I am working with a bilateral crossing geometry — two pyramids base-to-base at θ = π/8 — and the following expression arises naturally from the dimensional structure of the cascade: α⁻¹ = (9/2)π³ − √(2π) + 4/(9π³) ≈ 137.035951 Each term corresponds to a distinct geometric contribution. I am not asking for evaluation of the physical interpretation. My question is purely mathematical: does this three-term expression, or anything structurally similar combining π³ and √(2π), appear anywhere in existing mathematical literature?
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4$\begingroup$ Purely by virtue of the fact you are writing an expression approximating the fine structure constant using a formula for $\pi$, somehow I suspect that what you are after does not exist. I don't know what "cascade" is, what "bilateral crossing geometry is" or what pyramids have to do with the fine structure constant. A better question would describe purely the geometry, how the expression $\frac{9\pi^3}{2}-\sqrt{2\pi}+\frac{4}{2\pi^3}$ arises from that geometry, and if anyone recognises this constant. Mentioning $\alpha$ is a mathematical red herring, but does give a signal to the Q's origin. $\endgroup$David Roberts– David Roberts ♦2026-03-26 00:37:37 +00:00Commented Mar 26 at 0:37
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$\begingroup$ $1,3,7,0,3,5,9$ matches oeis.org/A005600 "Decimal expansion of reciprocal of fine-structure constant alpha." $\endgroup$Gerry Myerson– Gerry Myerson2026-03-26 01:27:54 +00:00Commented Mar 26 at 1:27
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$\begingroup$ Related question, same OP: mathoverflow.net/questions/509483/… $\endgroup$Gerry Myerson– Gerry Myerson2026-03-26 02:22:33 +00:00Commented Mar 26 at 2:22
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