aa r X i v : . [ m a t h . HO ] J a n The Golden Angle is not Constructible
Pedro J. FreitasDecember 14, 2020
The golden mean is usually defined with relation to a line segment. A linesegment is said to be divided according to the golden ratio if it is decomposedinto two segments, with lengths a > b , satisfying a + ba = ab (1)If this happens, the value of these two ratios is the golden number , ϕ =(1 + √ / ≈ . a = 1 ϕ ( a + b ) b = ( a + b ) − a = (cid:18) − ϕ (cid:19) ( a + b )The same can be done with a circle, instead of a line segment. A circleis divided into two arcs α and β according to the golden ratio if they satisfyequation (1), which leads to α = 1 ϕ π β = (cid:18) − ϕ (cid:19) π (2)The smaller angle β is called the golden angle and has some connectionsto plant growth and phyllotaxis, see [Th, ch. 14]. Its measure in degrees,approximated to two decimal points, is 137 . o .1 . o Figure 1. The golden angleIn this note we prove that the golden angle is not constructible with straight-edge and compass, by proving that its sine and cosine are transcendentalnumbers. Since all constructible numbers have to be algebraic, this is enoughto prove what we want.We recall that the algebraic numbers form a subfield of C , which is closedfor taking n -th roots. Lemma 1.
Given x ∈ R , we have that sin x and cos x are either bothalgebraic or both transcendental.Moreover, the number e ix is transcendental iff either cos x or sin x istranscendental (in which case, both are).Proof. If both sin x and cos x are transcendental, then the first statementis true. If one of them is algebraic, say sin x , then cos x = ± p − sin x isalso algebraic.For the second statement, if z = e ix is algebraic, then cos x = ( z + ¯ z ) / x . Conversely, if both cos x and sin x are algebraic, then e ix = cos x + i sin x is algebraic.We now make use of the Gelfond-Schneider theorem, which is a verypowerful tool to generate transcendental numbers (one could also use theLindemann–Weierstrass theorem). Theorem 2 (Gelfond-Schneider) . Let a and b be algebraic numbers, suchthat a / ∈ { , } and b ∈ C \ Q . Then a b is a transcendental number. α and β in equation (2). We wish to prove thatthe golden angle has transcendental sine and cosine. Proposition 3.
The golden angle has transcendental sine and cosine, andtherefore it is not constructible with straightedge and compass.Proof.
Since β = 2 π − α , it has the same cosine as α , and symmetric sine,we can prove that α = 2 π/ϕ has transcendental sine and cosine. For this weprove that z = e iα = e iπ/ϕ is transcendental, which is equivalent, accordingto the lemma.If z were algebraic, then, according to the Gelfond-Schneider theorem, z ϕ = e πi = 1 would be transcendental, which is false. Therefore, e π/ϕ istranscendental.This proves the non-constructibility of the golden angle. Nevertheless,it is possible to achieve very good approximations, using straightedge andcompass. Portuguese artist Almada Negreiros (1893–1970) devoted severalyears to finding geometric constructions which related to his own analysisof artistic artefacts (see [FC] for more information). One of his discoverieswas precisely an approximate construction for the golden angle, presentedin figure 2, based on the regular pentagram, which is constructible. A BC b a b bb b b
Figure 2. An approximate construction for the golden angle3n this drawing, point C is obtained through an arc of circle centred at A .The golden angle is approximated by circle arc BC . To compute its exactmeasure, one only needs to notice three facts—the two first ones are knownproperties of the regular pentagon. • The segments marked a and b are proportioned according to the goldennumber: a/b = ϕ . • Length a coincides with the side of the pentagon, that is, it is thechord of 2 π/ • Arc AC has chord b .To compute the value of arc BC , it is useful to use the chord as atrigonometric function of the angle. x Figure 3. The chord functionFigure 3 helps to deduce its expression, as well as that of its inversefunction: chr( x ) = 2 sin x x ) = 2 arcsin x a = 2 sin π b = 2 ϕ sin π AC = arcchr b = 2 arcsin (cid:18) ϕ sin π (cid:19) From this, we get that the measure of arc BC , in degrees, rounded to twodecimal values, is 137 . o . This represents an error of 0.08% with respectto the golden angle. 4 eferences [FC] Pedro J Freitas and Sim˜ao Palmeirim Costa, “Almada Negreiros andthe Geometric Canon,” Journal of Mathematics and the Arts , vol. 9nos. 1-2 (2015).[La] Serge Lang,
Algebra – Revised Third Edition , vol. 1, Springer, 2002.[Th] D’Arcy Wentworth Thompson,