Det Gyldene Snidt: A Φlosophy of τau…
“Geometry has two great treasures;
One is the Theorem of Pythagoras; the other,
The division of a line into extreme and mean ratio.
The first we may compare to a measure of gold;
The second we may name a precious jewel.”
Johannes Kepler (1571-1630)
As far back as I can remember, I have been fascinated by the mathematical relation referred to as the Golden Ratio (where Ratio may be substituted by Number, Cut, Mean, Angle, Arc, or Section). Can't say that I know why I've had this fascination... other than that it makes me think... makes me wonder about stuff... And it seems as if I'm not alone: On Amazon.com there are no less than twelve (12) different books available on this subject!
Vitruvian Man, Leonardo da Vinci (1452-1519)
I wrote a small spreadsheet hack (Microsoft Excel), a "numerical toy" for playing around with this enigmatic constant. You can download it from here, feel free to spread it around! It visualizes even more (and maybe better) the depth of the Golden Ratio mysteries!
The Golden Section is a line segment divided such that its longer part a + its shorter part b (which is, of course, the entire line) is to part a what part a is to part b. In other words, two quantities are in the Golden Ratio if the ratio of the sum of the quantities to the larger quantity equals the ratio of the larger quantity to the smaller. It is symbolized by the Greek letter phi, usually in lower case (φ), and its value is an irrational mathematical constant, ≈ 1.6180339887... Mathematicians study its many unique and interesting numerical properties. A picture may say it better:
Phi (φ), The Golden (Divine) Mean
φ = a : b = a+b : aφ = a / ( ( a2 + ( a/2 )2 )0.5 – ( a/2 ) )
Other names often seen used for the golden ratio are golden section (sectio aurea) and golden mean, as well as extreme and mean ratio, medial section, divine proportion, divine section (sectio divina), golden proportion, golden cut, golden number, and mean of Phidias (one of the greatest sculptors of ancient Greece (c.500–c.432 B.C.E.), also why the Greek letter phi was chosen to symbolize the golden mean). In them olden days, phi was called tau (τ), still used in some (especially mathematical) texts.
Algebraically, the geometric relationship defining phi is:
φ = a / b = ( a + b ) / a
Which, positively solved, yields the irrational number:
φ = ( 1 + √5 ) / 2 ≅ 1.6180339887…
As with π, its succession of non-repeating decimal patterns is endless. A unique property of φ (among positive numbers), is that:
1 / φ = φ – 1 (and the inverse: 1 / Φ = Φ + 1)
The Golden Rectangle
Artists (painters (see below for a couple of examples), architects, musicians) often proportion their works to φ—notably as the Golden Rectangle (where φ is the ratio of the long side to the short side)—a proportion that is considered aesthetically pleasing. At least by some people.
The Golden Ratio φ = (1+5½)/2 ≈ 1.618…
Φ = 1/φ, thus φ:1 = 1:Φ(1/φ)
φ = a/b = (a+b )/a
Golden Section: a is to b what a+b is to a
To construct a Golden Rectangle:
1. Construct a unit side square (red in the figure above).
2. Draw a line r from a top side center to an opposite corner.
3. Let r be the radius of an arc, so defining the longer side.
The Spira Mirabilis
Successively smaller Golden Rectangles form a logarithmic spiral called Spira Mirabilis (Wonderful Spiral). To generate a similar but subtly different spiral, use Golden Triangles instead of rectangles. For a particularly beautiful example of a natural occurrence, see the Chambered Nautilus, and many phyllotaxic patterns (another rather striking example is the Romanesco broccoli, pictured below). It can also be observed on a grand scale:
The Whirlpool Galaxy (M51) ~ 28 Mly distant, ~ 38 Kly across, mass ~ 160 GM⊙
The Golden Triangle
The Golden Triangle is an isosceles triangle ABC where a bisecting angle C produces a new triangle CXB, similar to the original. If angle BCX = α, then XCA = α (bisection) and CAB = α (similar triangles); ABC = 2α (original isosceles symmetry) and BXC = 2α (similarity). Angles in a triangle add up to 180°, so 5α = 180, giving α = 36°. The angles of the golden triangle are thus 36°–72°–72°. The angles of the remaining obtuse triangle AXC (called Golden Gnomon) are 36°–36°–108°.
Pentagons, Pentagrams: Golden Triangles
a + b is to a what a is to b
The pentagram includes thirteen isosceles triangles: six acute and seven obtuse. In all of them, the ratio of the longer side to the shorter is φ. The acute triangles are Golden Triangles, the obtuse are Golden Gnomon. Also, φ is the ratio of the length of the shorter segment to the segment bounded by the two intersecting edges (any side of the pentagram’s central pentagon (which "hides" two obtuse and one acute triangle!)).
This regular dodecahedron (twelve regular pentagon faces (left)) was drawn by Leonardo da Vinci (in his days this polyhedron was called a duodecedron), and another one was created by nature (see the microscopic Braarudosphaera bigelowii below):
The regular icosahedron (twenty equilateral triangle faces (middle)), has five triangles (5,3) centered at each vertex (the "reciprocal" of the dodecahedron, which has three pentagons (3,5) centered at each vertex).The vertices form the corners of three mutually perpendicular Golden Rectangles, whose edges form Borromean rings (right). In two dimensions, at the viewing angle used above, the edges of the three rectangles outline a hexagram (Mogein Dovid (light yellow shade))
Kepler's "Perfect" TriangleCharacterised by a short cathetus to hypotenuse ratio equal to the golden ratio:
Left: The 3:4:5 (units) triangle is the only right triangle with edge lengths in arithmetic +1 progression. It is close to, but not quite, the geometric progression (right): The Kepler triangle edge lengths are in geometric progression, combining the Pythagorean Theorem with the Golden Ratio ((φ)2 = (φ0.5)2 + (1)2).
Compared to the golden triangle, there is yet another arithmetic progression, in how the hypotenuse "leans" against the long cathetus: the golden triangle α ("top") angle is 36°, the 3:4:5 triangle "leans" at 37°, and the geometric triangle at 38°.
The Egyptian (Geometric) Triangle
(so named after found to be a (disputed) attribute of the Cheop's Pyramide)
The Golden Arc and the Golden Angle
The Golden Angle is the smaller of the two angles created by cutting the circumference of a circle c at the Golden Section; that is, into two arcs, a and b, such that the ratio of the length of arc a to the length of arc b is the same as the ratio of the length of the full circle c (arc a + arc b) to the length of arc a.
The aesthetically pleasing Φ relation is a likely reason why many watch (and clock) ads have the faces show the time nine minutes past ten o'clock (M09:H51). M50:H10 is a closer approximation to Φ for a clock face divided into 60 minutes (H49:M11 is even closer, but not possible with a proper watch where the hands are fixed in their relation to each other).
The golden angle Φ measures 360(1–(1/φ)) = 360(2–φ) = 360/φ2 = 180 (3–5½) degrees ≈ 137.51° (2π(1–(1/φ)) = 2π(2–φ) = 2π/φ2 = π(3–5½) ≈ 2.4 radians). Fraction f of the circle circumference covered by the Golden Arc (Φ divided by the angular measure) = b/c = b/ (b+a) = 1/(1+φ) ≈ 0.382. Since 1+φ=φ2, it follows that f =1/φ2 (that is, a circle has room for φ2 golden angles).
In physics, the fine structure constant alpha =
alpha = e2/ħc(4πε0) ≅ 7.297 352 537 6×10–3, and alpha–1 ≅ 137.036
alpha = sinc(1–(1/π( tanh(π/138)))), then alpha–1 ≅ 137.036And (coincidentally?) Φ ≅ (138+alpha–1)/2 ≅ 137.51 (the Golden Angle)
The Golden Mean in Nature (Wikipedia)Adolf Zeising (whose main interests were mathematics and philosophy) found φ expressed in the arrangement of branches along the stems of plants and of veins in leaves. He extended his research to the skeletons of animals and the branchings of their veins and nerves, to the proportions of chemical compounds and the geometry of crystals, even to the use of proportion in artistic endeavors. In these phenomena he saw the golden ratio operating as a universal law,
Braarudosphaera bigelowii The Chambered Nautilus
In connection with his scheme for φ-based human body proportions, Zeising wrote (1854) of a universal law “in which is contained the ground-principle of all formative striving for beauty and completeness in the realms of both nature and art, and which permeates, as a paramount spiritual ideal, all structures, forms and proportions, whether cosmic or individual, organic or inorganic, acoustic or optical; which finds its fullest realization, however, in the human form.” (See also "A Golden Brain Connection?" and Quantum Golden Ratio, last on this page!)
Fibonacci phyllotaxis: Many plants follow this or very similar growth patterns, involving the Golden Arc: Following the numbered leaves counterclockwise backwards (from #16, bottom centre), the angular distance between successively lower numbers is φ (222.5°). There are some interesting discussions about the golden mean and bonsai trees on http://www.bonsai4me.com.
Leonardo da Vinci: Mona Lisa.
This famous work of art indicates that Leonardo was familiar with the Golden Mean proportions.
Salvador Dali: The Sacrament of the Last Supper
The canvas dimensions are those of a golden rectangle. The large dodecahedron (made up of regular pentagons, as seen earlier) has edges in golden ratio to one another (no matter how I try and count, the number of apostles here add up to eleven... Were they not supposed to be twelwe? Or did Dali have Judas Iscariot skip his dinner invitation?).
φ = n:(n2 + (n/2)2)0.5 – (n/2) = n:b = (n+b):n
AB = b+c = n
AC = (n2+(n/2)2)0.5
BC = DC = a = n/2
Φ = AE/EB = AB/AE = (1+50.5)/2
EB = AB–AE = AE/φ = c = n/φ2
AE = AD = AC–DC = AB–EB = b = n/φ
Seven years ago (2003), Weiss and Weiss concluded—based on psychometric data and theoretical considerations—that the Golden Ratio underlies the clock cycle of brain waves. Five years later (2008) this was empirically confirmed by a group of neurobiologists. Phi-losophically interesting stuff!
 Weiss, Volkmar; Weiss, Harald (2003). “The golden mean as clock cycle of brain waves.” Chaos, Solitons and Fractals 18: 643–652.
 Roopun, Anita K. et al. (2008). “Temporal interactions between cortical rhythms.” Frontiers in Neuroscience 2: 145–154.
The double helix of the gigantic (in terms of number of atoms) DNA molecule (another "spira mirabilis," indeed!) completes a full helical cycle in ~34 Ångström (Å (1 Å = 100 picometer (pm))), with its width (~21 Å) at the golden cut. With the lengths of the "major" and "minor" grooves being ~21 Å and ~13 Å, these are also in the golden ratio.
Another interesting (but far less complicated) molecule is the C60 (built from exactly 60 carbon atoms) molecule known as the Buckminsterfullerene (a.k.a. the “buckyball”):
It looks a lot like a soccer ball, and also resembles the geodesic dome designed by Buckminster Fuller (whom it is named after). The C60 shape is a “truncated icosahedron,” ~7 Å in diameter. The alternating hexagons and pentagons provide the link to the Golden Ratio.
Interesting link: wikipedia.org/wiki/Molecular_Borromean_rings
ScienceDaily (Jan. 7, 2010) — Researchers from the Helmholtz-Zentrum Berlin für Materialien und Energie (HZB), in cooperation with colleagues from Oxford and Bristol Universities, as well as the Rutherford Appleton Laboratory, UK, have for the first time observed a nanoscale symmetry hidden in solid state matter. They have measured the signatures of a symmetry showing the same attributes as the golden ratio famous from art and architecture. Full Article Here
Other sites featuring the Golden Mean (in order of URL size ;):