Much has been written on 108 as a holy number. It may represent:

  • the 9 planets times the 12 constellations (9×12)
  • the 27 nakshatras with each of their four padas or parts (27×4)
  • much other goodness.

The other day another reason came to me.

One is the numerological representation for the Sun.

Eight is the numerological representation for Saturn.

Zero is the same as all, infinity, in a sense, since one divided by each approaches the other (assuming here that zero is being approached from the right, sinceVedic math does not tend to use negative numbers).

So, 108 represents all between Sun and Saturn, which is to say our whole solar system, unifying the domain and range of astrology with one’s body, one’s lineage, the Ganga, and divine geometry.

Jai Guru Dev.

Originally published at https://www.ayurastro.com/articles/yet-another-reason-why-108-is-a-holy-number

Postscript · 29 August 2026

The piece above is from 2015. This is a note from eleven years later.

Twenty-seven lines

In 1849 Arthur Cayley wrote to George Salmon about a curiosity. Take a cubic surface — the smooth, rippling sheet you get when a third-degree polynomial is set equal to zero — and ask how many perfectly straight lines lie flat inside all that curvature. Cayley proved the number has to be finite. Salmon pinned it down: exactly twenty-seven.1 Not roughly twenty-seven, not twenty-seven on average. Twenty-seven, on every smooth cubic surface there is.

Twenty-seven. The same count as the nakshatras.

The theorem has lately been given fresh life — in 2017 Jesse Kass and Kirsten Wickelgren rebuilt the count using motivic homotopy theory, so that the twenty-seven survives being carried out of the complex numbers and into other number systems entirely2 — and Quanta Magazine sent the film of it around again this month. I passed it along.

Quanta Magazine’s film of the twenty-seven lines.

Here is the speculation I cannot put down. A cubic surface is a two-dimensional skin curving through three dimensions, and it carries twenty-seven lines. Lift the same idea one dimension higher — a cubic form curving through four dimensions — and my guess is that it would want four times as many.

Four times twenty-seven. One hundred eight.

Which is, of course, the oldest reading of 108 there is, and the second bullet at the top of this page: the twenty-seven nakshatras, four padas each. The same count arriving from the far side of geometry, one hundred seventy-seven years after Cayley put his question to Salmon.

What is actually known

The hundred and eight above is a poet’s arithmetic and not a proof, and I would rather set the real numbers beside it than let it stand alone.

The lines on a smooth cubic do not stay finite in number. They form a family — the Fano variety of lines — whose dimension is 2d − 4, where d is the dimension of the cubic itself, for every d ≥ 3.3 The cubic surface falls just below that range, and it is the exception that lets the classical theorem exist at all: there the family is zero-dimensional, and a zero-dimensional family is a finite list — the twenty-seven points. Lift by one and the family opens out. A cubic threefold, the one that sits in four-dimensional projective space, carries a whole surface of lines; a cubic that is itself four-dimensional carries a four-dimensional variety of them, the holomorphic symplectic fourfold of Beauville and Donagi. Neither reading of “four dimensions” gives one hundred eight lines. Neither gives any finite number at all.

And if you insist on a finite count one dimension up, you must raise the degree as well: the lines on a general hypersurface of degree 2n − 3 in n-dimensional projective space are finite in number, and the counts run 1, 27, 2875, 698005, 305093061.4 The term after twenty-seven is two thousand eight hundred seventy-five.

And yet the twenty-seven does make the crossing. The surface of lines on a smooth cubic threefold — the Fano surface, the one Clemens and Griffiths used to prove that such a threefold is never rational — has Chern numbers K² = 45 and c2 = 27. Its topological Euler characteristic is exactly twenty-seven.5 (And forty-five is the number of tritangent planes of a cubic surface, the classical companion to the twenty-seven. Whether that is more than a coincidence I do not know.)

So the number does survive the lift. It simply stops being a count of lines and becomes a property of the shape the lines make. That is not the answer I guessed, and it is a better one — which is the usual way of it, when you go and check.

Jai Guru Dev.

References

  1. Cayley proved the number of lines finite and Salmon fixed it at twenty-seven, in correspondence in 1849; Salmon set out the collaboration in A Treatise on the Analytic Geometry of Three Dimensions (1865). See MacTutor, “Cubic surfaces”. ↩
  2. J. L. Kass and K. Wickelgren, “An arithmetic count of the lines on a smooth cubic surface”, arXiv:1708.01175 (2017); Compositio Mathematica 157 (2021), 677–709. Their count is 15⟨1⟩ + 12⟨−1⟩ in the Grothendieck–Witt ring: the rank, 15 + 12, returns Cayley and Salmon’s twenty-seven over the complex numbers, and the signature, 15 − 12, returns the real count of three. ↩
  3. For a smooth cubic hypersurface of dimension d ≥ 3, the Fano variety of lines has dimension 2d − 4; for d = 4 it is a holomorphic symplectic fourfold deformation-equivalent to a Hilbert scheme of two points on a K3 surface. S. Galkin and E. Shinder, “The Fano variety of lines and rationality problem for a cubic hypersurface”, arXiv:1405.5154 (2014); D. Huybrechts, The Geometry of Cubic Hypersurfaces, Cambridge University Press (2023), ch. 2. ↩
  4. The sequence 1, 27, 2875, 698005, 305093061, … counts the lines on a general hypersurface of degree 2n − 3 in n-dimensional projective space, computed here with Macaulay2’s Schubert2 package. The 2875 is the classical count of lines on a quintic threefold. See also arXiv:2402.13206, “Two formulas for the number of lines on complex projective hypersurfaces”. ↩
  5. X. Roulleau, “The Fano surface of the Fermat cubic threefold…” — the Fano surface is smooth with c12 = 45, c2 = 27 and irregularity 5. Its role in irrationality is H. Clemens and P. Griffiths, “The intermediate Jacobian of the cubic threefold”, Annals of Mathematics 95 (1972). ↩