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Sparavigna, A. & Baldi, M. (2016). A Mathematical Study of a Symbol: the Vesica Piscis of Sacred Geometry. PHILICA.COM Article number 560.

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A Mathematical Study of a Symbol: the Vesica Piscis of Sacred Geometry

Amelia Carolina Sparavignaunconfirmed user (Department of Applied Science and Technology, Politecnico di Torino)
Mauro Maria Baldiunconfirmed user (Department of Control and Computer Engineering, Politecnico di Torino)

Published in matho.philica.com

In this paper, we are proposing a study of the Vesica Piscis, a symbol of Sacred Geometry, starting from mathematics. This study aims to give a deeper comprehension of the hidden meanings of this icon. The paper is also showing that scientific and philosophical studies can be integrated, leading to very interesting results.

Article body

A Mathematical Study of a Symbol: the Vesica Piscis of Sacred Geometry


Amelia Carolina Sparavigna1 and Mauro Maria Baldi2 

1 Department of Applied Science and Technology, Politecnico di Torino, Torino, Italy

1 Department of Control and Computer Engineering, Politecnico di Torino, Torino, Italy


Abstract: In this paper, we are proposing a study of the Vesica Piscis, a symbol of Sacred Geometry, starting from mathematics. This study  aims to give a deeper comprehension of the hidden meanings of this icon. The paper is also showing  that scientific and  philosophical studies can be integrated, leading to very interesting results.


Keywords: Sacred Geometry, Geometry, Architecture.


Introduction: The Vesica Piscis is one of the most important symbols of the Sacred Geometry, a discipline ascribing symbolic meanings to geometric shapes and proportions, which has its roots in the study of nature and natural principles [1,2]. This Geometry is representing numbers in the tridimensional space, differently from the Euclidean geometry, because the concepts involved have symbolic values, used with the aim of facilitate “the evolution of the soul” [3]. In this manner, the symbols of the Sacred Geometry assume a peculiarity: they silently appear in several places, like in churches  and  temples, and in some cases, are ruling architectures. For example, the Vesica Piscis appears in the design of Bernini’s oval part of Saint Peter’s  Square  in Rome [4],  the  most  important place of the Catholic Church.

Also in the case of  Euclidean geometry, we have to consider that some geometric diagrams can play a crucial role in visualizing mathematical proofs, and therefore in the evolution of our knowledge of the world. In [5],  twenty of these icons of mathematics are presented and discussed. As proposed in this reference, each icon has a presence in real life and a primary mathematical characteristic, and it gives origin, with its visual proofs, to a wide range of mathematical facts. And then, from them, we can find the classical results of plane geometry, properties of numbers, trigonometric relations identities, theorems of mathematics, and so on. If we can, from icons discover mathematics, we can also try to find symbolic meanings of the icons through the  use  of mathematical  techniques, as we will see in the following discussion. In  this  paper,  the method will be applied  to the Vesica Piscis.  In the following, we will introduce the symbol, then the mathematical approach to it and role in arts and architecture, before the disclosure of the hidden  meaning of the symbol.


Figure 1: The Vesica Piscis formed by two identical  circumferences.


The symbol: The  Vesica Piscis, the "bladder of a fish" in Latin, is shown in the Figure 1. This  symbol  is created  by two circumferences, γ1 and γ2, having a radius of the same length. The center of each circumference lies on the other  circumference.  The overlapping  area of the two circles is called  “eye” ormandorla”,  almond  in Italian.  In Figure 1, the two centers are denoted  by letters  A and B.  The Vesica Piscis also includes segment AB and its perpendicular segment CD, having midpoint H .  Without loss of generality,  we can assume the radius r of both circumferences be equal of the unit length. In fact, if r = a, with a > 0 and a = 1, then  the resulting  symbol is a  homotetic  transformation of the  same  symbol  with  radius  1.  Thus,  the shape of the symbol does not depend on the given radius.

Mathematical study: Starting from the two circumferences γ1  and γ2, we want to compute  lengths AB  and  CD of the  respective  segments.  Let r1 and  r2 denote  respectively  the radii of circumferences  γ1  and γ2 , we have r1 = r2 = 1. Segment  AB  is, by construction, the  radius  of both  circumferences, and then its length is equal to unit.  Since H  is the midpoint of segment AB, the segment AH has a length equal to 1/2. By construction, AC is a radius  of circumference  γ1 , and the its length is equal to 1. Triangle ABC is equilateral. Triangle  AHC  is rectangular, being  segment  CH  perpendicular to AB.   Therefore,  we can apply  Pythagoras’ theorem  to triangle  AHC and compute  the length  of segment CH.  We have:



By the symmetry of the figure:



The ratio of the two main segments is:


 We have then, geometrically, the square root of 3, which is an irrational number.

However, since we referred to Pythagoras’ theorem, let us consider (3), written as:



And, for this equation, Archimedes reported the value (1351/780)2 > 3 > (265/153)2 [6].

The square root of 3 is also known as Theodorus' constant, named after Theodorus of Cyrene [7]. However, for what concerns the icon in the Figure 1, to the square root of 3 we have to add other numbers: as shown in [8], we can find the square root of 2, the square root of 5 and the Golden Ratio. However, it is number 3, the relevant one of this icon.

The icon in architecture and arts: In the introduction, we have already mentioned that the Vesica Piscis is appearing in the layout of the oval part of  Saint Peter’s Square, a square planned by Gian Lorenzo Bernini (1598-1680) [9,4] (see Figure 2).  The great artist and architect had probably followed the canons, in his planning of that part of the Square framed by an “ovato tondo”. This figure was the approximation of ellipse used in architecture.  The Vesica Piscis was recommended by Sebastiano Serlio (1475-c.1554), the Italian Mannerist architect that helped canonize the classical orders of architecture. This construction of “ovato tondo”, besides being the simplest and quickest manner to construct oval spaces, was also suggested for its beauty. In fact, it became the standard ellipse approximation used in architectural practise [10]. Let us note that the Vesica Piscis was already used as a proportioning system of Gothic architecture, as illustrated by Cesare Cesariano (1475-1543), in the first Italian-language version of Vitruvius’ De Architectura [11]. Cesariano called it  "the rule of the German architects". This geometric shape was used frequently in Gothic cathedrals then, and in fact, the construction appears in the Portfolio of Villard de Honnecourt, a 13th-century artist from Picardy in northern France [12].

The Vesica Piscis was known to Pythagoreans as the “Potential Logos”, symbolizing the Dyad that becomes a Triad, the harmony or Logos [13]. Later, this symbol migrated  to the Christianity [14]. In the Christian art, the Vesica Piscis often became the halos of light, that is, the aureoles surrounding God, Jesus Christ, Virgin Mary and Saints (see an example in the Figure 2). Also the seals of ecclesiastical organizations had this icon as a frame. The Vesica Piscis has been used as a symbol within Freemasonry too, and the proper shape for the enclosure of the seals of Masonic lodges [15].


Figure 2: On the left, the “ovato tondo” of Bernini’s Square in front of Saint Peter’s Basilica, seen in a Google Earth image. To evidence the geometry of this ovato, black and red numbers give the lengths (in pixels) of the corresponding lines. The obelisk is the center of a Vesica Piscis. On the right, we see  a Vesica Piscis in the central tympanum of the Royal portal of the Gothic cathedral of Chartres.


The hidden meaning of the icon: After the examples of Vesica Piscis given in the previous section, let us consider again the results we have obtained from calculus in Eqs.(1)-(4). In the Sacred  Geometry, God,  or the  One of the Universe (Unus  mundi,  in Latin)  or the  divinity, is represented as a circle with radius 1, because One  is the  concept  of unity, being God omnipresent and  the  creator  of the world.  In the Vesica Piscis, the  projection  of the same circumference  starting from the  initial  one, given the  concept  of Duality.  In a number of spiritual believes, a soul temporary separates from the  One (Unity) to know herself. Therefore,  we have a spiritual  journey of the soul from Unity to Duality and then back to Unity, so that a highest level of evolution is acquired.  However, the two segments AB and CD form a cross. This cross represents Christ and the concept of Trinity, as witnessed by the ratio of the squared lengths of these segments,  which is number 3, i.e., the number  of Trinity. In fact, as we have already mentioned, the Vesica Piscis migrated from the Pythagorean philosophy, where it was a Triad, to the Christianity becoming a Trinity [16].

Conclusions: In this paper, we have proposed a study of Vesica Piscis, starting from mathematics. This study  is fundamental for a deeper comprehension of hidden  meanings  of this icon,  revealing  at  least  two  important considerations. First, in the discipline of Sacred Geometry, geometry and calculus are not only human  and artificial  inventions,  but  are  symbolic means  through which the  divine  extends its  process of Creation and Life. Second, in a scholar approach, this work proves how scientific and  philosophical studies can be integrated, leading to very interesting results.



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[2] Lawlor, R. (1982). Sacred Geometry: Philosophy and Practice (Vol. 4). New York: Thames and Hudson. ISBN-10: 0500810303, ISBN-13: 978-0500810309

[3] Lundy, M. (2001). Sacred Geometry, Walker Books, ISBN-10: 0802713823, ISBN-13: 978-0802713827

[4] Sparavigna, A.C. (2015). Light  and  Shadows in  Bernini’s    Oval    of  Saint Peters  Square, PHILICA.COM, Article  number 540.

[5] Alsina, C. & Nelsen, R.B. (2011). Icons of Mathematics: An Exploration of Twenty Key Images, Mathematical Association of America. ISBN-10: 0883853523, ISBN-13: 978-0883853528

[6] Knorr, W.R. (1976), Archimedes and the Measurement of the Circle: A New Interpretation, Archive for History of Exact Sciences 15 (2): 115–140, DOI: 10.1007/bf00348496

[7] Weisstein, E.W. Theodorus's Constant. From MathWorld - A Wolfram Web Resource. http://mathworld.wolfram.com/ TheodorussConstant.html, retrieved 22 January 2016.

[8] See the image shown at web address (retrieved 22 January 2016) http://portal.groupkos.com/ index.php?title=POVRay_scene_Vesica_pisces.pov

[9] Kitao, T.K. (1974). Circle and Oval in the Square of Saint Peter’s. New York University Press. ISBN-10: 0814745571, ISBN-13: 978-0814745571

[10]  Rosin, P. L. (2001). On Serlio’s constructions of ovals. The Mathematical Intelligencer, 23(1), 58-69. DOI: 10.1007/bf03024523

[11] Cesariano's De Architectura on line, available at http://architectura.cesr.univ-tours.fr/Traite/Notice/BPNME276.asp?param=en

[12] Vv. Aa. (2016), Villard de Honnecourt, Wikipedia.

[13] Ralls, K. (2015). Gothic Cathedrals: A Guide to the History, Places, Art, and Symbolism,  Ibis Press, ISBN-10: 0892541733, ISBN-13: 978-0892541737

[14] Todorova, R.G. (2011). The Migrating Symbol: Vesica Piscis from the Pythagoreans to the Christianity. 1th International Conference “Harmony of Nature and Spirituality in Stone”, 17-18 March 2011, Kragujevac, Serbia. Stone Studio Association. Pages 217-228.

[15] Ward, J.S.M.  (1924). An Interpretation of Our Masonic Symbols, 1924, pages 34-35. Published also in 2010, by Kessinger Publishing, ISBN-10: 1162562846, ISBN-13: 978-1162562841

[16] French, K.L. (2014). Gateway to the Heavens: How Geometric Shapes, Patterns and Symbols Form Our Reality, Duncan Baird Publishers. ISBN.10: 1780287798, ISBN-13: 978-1780287799

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Sparavigna, A. & Baldi, M. (2016). A Mathematical Study of a Symbol: the Vesica Piscis of Sacred Geometry. PHILICA.COM Article number 560.

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