Nearly 2,500 years ago, the ancient philosopher Pythagoras proposed that all atomic structure was fundamentally based on five unique shapes. He noted that these forms nested within one another, akin to the nested Matryoshka dolls of Russian tradition (Stewart, 1998).
Buddha’s insights, garnered during his deep meditations, seemingly echoed Pythagoras’ findings. Buddha has been recorded describing the atom as eight-sided, using the term “Acta Kalapas,” which translates to “eight atoms”. This concept coincides strikingly with the eight corners of one of Pythagoras’s shapes – the Cube (Kak, 2006).
The “Acta Kalapas” concept goes beyond the atomic level, resonating with Buddha’s teachings on the interdependence and impermanence of all phenomena. According to Buddhism, everything in the physical and mental universe is in a constant state of flux, with components continually arising and passing away. The Acta Kalapas, being the smallest indivisible units, embody this fundamental idea, reinforcing the view of the universe as a network of interconnected, dynamic processes.
This article explores how Platonic Solids are manifest in the both the natural and built environment, their appearance in a religious context and their relevance to emerging Multiverse theories.
Origin of the Platonic Solids
The name ‘Platonic Solids’ originates from the ancient Greek philosopher Plato, a student of Socrates and teacher of Aristotle. In his work “Timaeus,” Plato associated each of the five solids with the classical elements of the world. The Tetrahedron, due to its sharpness and the number of faces, was associated with fire. The Cube, with its flat faces and stability, was linked with the earth. The Octahedron, with its resemblance to a flying kite, was associated with air, while the Icosahedron, with its many faces symbolizing flux, was attributed to water. Finally, the Dodecahedron, with its complexity, was designated to represent the universe or the cosmos (Stewart, 1998).
Conditions for Platonic Solids
To qualify as a Platonic Solid, a shape must satisfy three specific conditions. First, the shape should fit within a sphere such that all its vertices touch the inside of the sphere (Cromwell, 1997). Second, all of the shape’s faces, or polygons, must be identical. Consequently, this implies that all their angles are equal. The third condition dictates that every edge length within the shape should be identical. These three conditions are crucial in identifying and distinguishing the five Platonic Solids from other geometric structures.
Overview of the Five Platonic Solids
Detailed examination of the Platonic Solids reveals their respective uniqueness and uniformity. The Tetrahedron, the simplest of the solids, is made up of four equilateral triangles, has four vertices, and six edges. The Cube, which corresponds to Buddha’s “Acta Kalapas,” consists of six squares, eight vertices, and twelve edges. The Octahedron is formed from eight equilateral triangles, has six vertices, and twelve edges. The Icosahedron, one of the more complex Platonic Solids, consists of twenty equilateral triangles, twelve vertices, and thirty edges. Lastly, the Dodecahedron is composed of twelve regular pentagons, twenty vertices, and thirty edges (Coxeter, 1973)
The five Platonic Solids, despite being confined to a single category, exhibit a fascinating array of distinct properties while also adhering to a set of common criteria. This unique combination of uniformity and individuality adds to the intrigue of these special geometric structures.
The Tetrahedron
The Tetrahedron, the simplest of the solids, is composed of four equilateral triangles. This structure, boasting a total of four vertices (points where the edges meet), is interconnected by six edges. This is the only Platonic Solid where all faces meet at every vertex, which gives it a strong structural rigidity (Du Sautoy, 2008). Its simplicity and symmetry have made it a fundamental element in mathematical and philosophical thought.
The Cube
The Cube, or hexahedron, is a particularly familiar solid given its prevalence in everyday life. It is formed by six squares of equal size and consists of eight vertices and twelve edges. This Platonic Solid, attributed to Buddha’s “Acta Kalapas,” presents a level of symmetry that extends to its diagonals, where all space diagonals, face diagonals, and edges are mutually perpendicular (Weisstein, 2003). Its three-dimensional symmetry has made it a staple in architectural design and various scientific disciplines.
The Octahedron
The Octahedron, formed from eight equilateral triangles, has six vertices and twelve edges. Its configuration can be described as two square-based pyramids with their bases glued together. Interestingly, if lines are drawn from the center of the Octahedron to each of its vertices, a cube is formed, showing the close relationship between these two Platonic Solids (McRobie, 2018). The Octahedron’s fascinating geometric interrelationships have made it a subject of study in fields ranging from mathematics to crystallography.
The Icosahedron
The Icosahedron, arguably one of the more complex Platonic Solids, is composed of twenty equilateral triangles. It possesses twelve vertices, interconnected by thirty edges. This structure has been used to represent complex spherical objects, such as the geodesic domes popularized by Buckminster Fuller, and the configuration of viral particles (Bonner, 2013). Its complicated symmetry offers insights into spherical tessellation and is key to its numerous applications.
The Dodecahedron
The Dodecahedron is a unique entity, constructed from twelve regular pentagons. It has twenty vertices and thirty edges, making it the Platonic Solid with the most faces. Plato himself associated the Dodecahedron with the universe, reflecting its philosophical significance (Lindemann, 1997). Its unique geometry has been explored in fields ranging from game design to theoretical physics.
The Platonic Solids in a Religious Context
Judaism: The Cube and the Merkaba
In Judaic tradition, the cube holds a special significance. The Holy of Holies, the innermost sanctum of the Tabernacle, and later the Jerusalem Temple, was a perfect cube (1 Kings 6:20).
The Merkaba, a form of Jewish mystical meditation, also employs the image of interlocking tetrahedrons to form a three-dimensional Star of David. This forms a shape similar to a stellated octahedron, which is related to the Platonic Solids.
Christianity: The Dodecahedron
In Christian tradition, the dodecahedron has often been associated with the universe and the heavens, paralleling Plato’s association. St. Augustine, in “The City of God,” referred to the dodecahedron as a symbol for the universe (St. Augustine, The City of God, 426 A.D.). Additionally, in Renaissance art, the dodecahedron was used as a symbol of the divine. For instance, the “Melencolia I” engraving by Albrecht Dürer features a truncated dodecahedron in the background.
Islam: Geometric Art and Architecture
Islamic art and architecture do not explicitly reference the Platonic Solids, but they do make use of intricate geometric designs which can be derived from or linked to these shapes. The rich and complex patterns in Islamic tiling, mosaics, and architecture are mathematically precise, reflecting the belief in a divine universal order.
Whilst explicit references to the Platonic Solids in religious texts may not be abundant, their mathematical and aesthetic qualities have made them influential in religious philosophy, symbolism, art, and architecture across different cultures and eras.
Platonic Solids and Sacred Geometry
Sacred Geometry in Architecture and Art
The principles of sacred geometry, including the use of Platonic Solids, have been employed in the architecture and design of sacred spaces such as churches, temples, mosques, religious monuments, altars, and tabernacles. This is often intended to create spaces that are harmoniously aligned with the universe and to symbolize spiritual truths.
Notable examples include the pyramids of Egypt, which are tetrahedrons, and the complex geometrical patterns found in Islamic art and architecture, where the interplay of geometric forms is thought to reflect the language of the universe and spiritual truths. In Renaissance art, Platonic Solids were frequently used to symbolize various elements of the divine; the artist Albrecht Dürer, for example, depicted a disembodied hand holding a tetrahedron in his engraving “Melencolia I” (Dürer, 1514).
The Platonic Solids have held a deep spiritual and symbolic significance in the realm of sacred geometry due to their association with the classical elements and the cosmos. From the ancient Greeks to contemporary practices, these geometrical forms continue to be used as tools for understanding, expressing, and seeking harmony with the universe.
Sacred geometry, a philosophical and religious concept that ascribes symbolic and sacred meanings to certain geometric shapes, is an ancient practice shared by numerous cultures worldwide. Platonic Solids hold a pivotal place in this context due to their geometric harmony and the philosophical symbolism associated with them.
The Philosophical and Spiritual Significance
Each Platonic Solid is said to correspond to a fundamental element of life, a theory initially proposed by the Greek philosopher, Plato. The Tetrahedron is associated with the element of fire and is thought to represent change and conflict. The Cube (or Hexahedron) is linked with earth and symbolizes grounding and stability. The Octahedron, associated with air, embodies balance and intellect. The Icosahedron is connected with water, representing flow and creativity. Lastly, the Dodecahedron, which Plato assigned to represent the cosmos, is thought to provide a framework for the other elements and to symbolize the aspect of spirit or ether (Plato, Timaeus, 360 B.C.).
These associations have been widely incorporated in spiritual practices, meditation, and healing. For example, in the practice of crystal healing, different Platonic Solid shapes carved from various minerals are used to align or cleanse the energies of corresponding chakras or elements within the individual (Melchizedek, 1999).
Sacred Geometry in Architecture and Art
The principles of sacred geometry, including the use of Platonic Solids, have been employed in the architecture and design of sacred spaces such as churches, temples, mosques, religious monuments, altars, and tabernacles. This is often intended to create spaces that are harmoniously aligned with the universe and to symbolize spiritual truths.
Notable examples include the pyramids of Egypt, which are tetrahedrons, and the complex geometrical patterns found in Islamic art and architecture, where the interplay of geometric forms is thought to reflect the language of the universe and spiritual truths.
In Renaissance art, Platonic Solids were frequently used to symbolize various elements of the divine, for instance:
“Sacred Geometry” by Luca Pacioli (c. 1509): This is actually an illustration from a book rather than a standalone piece of artwork, but it’s a perfect example of the use of Platonic solids in Renaissance art. Pacioli was a mathematician, and his book “De divina proportione” discusses mathematical and artistic proportion, including the use of Platonic solids. The illustrations for the book were made by Leonardo da Vinci, including one that depicts a man holding a rhombicuboctahedron, an Archimedean solid. Though not a Platonic solid, this geometric figure exemplifies the period’s fascination with geometric shapes.
“Melancholia I” by Albrecht Dürer (1514): This engraving by the German Renaissance artist contains a magic square and a truncated rhombohedron. The complex symbolism in the piece has been the subject of much interpretation, with some suggesting the polyhedron represents the physical world and the magic square the spiritual.
“The Last Supper” by Leonardo da Vinci (1498): While this famous fresco does not feature explicit depictions of Platonic solids, da Vinci was well-versed in geometry and Platonic and Euclidean thought, which can be seen in the precise perspective and proportion he uses in the piece. Moreover, it’s argued that the placement of the apostles and the geometric layout of the painting aligns with certain Platonic solids’ attributes.
The Platonic Solids have held a deep spiritual and symbolic significance in the realm of sacred geometry due to their association with the classical elements and the cosmos. From the ancient Greeks to contemporary practices, these geometrical forms continue to be used as tools for understanding, expressing, and seeking harmony with the universe.
Platonic Solids in the Biological World
Remarkably, these Platonic Solids are not confined to theoretical mathematics and ancient philosophy; they exist concretely in the natural world. Single-celled marine organisms called Radiolarians possess exoskeletons that, upon death, solidify into the precise shapes of the Platonic Solids (Haeckel, 2005). These geometric structures hence provide a stunning illustration of the confluence between abstract mathematics, philosophy, and the tangible world.
Radiolarians
The exoskeletons of Radiolarians, single-celled marine organisms, provide a classic example of the appearance of Platonic Solids in biology. When these organisms die, their siliceous exoskeletons solidify and often maintain the precise geometric forms of the Platonic Solids. This phenomenon, extensively documented by the biologist Ernst Haeckel in the 19th century, reveals an astonishing relationship between intricate geometric structures and biological life forms (Haeckel, 2005).
Viruses
Another fascinating example of the appearance of Platonic Solids in biology can be observed in viruses. Many viruses, including the well-known Adenoviruses and the Human Immunodeficiency Virus (HIV), exhibit icosahedral symmetry in their capsid structure, essentially making the virus particle an icosahedron. This structure, made up of 20 triangular faces, provides an efficient and stable arrangement for the virus’s protein subunits (Crick & Watson, 1956).
Pollen Grains
The world of plants also offers examples of Platonic Solids. Certain types of pollen, such as from the plant genus Hibiscus, form exines (outer layers of the pollen grain) that resemble Platonic Solids. These symmetrical, multi-faceted structures assist in the adhesion of pollen grains to pollinators, enhancing the plant’s reproductive success (Franchi, 2012).
Biological Crystalline Structures
Biological crystalline structures are another instance where Platonic Solids find representation. The face-centered cubic crystalline structure, which is equivalent to a dodecahedron, can be seen in the arrangement of atoms in various biological mineral crystals, such as the calcite in mollusk shells and the hydroxyapatite in bone and teeth (Currey, 2002).
Through these examples, it becomes clear that the Platonic Solids, conceived in the realm of abstract thought, play a remarkable role in the organization of life on Earth. This underlines the confluence between seemingly disparate fields—mathematics, philosophy, and biology—and reinforces the intricate interconnectedness of the natural world.
The Utilization of Platonic Solids in Human Architecture, Engineering, and Construction
The concept of Platonic Solids has played a profound role in human construction and design, given their perfect symmetry and pleasing aesthetics. These shapes have found their way into various realms of human creation, from architecture and engineering to product design, gaming, and art.
Architecture and Engineering
Architectural and engineering designs throughout history have utilized the properties of Platonic Solids for structural integrity, spatial efficiency, and aesthetic appeal. The use of the cube, for instance, is ubiquitous in architecture due to its inherent stability and the efficient use of space it provides. It’s common in the design of buildings and rooms, as it provides a simple and efficient way to divide space.
In more complex applications, geodesic domes, a concept popularized by the architect Buckminster Fuller in the mid-20th century, are based on the Icosahedron (Fuller, 1975). These domes, constructed from a network of triangles, are structurally robust, efficient in terms of materials used, and can cover large spaces without internal supports, making them ideal for large enclosures like sports stadiums or exhibition spaces.
Satellite Construction
In the field of aerospace engineering, Platonic Solids, particularly the tetrahedron and icosahedron, have been utilized in the construction of satellites. Their geometric regularity and symmetry make them ideal for creating structurally sound, lightweight frames for spacecraft. For example, the Alouette-1, a Canadian satellite launched in 1962 (pictured right), had an icosahedral framework (Chapman, 2001).
Product Design and Gaming
Platonic Solids have also found use in product design and gaming. Role-playing games (RPGs), like Dungeons and Dragons, use dice shaped as each of the five Platonic Solids to generate random outcomes (Gygax, 1974). This is a direct application of their geometric uniformity; the equal size and shape of each face ensure an equal probability for each outcome.
Art and Sculpture
In art and sculpture, Platonic Solids are often used due to their aesthetic appeal and the philosophical symbolism associated with them. Artists like Leonardo da Vinci, who made detailed sketches of the solids, and Salvador Dali, who used the dodecahedron in his painting “The Sacrament of the Last Supper,” have drawn inspiration from these timeless shapes (Da Vinci, 1492; Dali, 1955).
The concepts of Platonic Solids, derived from ancient Greek mathematics and philosophy, continue to be influential in modern human construction and design. The practical utility of these shapes, combined with their aesthetic and philosophical appeal, ensures their enduring relevance.
Platonic Solids and the Multiverse
Quantum Gravity and Spin Networks
In quantum gravity, a theory that aims to describe the gravitational field using quantum mechanics, a concept known as spin networks has been developed. Spin networks are a way to visualize quantum states of the gravitational field. In particular, Roger Penrose and others have used spin networks where nodes correspond to volumes of space and links to areas. Intriguingly, the junctions in these networks tend to form the shapes of Platonic Solids (Rovelli & Upadhya, 1998). This suggests that the fundamental structure of space-time itself may be related to these shapes.
Multiverse Theories
In the context of Multiverse Theories, some speculative proposals have invoked Platonic Solids. For instance, it has been suggested that each of the universe’s fundamental forces – the strong force, the weak force, electromagnetism, and gravity – might correspond to a different Platonic Solid. Different universes within the multiverse might then be distinguished by the ways these forces (and their corresponding solids) interact. However, this idea remains largely speculative and has not been widely adopted in mainstream physics (Barrow, 2002).
Black Holes
In the realm of black holes, a recent study has suggested that rotating black holes might form a ‘ring singularity’ that could theoretically be shaped like a dodecahedron (Aichelburg & Sexl, 2017). While this idea remains very much theoretical, it provides another potential link between Platonic Solids and the structure of the universe.
Quantum Mechanics
In quantum mechanics, Platonic Solids are occasionally used to visualize the arrangement of electron orbitals, particularly in the context of the ‘shell model’ of the atom. For example, the d-orbitals in atoms form a pattern similar to a rotated cube, one of the Platonic Solids (Deza & Deza, 2012). This is another example of how Platonic Solids can be used to model and understand the fundamental structure of matter in the universe.
Overall, while the Platonic Solids are not directly present in our current fundamental theories of physics, they continue to inspire researchers as they seek to uncover the deepest structures of reality. As we continue to explore the universe at both the smallest and largest scales, it will be fascinating to see if these timeless shapes continue to make appearances.
References
- Aichelburg, P. C., & Sexl, R. U. (2017). On the Gravitational field of a massless particle. General Relativity and Gravitation, 49(4), 56.
- Barrow, J. D. (2002). The Constants of Nature: From Alpha to Omega–the Numbers That Encode the Deepest Secrets of the Universe. Pantheon Books.
- Bonner, J. T. (2013). The Self-Assembly of Biological Structures. In The Evolution of Complexity by Means of Natural Selection. Princeton University Press.
- Chapman, S. (2001). Alouette-ISIS Satellite Operations. The Canadian Encyclopedia.
- Coxeter, H. S. M. (1973). Regular Polytopes. Dover publications.
- Crick, F. H., & Watson, J. D. (1956). Structure of small viruses. Nature, 177(4506), 473-475.
- Cromwell, P. (1997). Polyhedra. Cambridge.
- Currey, J. D. (2002). Bones: Structure and Mechanics. Princeton University Press.
- Da Vinci, L. (1492). The Platonic Solids. Codex Atlanticus.
- Dali, S. (1955). The Sacrament of the Last Supper. National Gallery of Art.
- Deza, M., & Deza, E. (2012). Figurate numbers. World Scientific.
- Du Sautoy, M. (2008). Finding Moonshine: A Mathematician’s Journey Through Symmetry. Harper Perennial.
- Dürer, A. (1514). Melencolia I. Staatliche Kunsthalle Karlsruhe.
- Franchi, G. G., Piotto, B., Nepi, M., Baskin, C. C., Baskin, J. M., & Pacini, E. (2012). Pollen and seed desiccation tolerance in relation to degree of developmental arrest, dispersal, and survival. Journal of Experimental Botany, 63(15), 5267-5281.
- Fuller, R. B. (1975). Synergetics: Explorations in the Geometry of Thinking. Macmillan.
- Gygax, G. (1974). Dungeons & Dragons. TSR, Inc.
- Haeckel, E. (2005). Art Forms from the Ocean: The Radiolarian Atlas of 1862. Prestel.
- Kak, S. (2006). The Universe, Quantum Physics, and Consciousness. Subtle Energies & Energy Medicine Journal Archives, 15(3).
- Lindemann, M. (1997). Philosophy of Plato and Aristotle. Cornell University Press.
- McRobie, A. (2018). The Seduction of Curves: The Lines of Beauty That Connect Mathematics, Art, and The Nude. Princeton University Press.
- Melchizedek, D. (1999). The Ancient Secret of the Flower of Life. Light Technology Publishing.
- Plato. (360 B.C.). Timaeus.
- Rovelli, C., & Upadhya, P. (1998). Loop quantum gravity and quanta of space: A primer. arXiv preprint arXiv:gr-qc/9806079.
- St. Augustine. (426 A.D.). The City of God.
- Stewart, I. (1998). Life’s Other Secret: The New Mathematics of the Living World. Wiley.
- Weisstein, E. W. (2003). CRC Concise Encyclopedia of Mathematics, Second Edition. CRC Press.
- 1 Kings 6:20 (New International Version)
