Shape that is the intersection of two circles with the same radius
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.
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 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 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]
Several other artworks or designs have also featured this shape:
The cover of the
Chalice Well in
Glastonbury (United Kingdom) depicts a stylized version of the vesica piscis design.
Several mathematical sculptures by Susan Latham use a three-dimensional form obtained from the planar depiction of two circles forming the vesica piscis, deformed into as a curved surface with folds along the inner arcs of the vesica and with the two outer arcs meeting in a single curve. Its shape can be analyzed using the mathematics of
developable surfaces.[12]
Symbolism
Various symbolic meanings have been associated with the vesica piscis:
When arranged so that the lens is horizontal, with its two overlaid circles placed one above the other, it symbolizes the interface between the spiritual and physical worlds, represented by the two circles.[13][14] In this arrangement, it also resembles the
ichthys (fish) symbol for
Christ,[14] and has also been said to be a symbol of life, of "the materialization of the spirit", of Christ's mediation between heaven and earth, and of the
eucharist.[15]
When arranged so that the lens is placed vertically, and used to depict a halo or aureola, it represents
divine glory.[16]
When arranged so that the lens is placed vertically, it has also been said to be a depiction of the
vulva, and therefore symbolic of femininity and fertility.[3][17]
A diagram of Euclid's use of this diagram to construct an equilateral triangle, appearing with the vertical placement of the lens in
James Joyce's Finnegans Wake, has been said to be "emblematic of rational man", but overlaid onto a vaginal triangle again symbolizing femininity.[18]
^
abcBarrallo, 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,
S2CID122824246
^
abHouston, Madeleine Claire (2005), Co-respondance: Presence and praxis in land, life, myth (Ph.D. thesis), Pacifica Graduate Institute,
ProQuest305360245
^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,
ISBN978-3-319-00136-4
^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
^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