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The circle stands for unity, oneness, eternity, wholeness, and completeness. The only form that encloses all other radially symmetrical or regular figures, it may connote pre-form, the genesis of form, origins or beginnings.
The circle's circumference is closed and continuous and as such conveys continuous cycles of endings and beginnings. Circles may signify cycles of time such as: phases of the sun and moon; cycles of light and dark; and perpetual rhythms of sleeping and waking, birth and death, growth and decay, systole and diastole, and inhalation and exhalation. In its totality, the circle suggests the timeless whole. The moving point along the circle conveys the passage of time.
Circles are measured relative to the incommensurable value pi (p = 3.1415927 ). In contrast to the square, whose perimeter and area can be measured in finite whole numbers, the circle may symbolize heavenly, transfinite or transcendental realms.
The circle expresses justice and democracy, since all of its points are equally distant from the center. A communal form, it imparts no sense of social hierarchy. The circle may symbolize the collective -- the ring that "unites man through the infinite chain of hands."
Some Christian churches, tribal ritual spaces, and the instinctive way in which children gather to play -- all take the form of the circle, drawing upon its magical and protective qualities and its sense of center and place. As a sacred space, the circle orients to the horizon and to the cosmic edges of the universe. Mircea Eliade observes that one gathers in a circle to distinguish what is known (cosmos) from what is unknown (chaos); to "'found the world' and to live in a real sense" [Eliade 1959, 23].
The vesica piscis signifies the mediation of two distinct entities; the complementariness of polar opposites, as when two extremes complete and depend upon one another to exist. One circle may signify the breath of spirit, which is eternal; the other may signify the body physical, which is forever changing and adapting, The vesica piscis itself symbolizes that which mediates spirit and body; or the psyche or soul.
In another way, the vesica piscis may represent the phenomenon of color, which Goethe understands to mediate "light and its absence." He notes that we perceive "all the varieties of hues" when "the greatest brightness acts near the greatest darkness" [Goethe 1970, 5, 206].
"Vesica piscis" is the Latin vesica "bladder" + piscis "fish" [Hoad 1996]. Commonly used in Christian symbolism, it resembles the graphic symbol for Pisces, the twelfth and last sign of the Zodiac, which is signified by two partial circles that are distinct, yet bound by a line. Historically, the Christian Era coincides with the Piscean Age, when the sun's entry into the constellation Pisces marked the first day of spring and the new year.
The pisces symbol
As a Christian symbol, the vesica piscis may signify Christ Incarnate, who mediates heaven and earth, or humanity and the divine. In Medieval Christian art, Christ is commonly depicted emerging from a vesica piscis to portray the entry of transcendent form into the physical world and made flesh [Williams 2001]. The vesica piscis can also signify the womb -- in Christianity, the womb of the Virgin from which Christ emerges. The proportions of the vesica piscis appear in the Gothic arch and underlie rectangular floor plans of numerous churches and chapels, such as the Mary Chapel in Glastonbury Abbey.
Some vernacular cultures combine images of sun and moon in the form of a vesica piscis. The full disk of the sun, which radiates its own permanent golden light, represents the unifying state of "solar" consciousness in which reality is perceived as eternal and One. In contrast to the sun, the sharp and cutting edge of the crescent moon may signify the division of unity into different parts. The moon continually changes as it progresses through various phases. Its silvery light is the surface reflection of other bodies. Together, the sun and moon convey complementary polarities of self and other, sameness and difference, and one whole and many parts (Fig. 3).
EQUILATERAL TRIANGLE AND THE RATIO 1 : Ö3
The short, horizontal axis equals the radius of the vesica's generating circles. The short and long axes are in the ratio l : Ö3, or 1: 1.7320508 .. If the short axis (AB) equals 1, the long axis (CD) equals Ö3 (Fig. 4).
The vesica piscis shares a common geometry with the equilateral triangle and may signify the Holy Trinity and other triadic relationships.
If the side (AB) of the square is 1, the diagonal (BE) equals Ö2, or 1.4142135... (Fig. 6).
THE GOLDEN SECTION AND THE RATIO 1 : f
If the side (FH) is 1, the diagonal (HE) equals Ö5, or 2.236067... (Fig. 7).
If the short axis of the vesica (AB) is 1, segment AK equals phi (f = Ö5/2 + l/2) or 1.618034 . Segment BK equals the reciprocal of f; in other words 1/f or (Ö5/2 - l/2) or 0.618034 . The ratio 1 : f is known as the Golden Section, or the "extreme and mean" ratio (Fig. 8).
INCOMMENSURABLE RATIOS AND DYNAMIC SYMMETRY
Incommensurable ratios may organize space so that the same proportion persists continually through endless divisions. This quality of continuity, which Jay Hambidge calls "dynamic symmetry," is unique to the incommensurables and implies that every level of form, from the micro- to the macrocosmic, may be united through measure and proportion [Hambidge 1960; 1967].
HOW TO GENERATE A l : Ö3 PROPORTIONAL SYSTEM WITH A VESICA PISCIS
The short and long axes of the vesica piscis equal 1 and Ö3 (Fig. 10).
Draw a new vesica piscis from two circles of radius CD:
The short and long axes of the new vesica (CD and EF) equal Ö3 and 3 (Fig. 11).
Draw a new vesica piscis from two circles of radius EF:
The short and long axes of the new vesica (EF and GH) equal 3 and 3 Ö3 (Fig. 12).
The spiral that results follows a 1: Ö3 geometric progression (Fig. 14).
THE l : Ö3 RECTANGLE
The result is a rectangle (FGIK) with short and long sides in the ratio l : Ö3 (Fig. 15).
The result is a smaller rectangle (DFGL) in the ratio 1/Ö3 : 1 or l : Ö3. Rectangle DFGL is the reciprocal of the whole rectangle FGIK (Fig. 16).
The rectangles NDLM and KNMI that result are each in the ratio 1/Ö3 : 1 or l : Ö3. The major l : Ö3 rectangle FGIK divides into three reciprocals that are proportionally smaller in the ratio l : Ö3 (Fig. 17).
The result is a smaller rectangle (ADFP) in the ratio 1/3 : 1/Ö3 or l : Ö3 (Fig. 18).
The rectangles QAPR and LQRG that result are each in the ratio 1/3 : 1/Ö3 or l : Ö3. The major l : Ö3 (or 1/Ö3 : 1) rectangle DFGL divides into three reciprocals that are proportionally smaller in the ratio l : Ö3 (Fig. 19).
Such constructions illustrate how incommensurable ratios replicate consistently through endless spatial divisions, even as the identical ratio remains present in the relationship of one level to the next. A l x Ö3 rectangle of any size divides into three proportionally smaller reciprocals in the ratio l : Ö3. If the process continues indefinitely, the side lengths of successively larger rectangles form a perfect geometric progression (l ,Ö3 , 3, 3Ö3 ).
"Dynamic symmetry" is the name given by Jay Hambidge to this proportioning principle, which he finds in l x Ö3 and other root rectangles, and observes in the human figure, in plant life, and in classical Greek and other forms of art. Hambidge associates dynamic symmetry with our perception of beauty, noting "its power of transition and movement from one form to another...It produces the only perfect modulating process in any of the arts" [Hambidge 1967, xv-vxi].
Of symmetry, Hambidge says, "using the word in the Greek sense of analogy; literally it signifies the relationship which the composing elements of form in design, or in any organism of nature, bear to the whole. In design, it is the thing which governs the just balance of variety in unity" [Hambidge 1967, xii].
We will now explore how dynamic symmetry and l : Ö3 proportions manifest in other ways.
GD: IF :: 1: Ö3. The diagonals IF and GD intersect at 90° at point O (Fig. 20).
THE l : Ö3 EQUIANGULAR SPIRAL
An equiangular spiral is a spiral curve in which distinct radii vectors emanating from the pole at equal angles to one another are in continual proportion. Between any three consecutive radii vectors, the middle vector is the mean proportional or geometric mean of the other two. The spiral that results grows in size continuously without changing its shape. It is also called a logarithmic spiral or proportional spiral.
The radii vectors are separated by equal angles (90°). Their lengths increase in a 1: Ö3 geometric progression. Equiangular spirals such as IGFDA decrease continuously towards the pole, but never touch it (Fig. 21).
THE THEOREM OF THALES AND THE LAW OF SIMILAR TRIANGLES
The triangle CBA is a right triangle.
Line BO is the mean proportional or geometric mean of lines OA and OC (Fig. 22).
Thales of Miletus (ca. 624-547 B.C.), considered the first Greek mathematician, is thought to have learned geometry from the Egyptians. He is credited with the theorem that any triangle inscribed within a semi-circle is right-angled. The Theorem of Thales states that within a semi-circle, as in Fig. 22, a perpendicular line (BO) drawn from any point (B) along the perimeter to the diameter is the mean proportional or geometric mean of the two line segments (OA and OC) that result on the diameter, that is, OA: OB :: OB:OC.
Triangles are similar that have corresponding angles equal and corresponding segments proportional. The Law of Similar Triangles states that two triangles are similar if they have two angles and one side equal.
Equiangular spirals demonstrate the Theorem of Thales and the Law of Similar Triangles. In Fig. 21:
The triangle IGF is a right triangle.
Triangles GOF, IOG and IGF are similar. Line OG is the mean proportional or geometric mean of lines OF and OI (Fig. 23).
The triangle GFD is a right triangle.
Triangles FOD, GOF and GFD are similar. Line OF is the mean proportional or geometric mean of lines OD and OG (Fig. 24).
APPLICATION: BRAMANTE'S TEMPIETTO
Sebastiano Serlio's Trattato di architettura (On Architecture) presents an elevation of the Tempietto that appears to unfold from a vesica piscis proportioned to the temple's dome [Serlio 1544, III, xlviii, 48] (Fig. 25).
The bottom apex of the triangle locates the top of the balcony rail and the base of the second story.
The base of the vesica piscis locates the floor level of the
temple. The top of the vesica piscis locates the top of the dome.
 R. Schwarz, Von Bau der Kirche (The Church Incarnate), 24. Cited in [Norberg-Schultz 1972, 20]. return to text
 For proof, see Euclid, Book I, Prop. 1: "On a given finite straight line to construct an equilateral triangle" [Euclid 1956, I: 241-242]. return to text
 For proof, apply the Pythagorean theorem (OB2 + OC2 = BC2). return to text
 [Thompson 1992, 748-758]; [Hambidge 1967, 5-6]. D'Arcy Thompson describes equiangular spirals as "any plane curve proceeding from a fixed point (or pole), and such that the vectorial area of any sector is always a gnomon to the whole preceding figure" [1992, 763]. "Gnomon," the post that marks the time of day by the shadow it casts on a sundial, is from the Greek gnômôn, which means "indicator or "interpreter and specifically the pointer of the sundial or carpenters square [Liddell 1940, Simpson 2005]. In mathematics, the gnomon is the shape which, when added to a figure, produces the same figure, but larger; as when an "L" shape added to a square produces a larger square. In similar fashion, the equiangular spiral is a "growing structure built up of successive parts, similar in form, magnified in geometrical progression, and similarly situated with respect to a center of similitude" [Thompson 1992, 763]. return to text
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