Vesica piscis

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The vesica piscis is the intersection of two congruent disks, each centered on the perimeter of the other. Vesica piscis circles.svg
The vesica piscis is the intersection of two congruent disks, each centered on the perimeter of the other.

The vesica piscis is a type of lens, a mathematical shape formed by the intersection of two disks with the same radius, intersecting in such a way that the center of each disk lies on the perimeter of the other. [1] In Latin, "vesica piscis" literally means "bladder of a fish", reflecting the shape's resemblance to the conjoined dual air bladders (swim bladder) found in most fish. [2] In Italian, the shape's name is mandorla ("almond"). [3] A similar shape in three dimensions is the lemon.

Contents

The vesica piscis in Euclid's Elements Euclid-proof.svg
The vesica piscis in Euclid's Elements

This figure appears in the first proposition of Euclid's Elements, where it forms the first step in constructing an equilateral triangle using a compass and straightedge. The triangle has as its vertices the two disk centers and one of the two sharp corners of the vesica piscis. [4]

Mathematical description

Mathematically, the vesica piscis is a special case of a lens, the shape formed by the intersection of two disks.

The mathematical ratio of the height of the vesica piscis to the width across its center is the square root of 3, or 1.7320508... (since if straight lines are drawn connecting the centers of the two circles with each other and with the two points where the circles intersect, two equilateral triangles join along an edge). The ratios 265:153 = 1.7320261... and 1351:780 = 1.7320513... are two of a series of approximations to this value, each with the property that no better approximation can be obtained with smaller whole numbers. Archimedes of Syracuse, in his Measurement of a Circle , uses these ratios as upper and lower bounds: [5]

Area

The areas in blue - an equilateral triangle and a segment - form together a sector of one sixth of the circle (60deg) Vesica piscis2.svg
The areas in blue – an equilateral triangle and a segment – form together a sector of one sixth of the circle (60°)

The area of the vesica piscis is formed by two equilateral triangles and four equal circular segments. In the drawing, one triangle and one segment appear in blue.

One triangle and one segment form a sector of one sixth of the circle (60°). The area of the sector is then: .

Since the side of the equilateral triangle has length r, its area is .

The area of the segment is the difference between those two areas:

By summing the areas of two triangles and four segments, we obtain the area of the vesica piscis:

Applications

The modern cover of the Chalice Well with an artistic rendering of the vesica piscis Chalice Well Cover.jpg
The modern cover of the Chalice Well with an artistic rendering of the vesica piscis

The two circles of the vesica piscis, or three circles forming in pairs three vesicae, are commonly used in Venn diagrams. Arcs of the same three circles can also be used to form the triquetra symbol, and the Reuleaux triangle. [3]

In Christian art, some aureolas are in the shape of a vertically oriented vesica piscis, and the seals of ecclesiastical organizations can be enclosed within a vertically oriented vesica piscis (instead of the more usual circular enclosure). Also, the ichthys symbol incorporates the vesica piscis shape. Ecclesiastical heraldry of the Catholic Church appeared first in seals, nearly all vesica-shaped. [6] [7] The vesica piscis has been used within Freemasonry, most notably in the shapes of the collars worn by officiants of the Masonic rituals. [8] It was also considered the proper shape for the enclosure of the seals of Masonic lodges. [9] [10]

The vesica piscis is also used as a proportioning system in architecture, in particular Gothic architecture. The system was illustrated in Cesare Cesariano's 1521 version of Vitruvius's De architectura , which he called "the rule of the German architects". The vesica piscis was a leitmotif of architect Carlo Scarpa and is used as a "viewing device" in Tomba Brion (Brion Cemetery) in San Vito d'Altivole, Italy. [11]

Several other artworks or designs have also featured this shape:

Symbolism

Various symbolic meanings have been associated with the vesica piscis:

See also

Related Research Articles

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References

  1. Fletcher, Rachel (2004), "Musings on the Vesica Piscis", Nexus Network Journal, 6 (2): 95–110, doi: 10.1007/s00004-004-0021-8 .
  2. Norwood, J. W. (1912), "Fish and water symbols", The Open Court, 1912 (11): 662–672
  3. 1 2 3 Barrallo, Javier; González-Quintial, Francisco; Sánchez-Beitia, Santiago (May 2015), "An Introduction to the Vesica Piscis, the Reuleaux Triangle and Related Geometric Constructions in Modern Architecture", Nexus Network Journal, 17 (2): 671–684, doi: 10.1007/s00004-015-0253-9 , S2CID   122824246
  4. Heath, Sir Thomas L. (1956), The Thirteen Books of Euclid's Elements (2 ed.), New York: Dover Publications, pp.  241, ISBN   0486600904
  5. Heath, Thomas Little (1897), The Works of Archimedes, Cambridge University: Cambridge University Press., pp. lxxvii , 50, retrieved 2010-01-30
  6. Arthur Charles Fox-Davies "Ecclesiastical Heraldry"  , Catholic Encyclopedia , 1913
  7. Scanned reproduction of the article, with illustrations Archived 2014-02-24 at the Wayback Machine
  8. J. S. M. Ward, An Interpretation of Our Masonic Symbols, 1924, pp. 34–35.
  9. Albert G. Mackey, Encyclopaedia of Freemasonry, 1921 ed., vol. 2, p. 827.
  10. Shawn Eyer, "The Vesica Piscis and Freemasonry". Retrieved on 2009-04-18.
  11. Cannata, Mark (2007), "Carlo Scarpa and Japan: The influence of Japanese art and architecture in the work of Carlo Scarpa" (PDF), University of Lincoln, archived from the original (PDF) on 2010-04-01, retrieved 2010-02-14
  12. Mundilova, Klara; Wills, Tony (2018), "Folding the Vesica Piscis", in Torrence, Eve; Torrence, Bruce; Séquin, Carlo; Fenyvesi, Kristóf (eds.), Proceedings of Bridges 2018: Mathematics, Art, Music, Architecture, Education, Culture, Phoenix, Arizona: Tessellations Publishing, pp. 535–538, ISBN   978-1-938664-27-4
  13. Fletcher, Rachel (October 2004), "Musings on the Vesica Piscis", Nexus Network Journal, 6 (2): 95–110, doi: 10.1007/s00004-004-0021-8 , S2CID   122154094
  14. 1 2 Houston, Madeleine Claire (2005), Co-respondance: Presence and praxis in land, life, myth (Ph.D. thesis), Pacifica Graduate Institute, ProQuest   305360245
  15. Williams, Kim; Ostwald, Michael J. (2015), Williams, Kim; Ostwald, Michael J (eds.), Architecture and Mathematics from Antiquity to the Future, Springer International Publishing, pp. 68–69, 679, doi:10.1007/978-3-319-00137-1, ISBN   978-3-319-00136-4
  16. Todorova, Rostislava (January 2013), "Visualizing the divine: Mandorla as a vision of God in Byzantine iconography", IKON, 6: 287–296, doi:10.1484/j.ikon.5.102956
  17. Pearson, E. Ann (2002), Revealing and concealing: The persistence of vaginal iconography in medieval imagery. The mandorla, the vesica piscis, the rose, Sheela-na-gigs and the double-tailed mermaid (Ph.D. thesis), University of Ottawa, doi:10.20381/ruor-14701, hdl:10393/6133
  18. Bloomer, Jennifer (February 1988), "In the museyroom", Assemblage (5): 58–65, doi:10.2307/3171026, JSTOR   3171026