It is the mean 1.618 (segment B above) that is known as phi for it both cuts and builds, as shall be amply demonstrated herein, as "the fairest bond" among the fair, the most divine building block. It is also mathematically stated as: one plus the square root of five divided by two; or (1 + 2.236) / 2 = 1.618. It is portrayed in the following figure,⁴² where we can already see a relationship to the circle beginning to emerge:

Phi, The Gnomon and the Spiral

This brings us to the gnomon and what is called "gnomonic growth." Gnomon is a Greek word meaning "one who knows,"⁴³ and in its geometric usage seems to be defined in most dictionaries as "the part of a parallelogram remaining after a similar smaller parallelogram has been taken from one of its corners" (WNWCD; another definition is the column or pin of a sundial the shadow of which indicates the time of day-thus relating the gnomon to time, or creation). This definition can be seen in the following figures:⁴⁴

However, geometers, who have long been fascinated with gnomons, have defined the concept more broadly according to a definition that reminds us of "fractal." Huntley, echoing Hero of Alexandria (first century A.D.), says, "A gnomon is a portion of a figure which has been added to another figure so that the whole is of the same shape as the smaller figure. Hero of Alexandria showed that in any triangle ABC [see figure below], triangle ABD is a gnomon to triangle BCD if angle CBD equals angle A":

Now let us look at an elegant example of gnomonic addition, illustrated in the following figure.⁴⁵ We begin with the smallest rectangle (indicated by e at the lower right-hand corner),

In fact this spiral is a member of a family of spirals called equiangular or logarithmic spirals. Ghyka (GAL, pp. 91-92) explains that "the logarithmic spiral is the only plane curve in which two arcs are always 'similar' to each other, varying in dimension but not in shape (in the same spiral), and this property is extended to the surfaces determined by the vector radii limiting the arcs. . . . Every logarithmic spiral is directly connected to a characteristic geometric series or progression." The spiral above is of particular interest to us, for reasons that will become clear below, because it exhibits a phi progression.

The figure below (from DP, p. 171) demonstrates the logarithmic spiral of phi on the Pythagorean "golden triangle," an isosceles triangle ABC with base angles 72 degrees and apex angle 36 degrees. This triangle is inherent in the pentagon, being formed by drawing two diagonals from the same apex and connecting the base. In fact phi appears in many of the pentagon's proportions.

It is also well to know that phi gnomonic growth is popularly identified with the so-called Fibonacci ratio, named after a medieval Italian mathematician Leonardo Pisano, or Leonardo of Pisa, originally named Leonardo Fibonacci (see 7 Brit 279, "Leonardo Pisano"). In 1202 he wrote Liber abaci ("Book of the Abacus"), including the following problem from which his famous sequence is derived:

A certain man put a pair of rabbits in a place surrounded on all sides by a wall. How many pairs of rabbits can be produced from that pair in a year if it is supposed that every month each pair begets a new pair which from the second month on becomes productive?

The resulting number sequence is 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, … After the first few numbers, the ratio between succeeding numbers ad infinitum becomes ever closer to the golden mean, phi, or 1.618. In truth Fibonacci had merely happened upon the divine ratio given earthly expression in the Bible under the hermetic concept "As Above, So Below." If we plot this sequence as the (Cartesian) coordinates on a graph, we get the same spiral produced by the gnomonic growth figure above. Carolyn illustrates this with a golden spiral (see figure) that has radii in the phi proportion at right angles.⁴⁷

Though we have barely scratched the surface of the endless mathematical relationships to be found in phi,⁴⁸ these should suffice for us to move on to see its presence in creation. We can at once see that gnomonic growth is identical with the fractal nature of creation that we saw in the Creation essay and again in "As Above, So Below." We shall see below examples demonstrating this growth perhaps even more profoundly. The importance of the concept cannot be overemphasized.

This gnomonic growth ("As Above, So Below") according to the golden mean (or section) impresses itself as the dominant feature in nature's creation in time (and space). In GAL (p. 91), Ghyka writes, "A certain preference for pentagonal symmetry, a symmetry connected with the Golden Section and unknown in inanimate systems, seems to exist in the animal reign as well as in botany." We note the implied relationship above between the gnomon (the post of the sundial) and time. Time, as seasons, days and years, was planted into the etheric creative process back in Gen 1,14. Then an even larger frame of time reference was laid by "the stars" (the twelve zodiacal periods) in Gen 1,16. The symbol of transition from one era of time to another has been signified from ancient days by the zodiacal symbol of Cancer, the Crab, depicted by two separated but intertwining logarithmic spirals of precisely this gnomonic configuration (see I-81).

As in geometry any effort to identify all instances where this phi spiral occurs in the created world we know is doomed to failure simply because it shows up ever again in places newly investigated. It is found in the phylotaxis of plants (the helical or spiral form of leaf arrangement on the stem), pine and fir cones, the pattern of seeds in the sunflower, the trunk of the palm tree,⁴⁹ animal horns (as in the case of the Ram, the zodiacal symbol of Aries, the Cultural Era of the coming of Christ, the Lamb of God, the Creative Word; Jn 1,1-3 and I-19), the spiral form of galaxies, water running out of a drain, breaking ocean waves, or hurricanes (see "The Whirlwind" below), the dimensions of ancient temples, the Parthenon, the Great Pyramid, the most pleasing classical art, the human body (including the internal dimensions of its vertical stature, its countenance,⁵⁰ its pentagonal dimension of head and limbs as well as digits, its outer ear, the helical microscopic spiral in its DNA,⁵¹ and the tiny microtubules in the human brain⁵²), and on and on.

In The Kingdom of Childhood (KC), Lect. 5, Steiner spoke of the amazing nature of the theorem of Pythagoras, where "if I have a right-angled triangle here (see left diagram on following page ) the area of the square of the hypotenuse is equal to the sum of the other two areas, the two squares on the other two sides."

This figure can also be used to demonstrate another example related to gnomonic growth (see right digram).⁵³

One must marvel upon seeing that this diagramed theorem of Pythagoras is itself a fractal that when extended in diminishing scale grows both the logarithmic spiral of the Ram's horn and an outline of the human brain. It is illustrated by Hans Lauwerier in his Fractals (FRAC), Chap. 4, p. 70, in the following computer-drawn figure:

Interestingly, he terms it "a lopsided Pythagoras tree," and we may well see in it an expression of the "tree of knowledge" spawned by the descent of the human being and its separation from the "tree of life," (the Fall; Gen 3,22-24 frequently discussed in this connection in The Burning Bush).