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Topic: Mathematical beauty

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	Beauty - Wikipedia, the free encyclopedia
		Beauty involves the cognition of objects as having a balance and harmony with nature, which elicits in the viewer a sense and experience of attraction, affection, and pleasure.
		Understanding the nature and meaning of beauty is one of the key themes in the philosophical discipline known as aesthetics.
		The phrase "beauty is in the eye of the beholder," however, suggests that beauty is wholly subjective.
	en.wikipedia.org /wiki/Beauty (1599 words)

	Mathematical beauty - Wikipedia, the free encyclopedia
		Mathematicians see beauty in mathematical results which establish connections between two areas of mathematics that at first sight appear to be totally unrelated.
		Galileo Galilei is reported to have said "Mathematics is the language with which God wrote the universe", a statement which (apart from the implicit deism) is consistent with the mathematical basis of all modern physics.
		This viewpoint expresses the idea that mathematics, as the intrinsically true foundation on which the laws of our universe are built, is a natural candidate for what has been personified as God by different religious mystics.
	en.wikipedia.org /wiki/Mathematical_beauty (986 words)

	[No title]
		Beauty is the first test: there is no permanent place in the world for ugly mathematics.
		It may be very hard to define mathematical beauty, but that is just as true of beauty of any kind - we may not know quite what we mean by a beautiful poem, but that does not prevent us from recognizing one when we read it.
		Pure mathematics, on the other hand, seems to me a rock on all which all idealism founders: 317 is a prime, not because we think so, or because our minds are shaped in one way rather than another, but because it is so, because mathematical reality is built that way.
	sujith_v.tripod.com /quotes/apology.txt (2231 words)

	Learn more about Beauty in the online encyclopedia. (Site not responding. Last check: 2007-11-07)
		Beauty is visual pleasantness of a person, animal, object or scene, and also pleasantness of sound, especially music.
		The earliest theory of beauty can be found in the works of early Greek philosophers from the presocratic period, like Pythagoras.
		Another connection between mathematics and beauty which played a prominent role in Pythagoras's philosophy was the way in which musical tones can be arranged in mathematical sequences, which repeat at regular intervals called octaves.
	www.onlineencyclopedia.org /b/be/beauty.html (684 words)

	Beauty in Mathematics
		While it is true that mathematics is built on an axiomatic foundation, it seems to me that a strong case could be made for the ultimate foundation of mathematics being its beauty.
		In fact, mathematicians are so enamored of the beauty of mathematical systems and structures that if inconsistencies were found in the axiomatic foundations of mathematics, most mathematicians would probably prefer to change the axiomatic foundations than to give up the beauty of the body of mathematics.
		If we view beauty as central to mathematics we would be introducing ideas that, while they may have application or may build some tools, would certainly let students see a much bigger picture of mathematics at a much earlier stage in their mathematical development.
	www.westmont.edu /~howell/cccu/Projects/Stueckle.html (846 words)

	Science, Mathematics, and Beauty (Site not responding. Last check: 2007-11-07)
		The mathematician is fascinated with the marvelous beauty of the forms he constructs, and in their beauty he finds everlasting truth.
		Mathematics has beauties of its own -- a symmetry and proportion in its results, a lack of superfluity, an exact adaptation of means to ends, which is exceedingly remarkable and to be found only in the works of the greatest beauty.
		How thoroughly it is ingrained in mathematical science that every real advance goes hand in hand with the invention of sharper tools and simpler methods which, at the same time, assist in understanding earlier theories and in casting aside some more complicated developments.
	www.chemistrycoach.com /science_mathematics_and_beauty.htm (1990 words)

	Mathematics Curriculum Framework: Achieving Mathematical Power - January 1996 - The Beauty and Power of Mathematics - ... (Site not responding. Last check: 2007-11-07)
		Mathematics is a universal language of numbers that is spoken in all cultures.
		Mathematics is beautiful patterns in nature and art, pleasing proportions in architecture.
		Mathematics as a field of study is characterized as much by particular types of reasoning as by particular types of content.
	www.doe.mass.edu /frameworks/math/1996/beauty.html (1021 words)

	DISF - Dizionario Interdisciplinare di Scienza e Fede \| Beauty
		Beauty is defined as that which pleases the senses, the useful or the one matching the goal, the good, the true, the finding of an idea or its appearing, the divine and its epiphany, but also the harmonious, the proportionate, the one in the multiplicity.
		If mathematics, apparently, does not speak of beauty, he says, in reality it takes into consideration those same supreme conditions that constitute the forms of beauty: «[...] those who assert that the mathematical sciences say nothing of the beautiful or the good are in error.
		Beauty in mathematics and geometry may be recognized in some regular forms occurring in nature.
	www.disf.org /en/Voci/34.asp (8412 words)

	ipedia.com: Mathematical beauty Article (Site not responding. Last check: 2007-11-07)
		Given the utility of mathematics in science and engineering, it is likely that any technological society willl actively cultivate these aesthetics, certainly in its philosophy of science if nowhere else.
		At one stage in his life, Johannes Kepler believed that the proportions of the orbits of the then-known planets in the Solar System had been arranged by God to correspond to a concentric arrangement of the five Platonic solids, each orbit lying on the circumsphere of one polyhedron and the insphere of another.
		An extreme tendency to focus on the elegance, beauty or simplicity of a theory rather than its empirical use in applications is sometimes described as mathematical fetishism or scientism.
	www.ipedia.com /mathematical_beauty.html (775 words)

	The Most Seductive Equation in Science: Beauty Equals Truth (Site not responding. Last check: 2007-11-07)
		The experience went a long way toward convincing Einstein that mathematics could be a telegraph line to God, and he spent most of the rest of his life in an increasingly abstract and ultimately fruitless pursuit of a unified theory of physics.
		The book is partly a meditation on mathematical beauty, possibly a difficult concept for many Americans right now as they confront their tax forms.
		Just as faces and snowflakes are prettier for their symmetrical patterns, so physical laws are considered more beautiful if they keep the same form when we change things by, for example, moving to the other side of the universe, making the clocks run backward, or spinning the lab around on a carousel.
	www.nytimes.com /glogin?URI=http://www.nytimes.com/2002/03/26/science/26MATH.html&OQ=_rQ3D3Q26orefQ3DsloginQ26orefQ3Dslogin&OP=7659b62aQ2F_E(J_!gQ7D8Q51ggzp_pDDp_DQ3C_pe_8Q7Do(iQ7D(_peKVNQ3A-Q22zIs (1032 words)

	Notebook (Site not responding. Last check: 2007-11-07)
		Of course, mathematics too has its beauty but we have to traverse a long road of reflection before we perceive it.
		This, of course, does not mean that the beauty of a work of art cannot remain hidden as long as desired under certain conditions--longer, perhaps, than the sense of mathematics to a mathematician.
		It is possible to help someone become permeable to this beauty, but we certainly cannot explain it to him, for there is nothing to explain.
	www.noteaccess.com /RELATIONSHIPS/GNumber.htm (456 words)

	Faculty Forum Weblog: Does Mathematical Beauty Pose Problems for Naturalism?
		More precisely, it asks whether aspects of mathematical theorizing, based mostly on notions of beauty and symmetry, and the subsequent success of mathematics in the natural sciences, cause difficulties for a naturalistic worldview.
		Regarding Wigner’s first point, he concedes that much of mathematics, such as Euclidean Geometry, has been developed because its axioms were modeled on what appeared to be true of the world.
		The theories in mathematics that Hardy deems “important” are precisely the ones that satisfy these standards.
	www.cslewis.org /ffblog/archives/2005/09/does_mathematic.html (3356 words)

	Applications of Mathematics (Site not responding. Last check: 2007-11-07)
		Mathematics, as much as music or any other art, is one of the means by which we rise to a complete self-consciousness.
		Mathematics was born and nurtured in a cultural environment.
		To isolate mathematics from the practical demands of the sciences is to invite the sterility of a cow shut away from the bulls.
	www.chemistrycoach.com /applications_of_mathematics.htm (4095 words)

	MATH 125: Mathematical Perspectives
		The general objective of Mathematical Perspectives is to provide students with experiences characteristic of the mathematical enterprise, and to do so at a depth that allows successful students to appreciate the aesthetic beauty of mathematical truth as well as its timeless and user-independent nature.
		The overarching objective of Mathematical Perspectives is to provide students with experiences characteristic of the mathematical enterprise, and to do so at a depth that allows successful students to appreciate the aesthetic beauty of mathematical truth and its timeless, user-independent nature.
		This is one of the fundamental themes of mathematics as a whole, and the unifying thread of the investigations in Dimensionality.
	www.cs.xu.edu /math/math125.html (1326 words)

	Mathematics Archives by Chris Abraham - Because the Medium is the Message (Site not responding. Last check: 2007-11-07)
		Those are the discoveries of the philosophers and scientists who spend their lives exploring the mathematical world and revealing its great wonders.
		In the same way that an ordinary photograph is a snapshot of an area of outstanding natural beauty, a mathematical photograph is a snapshot of mathematical beauty.
		But while the notion of mathematical beauty, and indeed ugliness, is well established, mathematics and mathematical physics can inspire (for me at least) an extraordinary mix of other emotions and ideas.
	www.chrisabraham.com /mathematics (432 words)

	Christianity and Mathematics
		Pythagoras saw the beauty in the theory of numbers and he saw this mathematical beauty translated into musical beauty.
		He saw mathematics as providing the most fundamental of all ideas and the deductive reasoning of mathematics was seen as the ideal way of achieving knowledge.
		The point at issue was whether Copernicus had simply put forward a mathematical theory which enabled the calculation of the positions of the heavenly bodies more simply or whether he was proposing a physical reality.
	www-groups.dcs.st-and.ac.uk /~history/HistTopics/Heliocentric.html (4720 words)

	Noah's Class Reviews, 2004 (Site not responding. Last check: 2007-11-07)
		When I saw the cover of this issue in the library which said "The Beauty of Geometry", I was ecstatic as I had not yet seen an article on mathematical beauty in an NCTM periodical.
		The purpose of this publication by the NCTM is to explore in depth the idea of "Connecting Mathematics" in mathematics education at the high school level.
		Mathematics is expressed through a whole separate language, and just as it is impossible for me to understand a sentence in swahili if I don't know what the words are, it is impossible to understand a mathematical statement if you can't read the language and don't know what is being said.
	www.stolaf.edu /people/wallace/Courses/MathEd/Reviews/Reviews2004/Noah.html (6968 words)

	Page Title (Site not responding. Last check: 2007-11-07)
		Others debate this, but it remains true that mathematics is steeped in patterns of number, quantity, language, symbology, and language.
		Mathematicians can be so enamored of the beauty of mathematical systems and structures that if inconsistencies were found in the axiomatic foundations of mathematics, these mathematicians would probably prefer to change the axiomatic foundations than to give up the beauty of the body of mathematics.
		I believe that with exposure to more beautiful areas of mathematics students will have much less hatred for mathematics and a greater appreciation for the world around them.
	www.maththatcounts.com /page62.html (540 words)

	The Eyes of the Beholder
		Philosophy is in the form and pattern of the beauty of mathematics.
		everything in the universe followed these mathematical laws and the universe was created out of the geometrical relationships of the numbers, and thus constituted the true basis of reality.
		In terms of mathematics, it is to marvel at the form and the shape of all things, and how they relate in simple harmonies, ratios, or juxtapositions.
	www.halexandria.org /dward782.htm (649 words)

	Divine Proportion
		A small and elegant book with a persuasive argument that beauty is a function, not only of consciousness, but of precise mathematical proportions that is hard-wired into the universe.
		The power to appreciate beauty appears to be a human endowment and this suggests that we should seek its origin and its purpose in human nature-in that nature which distinguishes us from the animal creation.
		And we have interpreted the mystery of the nature and purpose of beauty by recalling the familiar fact that the inborn faculty of aesthetic appreciation constitutes the motive for the creation of objects of beauty.
	www.new-universe.com /pythagoras/divine.htm (1390 words)

	ABOUT PAUL ERDÖS
		While the pursuit of mathematical beauty was Erdös's only goal, his ideas inevitably have found practical applications.
		One of the small ironies of Erdös's life is that, although he never owned or used a computer, mathematics that he invented is the basis for modern computer science; although he never had a secret, his mathematics is used by those who invent secret codes.
		He edits a score of mathematical journals, is the head of mathematical research at Bell labs and, in his spare time has taken up the piano and mastered Chinese.
	bookbuzz.com /MBIO_About_Erdos.htm (3759 words)

	The Numbers of Life
		More important, I realized that mathematical beauty exists not only in mere numbers--it is also an intrinsic feature of the living world.
		The claws of a lion, the horns of a ram, the tusks of an elephant, the beak of a parrot and the shell of a snail all obey the rules of the golden spiral.
		In fact, the segmentation of their bodies matches the law of beauty: A fruit fly has two or three head segments (hardly visible), three thoracic segments (where the legs and wings are attached) and eight abdominal segments (with no legs), all of them Fibonacci numbers.
	scicom.ucsc.edu /SciNotes/9402/Numbers.html (1251 words)

	The Divine Proportion A Study in Mathematical Beauty by H. E. Huntley
		Using simple mathematical formulas, most as basic as Pythagora's theorem and requiring only a very limited knowledge of mathematics, Professor Huntley explores the fascinating relationship between geometry and aesthetics.
		Poetry, patterns like Pascal's triangle, philosophy, psychology, music, and dozens of simple mathematical figures are enlisted to show that the "divine proportion" or "golden ratio" is a feature of geometry and analysis which awakes answering echoes in the human psyche.
		When we judge a work of art aethetically satisfying, according to his formulation, we are making it conform to a pattern whose outline is laid down in simple geometric figures: and it is the analysis of these figures which forms the core of Professor Huntley's book.
	www.seekerbooks.com /prod/0486222543 (457 words)

	New Scientist Premium- Mathematical photography - beyond beauty - Comment and Analysis (Site not responding. Last check: 2007-11-07)
		The elegance of mathematics is rarely celebrated as part of our culture, but equations can be an art form in themselves, says Justin Mullins
		But mathematical beauty is barely recognised beyond the confines of academia, and it is never celebrated.
		This seems curious, since it is clear that artists have long found inspiration in mathematics.
	www.newscientist.com /channel/opinion/mg18925365.800.html (301 words)

	Frank Cleary (Site not responding. Last check: 2007-11-07)
		Mathematics is often regarded as the most objective of all academic subjects.
		His idea is solid, but he applies it only to mathematics he considers beautiful, ignoring those he considers to be not beautiful.
		VIIIa) The implications for the funding of mathematics.
	www.cam.cornell.edu /~sharad/infinity/assignments/3b/fbc-3b.htm (880 words)

	Student shares love of symmetry in Pinchpenny Press book
		Leigh’s book provides examples of beauty found in mathematical patterns and designs – combining simple, hand-drawn symmetrical figures with lessons in symmetry groups, including cyclic, dihedral, frieze and wallpaper.
		Leigh hopes that readers of Mathematical Beauty will be inspired to create their own symmetrical forms and become more aware of the abundance of symmetry in the surrounding world.
		Said Birky, “It’s a beautiful book – the images that she developed as well as her clear explanations of complicated mathematical concepts are wonderful.”
	www.goshen.edu /news/pressarchive/09-13-04-ppp-leigh.html (419 words)

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