Sargon's number (k) is a mathematical constant that can be used to easily calculate arc lengths and chord lengths of circles, without the involvement of pi (π).
Arc length = k * chord length
Chord length = arc length / k
Similar to π, Sargon's number (k) for a given circular segment can be taken as approximates for real-world calculations:
180° (π/2) radians = π/2 = 1.57079632,
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90° (π/4) radians = (π/2)/sqrt(2) = 1.1107...,
60° (π/6) radians = sqrt(3)/2 = 1.2247...,
45° (π/8) radians = sqrt(2) = 1.4142...,
30° (π/12) radians = 2sqrt(3)/3 = 1.1547...
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1° (π/360) radians = (π/180)/sqrt(0.0003045) = 1.000116...
The constant can be used to find the length of the arc of a circle without resolving π, which is an irrational number.
The constant makes calculations involving circles, spheres, and cones more optimal by eliminating the need for a trigonometric solution using the arc-length formula and by avoiding π like irreconcilable infinite series.
Definition
The following can be derived from the special properties of right angled triangles described by Archimedes in his publications - On the measurement of a circle and the special properties of a right-angled triangle (case. inscribed in a circle) [1], [2] or The Madhavan-Leibniz series based on their 14th and 17th century proofs for the infinite series π/4 = 1− 1/2 + 1/3 − 1/5 + 1/7 − 1/9 + .... + ∞ [3], [4]
Madhavan-Leibniz series proof -
TBD
Trigonometric proof -
For a circle of radius r, consider an arc of length s subtended by a central angle θ. Let c be the length of the chord corresponding to this arc. By inscribing a right angled triangle in the circle with the chord c as the hypotenuse, the length of the adjacent side is r*sin(θ).
Sargon's constant k is defined as:
k = s/c
Using the theorem relating arc length, radius and subtended angle, this can be expressed as:
With θ in radians, the constant k has a value of approximately 1.11072, 1.547, 1.4142, 1.7321 at 90, 60, 45, and 30 degrees respectively.
Applications
Sargon’s number is true for all integers, rational and irrational numbers, and imaginary and complex numbers.
For spheres and cones, Sargon's constant enables the computation of arc lengths along great circles and generators without resorting to irrational numbers:
Arc length of great circle on a sphere = k * chord length
Slant height of a cone = k * radius * θ
Measurement of almost spherical objects (ex. Earth)
Radius of curvature in the equatorial plane: R_e = a / sqrt(1 - e^2 * sin^2(latitude))
Radius of curvature along the meridian: R_m = a * (1 - e^2) / (1 - e^2 * sin^2(latitude))^1.5
Higher Dimensions
While this symmetry seems to hold in higher dimensions, its accuracy dips as the angle increases. For example, at 170 degrees, the Sargon's number for a circle of radius 3 units deviates from pi/2 by 0.0062, which is 0.4% of pi/2.
This is because the Sargon’s number is still an approximation and at higher angles and higher dimensions. Further derivations are needed to prove that existing equations for hyper-curves are consistent with their trigonometric proofs.
History
The constant was first introduced in a 2023 paper Defining Sargon's constant - Constants in curved n-dimensional space - by Sargon Alik (Bhargav Pandravada) [5], who named it in honor of the ancient king Sargon of Akkad. Archaeological evidence suggests that Sargon's surveyors may have used an approximation of this constant in cartography like the maps from Girsu showing newly dug watercourses [6] and maps of the fields of Gusur [7]. The integrity of the mathematical examples is crucial.
References
^ Archimedes (1897), "Measurement of a circle", in Heath, T. L. (ed.), The Works of Archimedes, Cambridge University Press
^ Coolidge, J. L. (February 1953). "The Lengths of Curves". The American Mathematical Monthly. 60 (2): 89–93. doi:10.2307/2308256. JSTOR 2308256.
^ Roy, Ranjan (1990). "The Discovery of the Series Formula for π by Nilakantha, Leibniz, and Gregory" (PDF). Mathematics Magazine. 63 (5): 291–306. doi:10.1080/0025570X.1990.11977541..
^ Plofker, Kim (November 2012), "Tantrasaṅgraha of Nīlakaṇṭha Somayājī by K. Ramasubramanian and M. S. Sriram", The Mathematical Intelligencer, 35 (1): 86–88, doi:10.1007/s00283-012-9344-6, S2CID 124507583
^ Pandravada, B. (2023, October 18). Sargon’s number - constants in curved n-dimensional space - by Sargon Alik (Bhargav Pandravada). MeshAI. https://www.mymesh.ai/post/sargon-s-number-constants-in-curved-n-dimensional-space-by-sargon-alik-bhargav-pandravada
^ Foster, B. R. (2015, December 14). The Age of Agade, pg.216. Google Books. https://books.google.com/books?id=O680CwAAQBAJ&pg=PA3#v=snippet&q=cartography&f=false%3C
^ Foster, B. R. (2015, December 14). The Age of Agade, pg.217. Google Books. https://books.google.com/books?id=O680CwAAQBAJ&pg=PA3#v=snippet&q=cartography&f=false%3C
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