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GoldenAngle

Status: Stable

documented, exercised by the test suite and/or worked examples, with no known limitations recorded.

Description

GoldenAngle

is the golden angle (3 - Sqrt[5]) Pi = 2 Pi / GoldenRatio^2, with numerical value ~= 2.39996 radians (~= 137.5 degrees).

N[GoldenAngle, prec]

evaluates it to any precision.

Notes GoldenAngle is a mathematical constant: it has attributes Constant and Protected, NumericQ\[GoldenAngle\] is True, and D\[GoldenAngle, x\] is 0.

Examples (4)

Every input below was run against the current Mathilda build and its output recorded.

Applications (4)

In[1]:= N[GoldenAngle]
Out[1]= 2.39996

In[2]:= N[GoldenAngle, 40]
Out[2]= 2.399963229728653322231555506633613853125

In[3]:= N[GoldenAngle/Degree, 30]
Out[3]= 137.5077640500378546463487396284

In[4]:= N[GoldenAngle - (3 - Sqrt[5]) Pi, 40]
Out[4]= 3.9827297778311306925722009944192795139778e-59

Implementation notes

Definition. GoldenAngle is the golden angle, the closed form (3 - Sqrt[5]) Pi = 2 Pi / GoldenRatio^2 ≈ 2.39996 radians (≈ 137.5°) — the smaller of the two arcs that divide a circle in the golden ratio, the spacing that appears in phyllotaxis. It is a constant symbol: row { "GoldenAngle", 2.3999632297…, fill_mpfr_goldenangle } of kConstants[] in src/numeric.c supplies its value and the numericalize_init loop stamps it Constant | Protected, so Attributes[GoldenAngle] is {Constant, Protected}. Interned name SYM_GoldenAngle (src/sym_names.c), docstring in src/info.c. It carries its defining relation numerically — GoldenAngle == 2 Pi / GoldenRatio^2 returns True — and D[GoldenAngle, x] -> 0 follows from Constant.

Representation & numeric value. A bare EXPR_SYMBOL that stays unevaluated (GoldenAngle prints as itself). N[GoldenAngle] returns the tabled machine double; N[GoldenAngle, k] runs fill_mpfr_goldenangle, which forms (3 - sqrt(5)) * pi in MPFR at k + 20 guard bits (using mpfr_const_pi) and rounds down to the target precision. Machine value 2.39996; N[GoldenAngle, 50] = 2.39996322972865332223155550663361385312499901105812.

Usage & limits. GoldenAngle is primarily a geometric constant; NumericQ[GoldenAngle] is True (whitelisted in is_numeric_quantity, src/core.c). Like the other Pi-derived constants it is irrational and is kept symbolic until N materialises its digits.

  • Attributes Constant, Protected. Attributes[GoldenAngle] = {Constant, Protected}; the symbol cannot be reassigned.
  • Propagated as an exact, unevaluated symbol; NumericQ[GoldenAngle] is True and D[GoldenAngle, x] = 0.
  • N[GoldenAngle] gives the machine value 2.39996; N[GoldenAngle, prec] gives any precision (computed in MPFR from the closed form (3 - Sqrt[5]) Pi), e.g. N[GoldenAngle, 50] = 2.3999632297286533222315555066336138531249990110581.

Attributes: Constant, Protected.

References

Notes & additional examples

Notes

GoldenAngle is (3 - Sqrt[5]) Pi = 2 Pi / GoldenRatio^2, the angle that divides a full turn in the golden ratio — about 137.5 degrees, the divergence angle that governs optimal phyllotactic spiral packing in plants. Dividing by Degree shows the familiar 137.5..., and subtracting the closed form (3 - Sqrt[5]) Pi numerically returns 0.0. It is a protected Constant (so D[GoldenAngle, x] is 0) that N evaluates to any precision.