sábado, 31 de março de 2007
Circle
Spiral
In modern times, the spiral is still spiritually significant. It is the symbol of spirit in Wicca, an emblem of the Goddess.
Vesica Pisces
The shape of the vesica pisces is derived from the intersection of two circles, the Pythagorean "measure of the fish" that was a mystical symbol of the intersection of the world of the divine with the world of matter and the beginning of creation. To the Pythagoreans, the whole of creation was based on number, and by studying the properties of number, they believed one could achieve spiritual liberation. The vesica pisces was the symbol of the first manifestation, the dyad (reflection) that gives birth to the entire manifest universe. Within the vesica can be found the triangle, the tetrad, the square, the pentacle, and many more polygons, making the vesica a true symbolic womb.
The wellspring at Glastonbury is considered to be one of England's most Holy sites. The well itself dates back over two thousand years, and was sacred to both early Pagans and the Christians who later built an abbey on the site. Today, the well is a place of pilgrimage for Christians and Pagans alike, many of whom believe the red, iron rich water has healing and miraculous properties. The vesica Pisces theme is repeated in the shape of the pool at the base of the hill, where the water from the spring flows.
Triquera, triqueta
The triquetra also appears on the motion picture Constantine as a charm used by a catholic priest to protect against demons.
Triskeles are one of the most common elements of Celtic art; they are found in a variety of styles in both ancient and modern Celtic art, especially in relation to depictions of the Mother Goddess. They also evoke the Celtic concept of the domains of material existence- earth, water, and sky, and their interrelations.
The Medusa in the center implies the protection of the Goddess Athena, the Patron Goddess of the Isle. (In early mythology, Medusa was the destructive aspect of Athena, later, a monster slain by the hero Perseus, who adorned Athena's shield.)
Cross
The Ankh is an ancient Egyptian symbol of life. Also known as an Ansata cross or crux ansata, this a visual representation of a sandal strap. The horizontal and vertical bars of the lower tau cross represent the feminine and masculine energy, respectively. This combination of male and female symbols (the cross and circle) in the ankh suggest fertility and creative power. The top loop also symbolises the sun on the horizon, and suggests reincarnation and rebirth. The ankh appears frequently in Egyptian writings about rebirth, and this symbolism was adopted by Coptic Christians, especially gnostic sects, to symbolize the resurrection of Christ and the heavenly marriage. The ankh was an earlier form of the cross than the later "latin" cross.
Swastika (Fylfot Cross)
The swastika used in Buddhist art and scripture is known as a manji (whirlwind), and represents Dharma, universal harmony, and the balance of opposites. It is derived from the Hindu religious swastika, but it is not identical in meaning.
In Zen Buddhism, the Manji represents the ideal harmony between love and intellect.
sexta-feira, 30 de março de 2007
Celtic Shield Knots
Golden spiral
Specifically, a golden spiral gets wider by a factor of φ every quarter-turn it makes, which means it gets wider by a factor of φ4 (about 6.854) every full turn.
Approximate and true Golden Spirals. The green spiral is made from quarter-circles tangent to the interior of each square, while the red spiral is a Golden Spiral, a special type of logarithmic spiral. Overlapping portions appear yellow. The length of the side of a larger square to the next smaller square is in the golden ratio. (A Fibonacci spiral is not shown, but could be constructed from a similar "whirling rectangle diagram", in which the ratios of the rectangles were based on the terms in the Fibonacci series, rather than phi.)
Fibonacci number
That is, after two starting values, each number is the sum of the two preceding numbers. The first Fibonacci numbers also denoted as Fn, for n = 0, 1, … , are:
0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, 4181, 6765, 10946, 17711, 28657, 46368, 75025, 121393, 196418…
Sometimes this sequence is considered to start at F1 = 1, but it is more common to include F0 = 0. The Fibonacci numbers are named after Leonardo of Pisa, known as Fibonacci, although they had been described earlier in India.
A tiling with squares whose sides are successive Fibonacci numbers in length.
A Fibonacci spiral, created by drawing arcs connecting the opposite corners of squares in the Fibonacci tiling shown above.
Golden rectangle
According to astrophysicist and math popularizer Mario Livio, since the publication of Luca Pacioli's Divina Proportione in 1509, when "with Pacioli's book, the Golden Ratio started to become available to artists in theoretical treatises that were not overly mathematical, that they could actually use," many artists and architects have proportioned their works to approximate the form of the golden rectangle, which has been considered aesthetically pleasing.
A method to construct a golden rectangle. The resulting dimensions are in the ratio 1:φ, the golden ratio.
- Construct a simple square
- Draw a line from the midpoint of one side of the square to an opposite corner
- Use that line as the radius to draw an arc that defines the height of the rectangle
- Complete the golden rectangle
Golden canon of page construction
Medieval manuscript framework according to Tschichold, in which a text proportion near the golden ratio is realized on a 2:3 page proportion.
Tschichold's drawing of an octavo-format page proportioned in the golden ratio or golden section "34:21". The text area and margin proportions are determined by the starting page proportions.
Tschichold's drawing of an octavo-format page proportioned in the golden ratio or golden section "34:21". The text area and margin proportions are determined by the starting page proportions.
These page proportions based on the golden section or golden ratio, are usually described through its convergents such as 2:3, 5:8, and 21:34.
Tschichold says that common ratios for page proportion used in book design include as 2:3, 1:√3, and the golden section. The image with circular arcs depicts the proportions in a medieval manuscript, that according to Tschichold feature a "Page proportion 2:3. Margin proportions 1:1:2:3. Text area in accord with the Golden Section. The lower outer corner of the text area is fixed by a diagonal as well." By accord with the golden section, he does not mean exactly equal to, which would conflict with the stated proportions.
Tschichold refers to a construction equivalent to van de Graaf's or Rosarivo's with a 2:3 page ratio as "the Golden Canon of book page construction as it was used during late Gothic times by the finest of scribes." For the canon with the arc construction, which yields a text area ratio closer to the golden ratio, he says "I abstracted from manuscripts that are older yet. While beautiful, it would hardly be useful today."
Of the different page proportions that such a canon can be applied to, he says "Book pages come in many proportions, i.e. relationships between width and height. Everybody knows, at least from hearsay, the proportion of the Golden Section, exactly 1:1.618. A ratio of 5:8 is no more than an approximation of the Golden Section. It would be difficult to maintain the same opinion about a ratio of 2:3."
And he expresses a preference for certain ratios over others: "The geometrically definable irrational page proportions like 1:1.618 (Golden Section), 1:√2, 1:√3, 1:√5, 1:1.538, and the simple rational proportions of 1:2, 2:3, 5:8 and 5:9 I call clear, intentional and definite. All others are unclear and accidental ratios. The difference between a clear and an unclear ratio, though frequently slight, is noticeable. ... Many books show none of the clear proportions, but accidental ones."
John Man's quoted Gutenberg page sizes are in a proportion not very close to the golden ratio, but Rosarivo's or van de Graaf's construction is applied by Tschichold to make a pleasing text area on pages of arbirary proportions, even such accidental ones.