My research has been focused on the mathematical ratios and sequences that explain the geometric patterns of our natural world. Through the discoveries and writings of our ancient philosopher Plato, we have come to find that there five solids known as the Platonic Solids. Plato associates these solids with natural elements such as earth, air, water, fire, and ether, and these solids are what define all matter within the universe. Through exploring the properties of these solids, and patterns created by them, scientists, spiritualists, and philosophers have come to find reoccurring mathematical symmetries and sequences such as Phi, The Golden Ratio, and Fibbonnacci Sequence, that appears on both a macro and microscopic level of physical reality. These patterns can be seen and experienced through, movement, light, sound, and vision, and they repeat from the molecules in our bodies, to the stars in the galaxy. Life manifests itself in a series of patterns that tessellate and grow in an efficient and beautifully geometric form (Wilzeck 2015). Through my research, I seek to create an awareness of these symmetries and patterns to provide a connection to our environment and provide an understanding that we as humans are also the environment.
Euclid (325-265 B.C.E.) of Alexandria (Egypt) wrote the book Elements which is one of our first recorded and most basic teaching of geometry (it is still used today). This book marks a movement in history in the way that we think as a species, because it proposed a new way of thinking based on empirical thought; where ideas could be reality based on just an individuals thought. This is also the first record of the knowledge of the divine proportion. The Divine Proportion (Phi) is a ratio in which the whole is to the larger in exactly the same proportion as the larger is to the smaller. Or, a pattern of numbers that increases by adding the two previous numbers. This ratio can be reduced to 1.618 (but is continuous) and has multiple names such as Phi, Golden Mean, Golden Ratio, and Divine Proportion. It is a mathematical representation of the structure and framework in which life is manifested. Phi is present in both micro and macrocosms of all life forms on this planet, and beyond. (Hemenway, 2005).
Ancient civilizations relied on geometry and used it as there tool and blueprint for understanding the physical world and how it functions. People recognized geometric patterns, proportions, and ratios that related from part to whole, and saw that these were not only apparent in everyday natural life, but that they could also apply this knowledge in order to create.
Phidias (490-430 B.C.E.) was a Greek sculptor and mathematician who is known to have applied the Divine Proportion to the rectangles in the architecture of the Parthenon in Greece. The building was constructed in parallel lines in which the height and width of the columns are spaced in the ration of phi. Phidias worked on the Parthenon with Pythagoras, a Greek philosopher and mathematician who is a founder of Pythagoreanism. Pythatgoreanism is a school of teachings that are based upon “ideas that the metaphysics of number and conception that reality (including musical and astronomy) is, at its deepest level mathematical in nature (Wilzeck, 2015).” He is also recognized for the theory of Harmonia Kosmos translates as “fitting together, the beautiful order of things” (Hemenway 2005).
Plato (427-347 B.C.E) was next to recognize the divine proportion. He described five regular solids known as the platonic solids. These solids apply the divine proportion within their formations, and are said to be the basis for the structure of all matter in our universe. Platonic solids are shapes where each face is same regular polygon and the same number of polygons meets at each vertex. The platonic solids have been identified with the four basic elements-the hexahedron with earth, the tetrahedron with fire, the octahedron with air, and the icosahedron with water. In ancient Greek philosophy, the people believed Platonic solids to have a spherical property, where one Platonic Solid fits in a sphere, which alternately fits inside another Platonic Solid, again fitting in another sphere. This organization of spherical property is similar to our modern atom which shows a nucleus surrounded by electrons in orbit, which create a sphere of energy. This goes on infinitely large and small, as everything connects. Phi occurs in each of the platonic solids, within the geometry of the surface area of the faces of each three dimensional solid. Finally, in Plato's view, the regular polyhedral shapes are the building-blocks not only of the inorganic world, but of the entire physical universe, including organic and inorganic matter. (Hemenway 2005).
The Fibonnacci Sequence is a series of numbers that proceeds by adding the previous two numbers together, and it is a sequence of numbers that goes on infinitely. This mathematical sequence correlates with spirals that are found in art, science, and nature. Leonardo Fibonacci is the founder of this sequence and it first showed up in his book Liber Abaci. The Fibonacci sequence has a direct correlation and has a mathematical relationship to phi. When you take two numbers in the sequence and divide the first number by the second number, the decimal is always 1.60 which is approximately equal to Phi. This sequence is commonly applied to analyze growth patterns of plant and human life. Leonardo Da Vinci even used this sequence to construct the Vitruvian Man. (Hemenway 2005).
Next we have Johannes Kepler, who discusses the harmony of the world and the laws of planetary motion, in which he explains how planets move in ellipses and not circles. His view of the universe is linked to the platonic solids, as he believed the planets were aligned in the shapes of the platonic solids. He even created a platonic solid model of the solar system, which he believed was the blueprint of the universe. He stated that the five platonic solids could be applied to six planets if the solids bounding surfaces marked the spaces between the planetary spheres (Hemenway 2005)(Wilzeck 2015).
In the history of the human species, there was a deeper connection between human life and the earth. Cycles and patterns of life were commonly recorded, in order to gain a deeper understanding of the world and of our place as humans within it all. As humans evolved as a species, I think we have lost some of this connection, and created an imbalance in our practices. In today’s society I find that people are heavily relying on technology and so distracted by its wonders, that they forget where everything is rooted from, which is nature. By becoming aware of how our world is synchronized and the harmonic scale levels in which we reside, we can begin to use the divine proportion and sacred geometry to define how we internally perceive the world. When we apply this information to ourselves, to our bodies, we can begin to think about our place in the equation of the divine proportion, and see that we are both the larger and smaller parts in relation to the universe. This awareness to the natural and “beautiful” ideas that the world embodies enlightened me to the idea that everything can be reduced to patterns, and has an underlying structure. I think this is sort of the birth in my obsession with both recording, and creating patterns. By re-establishing a relationship with Mother Nature, we as a species will create a deeper understanding of our inner and outer selves, and what are place is in the universe (Dalai Lama, 2005)(Wilzeck, 2015).
More recently, I have been looking into the studies of Rudolf Laban by reading his book “Choreutics.” Rudolf Laban is a dance theorist who draws a lot of his information from the Pythagorean community and theories of Plato. Laban is known for the theory of space harmony (also known as Choreutics). This theory is based upon universal patterns of nature, and man as a part of a universal design. Being a dancer and choreographer, Laban recognized patterns in the movements in human beings. He recognized a spacial pattern in human movements and saw the shapes of the platonic solids within these patterns. He was then able to reduce these movement patterns to choreutic scales through looking at inclinations, pathways, levels, and directions of movement. This determined the pattern of movement through the shapes of polyhedrons and the platonic solids. His theory is called space harmony for its relationship to harmonics and vibrations that can be seen in other theories such as the Harmony of the Spheres suggested by pythagoreous, and then later Johannes Kepler. (Laban, 1966)
In a more contemporary practice, the platonic solids, and phi are applied to a lot of common eastern knowledge. In Yogic philosophy there is a relationship between the platonic solids and the chakra energy systems. Each chakra corresponds with a platonic solid, as well as an element of each solid, as determined by Plato. In the book “The Universe in a Single Atom- the Convergence of Science and Spirituality”, the 14th Dalai Lama speaks a lot about the commonalities between Buddism and science. Buddhists commonly deconstruct thought by a combination of introspective thinking and . sensory perception viewed through a body. Modern Western science is primarily based just on thought rather than an experience. The Dalai Lama compares different scientific advancements in genetics, consciousness, and cosmology and quantum physics from the western society, to the more peaceful eastern practice. He believes that it is with a balance of both of these knowledges that humans can live in harmony with the rest of the world (Dalai Lama, 2005).
What I have learned is that on a physical level, everything can be reduced to a particle…. Or a wave. Sound, light, movement… EVERYTHING can be reduced to a simple frequency. Everything infinitely large, or small, is connected and is always constantly vibrating or moving. This to me means that movement is a fundamental aspect of perception, and that space itself, is a combination of synchronal movements. Each movement has a structure, yet all structures are composed of movements.
Overall I am convinced that our perception is made up of both what we do, and what we do not see, but just because we cannot see it, does not mean that it doesn’t exist. I specifically used glass to make this piece because I think that these geometric patterns, movements, and proportions are commonly taken for granite, or even overlooked. By utilizing the transparency of clear glass, I showed the shape of space (something intangible and unseen) and made it tangible.
Leonardo Da Vinci is one of my artistic influences because he is one of the first people to actually draw one of the platonic solids. His artwork was heavily impacted by the mathematical discoveries that were taking place during his lifetime. Da Vinci was always fascinated with all things science and his art work and philosophy reflect this interest. He filled sketch books with scientific nature studies, as well as architectural, and human form studies that in some way, all related back to a geometric foundation. Da Vinci, believed that art was connected with science and nature and his notebooks were filled of thoughts and drawings on how things are connected. The Italian Renaissance period (the time in which Da Vinci was an active artist), was a time in which artists were viewed as high members of society, and their practices were thought of to be similar to a practice of a scientist. Artists like Leonardo and Michelangelo followed the philosophies of Plato and Aristotle, and even depicted them in some of their later paintings. I find that my work and Leonardo’s work focus similarly on the need to understand something on a deeper level, and allowing math, science and nature, to lead the way to this comprehension.
All in all, I think the ideas I am researching are ideas that I can continue to explore for years into my future. I know that I am both an analytical as well as spiritual person, and I find that the more that I continue to explore these mathematical sequences in my research, yoga/dance, and artistic practice, the more reoccurring, and interconnected these ideas appear. I am very interested in continuing to explore the word space, and what it means be a living being that occupies and alters space. Whether I continue in glass, printmaking, performance art, or writing, I know that these topics will always be on the surface of my thought processes. My intention in all of my efforts is to create a moment of interconnectedness where the viewer recognizes these mathematical patterns as a framework that provides the structure for how we experience everyday life. I would like to create a moment of transcendence where the viewer steps outside of themself and sees themself as a part of something larger (or smaller), and how everything is inevitably connected through a system of complex energy networks.
Sources:
Books:
Articles:
-EK
Euclid (325-265 B.C.E.) of Alexandria (Egypt) wrote the book Elements which is one of our first recorded and most basic teaching of geometry (it is still used today). This book marks a movement in history in the way that we think as a species, because it proposed a new way of thinking based on empirical thought; where ideas could be reality based on just an individuals thought. This is also the first record of the knowledge of the divine proportion. The Divine Proportion (Phi) is a ratio in which the whole is to the larger in exactly the same proportion as the larger is to the smaller. Or, a pattern of numbers that increases by adding the two previous numbers. This ratio can be reduced to 1.618 (but is continuous) and has multiple names such as Phi, Golden Mean, Golden Ratio, and Divine Proportion. It is a mathematical representation of the structure and framework in which life is manifested. Phi is present in both micro and macrocosms of all life forms on this planet, and beyond. (Hemenway, 2005).
Ancient civilizations relied on geometry and used it as there tool and blueprint for understanding the physical world and how it functions. People recognized geometric patterns, proportions, and ratios that related from part to whole, and saw that these were not only apparent in everyday natural life, but that they could also apply this knowledge in order to create.
Phidias (490-430 B.C.E.) was a Greek sculptor and mathematician who is known to have applied the Divine Proportion to the rectangles in the architecture of the Parthenon in Greece. The building was constructed in parallel lines in which the height and width of the columns are spaced in the ration of phi. Phidias worked on the Parthenon with Pythagoras, a Greek philosopher and mathematician who is a founder of Pythagoreanism. Pythatgoreanism is a school of teachings that are based upon “ideas that the metaphysics of number and conception that reality (including musical and astronomy) is, at its deepest level mathematical in nature (Wilzeck, 2015).” He is also recognized for the theory of Harmonia Kosmos translates as “fitting together, the beautiful order of things” (Hemenway 2005).
Plato (427-347 B.C.E) was next to recognize the divine proportion. He described five regular solids known as the platonic solids. These solids apply the divine proportion within their formations, and are said to be the basis for the structure of all matter in our universe. Platonic solids are shapes where each face is same regular polygon and the same number of polygons meets at each vertex. The platonic solids have been identified with the four basic elements-the hexahedron with earth, the tetrahedron with fire, the octahedron with air, and the icosahedron with water. In ancient Greek philosophy, the people believed Platonic solids to have a spherical property, where one Platonic Solid fits in a sphere, which alternately fits inside another Platonic Solid, again fitting in another sphere. This organization of spherical property is similar to our modern atom which shows a nucleus surrounded by electrons in orbit, which create a sphere of energy. This goes on infinitely large and small, as everything connects. Phi occurs in each of the platonic solids, within the geometry of the surface area of the faces of each three dimensional solid. Finally, in Plato's view, the regular polyhedral shapes are the building-blocks not only of the inorganic world, but of the entire physical universe, including organic and inorganic matter. (Hemenway 2005).
The Fibonnacci Sequence is a series of numbers that proceeds by adding the previous two numbers together, and it is a sequence of numbers that goes on infinitely. This mathematical sequence correlates with spirals that are found in art, science, and nature. Leonardo Fibonacci is the founder of this sequence and it first showed up in his book Liber Abaci. The Fibonacci sequence has a direct correlation and has a mathematical relationship to phi. When you take two numbers in the sequence and divide the first number by the second number, the decimal is always 1.60 which is approximately equal to Phi. This sequence is commonly applied to analyze growth patterns of plant and human life. Leonardo Da Vinci even used this sequence to construct the Vitruvian Man. (Hemenway 2005).
Next we have Johannes Kepler, who discusses the harmony of the world and the laws of planetary motion, in which he explains how planets move in ellipses and not circles. His view of the universe is linked to the platonic solids, as he believed the planets were aligned in the shapes of the platonic solids. He even created a platonic solid model of the solar system, which he believed was the blueprint of the universe. He stated that the five platonic solids could be applied to six planets if the solids bounding surfaces marked the spaces between the planetary spheres (Hemenway 2005)(Wilzeck 2015).
In the history of the human species, there was a deeper connection between human life and the earth. Cycles and patterns of life were commonly recorded, in order to gain a deeper understanding of the world and of our place as humans within it all. As humans evolved as a species, I think we have lost some of this connection, and created an imbalance in our practices. In today’s society I find that people are heavily relying on technology and so distracted by its wonders, that they forget where everything is rooted from, which is nature. By becoming aware of how our world is synchronized and the harmonic scale levels in which we reside, we can begin to use the divine proportion and sacred geometry to define how we internally perceive the world. When we apply this information to ourselves, to our bodies, we can begin to think about our place in the equation of the divine proportion, and see that we are both the larger and smaller parts in relation to the universe. This awareness to the natural and “beautiful” ideas that the world embodies enlightened me to the idea that everything can be reduced to patterns, and has an underlying structure. I think this is sort of the birth in my obsession with both recording, and creating patterns. By re-establishing a relationship with Mother Nature, we as a species will create a deeper understanding of our inner and outer selves, and what are place is in the universe (Dalai Lama, 2005)(Wilzeck, 2015).
More recently, I have been looking into the studies of Rudolf Laban by reading his book “Choreutics.” Rudolf Laban is a dance theorist who draws a lot of his information from the Pythagorean community and theories of Plato. Laban is known for the theory of space harmony (also known as Choreutics). This theory is based upon universal patterns of nature, and man as a part of a universal design. Being a dancer and choreographer, Laban recognized patterns in the movements in human beings. He recognized a spacial pattern in human movements and saw the shapes of the platonic solids within these patterns. He was then able to reduce these movement patterns to choreutic scales through looking at inclinations, pathways, levels, and directions of movement. This determined the pattern of movement through the shapes of polyhedrons and the platonic solids. His theory is called space harmony for its relationship to harmonics and vibrations that can be seen in other theories such as the Harmony of the Spheres suggested by pythagoreous, and then later Johannes Kepler. (Laban, 1966)
In a more contemporary practice, the platonic solids, and phi are applied to a lot of common eastern knowledge. In Yogic philosophy there is a relationship between the platonic solids and the chakra energy systems. Each chakra corresponds with a platonic solid, as well as an element of each solid, as determined by Plato. In the book “The Universe in a Single Atom- the Convergence of Science and Spirituality”, the 14th Dalai Lama speaks a lot about the commonalities between Buddism and science. Buddhists commonly deconstruct thought by a combination of introspective thinking and . sensory perception viewed through a body. Modern Western science is primarily based just on thought rather than an experience. The Dalai Lama compares different scientific advancements in genetics, consciousness, and cosmology and quantum physics from the western society, to the more peaceful eastern practice. He believes that it is with a balance of both of these knowledges that humans can live in harmony with the rest of the world (Dalai Lama, 2005).
What I have learned is that on a physical level, everything can be reduced to a particle…. Or a wave. Sound, light, movement… EVERYTHING can be reduced to a simple frequency. Everything infinitely large, or small, is connected and is always constantly vibrating or moving. This to me means that movement is a fundamental aspect of perception, and that space itself, is a combination of synchronal movements. Each movement has a structure, yet all structures are composed of movements.
Overall I am convinced that our perception is made up of both what we do, and what we do not see, but just because we cannot see it, does not mean that it doesn’t exist. I specifically used glass to make this piece because I think that these geometric patterns, movements, and proportions are commonly taken for granite, or even overlooked. By utilizing the transparency of clear glass, I showed the shape of space (something intangible and unseen) and made it tangible.
Leonardo Da Vinci is one of my artistic influences because he is one of the first people to actually draw one of the platonic solids. His artwork was heavily impacted by the mathematical discoveries that were taking place during his lifetime. Da Vinci was always fascinated with all things science and his art work and philosophy reflect this interest. He filled sketch books with scientific nature studies, as well as architectural, and human form studies that in some way, all related back to a geometric foundation. Da Vinci, believed that art was connected with science and nature and his notebooks were filled of thoughts and drawings on how things are connected. The Italian Renaissance period (the time in which Da Vinci was an active artist), was a time in which artists were viewed as high members of society, and their practices were thought of to be similar to a practice of a scientist. Artists like Leonardo and Michelangelo followed the philosophies of Plato and Aristotle, and even depicted them in some of their later paintings. I find that my work and Leonardo’s work focus similarly on the need to understand something on a deeper level, and allowing math, science and nature, to lead the way to this comprehension.
All in all, I think the ideas I am researching are ideas that I can continue to explore for years into my future. I know that I am both an analytical as well as spiritual person, and I find that the more that I continue to explore these mathematical sequences in my research, yoga/dance, and artistic practice, the more reoccurring, and interconnected these ideas appear. I am very interested in continuing to explore the word space, and what it means be a living being that occupies and alters space. Whether I continue in glass, printmaking, performance art, or writing, I know that these topics will always be on the surface of my thought processes. My intention in all of my efforts is to create a moment of interconnectedness where the viewer recognizes these mathematical patterns as a framework that provides the structure for how we experience everyday life. I would like to create a moment of transcendence where the viewer steps outside of themself and sees themself as a part of something larger (or smaller), and how everything is inevitably connected through a system of complex energy networks.
Sources:
Books:
- Hemenway, Priya. Divine Proportion: Phi in Art, Nature, and Science. New York: Published by Sterling Pub., 2005. Print.
- Lama, Dalai. The Universe in a Single Atom How Science and Spirituality Can Serve Our World. London: Abacus, 2005. Print
- Laban, Rudolf Von. Choruetics. N.p.: Lisa Ullmann, n.d. Print.
- Wilczek, Frank. A BeauHage, Stephen J. Let There Be Light: Physics, Philosophy & The Dimensional
- Hage, Stephen J. Let There Be Light: Physics, Philosophy & The Dimensional Structure of Consciousness. N.p.: n.p., 2013. Print
- Judith, Anodea. The Chakra System. N.p.: n.p., 2015. Print.
Articles:
- Ahmed, Sara. "Orientation Matters." New Materialism- (2010): n. pag. Print
- Mauss, Marcel. "Techniques of the Body." Techniques of the Body- Economy and Society (1973): 70-88. Print.
-EK