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Vesica Piscis

VesicaPiscis

The term "vesica piscis," meaning "fish bladder" in Latin, is used for the particular symmetric lens formed by the intersection of two equal circles whose centers are offset by a distance equal to the circle radii (Pedoe 1995, p. xii). The height of the lens is given by letting d=r=R=a in the equation for a circle-circle intersection

h=a/dsqrt(4d^2R^2-(d^2-r^2+R^2)^2),
(1)

giving

h=asqrt(3).
(2)

The vesica piscis therefore has two equilateral triangles inscribed in it as illustrated above.

The area of the vesica piscis is given by plugging d=R into the circle-circle intersection area equation with r=R,

A=2R^2cos^(-1)(d/(2R))-1/2dsqrt(4R^2-d^2),
(3)

giving

A=1/6(4pi-3sqrt(3))a^2
(4)
=1.22837...a^2
(5)

(OEIS A093731). Since each arc of the lens is precisely 1/3 of a circle, perimeter is given by

p=4/3pia.
(6)

Renaissance artists frequently surrounded images of Jesus with the vesica piscis (Pedoe 1995, p. xii; Rawles 1997).

See also

Arc, Circle, Circle-Circle Intersection, Circular Sector, Circular Segment, Double Bubble, Flower of Life, Goat Problem, Kapteyn Series, Kepler's Equation, Laplace Limit, Lemon Surface, Lens, Lune, Mohammed Sign, Moss's Egg, Reuleaux Triangle, Seed of Life, Semicircle, Triangle Arcs, Triquetra, Venn Diagram

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References

Pedoe, D. Circles: A Mathematical View, rev. ed. Washington, DC: Math. Assoc. Amer., 1995.Rawles, B. Sacred Geometry Design Sourcebook: Universal Dimensional Patterns. Nevada City, CA: Elysian Pub., p. 11, 1997.Sloane, N. J. A. Sequence A093731 in "The On-Line Encyclopedia of Integer Sequences."

Cite this as:

Weisstein, Eric W. "Vesica Piscis." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/VesicaPiscis.html

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