Visualization of Polyhedra with Virtual Reality (VR) in A-frame
author: Paulo Henrique Siqueira - Universidade Federal do Paraná
contact: paulohscwb@gmail.com
versão em português
Vesica piscis and Trinexus
The Vesica Trinexus is a geometric prototype created by Ramiro Franceschin based on a simple question: what happens if we take the Vesica piscis to its next three-dimensional octave?
Instead of using the construction with two flat circles, two intersecting icosahedrons are used. This intersection generates the seed, the zero point of the construction. From there, the geometry expands into successive octaves.
This work shows the polyhedral compositions that form the Vesica Trinexus, as well as some Vesica piscis constructions using polyhedrons, all modeled for viewing in Virtual Reality.
3D models
1. Vesica Trinexus: 1st octave
Instead of using a construction with two flat circles, two intersecting icosahedrons are used to form the first stage of the Vesica Trinexus. This intersection of icosahedrons generates the seed, the zero point of the construction. From there, the geometry expands in successive octaves.
2. Vesica Trinexus: 2nd octave
In the second octave of the Vesica Trinexus, truncated icosidodecahedrons appear, along with an equatorial belt of cubes. This arrangement resembles a system of 12 centers or chakras around a 13th central node, consistent with multiple symbolic traditions.
3. Vesica Trinexus: 3rd octave
In the third octave of the Vesica Trinexus, antiprisms and pentagonal domes appear, resembling arrows that direct energy first toward the root and crown and finally back to the center. The pattern of these polyhedrons reveals a cycle of expansion and return.
4. Vesica Trinexus: 4th octave
In the fourth octave of the Vesica Trinexus, antiprisms, pyramids, and pentagonal domes appear. The pattern of these polyhedrons reveals a cycle of expansion and return. The result is a geometry that:
1. Integrates Platonic and Archimedean solids in harmonic progression;
2. Suggests a three-dimensional energy system (12+1) that dialogues with traditions such as Kabbalah, Mayan symbolism, and the thought of Rudolf Steiner; and
3. Shows how energy circulates in octaves: from the center to the poles and from the poles back to the center.
5. Vesica piscis with triakis icosahedrons
The Vesica piscis is a geometric figure formed by the intersection of two circles with the same radius, where the center of each circle is located on the circumference of the other. In this adaptation, we have the Vesica piscis constructed with two triakis icosahedrons.
6. Vesica piscis with pentakis dodecahedrons
The Vesica piscis is used in a variety of fields, including mathematics (the construction of an equilateral triangle), art, architecture, and symbolism, representing union, creation, and duality. In this adaptation, the Vesica piscis is constructed with two pentakis dodecahedrons.
7. Vesica piscis with rhombicosidodecahedrons
The Vesica piscis can be found in architecture, such as in the arches and vaulted ceilings of Gothic cathedrals, and in artistic representations, such as the outlines of images of saints. In this adaptation, we have the Vesica piscis constructed with two rhombicosidodecahedrons.
8. Vesica piscis with rhombicuboctahedrons
The Vesica Piscis symbol represents the creation of the universe and the origin of life from duality. In this adaptation, the Vesica Piscis is constructed with two rhombicuboctahedrons.
9. Vesica piscis with small stellated dodecahedrons
The Vesica Piscis symbol represents the union of opposites: masculine and feminine, spiritual and material. In this adaptation, the Vesica Piscis is constructed with two small stellated dodecahedrons.
10. Vesica piscis with rhombidodecadodecahedrons
The Vesica Piscis symbol represents the alchemical marriage in the search for spiritual wholeness. In this adaptation, the Vesica Piscis is constructed of two rhombidodecadodecahedrons.
11. Vesica piscis with small icosicosidodecahedrons
The Vesica piscis is a geometric figure formed by the intersection of two circles with the same radius, where the center of each circle is located on the circumference of the other. In this adaptation, we have the Vesica piscis constructed with two small icosicosidodecahedrons.
12. Vesica piscis with dodecadodecahedrons
The Vesica piscis is used in a variety of fields, including mathematics (the construction of an equilateral triangle), art, architecture, and symbolism, representing union, creation, and duality. In this adaptation, the Vesica piscis is constructed with two dodecadodecahedrons.
13. Vesica piscis with great icosicosidodecahedrons
The Vesica piscis can be found in architecture, such as in the arches and vaulted ceilings of Gothic cathedrals, and in artistic representations, such as the outlines of images of saints. In this adaptation, we have the Vesica piscis constructed with two great icosicosidodecahedrons.
14. Vesica piscis with great icosidodecahedrons
The Vesica Piscis symbol represents the creation of the universe and the origin of life from duality. In this adaptation, the Vesica Piscis is constructed with two great icosidodecahedrons.
15. Vesica piscis with truncated great dodecahedrons
The Vesica Piscis symbol represents the union of opposites: masculine and feminine, spiritual and material. In this adaptation, the Vesica Piscis is constructed with two truncated great dodecahedrons.
16. Vesica piscis with small stellapentakis dodecahedrons
The Vesica Piscis symbol represents the alchemical marriage in the search for spiritual wholeness. In this adaptation, the Vesica Piscis is constructed of two small stellapentakis dodecahedrons.
17. Vesica piscis with great truncated icosidodecahedrons
The Vesica piscis is a geometric figure formed by the intersection of two circles with the same radius, where the center of each circle is located on the circumference of the other. In this adaptation, we have the Vesica piscis constructed with two great truncated icosidodecahedrons.
18. Vesica piscis with snub icosidodecadodecahedrons
The Vesica piscis is used in a variety of fields, including mathematics (the construction of an equilateral triangle), art, architecture, and symbolism, representing union, creation, and duality. In this adaptation, the Vesica piscis is constructed with two snub icosidodecadodecahedrons.
19. Vesica piscis with Möbius hexakis icosahedrons
The Vesica Piscis symbol represents the creation of the universe and the origin of life from duality. In this adaptation, the Vesica piscis is constructed with two Möbius hexakis icosahedrons.
20. Vesica piscis with Möbius 10-akis dodecahedrons
The Vesica piscis is a geometric figure formed by the intersection of two circles with the same radius, where the center of each circle is located on the circumference of the other. In this adaptation, the Vesica piscis is constructed with two Möbius 10-akis dodecahedrons.
21. Vesica piscis with small icosacronic hexecontahedrons
The Vesica piscis is used in a variety of fields, including mathematics (the construction of an equilateral triangle), art, architecture, and symbolism, representing union, creation, and duality. In this adaptation, the Vesica piscis is constructed with two small icosacronic hexecontahedrons.
22. Vesica piscis with small dodecacronic hexecontahedrons
The Vesica piscis can be found in architecture, such as in the arches and vaulted ceilings of Gothic cathedrals, and in artistic representations, such as the outlines of images of saints. In this adaptation, the Vesica piscis is constructed with two small dodecacronic hexecontahedrons.
23. Vesica piscis with small hexagonal hexecontahedrons
The Vesica Piscis symbol represents the union of opposites: masculine and feminine, spiritual and material. In this adaptation, the Vesica piscis is constructed with two small hexagonal hexecontahedrons.
24. Vesica piscis with icositruncated dodecadodecahedrons
The Vesica Piscis symbol represents the alchemical marriage in the search for spiritual wholeness. In this adaptation, the Vesica piscis is constructed with two icositruncated dodecadodecahedrons.
25. Vesica piscis with great truncated icosahedrons
The Vesica piscis is used in a variety of fields, including mathematics (the construction of an equilateral triangle), art, architecture, and symbolism, representing union, creation, and duality. In this adaptation, the Vesica piscis is constructed with two great truncated icosahedrons.
26. Vesica piscis with truncated dodecadodecahedrons
The Vesica Piscis symbol represents the creation of the universe and the origin of life from duality. In this adaptation, the Vesica piscis is constructed with two truncated dodecadodecahedrons.
27. Vesica piscis with small snub icosicosidodecahedrons
The Vesica piscis is a geometric figure formed by the intersection of two circles with the same radius, where the center of each circle is located on the circumference of the other. In this adaptation, the Vesica piscis is constructed with two small snub icosicosidodecahedrons.
28. Vesica piscis with rhombicosidodecahedrons of kites
The Vesica piscis is used in a variety of fields, including mathematics (the construction of an equilateral triangle), art, architecture, and symbolism, representing union, creation, and duality. In this adaptation, the Vesica piscis is constructed with two rhombicosidodecahedrons of kites.
29. Vesica piscis with rhombicuboctahedrons of kites
The Vesica piscis can be found in architecture, such as in the arches and vaulted ceilings of Gothic cathedrals, and in artistic representations, such as the outlines of images of saints. In this adaptation, the Vesica piscis is constructed with two rhombicuboctahedrons of kites.
30. Vesica piscis with icosidodecahedrons
The Vesica Piscis symbol represents the union of opposites: masculine and feminine, spiritual and material. In this adaptation, the Vesica piscis is constructed with two icosidodecahedrons.
31. Vesica piscis with icosidodecahedrons of kites
The Vesica piscis is used in a variety of fields, including mathematics (the construction of an equilateral triangle), art, architecture, and symbolism, representing union, creation, and duality. In this adaptation, the Vesica piscis is constructed with two icosidodecahedrons of kites.
32. Vesica piscis with deltoidal hexecontahedrons
The Vesica piscis can be found in architecture, such as in the arches and vaulted ceilings of Gothic cathedrals, and in artistic representations, such as the outlines of images of saints. In this adaptation, the Vesica piscis is constructed with two deltoidal hexecontahedrons.
33. Vesica piscis with disdyakis triacontahedrons
The Vesica Piscis symbol represents the union of opposites: masculine and feminine, spiritual and material. In this adaptation, the Vesica piscis is constructed with two disdyakis triacontahedrons.
34. Vesica piscis with deltoidal icositetrahedrons
The Vesica Piscis symbol represents the alchemical marriage in the search for spiritual wholeness. In this adaptation, the Vesica piscis is constructed with two deltoidal icositetrahedrons.
35. Vesica piscis with truncated icosahedrons
The Vesica piscis is used in a variety of fields, including mathematics (the construction of an equilateral triangle), art, architecture, and symbolism, representing union, creation, and duality. In this adaptation, the Vesica piscis is constructed with two truncated icosahedrons.
36. Vesica piscis with snub cubes
The Vesica Piscis symbol represents the creation of the universe and the origin of life from duality. In this adaptation, the Vesica piscis is constructed with two snub cubes.
37. Vesica piscis with truncated icosahedrons of kites
The Vesica piscis is a geometric figure formed by the intersection of two circles with the same radius, where the center of each circle is located on the circumference of the other. In this adaptation, the Vesica piscis is constructed with two truncated icosahedrons of kites.
38. Vesica piscis with snub cubes of kites
The Vesica piscis is used in a variety of fields, including mathematics (the construction of an equilateral triangle), art, architecture, and symbolism, representing union, creation, and duality. In this adaptation, the Vesica piscis is constructed with two snub cubes of kites.
39. Vesica piscis with joined truncated cuboctahedron
The Vesica piscis is used in a variety of fields, including mathematics (the construction of an equilateral triangle), art, architecture, and symbolism, representing union, creation, and duality. In this adaptation, the Vesica piscis is constructed with two joined truncated cuboctahedrons.
40. Vesica piscis with joined truncated icosahedron
The Vesica piscis is used in a variety of fields, including mathematics (the construction of an equilateral triangle), art, architecture, and symbolism, representing union, creation, and duality. In this adaptation, the Vesica piscis is constructed with two joined truncated icosahedrons.
41. Vesica piscis with joined truncated dodecahedron
The Vesica piscis is used in a variety of fields, including mathematics (the construction of an equilateral triangle), art, architecture, and symbolism, representing union, creation, and duality. In this adaptation, the Vesica piscis is constructed with two joined truncated dodecahedrons.
42. Vesica piscis with joined truncated icosidodecahedron
The Vesica piscis is used in a variety of fields, including mathematics (the construction of an equilateral triangle), art, architecture, and symbolism, representing union, creation, and duality. In this adaptation, the Vesica piscis is constructed with two joined truncated icosidodecahedrons.
Polyhedral Compound - Vesica piscis and Trinexus: visualization with Virtual Reality by Paulo Henrique Siqueira is licensed with a license Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International.
How to cite this work:
Siqueira, P.H., "Polyhedral Compound - Vesica piscis and Trinexus: visualization with Virtual Reality". Available in: <https://paulohscwb.github.io/polycompound/vesica/>, October 2025.
References:
Weisstein, Eric W. “Polyhedron Compound” From MathWorld-A Wolfram Web Resource. https://mathworld.wolfram.com/PolyhedronCompound.html
Pardesco. “Sacred Geometry Art, Symbols & Meanings”. https://pardesco.com/blogs/news/sacred-geometry-art-symbols-meanings
McCooey, David I. “Visual Polyhedra”. http://dmccooey.com/polyhedra/