Chapter 2 · Why It Cannot Be Drawn by Eye

In 1990 a French computer scientist named Gérard Huet sat down to draw a Śrī Yantra.

Huet was not an amateur. He was a senior researcher at INRIA, one of the significant figures in twentieth-century computer science, whose work on higher-order unification and on the Coq proof assistant is still foundational. He was also a Sanskritist of real standing, who would go on to build the Sanskrit Heritage Site and its computational morphology engine. If anyone had the equipment to draw a Sanskrit sacred diagram — the mathematics, the languages, the patience, the access to sources — it was him.

He records what happened in the first section of his paper, and the sentence is worth quoting exactly, because there is no ego in it at all:

Our first approach was completely experimental: the author tried to draw Śrī Yantra in free hand, and failed. A more systematic attempt with a computer drawing system failed too.

That is the whole problem in two lines. Freehand failed. Then a computer drawing package failed. Huet eventually got a correct figure only by measuring a printed plate with calipers, feeding the measurements in as a starting guess, and running Newton's method on a system of simultaneous equations.

This chapter explains why.

2.1 The triple intersections

Take any two of the nine triangles. Their edges cross. That is unremarkable — any two overlapping triangles cross.

Now take three. In the Śrī Yantra there are places where the edges of three different triangles are required to pass through one single point. Not near each other. Through the same point.

These are the triple intersections, and there are twenty-four of them.

This is the entire difficulty, and it is worth sitting with for a moment, because it is easy to read past. Three straight lines drawn at random in a plane do not meet at a point. They form a small triangle. To make three lines concurrent you must impose a condition — and every condition you impose costs you a degree of freedom somewhere else in the figure.

The Śrī Yantra requires this twenty-four times over.

The consequence, stated plainly:

You cannot move one triangle.

If you nudge U2 slightly to improve one crossing, you have broken every other crossing that U2 participates in. To repair those you must move D3 and D5, which breaks their other crossings, which requires moving U1 and D2, and so on around the figure until you are back where you started, having achieved nothing. The nine triangles form a single rigid mechanism, and there is no local adjustment. Every fix is a global fix.

This is why freehand fails, why a drawing package fails, and why the figure resisted a complete treatment until people started pointing computers at it. It is not a drawing problem. It is a simultaneous-equation problem that happens to be presented as a drawing.

2.2 What goes wrong, concretely

When concurrency fails, the failure is visible if you know where to look, and invisible if you do not.

Three lines that should pass through one point but do not will instead form a tiny triangle at the place where the point ought to be. Practitioners of the drawing tradition have a name for these: they are extraneous or secondary triangles — regions that exist in the drawing but not in the figure.

Their consequences are exact:

The region count breaks. Chapter 1 established that a correct figure has forty- three chambers. Every failed triple intersection adds one small spurious region. A figure with three bad crossings has forty-six regions, not forty-three, and the navāvaraṇa structure — 14 + 10 + 10 + 8 + 1 — no longer describes it.

The enclosure boundaries stop being closed. A chamber that should be a clean triangle becomes a quadrilateral with a sliver cut off it. The bands stop being bands.

It is easy to hide. Draw the lines thick enough and a crossing that misses by a millimetre disappears under the ink. This is a completely standard practice and it is not dishonest — a hand-drawn yantra on cloth or copper has line widths of a millimetre or more, and errors below that threshold are simply invisible. But it means visual inspection of a finished plate tells you almost nothing about its accuracy.

That last point deserves emphasis, because it explains an otherwise puzzling feature of the literature. Many published Śrī Yantras look flawless and are not. Kulaichev, who in 1984 measured a large number of surviving specimens, found errors he classified into distinct types; his correspondents' attempts at accurate construction ranged from about 0.05% of the radius to about 2%. At the scale of a printed page, 2% of the radius is under a millimetre. You cannot see it. You can only measure it.

2.3 The false Śrī Yantra

There is a version of the figure — extremely widespread, reproduced in serious academic books, on the cover of at least one classic study — that fails in a more fundamental way than a mistimed crossing.

Huet called it the False Śrī Yantra. Its defining property is this: in the correct figure, the apex of nearly every triangle lands on the base line of another triangle. In the false version, the apexes float. They stop wherever they happen to stop.

This makes the figure enormously easier to draw — you have removed most of the coupling that made the problem hard — and it produces something that reads, at a glance, as a Śrī Yantra. It has nine triangles. It has the right silhouette. It has a bindu.

It does not have forty-three chambers, and it is not the figure the tradition describes.

The transmission history is worth knowing because it explains why the error is so persistent. Heinrich Zimmer's Kunstform und Yoga im indischen Kultbild (1926) contained diagrams that Huet describes as clearly erroneous. The later English translation added a frontispiece — credited to a source added by the translators and absent from Zimmer's original — which is a correct figure. So the book contains both a correct plate and incorrect diagrams, and they disagree with each other. Zimmer's posthumous Myths and Symbols in Indian Art and Civilization, edited by Joseph Campbell, reproduces a false version, credited to Sir John Woodroffe (Arthur Avalon), and it can be found on the cover of Woodroffe's 1914 volume. From there it propagated: into Campbell, into Jung — who reproduced it upside down — and into the general symbolic literature, where it remains.

Two morals, and they will both recur:

First, a figure's provenance tells you nothing about its geometry. The false version descends from Woodroffe and Zimmer, who were serious scholars. Descent from a good source is not evidence of correctness.

Second, and more uncomfortably: the false Śrī Yantra is often prettier. Huet noticed this and was honest about it. The correct figure has an awkward slope in the innermost downward triangle; the false one is smoother. When we come to argue in Part IV about which figure is right, we will have to be very careful that "harmonious" is not doing work that "correct" should be doing.

2.4 The drawing instructions do not work

If the figure is this hard, how did anyone ever draw it?

The tradition does provide instructions. Several sets, in fact. The best known is the "inside-out" method attributed to Bhāskararāya's Nityāṣoḍaśikārṇava, which builds the figure from the centre outward.

Huet examined it and reported a specific, checkable objection:

The inside-out instructions, attributed to Bhāskararāya's Nityāṣoḍaśikārṇava, are clearly misleading, since there is no hope, except by extraordinary luck, to get points J and Q on the circle determined by its diameter 0T.

The objection is precise. If you build outward from the centre, the outermost triangles are the last thing determined — and the construction requires their base vertices to land exactly on the enclosing circle. Nothing in an inside-out procedure forces that. You build the whole figure, arrive at the boundary, and discover you have missed. There is no step at which you can correct it without starting again.

Huet's conclusion was that the text should be read as a description of the Śrī Yantra rather than as a construction of it — a way of remembering a figure you already have, not a way of generating one you do not.

I think this is nearly right and importantly incomplete, and the gap is where this book does its work.

The inside-out instructions are indeed not a construction. But there is a second family of traditional instructions — the ones descending from the Vyāse devīkṛte verse and its commentaries, which work outside-in, starting from the circle and its diameter divided into forty-eight parts. Those instructions do not have Huet's defect, because the circle is where they begin rather than where they end. They are also far more specific than anyone in the modern literature has given them credit for, and in Chapter 5 and Chapter 6 we are going to read them extremely carefully and then measure exactly what they produce.

Huet cannot be faulted for this. He was examining the source that the secondary literature put in front of him. But the conclusion "the tradition cannot construct the figure" was drawn from one family of instructions and then generalised, and it has been repeated ever since by authors who did not go back to check. It is one of several places in this field where a reasonable judgment hardened into a received fact.

2.5 The shape of the real problem

Strip away the drawing and here is what remains.

Some number of quantities describe the figure — we do not yet know how many, and Chapter 3 will settle it. Against those quantities stand the conditions: twenty-four triple intersections, the requirement that certain vertices lie on the circle, and whatever else the tradition demands.

Three outcomes are possible, and until 1990 nobody knew which one obtained:

Over-determined. More conditions than freedoms. No exact figure exists, and every Śrī Yantra ever drawn is an approximation to something impossible.
Exactly determined. Conditions equal freedoms. Exactly one figure exists. There is a true Śrī Yantra and everything else is error.
Under-determined. Fewer conditions than freedoms. Infinitely many exact figures exist, and "which one" is a further question that the conditions do not answer.

Huet's paper settled it, and the answer is the third. We will state his theorem properly in Chapter 11. But the shape of the answer is worth carrying with you from here, because it reframes everything:

The problem is not that the Śrī Yantra is too hard to draw exactly. The problem is that there are infinitely many exact Śrī Yantras, and the classical conditions do not tell you which one is meant.

Everything difficult about this subject follows from that. The disagreements in the literature are not disagreements about arithmetic. They are disagreements about which member of an infinite family deserves the name — conducted, for the most part, by authors who did not realise that was the question they were answering.

What this chapter established

Twenty-four triple intersections couple the nine triangles into one rigid mechanism. No triangle can be adjusted alone.
Failed concurrency creates extraneous triangles, breaks the count of forty-three, and is easily hidden under line width — so visual inspection is nearly worthless as a test.
The false Śrī Yantra — floating apexes — is widespread, descends from respectable scholarly sources, and is frequently more attractive than the correct figure. Beauty is not evidence.
The famous inside-out instructions genuinely do not construct the figure. But a second, outside-in family of traditional instructions exists which does not share that defect, and which the modern literature has not examined properly. That is Chapters 5 and 6.
The real problem is not impossibility but under-determination.

Next: how many numbers does it actually take to specify a Śrī Yantra?