share one vertical axis: five point down, four point up. The conditions that make the figure a Śrī Yantra are all about coincidence — apexes that must land on other triangles' bases, and triples of lines that must meet at one point rather than three. Solving them is a non-linear system with no ruler-and-compass shortcut.
Tradition counts 43 triangles in five enclosures. The nine triangles actually cut the figure into 74 regions — the 43 are the ones an odd number of triangles covers. Under that rule every survivor is a triangle and the counts fall out exactly.
| i | base y | half-width | apex y |
|---|
In units of the radius of circle E, which circumscribes t₃ and t₇. Origin at its centre, y upwards. The JSON carries 40 significant digits.
Mahā Meru · the mountain
Give each of the nine enclosures its own terrace and the diagram becomes a stepped mountain with the bindu at the summit. Every terrace outline is measured off the solved figure, not modelled by hand — which is why the star-shaped ledges are exactly the triangle enclosures.
The maths
Number the triangles t₁…t₉ by the height of their base, top down; t₁…t₅ point down, t₆…t₉ point up. Each is fixed by three numbers: the height of its base, the half-width of that base, and the height of its apex. The conditions, following Chiodo (2021):
That is 21 independent equations in 27 unknowns: seven from (ii), twelve from (iii), and two from (i), since sharing a circumcircle means equal centre and equal radius. Solutions therefore form a six-parameter family. Fixing E as the unit circle spends the two similarity freedoms, scale and vertical shift, leaving four real parameters: the base heights of t₃, t₆, t₇ and t₉. Choose those and every other number in the figure is forced.
All seven pairs in (ii) are Chiodo's; his definition lists them in full, (t₁,t₆) included. It earns a note because it is the one a reader can lose without noticing: drop it and nothing looks broken, the construction still runs, yet the solution set gains a dimension (Jacobian rank 21 with seven pairs, 20 with six, checked numerically at the solution) and the figure stops being determined by the four parameters. The construction uses the pair as “a ray r stemming from Q” in §2.3.2, with Q the base point of t₆ per §2.2.2. With all seven, every apex except those of t₃ and t₇, which sit on E, is the base point of another triangle. The verification here asserts all seven, so a transcription that loses one fails loudly. An earlier version of this page claimed the pair was missing from Chiodo's published list; it is not, and the error was ours.
Rather than throw 27 unknowns at a solver, the conditions are applied in the order Chiodo's straightedge construction resolves them. Each step pins one more quantity, and the chain consumes every condition but one: ((iii); t₄,t₆,t₉). That leftover becomes a single scalar equation in a single unknown — the height of t₁'s base — closed by a one-dimensional root find at 60 digits.
Every condition is then re-derived from the final coordinates alone, never from the chain that produced them: each triple of lines becomes a 3×3 determinant of normalised line equations, each circle condition a distance. This table is that check, run fresh in the build.
The nine triangles cut the figure into 74 regions, not 43. Fill them by the even-odd rule — a region counts when an odd number of the nine triangles covers it — and exactly 43 survive, every one of them a triangle, with cover counts 9, 7, 5, 3, 1 giving the trikoṇa, aṣṭakoṇa, antardaśāra, bahirdaśāra and caturdaśāra in turn. All 31 leftovers have an even cover count and 21 of them are quadrilaterals — which are genuinely present in the classical figure too, plainly visible at the far left and right of Chiodo's own plate. This is why traditional Śrī Yantras are painted with alternating filled and open cells.
Nothing in the concurrency conditions constrains the lotuses, the three circles or the bhūpura. Those follow the traditional canon set out in Geometry of Srichakra, which sizes everything against an inner-most circle of diameter 108: the eight-petalled lotus at 127, the sixteen at 144, the outer circle of the trivalaya at 108√2, and the bhūpura enclosed in a circle of 224. Its twelve outer corners sit on the 30°, 45° and 60° lines; each gate is a T whose neck meets the wall on the 15° and 75° lines.
Download
Everything here is generated by the solver in this repo and is free to use.