By John F. Sase
It seems that earlier Greek monocentric cosmologies, such as those of Anaximander and Parmenides, inspired Plato’s monocentric urban models. Furley presents the Greek cosmologies of Anaximander and Parmenides that provide models for understanding modern social and economic relations within ancient Greek culture. However, Plato applies these same models in conjunction with the mathematics communicated by Pythagoras to create his mathematical allegories of the perfect city-state.
Essentially, Plato’s models provide mathematical allegories. Professor Brumbaugh states that much of the work which the early Greek philosophers and scientists thought of as mathematics is not “mathematics” in its twentieth-century form at all. However, their fundamental concepts of algebra, proportions, and geometry form the basis of contemporary mathematics.
It appears that Pythagoras introduced the analytical tools used by Plato to ancient Greece. The germane aspects of his teachings can be traced back to the Sumero-Babylonian age of the previous millennium. The ratio theories of the Academy of Plato translate into modern exponential functions and often represent special cases of general formulae. This body of ancient mathematics suffices for the construction of the allegorical city-states of Plato.
Plato constructs four allegorical city-states: Ancient Athens, an ideal one; Modern Athens (Calliopolis), also ideal but inhabited by only an essential population; Atlantis, a luxurious one of destructive excesses; and Magnesia, the practicable city-state.
Professor McClain states that Plato constructs his all-estate models found in later writings. In his book of The Republic, McClain presents allegorical city-states from abstract material developed in earlier mathematical allegories. For example, The Myth of Er contains the cosmological model of Plato embodied in the Spindle of Necessity. The allegory of the Spindle expounds on mathematical concepts, essential for understanding Plato’s four city plan.
In this, he describes the Spindle of Necessity as having a center column which appears as a straight shaft of light that stretches from above throughout heaven and earth. The Spindle of Necessity represents the dynamics of the universe.
In our current century, efforts have been made to search for common-sense implications in the mathematical allegories of Plato. The work of various leading Platonists, including
Adam, Taylor, and Cornford, led Brumbaugh to advance the theory that the “mathematical” passages in the works of Plato (which scholars had previously considered as nonsense or riddles in earlier centuries) actually describe the diagrams designed by the philosopher, as accompaniment to clarify his text.
Furthermore, Plato had described a set of eight bowl-shaped whorls (convex hulls). These whorls spin about the vertical shaft of light that extends infinitely upward and downward through the bottom center of all the bowls.
The mathematical allegory of the Spindle of Necessity resembles the cosmological image which Professor Eliade (introduced in part two) associates with the village construction rituals of primitive and traditional societies.
This Spindle of Necessity embodies a curvilinear mathematical function, fundamental to Plato’s city-state allegories, that approximates the negative-exponential function used by Mills and other modern age builders. The bridge between the Spindle and the negative exponential functions is the diatonic musical scale such that musical pitches, expressed as cycles per second, graph as a convex curvilinear function.
Professor McClain details the correlations of specific cases of general formulae. This body of ancient mathematics suffices for the construction of the allegorical city-states described by Plato.
Plato constructs four allegorical city-states: Ancient Athens, an ideal one; Modern Athens (Calliopolis), also ideal but inhabited by only an essential population; Atlantis, a luxurious one of destructive excesses; and Magnesia, the practicable city-state.
Professor McClain states that Plato constructs his allegorical state models found in later writings. In Plato’s Republic allegorical city-states developed from abstract material in his earlier mathematical allegories.
The Myth of Er contains Plato’s cosmological model embodied in the Spindle of Necessity.
The Allegory of the Spindle expounds on mathematical concepts, essential for understanding the four-city concept of Plato.
Plato describes the Spindle of Necessity as having a center column which appears as a straight shaft of light that stretches from above throughout heaven and earth. The Spindle of Necessity represents the dynamics of the universe.
During the current century, efforts have been made to search for common-sense implications in Plato’s mathematical allegories. The work of various leading Platonists, including
Adam, Taylor, and Cornford, led Professor Brumbaugh to advance the theory that the “mathematical” passages in Plato’s work, which scholars had considered nonsense or riddles in earlier centuries, actually describe the numerous diagrams, designed by Plato, that were intended to accompany and clarify his text.
Furthermore, Plato describes a set of eight bowl-shaped whorls (convex hulls). These whorls spin about the vertical shaft of light that extends infinitely upward and downward through the bottom center of all the bowls.
The mathematical allegory of the Spindle of Necessity resembles the cosmological image which Professor Eliade associates with the village construction rituals of primitive and traditional societies.
This Spindle of Necessity embodies a curvilinear, mathematical function, fundamental to Plato’s city-state allegories, that approximates the negative-exponential function used by Professor Mills and other modern age model builders.
The bridge between the Spindle and the negative exponential functions is the diatonic musical scale. As a result, musical pitches expressed as cycles per second graph as a convex curvilinear function. Professor McClain details the correlations between the musical scale and the nested whorls in the Spindle of Necessity in respect to the shape of the whorls, the width of the rims, the patterns of the colors, the speeds of the whorls, and other relevant aspects.
This body of harmonic mathematics forms the essential tools used by Plato to construct his allegories. The concept of “monocentricity” remains germane to the urban allegories of Plato urban allegories. He establishes this concept in other through the ideas of circularity and axis.
In his writing of Timaeus, Plato reveals to us that the circle constitutes his own primary image. From this observation, it follows that the cities pf Plato exist as circular. In his writing of the Republic, Plato commences his development of an ideal model state with a one-dimensional line which evolves into a two-dimensional circle.
Professor James Adam interpreted these passages as the state growing like a circle drawn with a compass. As a result, the state as a circle forms the fundamental ground plan for all of the city-state models of Plato.
As a circle drawn with a compass, pivoting at the center, all of these models exist as monocentric. In effect, Plato creates four variations of a monocentric city-state.
Each has a seat of power at its center:
Both ancient and new Athens have a temple of Zeus;
Atlantis has the palace of Poseidon;
and Magnesia has its capital city, the seat of its government.
Essentially, throughout ancient times, land transportation remained limited to foot and horse travel. As a result of existing transportation modes that appear expensive in both physical and time costs, the preference for residence and occupation tended to be close to the city center. Due to these high physical and time costs, employment and housing sectors tended to be integrated in the sense that citizens lived where they worked.
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John F. Sase earned an MBA. and a MA in Economics at the University of Detroit, and, a Ph.D. in Economics from Wayne State University. His primary research and teaching interests focus on Urban and Regional Economics and Industrial Organization. He won the Levin Award in Economics in 1989. Also, he has served as the section chairperson in Economics for the Michigan Academy of Science, Arts, and Letters.
It seems that earlier Greek monocentric cosmologies, such as those of Anaximander and Parmenides, inspired Plato’s monocentric urban models. Furley presents the Greek cosmologies of Anaximander and Parmenides that provide models for understanding modern social and economic relations within ancient Greek culture. However, Plato applies these same models in conjunction with the mathematics communicated by Pythagoras to create his mathematical allegories of the perfect city-state.
Essentially, Plato’s models provide mathematical allegories. Professor Brumbaugh states that much of the work which the early Greek philosophers and scientists thought of as mathematics is not “mathematics” in its twentieth-century form at all. However, their fundamental concepts of algebra, proportions, and geometry form the basis of contemporary mathematics.
It appears that Pythagoras introduced the analytical tools used by Plato to ancient Greece. The germane aspects of his teachings can be traced back to the Sumero-Babylonian age of the previous millennium. The ratio theories of the Academy of Plato translate into modern exponential functions and often represent special cases of general formulae. This body of ancient mathematics suffices for the construction of the allegorical city-states of Plato.
Plato constructs four allegorical city-states: Ancient Athens, an ideal one; Modern Athens (Calliopolis), also ideal but inhabited by only an essential population; Atlantis, a luxurious one of destructive excesses; and Magnesia, the practicable city-state.
Professor McClain states that Plato constructs his all-estate models found in later writings. In his book of The Republic, McClain presents allegorical city-states from abstract material developed in earlier mathematical allegories. For example, The Myth of Er contains the cosmological model of Plato embodied in the Spindle of Necessity. The allegory of the Spindle expounds on mathematical concepts, essential for understanding Plato’s four city plan.
In this, he describes the Spindle of Necessity as having a center column which appears as a straight shaft of light that stretches from above throughout heaven and earth. The Spindle of Necessity represents the dynamics of the universe.
In our current century, efforts have been made to search for common-sense implications in the mathematical allegories of Plato. The work of various leading Platonists, including
Adam, Taylor, and Cornford, led Brumbaugh to advance the theory that the “mathematical” passages in the works of Plato (which scholars had previously considered as nonsense or riddles in earlier centuries) actually describe the diagrams designed by the philosopher, as accompaniment to clarify his text.
Furthermore, Plato had described a set of eight bowl-shaped whorls (convex hulls). These whorls spin about the vertical shaft of light that extends infinitely upward and downward through the bottom center of all the bowls.
The mathematical allegory of the Spindle of Necessity resembles the cosmological image which Professor Eliade (introduced in part two) associates with the village construction rituals of primitive and traditional societies.
This Spindle of Necessity embodies a curvilinear mathematical function, fundamental to Plato’s city-state allegories, that approximates the negative-exponential function used by Mills and other modern age builders. The bridge between the Spindle and the negative exponential functions is the diatonic musical scale such that musical pitches, expressed as cycles per second, graph as a convex curvilinear function.
Professor McClain details the correlations of specific cases of general formulae. This body of ancient mathematics suffices for the construction of the allegorical city-states described by Plato.
Plato constructs four allegorical city-states: Ancient Athens, an ideal one; Modern Athens (Calliopolis), also ideal but inhabited by only an essential population; Atlantis, a luxurious one of destructive excesses; and Magnesia, the practicable city-state.
Professor McClain states that Plato constructs his allegorical state models found in later writings. In Plato’s Republic allegorical city-states developed from abstract material in his earlier mathematical allegories.
The Myth of Er contains Plato’s cosmological model embodied in the Spindle of Necessity.
The Allegory of the Spindle expounds on mathematical concepts, essential for understanding the four-city concept of Plato.
Plato describes the Spindle of Necessity as having a center column which appears as a straight shaft of light that stretches from above throughout heaven and earth. The Spindle of Necessity represents the dynamics of the universe.
During the current century, efforts have been made to search for common-sense implications in Plato’s mathematical allegories. The work of various leading Platonists, including
Adam, Taylor, and Cornford, led Professor Brumbaugh to advance the theory that the “mathematical” passages in Plato’s work, which scholars had considered nonsense or riddles in earlier centuries, actually describe the numerous diagrams, designed by Plato, that were intended to accompany and clarify his text.
Furthermore, Plato describes a set of eight bowl-shaped whorls (convex hulls). These whorls spin about the vertical shaft of light that extends infinitely upward and downward through the bottom center of all the bowls.
The mathematical allegory of the Spindle of Necessity resembles the cosmological image which Professor Eliade associates with the village construction rituals of primitive and traditional societies.
This Spindle of Necessity embodies a curvilinear, mathematical function, fundamental to Plato’s city-state allegories, that approximates the negative-exponential function used by Professor Mills and other modern age model builders.
The bridge between the Spindle and the negative exponential functions is the diatonic musical scale. As a result, musical pitches expressed as cycles per second graph as a convex curvilinear function. Professor McClain details the correlations between the musical scale and the nested whorls in the Spindle of Necessity in respect to the shape of the whorls, the width of the rims, the patterns of the colors, the speeds of the whorls, and other relevant aspects.
This body of harmonic mathematics forms the essential tools used by Plato to construct his allegories. The concept of “monocentricity” remains germane to the urban allegories of Plato urban allegories. He establishes this concept in other through the ideas of circularity and axis.
In his writing of Timaeus, Plato reveals to us that the circle constitutes his own primary image. From this observation, it follows that the cities pf Plato exist as circular. In his writing of the Republic, Plato commences his development of an ideal model state with a one-dimensional line which evolves into a two-dimensional circle.
Professor James Adam interpreted these passages as the state growing like a circle drawn with a compass. As a result, the state as a circle forms the fundamental ground plan for all of the city-state models of Plato.
As a circle drawn with a compass, pivoting at the center, all of these models exist as monocentric. In effect, Plato creates four variations of a monocentric city-state.
Each has a seat of power at its center:
Both ancient and new Athens have a temple of Zeus;
Atlantis has the palace of Poseidon;
and Magnesia has its capital city, the seat of its government.
Essentially, throughout ancient times, land transportation remained limited to foot and horse travel. As a result of existing transportation modes that appear expensive in both physical and time costs, the preference for residence and occupation tended to be close to the city center. Due to these high physical and time costs, employment and housing sectors tended to be integrated in the sense that citizens lived where they worked.
————————
John F. Sase earned an MBA. and a MA in Economics at the University of Detroit, and, a Ph.D. in Economics from Wayne State University. His primary research and teaching interests focus on Urban and Regional Economics and Industrial Organization. He won the Levin Award in Economics in 1989. Also, he has served as the section chairperson in Economics for the Michigan Academy of Science, Arts, and Letters.
