## Institute for Mathematics and its Applications

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### Teaching Mathematics and its ..

The intersections between origami, mathematics, and science occur at many levels and include many fields of the latter. We can group these intersections into roughly three categories: Origami mathematics, which includes the mathematics that describes the underlying laws of origami; Computational origami, which comprises algorithms and theory devoted to the solution of origami problems by mathematical means; Origami technology, which is the application of origami (and folding in general) to the solution of problems arising in engineering, industrial design, and technology in general. One genre blends into another. Origami math defines the "ground rules" for computational origami's goal of solving origami design problems (and quantifying their difficulty). The results of computational origami, in turn, can be (and have been) pressed into service to solve technological problems ranging from consumer products to the space program. Origami, like music, also permits both composition and performance as expressions of the art. Over the past 40 years, I have developed nearly 600 original origami compositions. About a quarter of these have been published with folding instructions, which, in origami, serve the same purpose that a musical score does: it provides a guide to the performer (in origami, the folder) while allowing the performer to express his or her own personality website includes galleries of my designs, crease patterns, schedule of my lectures, appearances and exhibitions, commissioned works, and more on the science of origami.

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### Mathematics ___ Art and its twin ..

The decad--10--according to the Pythagoreans, is the greatest of numbers, not only because it is the tetractys (the 10 dots) but because it comprehends all arithmetic and harmonic proportions. Pythagoras said that 10 is the nature of number, because all nations reckon to it and when they arrive at it they return to the monad. The decad was called both heaven and the world, because the former includes the latter. Being a perfect number, the decad was applied by the Pythagoreans to those things relating to age, power, faith, necessity, and the power of memory. It was also called unwearied, because, like God, it was tireless. The Pythagoreans divided the heavenly bodies into ten orders. They also stated that the decad perfected all numbers and comprehended within itself the nature of odd and even, moved and unmoved, good and ill. They associated its power with the following deities: Atlas (for it carried the numbers on its shoulders), Urania, Mnemosyne, the Sun, Phanes, and the One God.

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They use evolutionary theory for the same reason that they use mathematics — because their experience has shown them that it is an indispensable tool in their own area of study."

"Nothing in biology makes sense except in the light of evolution."
"You care for nothing but shooting, dogs, and rat-catching."
"Creationism is not a scientific alternative to natural selection any more than the stork theory is an alternative to sexual reproduction."
"People who believe the earth was created 6000 years ago, when it's actually 4.5 billion years old, should also believe the width of North America is 8 yards.

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## Famous Quotes on Mathematics - Space and Motion

Unfortunately, some students end upwith the impression that it is not necessary to check your work --just write it up once, and hope that it's correct.

## Why Teach Mathematics? - National Council of …

Examples of N7Tathen~atics in Use115number of such instances is steadily increasing, and the boundarylines between mathematical sciences and sciences that use mathe-matics are often difficult to draw. The increasing use of mathematical methods in the biologicalsciences was pointed out earlier in the section on The Mathematiza-tion of Culture (see page 3~; and the essay by Hirsh Cohen inreference 7 discusses in more detail a variety of biomedical applica-tions of mathematics. A 1967 compilation by Thrall et al.~4 pro-vides extensive further illustration of applications of mathematicalmodels in biology. An important omission in our discussion is the burgeoning fieldof mathematical psychology. A comprehensive survey of this held canbe found ire reference 13. Another important example of the pene-tration of mathematical methods into hitherto unmathematizedareas is in the young science of mathematical linguistics, which ap-plies mathematical methods arid the mathematical way of thinkingto the study of living languages. (See, for instance, the essay byHarris in reference 7.) All great goods spawn small evils. Every new and powerful tool ismisused as well as used wisely. This leas true of printing and me-chanical power when these tools were new. Today, in any held ofendeavor where mathematics or statistics or computing is new, therewill be those who use these tools inadvisedly, as a means of persuasion when the evidence is incomplete or even incorrect, or as a meansof "blessing" conclusions that do not deserve support. All fields nowwell mathematized or well statisticized or shell computerized havesuffered through these difficulties. Those in process now, or to be inprocess in the near future, will have to suffer too. Such difficultiesoften slow down the incorporation of mathematics or statistics orcomputing into the heart of a new field of application. These delaysare, we {ear, inevitable. The one antidote that has proved effective is an increased amountof mathematical or statistical or computing literacy for the majorityof those who work in the field. This increase comes in two parts,separable but usually joined: on the one hand, enough literacy aboutthe mathematics involved to understand the meaning, perhaps eventhe details, of the manipulations required; on the other, often evenmore important, art understanding of how mathematics or statisticsor computing fits into actual problems in similar areas. This latterincludes an appreciation of one of the skills of an effective user: theability to be usually sound as to what must be taken into account in

## The Genius Nicola Tesla and Mathematics - by Liliana …

Examples of Mathematics in Use103that have been introduced into physics in recent years are found inthe essays by Dyson and Wightman in reference 7. We quote froman article by the great physicist P. A. M. Dirac tProc. Roy. Soc., 133,66 (1931~:The steady progress of physics requires for its theoretical formulation amathematics that gets continually more advanced. This is only natural andto be expected. What, however, was not expected by the scientific workersof the last century was the particular form that the line of advancement othe mathematics would take, namely, it was expected that the mathematicswould get more and more complicated, but would rest on a permanent basisof axioms and definitions, while actually the modern physical developmentshave required a mathematics that continually shifts its foundations andgets more abstract. Non-Euclidean geometry and noncommutative algebra,which were at one time considered to be purely fictions of the mind andpastimes for logical thinkers, have been found to be very necessary for thedescription of general facts of the physical world. It seems likely that thisprocess of increasing abstraction will continue in the future and that ad-vance in physics is to be associated with a continual modification and gen-eralization of the axioms at the base of the mathematics rather than with alogical development of any one mathematical scheme on a fixed foundation. Many physicists believe that the central problem they face today,namely the structure of atomic nuclei and their constituent parts(also known as high-energy physics), may well be solvable only uponthe introduction of mathematical concepts not hitherto used inphysics and perhaps as yet unknown to mathematicians. Be this asit may, it has been repeatedly demonstrated that a sense of form andan appreciation of elegance, abstraction, and generalization, whichare the hallmarks of good mathematical development, are often alsothe characteristics of the new breakthroughs in physical insight. Infact, what one refers to as physical ideas often derive from propertiesof abstract mathematical concepts, which turn out to have wide-spread and deep-rooted applicability in natural phenomena. In re-viewing the interplay between mathematics and one branch ofphysics, M. J. Lighthill A. Roy. Aeronaut. Soc., 64, 375 (1960) ~observed that an important task of mathematics is to generate newphysical ideas, that is,. . . ideas which have been originated by mathematical investigation butwhich later become amenable to almost exclusively physical description, andwhose properties, although first derived mathematically, become familiarand are commonly described in purely physical terms. The value of physicalideas in practical work, of course, is their elasticity. Provided that they aresound ideas, such as those thrown up as the genuinely appropriate physical