Chapter 3 · Five Numbers

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Chapter 2 ended with a question that sounds technical and is actually the crux of the whole subject: how many numbers does it take to specify a Śrī Yantra?

The answer is five. This chapter shows why, sets up the machinery we will use for the rest of the book, and resolves a disagreement in the literature that has confused readers for thirty years.

If you want the mathematics, it is all here. If you do not, read §3.1, §3.6 and the summary, and you will have everything you need to follow the argument in Parts II through IV.

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3.1 The idea, before any algebra

Here is the observation that makes the whole thing tractable.

Every one of the nine triangles is isosceles, with its apex on the vertical axis and its base horizontal. Look at any correct Śrī Yantra and you will see it: nine horizontal lines, and every triangle sitting on one of them.

That is a very strong constraint, and it does most of the work for us. Because if the base is horizontal and the apex is on the axis, then a triangle is completely specified by just two numbers: the height of its base line, and the height of its apex.

So the whole figure is determined by a set of horizontal levels.

Now the second observation, which is the one that closes the problem: the apexes are not free. In a correct Śrī Yantra, the apex of nearly every triangle sits on the base line of another triangle. That was exactly what the false Śrī Yantra of §2.3 threw away. So the apex heights are not new information — they are the base heights again, reused.

Nine triangles, nine base lines. But the two largest triangles have their apexes at the top and bottom of the circle, and the remaining seven apexes are all borrowed from other triangles' bases.

That leaves the nine base heights as the only quantities in play — and, as we will see, only five of those can be chosen freely. The other four are forced.

Five numbers. That is the whole figure.

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3.2 Setting up

Fix the enclosing circle: radius r = 1, centre at the origin. The top of the circle is at y = +1, the bottom at y = −1.

Now define five vertical spacings. Following C. S. Rao — whose notation the entire modern literature uses, generally without saying so — call them

b,\quad c,\quad d,\quad e,\quad g

They are measured as gaps between consecutive horizontal lines, not as absolute heights, and they sit as follows:

`` y = +1 top of circle apex of U1 ↕ a y = +d + e base of D5 apex of U2 ↕ e y = +d base of D1 apex of U4 ↕ d y = 0 ─── centre ─── ↕ g y = −g base of U3 apex of D5 ↕ y = −c base of U1 apex of D4 ↕ b y = −(b+c) base of U4 apex of D2 ↕ a y = −1 bottom of circle apex of D1 ``

Two further quantities appear so often that they get names:

a = r - (b + c), \qquad f = r - (d + e)

a is the gap from the base of U4 down to the bottom of the circle; f is the gap from the base of D5 up to the top. Neither is independent — both are determined by the five.

So: five numbers, five horizontal lines fixed. The other four base lines — those of D2, D3, D4 and U2 — are not free. They are determined by the requirement that the triple intersections work out. We will see exactly how in §3.4.

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3.3 From the five numbers to every vertex

Each triangle needs a half-width: how far its base extends either side of the axis.

For the two largest triangles this is immediate, because their base vertices lie on the circle. A horizontal line at height y meets the unit circle at x = ±√(1 − y²), so:

X_1 = \sqrt{1 - c^2} \qquad\text{(half-base of U1, at } y = -c) X_2 = \sqrt{1 - d^2} \qquad\text{(half-base of D1, at } y = +d)

Those are the only two that come directly from the circle. Everything else is obtained by walking the figure: each remaining half-width is found by intersecting a horizontal line with an edge that has already been determined.

The mechanism is elementary similar triangles. If a line runs from apex (0, y_A) to a base vertex (X, y_B), then at any intermediate height y its horizontal distance from the axis is

x(y) = X \cdot \frac{y_A - y}{y_A - y_B}

Apply this repeatedly and the figure unfolds. For example:

X_3 = \frac{r-c}{r+d}\,X_2, \qquad X_4 = \frac{r-d}{r+c}\,X_1

X_5 = \frac{b}{b+c+d}\,X_4, \qquad X_6 = \frac{e}{c+d+e}\,X_3

and so on, through a chain of about thirty intermediate quantities, until every vertex of every triangle is expressed as an explicit algebraic function of b, c, d, e, g.

The full chain is set out in Appendix B. It is long but entirely mechanical — there is no cleverness in it, only bookkeeping. What matters here is the structural fact:

Given b, c, d, e, g, every one of the twenty-seven vertices of the nine triangles is determined. No choices remain.

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3.4 Where the four forced lines come from

Four base heights were not in our list of five: those of D2, D3, D4 and U2. They emerge from the walk, and it is worth seeing how one of them appears, because this is where the figure's rigidity actually lives.

Consider the base of D2. Working through the chain, one arrives at an intermediate quantity — call it U_8 — and the base of D2 turns out to sit at height

y_{D2} = d + v_8, \qquad \text{where } v_8 = r - U_8 - d

and U_8 itself is

U_8 = \frac{r+g}{Q_8 + 1}, \qquad Q_8 = \frac{d+g}{r+c}\cdot\frac{X_1}{X_6}

The details do not matter. What matters is what the expression contains: X_1 depends on c, X_6 depends on c, d and e, and the whole thing depends on g as well. The height of D2's base is a function of four of the five parameters at once.

That is the coupling of Chapter 2, written down. You cannot move D2 without moving everything, because the position of D2 is not a free choice at all — it is an output.

The same holds for D3, D4 and U2. Nine base lines; five chosen; four forced.

Remember this number. Chapter 6 is going to observe that the traditional construction specifies all nine — and then measure what happens.

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3.5 Conditions, and what they cost

We now have a five-dimensional space. Every point in it — every choice of (b, c, d, e, g) — gives a figure with nine triangles in a circle.

Almost all of them are wrong. Most are not even recognisable.

To pick out the good ones we impose conditions: equations that a correct figure must satisfy. The most important is concurrency — the requirement that the triple intersections actually meet. It turns out that if you construct the figure in the right order, all twenty-four crossings close automatically except one, and concurrency reduces to a single equation. In the notation of Appendix B it is

C_1 : \quad X_{11} - X_{11a} = 0

where X_{11} and X_{11a} are two independent routes to what ought to be the same point. If they differ, the lines miss.

And now the accounting that governs this entire book:

Five parameters. Each condition you impose costs you one.

To pin down one figure you need five conditions. Concurrency is one. The rest of this book is about what the other four should be — and it will turn out that no published account has ever supplied all five.

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3.6 A disagreement that is not a disagreement

Readers who go to the sources hit an apparent contradiction immediately, and it has confused people for years. Set the claims side by side:

Four or five?

Both, and they are counting at different moments.

Huet chooses five parameters and then imposes his closing condition — his requirement that a particular point land on a particular line, which is concurrency. Five minus one is four. He is counting after.

Rao and the Optimal article count the variables needed to describe the figure at all, before any condition is applied. That is five. They are counting before.

There is no dispute. The clean way to say it, which we will use throughout:

Five parameters describe the figure. Concurrency costs one, leaving four. Every further condition costs one more.

Chiodo's 2021 derivation is worth noting because it comes at the problem from a completely different direction — a six-parameter family, reduced by two conditions to land on four — and agrees. When two independent routes reach the same count, the count is probably right.

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3.7 Why the parameterisation matters more than it looks

One consequence of §3.3 is easy to miss and is, practically speaking, the most useful thing in this chapter.

Because every figure reduces to five numbers in a fixed frame, any two Śrī Yantras can be compared directly. Take a figure published in 1984 in Moscow, another from a 1998 Indian journal, a third from a 1990 French preprint, a fourth read off a Sanskrit verse of uncertain date — put each into (b, c, d, e, g) and they become five-element vectors in the same space. You can subtract them. You can measure the distance between them. You can ask of any of them whether it satisfies any given condition, and get a number.

Nobody has done this. The literature on the Śrī Yantra consists of authors who each built their own apparatus, published a figure in their own coordinates, and did not check their result against anyone else's. Kulaichev's figure is stated as a root of an eighth-degree polynomial. Huet's is stated as eleven Y-coordinates measured from the bottom of the circle with the diameter as unit. Rao's is a table of nonlinear-programming outputs. The traditional figure is stated as integers on a diameter of forty-eight.

They are all the same kind of object, and once you see that, they can all be laid on one table.

That table is Chapter 16, and it is the first time it has been assembled.

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What this chapter established

• Every triangle is isosceles, apex on the axis, base horizontal — so the figure is a set of horizontal levels.
• The apexes are not independent: they sit on other triangles' base lines. That coupling is exactly what the false Śrī Yantra discards.
• Nine base lines; five free, four forced by the geometry.
• The five are b, c, d, e, g — vertical spacings on a unit circle — and from them every one of the twenty-seven vertices follows explicitly.
• Five parameters; each condition costs one. Concurrency leaves four. Pinning down a single figure requires five conditions in total.
• The "four versus five degrees of freedom" disagreement in the literature is an artefact of counting before or after concurrency. There is no real dispute.
• Because all figures reduce to the same five numbers, every published Śrī Yantra can be put on one table and compared. Part III does the reading; Chapter 16 does the comparing.

Next, Part II: what the tradition actually says — beginning with a verse from the Saundaryalaharī that counts the nine triangles and tells us which of their properties the tradition thought mattered.