Month: February 2017

Exploring the Wonders of Geometry in “Poetry of the Universe” by Robert Osserman: A Journey into the Mathematical Beauty of the Cosmos

• The book “Poetry of the Universe” by Robert Osserman explores the connection between mathematics and the understanding of the universe, specifically in the field of cosmology.
• It discusses the unknown aspects of the universe that are yet to be discovered and the role of mathematics in unraveling its mysteries.
• The book highlights important events in mathematical history and their significance in shaping our understanding of the universe.
• It explores themes and controversies in mathematics and cosmology, such as the debate between discovery and invention in mathematical proofs, the relationship between theory and application, and the emergence of new geometries.
• The book emphasizes the interconnectedness of mathematical understanding across different cultures and time periods, with examples from ancient civilizations like Egypt, Babylon, and Greece.
• It mentions key figures like Pythagoras, Euclid, Eratosthenes, Ptolemy, Mercator, Gauss, Lobachevsky, and Riemann and their contributions to mathematics and cosmology.
• The text also touches on topics like negative curvature, geodesic triangles, spherical geometry, elliptic geometry, and differential geometry.
• The geometry of M. C. Escher’s circle-Limit-Woodcuts Die Geometrie von M.C. Eschers Kreislimit-Holzschnitten Peter Herfort Dr. Zentralblatt fü Didaktik der Mathematik volume 31, pages 144–148 (1999)

“Poetry of the Universe” successfully attempts a written demonstration of how mathematical threads, both thematic and controversial moved human inquisition and questions about the Universe toward the established practice and modern Science of Cosmology. The Tangible & Intangible Realms… of space, shape, and time that define our world infer what lies beyond and within them, still shrouded in the unknown, ready to be discovered. Therein, the Universe is a vast expanse of the little known or the great unknown. In the thesis, there is a retelling of important events in mathematical history that lend meaning to our understanding of the Universe and better yet, how truly unique our place is within it. We ask questions because we want to know… What are the stars made of?  Are there other planets?  What is a planet? We’ve sent men to walk on the moon. We can view multiple galaxies beyond our own through superpowered telescopes like the Hubble (only to see an age-old reflection of the milky way galaxy.) These are just some of the accomplishments that have become of our inquiries. As a whole, science, and math will not find lasting satisfaction with one problem solved or one answer determined but will continue to plug in each to the other, discovering more and more of this great unknown, the Universe. Luminosly radiating down on us from above in the night sky are some glittering stars. Those stars appear to the unaided human eye -quite small- though they were the guiding light of our ancestors before the time of space travel and before the time of far gazing telescopes. Our ancestors would discover that those stars moved in patterns or groups called constellations. They would look to the stars for navigational purposes. They would build megalithic structures in alignment with them. The ancients would even develop spiritual belief systems concerning the “heavens” above. The sun in the day giving life to all plants, sea creatures, animals, and all of mankind, could and should only be marveled at in awe-inspiring question and confirmation to the inquiring mind that there is something beyond what we are in immediate contact with. The book begins with the idea that the Universe is immeasurable. Osserman writes about the enigma in 1992 when the “big bang” was viewed as a flash of light by the Hubble Space telescope. That image that surprised scientists, marked the time that our planet is thought to have formed and our universe to have begun expanding. But what is of tantamount importance is that when this mirror effect happened to flashback in time through a speed-of-light reflection, it cleared all refractory evidence of the universe before that time. So based on where we are in relation to the great expanse of the universe, time is actually a quantum consideration and exponential function of space, and what is reflected toward us are images from long ago! We cannot re-create how the universe looked prior to that time, but perhaps in time, imagination will come around again to reveal more about a primordial universe that we now have no way of observing. For now, shapes, curvature, dimensions both real and imaginary, and comparison of these elements pertaining to the Earth and other celestial bodies of evidence gives us some vital points to verse the universe by. There are several common Themes and Controversies evident in the Poetry of the Universe. As a class, we have come to know the backgrounds and arguments of several Mathematical themes and controversies, so I need not explain what they are as much as I should share how they were evident in this book. Discovered or Invented?When we’re talking about the Universe, particularly when Mathematicians are referring to the Universe, they’re referring to something intangible, something perceptively vast and indicative. We’ve established that – So it makes perfect sense that man should come to explore theoretical possibilities and that mathematicians would move even beyond the math that defined space relatively (such as in Euclidean Geometry) to explore themes of abstraction and concrete mathematical proofs- and they did but not without encountering obstacles. Where proof and discovery later came, it could be surmised that there is an element of invention in the axiomatic method of the proof itself. Moreover, though, the math and science of cosmology exist in theory and had been discovered by many great minds in what could be considered mathematical threads. Mathematical threads are another theme that spans beyond one controversy into others but in the cape of discovery, other such themes coincide. Patterns to shapes that were applied were versed in relativity to the controversial theories and patterns with no direct application. The patterns that were discovered were availed by a number of theories, sets, and progressions that only then led to images such as Eshers, “Angels and Devils” (of “Circle Limit IV” p. 74) and other postulates in support of the theories to which they converge and disperse, tying in together some answers but ever leaving more questions, and since the questions would remain where there come to be only theoretical answers…. it is in great part because the Universe is greatly untouched and limitedly observed so that not all theories can be applied with the proof concretely offered by the application. So discovery remains more so creative. The creative process of Pure Mathematicians had been dually built upon certain applications in more than one instance and so I think neither can be ruled out as taking precedence over the other in absolution. Certainly, the prospectus in this book leans further in the spectrum, however, toward discovered theory and then invention. That is, we didn’t invent this stuff, just discovered it to only then invent a language to explain it all. Theory or Application? The theory had been tossed over toward application where Euclidean proofs offered answers and, likewise, problems. In some incidences, the Universe was not to be adequately defined by Euclidean geometry, so new geometries were developed. Elliptic Geometry came from consideration of negative numbers and the theoretical applications to learning the shape of the earth, in the shape of a pseudosphere which led mathematicians such as David Hilbert and Henri Poincare and Lamberts consideration of replacement to Euclidean parallel postulates, with all the above to coincidence as a culmination of answers to the problems posed by Lobachevsky’s geometry. This imaginary geometry that came in converse to Euclidean geometry came to be known as “non-euclidean geometry.” This Math can be traced to mathematical themes where Minding and Lobachevsky, two different mathematicians who never had the chance to opportunely cross paths with each other, still mirrored each other’s works. Other mathematicians, in addition to the few key mathematicians listed above, did, in fact, put the pieces laid forth by both Minding and Lobachevsky together in the scheme of a bigger puzzle. Enter Emanuel Kant, who precluded that Euclidean geometry and its parallel postulates were universally understood as innate to the human intellect. But the book also points out that when considering the earth as a multi-dimensional space and not just two-dimensional Euclidean geometry, we no longer could work to give the pattern, design, and definition of the space, and so spherical geometry could be used. But the earth was then again not a perfect sphere, and so Spherical geometry moved toward elliptical geometries as marked below. Then came Differential Geometry from Georg Friedrich Bernhard Riemann, later built upon by the Poincaré conjecture that moved topology into broader applications when showing that all points on a sphere could be reduced to another sphere, so all points coincided. In this way Riemann’s differential geometry sort of relates to Newtonian physics that established that earth had an elliptical bulge in concern to the measures by topology. In the book and in our studies concerning the subversive emergence of negative numbers, or in this case, negative curvatures and so on, initial rejection centered around much of theoretical mathematics in relation to the new geometries. The design and beauty not fully understood had been appreciated enough by others- mathematicians, philosophers, scientists such as physicists and cosmologists, alike- enough so that the movement toward answers to questions like, “What is the shape of the Universe” were nevertheless approached with the repeated paradox of understanding and new questions. In exploring the themes and controversies of cosmology alone, there is significant overlapping to the point of near redundancy. Theory versus application as controversial to one another may have impacted Gauss, who we know, worked on his theories without exclusion to the creativity of the establishment but also with great concern for application and strove to perfect his work. This is why he has been considered both an applied mathematician and a Pure mathematician. Carl Frederick Gauss gave us Gauss curvature and Gaussian distribution or the “Bell-shaped Curve,” which allowed substantiated proof in consideration of probability and thus the statistics in part of a theory. In this case, his theories were considerate of his scientific preoccupation with astronomy and physics. He explored geodesy in a way Mercator had theorized previously, where geography and its large-scale expanse met patterns (theorized) and methods (applications) for definition. Subsequently, he spent much time on what would become milestones in mathematics and spawn the discovery of new mathematical and scientific theories to be undertaken by others who succeeded him. Nonetheless, his work was marked by a concern for theory and application that paved the way for alternate geometries, several of which were particularly emphasized in “Poetry of the Universe.” The universal synthesis of Mathematical understanding through time and cross-cultural exploration lends to further discussion of the cultural diffusion that occurred in the educational centers of ancient Alexandria, Egypt. Here a great library served as the crossroads for Roman, Greek, Persian, Asian, and Arabian scholars. The exchange of information could occur though there were language barriers that led to actual and procedural differences in the mathematical applications even where the underlying philosophies were in line with one another. During those times, Descartes, Ptolemy, and others gave poetic literation of the ideas others expressed with numbers. As populations grew, cultural diffusion and the crossing over of ideas occurred more frequently. With the establishment of politics, religion, currency, technology, and more, came the increased probability that ideas would spread. This is exhibited by our history, both European and non-European, in conjunction with science, math, and cosmology. The principles and practices used for mapping the observable world around us indubitably lent to the practices and principles used to infer the unobservable world. Observations from ancient people may have inclined the imagination to fathom the earth as round, even though later on in the middle ages, Europeans would believe the earth was flat. But horizon lines, objects in the sky, the sun and moon, and even objects found innate to the environment gave clues that there was something of natural consequence characteristically circular or spherical. Ancient Egyptians obviously left their mark of evidence that they had a great understanding of celestial alignments in their engineering of great buildings that lined up with the stars. Not to mention that architecture requires an understanding of geometry. Osserman notes that Egyptians used pegs that they put into the ground where they had essentially plotted points on a plane of space and attached and connected ropes from one peg to the next to form accurate lines of measure. Ancient Babylonians evidently had knowledge of mathematics, including algebra, long before other civilizations would have a deep understanding of geometry. The Greek Pythagoras introduced into the world the Pythagorean Theorem, which became a foundational basis for mathematics to evolve and the first example of a mathematical formula that employed a process of giving proof for solutions and proofs to be used in form with particular shapes and angular measurements. Euclid would give the world “Geometry, meaning literally: the measure of the earth.” From the record of his work in “The Elements,” he wrote us the guides to the measurement of space, particularly two and three-dimensional space. Respectively, a circle and a sphere; Conic sections. This offered what was necessary to map the world more precisely. The Greek Philosopher, Aristotle, observed the lunar phases during eclipses and suspected that the Earth was a sphere based on the shadowing of the moon during the event. Eratosthenes, a native Egyptian, would use a “gnomon” (a stick stuck upright into the land that would cast a shadow upon the sun’s movement) to tell time. Notedly, Eratosthenes lived in a geographical location that was quite directly near the Tropic of Cancer, where he noted that there were precise moments in which the gnomon would cast no shadow onto the ground. It actually leads him to conclude the circumference of the earth, at an estimate of 150,000,000 feet. He offered us knowledge of latitudinal and longitudinal systems. Ptolemy, who wrote the “Almagest,” paved the way with his knowledge of the universe by consideration of geography. Next, the geography of Ptolemy would influence Mercator. Mercator would devise a projection of the world map that Europeans, in particular, would follow for centuries to come. The Mercator projection would make its initial mark during the 16th century. He created a map by stretching the two-dimensional placement of what the world was known to encompass into latitude and longitude degrees of equal measure. Ultimately, this led back to the idea that the Earth was spherical. Pseudospheres (sketch) emerged as an example of negative curvature – a sort of reverse quadrilateral form in which functions would meet the characteristics of being geometric forms, all concerning positive integers in Euclidean geometry; A pseudosphere was a form where all points are comparable in precise circumference as the curvature was entirely negative – or constantly negative. (p. 58) Negative integers were related to degrees on a sphere or unit circle. Theoretically, the degrees (0-90, 90-180) would cover the surface points on any size sphere, but 0-180 degrees would define half a circle or hemisphere. The rest of the integers on a unit circle would be 180-270 and 270-360. These hemispherical relationships relate to what has been referred to as a geodesic triangular. This occurs where a positive number line crosses with another axis – coordinate geometry (x, y-axis), and you can place a circle with the midpoint zero, and half of the circle would be plotted along negative integer coordinates. Today, physicists working in the area of quantum mechanics take positive integers and, in combining them, observe continual decimals and other infinite numbers that do not seem to reveal a conceivable pattern, although such a pattern should, by the nature of all things, measurable, be existential. This is where relationships between geodesy and non-euclidean Geometry become interesting. Apparently, Gauss had explored the relationship between these areas and found his results to NOT be conducive to his deductions- While Lobachevsky discovered that one coordinate of a parallel plane had to be negative in the characteristics of a sphere in his hyperbolic geometry (“inverse geometry of hyperbolic”) Riemann took Lobachevsky’s geometry and Euclid’s to another level by considering both. He understood that space is best represented by consideration of curvature because he proposed space as “being” in a state of constant positive curvature. Therefore, he created a new geometry that applied the one-dimensional conceptions of space into a spherical model. In these applications, Reimann broadened the notion that spherical space and its progressively positive nature were, therefore, applicable to the model of the universe. With new Geometry came new Discoveries. The result was that the Universe came to be viewed in terms of negative curvature, non-euclidean and hyper-spherical. The Mathematicians who intuitively considered the qualitative values of their surroundings and beyond, along with the philosophical poets who offered their own interpretations, have left in their wake a legacy of contributions to both the mathematical and scientific fields at large. Osserman, clearly aware and knowing of these foundational and historical contributions, was able to string them together in appropriate significance in his compelling and provocative “Poetry of the Universe.”