Mathematics

"Yet, though few, if any—Clerk-Maxwell perhaps only excepted—ever possessed the same almost magical quality of physical insight, none could be more strict than Lord Kelvin in requiring demonstration freed from untenable assumptions or undemonstrable hypotheses. Daring as he was, at least in his earlier days, in the application of analytical methods to the phenomena of nature, he was in several ways very conservative. For example, he never would countenance the use in physics of the method of quaternions. At the British Association Meeting at Cambridge in 1845, he had met Hamilton, who there read his first paper on Quaternions. One might have thought that the young enthusiast would have readily welcomed a new and ingenious method of symbolic analysis: but it was not so. He would not use quaternion notation or quaternion methods himself, nor did he admit the into his work."

"'(1). The word "Quaternion" requires no explanation, since... it occurs in the Scriptures and in Milton. Peter was delivered to "four quaternions of soldiers" to keep him; Adam, in his morning hymn, invokes air and the elements, "which in quaternion run." The word (like, the Latin "quaternio," from which it is derived) means simply a set of four, whether those "four" be persons or things."

"[O]f possible quadruple algebras the one... by far the most beautiful and remarkable was practically identical with quaternions, and... it [is] most interesting that a calculus which so strongly appealed to the human mind by its intrinsic beauty and symmetry should prove to be especially adapted to the study of natural phenomena. The mind of man and that of Nature’s God must work in the same channels."

"The familiar proposition that all A is B, and all B is C, and therefore all A is C, is contracted in its domain by the substitution of significant words for the symbolic letters. The A, B, and C, are subject to no limitation for the purposes and validity of the proposition; they may represent not merely the actual, but also the ideal, the impossible as well as the possible. In Algebra, likewise, the letters are symbols which, passed through a machinery of argument in accordance with given laws, are developed into symbolic results under the name of formulas. When the formulas admit of intelligible interpretation, they are accessions to knowledge; but independently of their interpretation they are invaluable as symbolical expressions of thought. But the most noted instance is the symbol called the impossible or imaginary, known also as the square root of minus one, and which, from a shadow of meaning attached to it, may be more definitely distinguished as the symbol of semi-inversion. This symbol is restricted to a precise signification as the representative of perpendicularity in quaternions, and this wonderful algebra of space is intimately dependent upon the special use of the symbol for its symmetry, elegance, and power."

"[T]he common solution, using three Euler's angles interpolated independently, is not ideal. The more recent (1843) notation of quaternions is proposed instead, along with interpolation on the quaternion unit sphere. Although quaternions are less familiar, conversion to quaternions and generation of in-between frames can be completely automatic, no matter how key frames were originally specified, so users don't need to know—or care—about inner details. The same cannot be said for Euler's angles, which are more difficult to use."

"I had been wishing for an occasion of corresponding a little with you on Quaternions: and such now presents itself, by your mentioning in your note... that you "have been reflecting on several points connected with them"... "particularly on the Multiplication of Vectors. ...No more important, or ...fundamental question, in the whole theory of Quaternions, can be proposed than that which thus inquires What is such Multiplication? What are its Rules, its Objects, its Results? What Analogies exist between it and other Operations, which have received the same general Name? And finally, what is (if any) its Utility? ...[R]eferring to an ante-quaternionic time, when you were a mere child, but had caught from me the conception of a Vector, as represented by a Triplet... I happen to be able to put the finger of memory upon the year and month—October, 1843—when... the desire to discover the laws of the multiplication referred to regained with me a certain strength and earnestness, which had for years been dormant, but was then on the point of being gratified, and was occasionally talked of with you. Every morning in the early part of the... month, on my coming down to breakfast, your (then) little brother William Edwin, and yourself, used to ask me, "Well, Papa, can you multiply triplets"? Whereto I was always obliged to reply, with a sad shake of the head: "No, I can only add and subtract them." But on the 16th day of the same month… I was walking… and your mother was walking with me, along the … and although she talked with me now and then, yet an under-current of thought was going on in my mind, which gave at last a result, whereof... I felt at once the importance. An electric circuit seemed to close; and a spark flashed forth, the herald (as I foresaw, immediately) of many long years to come of definitely directed thought and work, by myself if spared, and at all events on the part of others, if I should even be allowed to live long enough distinctly to communicate the discovery. Nor could I resist the impulse—unphilosophical as it may have been—to cut with a knife on a stone of , as we passed it, the fundamental formula with the symbols, i, j, k; namely,i^2 = j^2 = k^2 = ijk = -1,"

"I have not been exclusively occupied by my Quaternions, but confess that they have been growing in interest upon me, and that I more and more believe they will one day justify a hope which I ventured to express... that they will constitute nothing less than "a new algebraical geometry.""

"all the numbers that have been derived from the genus are four; but number is the indefinite genus, from which was constituted, according to them, the perfect number, viz. the decade. For one, two, three, four, become ten, if its proper denomination be preserved essentially for each of the numbers. Pythagoras affirmed this to be a sacred quaternion, source of everlasting nature, having, as it were, roots in itself; and that from this number all the numbers receive their originating principle."

"While translations are well animated by using vectors, rotation animation can be improved by using the progenitor of vectors, quaternions. ...By an odd quirk of mathematics, only systems of two, four, or eight components will multiply as Hamilton desired; triples had been his stumbling block."

"Mr. McAulay asks: "What is the first duty of the physical vector analyst quâ physical vector analyst?" The answer is... to present the subject in such a form as to be most easily acquired, and most useful when acquired. ...What then is the cause of the fact ...all of us deplore? ...We need only a glance at the volumes in which Hamilton set forth his method. No wonder that physicists and others failed to perceive the possibilities of simplicity, perspicuity, and brevity... in a system presented... in ponderous volumes of 800 pages. ...[I]f we turn to his earlier papers on Quaternions in the Philosophical Magazine... we find... "On Quaternions; or on a New System of Imaginaries in Algebra," and in them we find a great deal about imaginaries and very little of a vector analysis. To show how slowly the system of vector analysis developed itself in the quaternionic nidus, we need only say that the symbols S, V, and ∇ do not appear until two or three years after the discovery of quaternions. In short it seems to have been only a secondary object with Hamilton to express the geometrical relations of vectors... it was never allowed to give shape to his work. ...[I]s it not discouraging to be told that in order to use the quaternionic method one must give up the progress which he has already made in the pursuit of his favourite science and go back to the beginning and start anew on a parallel course? ...Whatever is special, accidental, and individual, will die, as it should; but that which is universal and essential should remain as an organic part of the whole intellectual acquisition. If that which is essential dies with the accidental, it must be because the accidental has been given the prominence which belongs to the essential. ...In Italy they say all roads lead to Rome. In mechanics, , astronomy, physics, all study leads to the consideration of certain relations and operations. These are the capital notions; these should have the leading parts in any analysis suited to the subject."

"My own introduction to quaternionics took place in quite a different manner. Maxwell exhibited his main results in quaternionic form in his treatise. I went to Prof Tait's treatise to get information, and to learn how to work them. I had the same difficulties as the deceased youth, but by skipping them, was able to see that quaternionics could be employed consistently in vectorial work. But on proceeding to apply quaternionics to the development of electrical theory, I found it very inconvenient. Quaternionics was in its vectorial aspects antiphysical and unnatural, and did not harmonise with common scalar mathematics. So I dropped out the quaternion altogether, and kept to pure scalar and vectors, using a very simple vectorial algebra in my papers from 1883 onward. The paper at the beginning of vol. 2 of my Electrical Papers may be taken as a developed specimen; the earlier work is principally concerned with the vector differentiator ∇ and its applications, and physical interpretations of the various operations. Up to 1888 I imagined that I was the only one doing vectorial work on positive physical principles; but then I received a copy of Prof. Gibbs's Vector Analysis (unpublished, 1881-4)."

"Prof. Tait has spoken of the calculus of quaternions as throwing off in the course of years its early Cartesian trammels. I wonder that he does not see how well the progress in which he has led may be described as throwing off the yoke of the quaternion. A characteristic example is seen in the use of the symbol ∇. Hamilton applies this to a vector to form a quaternion, Tait to form a linear vector function. ...Now I appreciate and admire the generous loyalty toward one whom he regards as his master which has always led Prof. Tait to minimise the originality of his own work in regard to quaternions and write as if everything was contained in the ideas which flashed into the mind of Hamilton at the classic . But... we owe duties to our scholars as well as to our teachers, and the world is too large, and the current of modern thought is too broad, to be confined by the ipse dixit [he says] even of a Hamilton"

"Frobenius' Theorem. Over the real number field there exist precisely three associative s, namely the real numbers, the complex numbers, and the real quaternions."

"More than a third part of a century ago, in the library of an ancient town, a youth might have been seen tasting the sweets of knowledge to see how he liked them. He was of somewhat unprepossessing appearance, carrying on his brow the heavy scowl that the "mostly-fools" consider to mark a scoundrel. In his father's house were not many books, so it was like a journey into strange lands to go book-tasting. Some books were poison; theology and metaphysics in particular they were shut up with a bang. But scientific works were better; there was some sense in seeking the laws of God by observation and experiment, and by reasoning founded thereon. Some very big books bearing stupendous names, such as Newton, Laplace, and so on, attracted his attention. On examination, he concluded that he could understand them if he tried, though the limited capacity of his head made their study undesirable. But what was Quaternions? An extraordinary name! Three books; two very big volumes called Elements, and a smaller fat one called Lectures. What could quaternions be? He took those books home and tried to find out. He succeeded after some trouble, but found some of the properties of vectors professedly proved were wholly incomprehensible. How could the square of a vector be negative? And Hamilton was so positive about it. After the deepest research, the youth gave it up, and returned the books. He then died, and was never seen again. He had begun the study of Quaternions too soon."

"I certainly admit that vectors may be used in connection with and to represent rotations. I have no objection to calling them in such cases versorial. In that sense Lagrange and Poinsot... used versorial vectors. But what has this to do with quaternions? Certainly Lagrange and Poinsot were not quaternionists. ...Does it follow that I have used a quaternion? Not at all. A quaternionic expression may represent a number. Does everyone who uses any expression for that number use quaternions? A quaternionic expression may represent a vector. Does everyone who uses any expression for that vector use quaternions? A quaternionic expression may represent a linear vector operator. If I use expression for that linear vector operator do I therefore use quaternions? My critic is so anxious to prove that I use quaternions that he uses arguments which would prove that quaternions were in common use before Hamilton was born."

"Closely akin to his third and fourth propositions is Riemann's fifth proposition, that continuous quantities are coördinate with discrete quantities, both being in their nature multiples or aggregates, and therefore species of the same genus. This pernicious fallacy is one of the traditional errors current among mathematicians, and has been prolific of innumerable delusions. It is this error which has stood in the way of the formation of a rational, intelligible, and consistent theory of irrational and imaginary quantities, so called, and has shrouded the true principles of the doctrine of "complex numbers" and of the calculus of quaternions in an impenetrable haze."

"I do think... that you would find it would lose nothing by omitting the word "vector" throughout. It adds nothing to the clearness or simplicity of the geometry, whether of two dimensions or three dimensions. Quaternions came from Hamilton after his really good work had been done; and, though beautifully ingenious, have been an unmixed evil to those who have touched them in any way, including Clerk Maxwell."

"The real numbers are the dependable breadwinner of the family, the complete ordered field we all rely on. The complex numbers are a slightly flashier but still respectable younger brother: not ordered, but algebraically complete. The quaternions, being noncommutative, are the eccentric cousin who is shunned at important family gatherings. But the octonions are the crazy old uncle nobody lets out of the attic: they are nonassociative."

"Quantum theory may be formulated using s over any of the three associative normed division algebras: the real numbers, the complex numbers and the quaternions. ...[P]roblems can be resolved if we treat real, complex and quaternionic quantum theory as part of a unified structure. Dyson called this structure the "three-fold way". ... This three-fold classification sheds light on the physics of time reversal symmetry, and it already plays an important role in particle physics. ...There are precisely four 'normed division algebras': the real numbers \mathbb{R}, the complex numbers \mathbb{C}, the quaternions \mathbb{H} and the octonions \mathbb{O}. Roughly speaking, these are the number systems extending the reals that have an ‘absolute value’ obeying the equation |xy| = |x| |y|. Since the octonions are nonassociative [their use] proves difficult... except in a few special cases. ...[I]nstead of being distinct alternatives, real, complex and quaternionic quantum mechanics are three aspects of a single unified structure."

"It is a curious fact in the history of mathematics that discoveries of the greatest importance were made simultaneously by different men of genius. The classical example is the... development of the infinitesimal calculus by Newton and Leibniz. Another case is the development of vector calculus in Grassmann's Ausdehnungslehre and Hamilton's Calculus of Quaternions. In the same way we find analytic geometry simultaneously developed by Fermat and Descartes."

"[Q]uaternions form the appropriate algebraic basis for a description of nature whenever we have to deal either with pseudoreal group representations or with co-representations of Wigner's Type II. The context in which quaternions arose historically, in a study of the three-dimensional rotation group, can now be seen to be an extremely special case of this general principle. Every group which admits pseudoreal representations equally admits a natural description in terms of real quaternions."

"'(2). But the question arises, what special connexion has the number Four with mathematics generally, or with that branch of mathematical science in particular, to which the "Lectures on Quaternions" relate?"

"The next grand extensions of mathematical physics will, in all likelihood, be furnished by quaternions."

"The history of geometry may be conveniently divided into five periods. The first extends from the origin of the science to about A. D. 550, followed by a period of about 1,000 years during which it made no advance, and in Europe was enshrouded in the darkness of the middle ages; the second began about 1550, with the revival of the ancient geometry; the third in the first half of the 17th century, with the invention by Descartes of analytical or modern geometry; the fourth in 1684, with the invention of the differential calculus; the fifth with the invention of by Monge in 1795. The quaternions of Sir William Rowan Hamilton, the Ausdehnungslehre of Dr. Hermann Grassmann, and various other publications, indicate the dawn of a new period. Whether they are destined to remain merely monuments of the ingenuity and acuteness of their authors, or are to become mighty instruments in the investigation of old and the discovery of new truths, it is perhaps impossible to predict."

"Symmetrical equations are good in their place, but "vector" is a useless survival, or offshoot, from quaternions, and has never been of the slightest use to any creature. Hertz wisely shunted it, but unwisely he adopted temporarily Heaviside’s nihilism. He even tended to nihilism in dynamics, as I warned you soon after his death. He would have grown out of all this, I believe, if he had lived. He certainly was the opposite pole of nature to a nihilist in his experimental work, and in his Doctorate Thesis on the impact of elastic bodies."

"'(3). One general form of answer... is... that in the mathematical quaternion is involved a peculiar synthesis, or combination, of the conceptions of space and time; and that while TIME is usually pictured or represented by metaphysicians under the figure of a line—a single stream with its ONE current—an unique axis of progression, SPACE is, on the contrary, imagined or conceived in connexion with THREE distinct axes, three lines at right angles to each other... height, length, and breadth. In time, we have only the forward and the backward, looking before and after. In space, there is not merely the contrast between the directions of upward and downward, but also between those of southward and northward, and again between westward and eastward. Time is said to have only one dimension, and space to have three dimesions. The former is an unidimensional, the latter a tridimensional progression. The mathematical quaternion partakes of both these elements; in technical language it may be said to be "time plus space," or "space plus time": and in this sense it has, or at least it involves a reference to, four dimensions. In an unpublished sonnet to Sir John Herschel, entitled "The "(...Greek ...equivalent to the Latin Quaternio), the author of the Lectures introduced the two following lines... an expression of the view... in the foregoing remarks..:"And how the One of Time, of Space the Three, Might in the Chain of Symbol girdled be.""

"... Ještěd in Czechia, Guangzhou in China... and Khan Shatir Shopping-Entertaining Center in Kazakhstan, Aspire Tower in Qatar... are vivid examples of using... the hyperboloid principle... [in] modern buildings."

"In 1899... V. Shukhov... patented the principle of constructing hyperboloid gridshell structures based on a hyperboloid of revolution. ...A one-sheet hyperboloid is a connected surface having negative at each point. Through any point of this surface, two intersecting [straight] lines can be drawn, completely belonging to it. Thus the... surface can be formed by the set of straight lines. It the steel beams would be placed along these lines the shape... could be retained under... external loading... [I]n Shukhov's towers horizontal rims [rings] were used... located at different levels... The stability... was ensured by numerous riveted connections... During the installation... the straight angles of the metal profile was somewhat deformed... [at] the... intersecting elements in order to ensure maximum contact... by rivets."

"The main hazard for high-rise structures is the loads caused by wind gusts. The grid shell design... minimize[d] their influence. Open-work design... ensured... sufficient strength, high stability and low metal consumption. ...[C]onsumption of metal per unit height... was three times less than... the ..."

"191 of the Shukhov's Towers (from known near 200...) have been irretrievably lost during... the 20th century... [Some] towers were destroyed because their continued use for water supply has become impractical. To use them for another purpose... large investments were necessary."

"Single curved surfaces, for example cylinders, have strengths but also weaknesses. Double curved surfaces... are curved in two directions and thus avoid... weak directions."

"[T]he structure of the lattice tower was a spatial system, where the load was equally spread along the surface. ...Aiming to optimise the design process, soon after building the tower Shukhov presented the standardised elements of the tower structure in a table format... with the aid of which it became possible to design a new water tower according to a client’s requirements in twenty-five minutes..."

"Grigory Kovelman writes that Shukhov told him he had been thinking about the properties of hyperboloid structures for a long time, that he had studied hyperboloid forms at the Technical School, and that apparently the moment of enlightenment came about when he saw an up-ended wicker wastepaper basket with a focus on top of his desk. According to Shukhov, this was when he understood clearly how a hyperboloid structure with its curved surface [was] generated by straight rods..."

"[T]his is the magical part... despite the surface being curved in two directions, it is made entirely of straight lines. Apart from the cost savings of avoiding curved beams or shuttering, they are far more resistant to buckling because the individual elements are straight..."

"When designing... cooling towers, engineers are faced with two problems: I. The structure must be able to withstand high winds and II. They should be built with as little material as possible... The hyperbolic form solves both..."

"Shukhov’s lattice-suspended and vaulted structures represented a carrying surface, which could be shaped in any form. ...The density of the grid made it possible to attach it to the shell without additional structures. ...[T]he grids were two to three times lighter than roofs with conventional frames..."

"As hyperboloid structures are double curved, that is simultaneously curved in opposite directions, they are very resistant to . This means that you can get away with far less material than you would otherwise need..."

"The Radio Tower in Moscow is a gridshell which aims for structural efficiency. Its minimal surface and open lattice structure help in reducing the wind load, one of the main challenges in high-rise building design. ...Shukhov's design logic focuses on structural . The rings between the different segments offer additional reinforcement to create an equilibrium between minimal material consumption, structural efficiency and geometry."

"In 2010, the journal Detail published an analysis of Shukhov’s constructions, calling his approach to design 'an early example of ’..."

"The final and most unusual of the gridshell structures presented at the Exhibition was the 32-metre-tall lattice hyperboloid water tower. Everything was amazing in that first Shukhov tower—everything in it was some structural and geometric puzzle: straight rods and the external silhouette double curvature, the openwork lightness below and the solid heaviness above."

"Vaulted gridshell constructions... were formed with thin metal arches turned away from the frontal position at a particular angle. They thus worked as one continuous resilient truss. ...Each arch was made with rigid metal strips of equal length... during the assembling process, each piece was bent equally. ...It was the first time in the world’s building practice that double-curved spatial vaults were created with single type rod elements ..."

"Alongside Shukhov’s drawings and sketchpads... a typescript produced by Shukhov’s former employee Grigory Kovelman... put together an extensive overview of Shukhov’s inventions and projects, both as a biographer and as a specialist who had worked with Shukhov.... Elena Shukhova... grand-daughter... presents extensive biographical details in Vladimir Grigorevich Shukhov. The First Engineer in Russia... Art of Construction... is a valuable collection of articles about Shukhov and his various inventions, written by... specialists. It includes... examples of his calculations. Shukhov’s own book Rafters... discusses the mathematical investigations which led him to conceptualise the spatial lattice structure, describing it as 'an optimisation process'."

"The water tower was a unique structure of its time... According to Cooper, the idea... came directly from an imaginary hyperboloid geometry, invented by... Lobachevski in 1829..."

"This is an interesting paradox: you get the best local buckling resistance because the beams are straight and the best overall buckling resistance because the surface is double curved."

"For a given diameter and height of a tower and a given strength, this shape requires less material than any other form. ...Hyperboloidal towers can be built from or as a steel lattice, and is the most economical such structure for a given diameter and height."

"We present a new mechanical model of interatomic bonds, which can be used to describe the elastic properties of the carbon allotropes, such as , diamond, , and s. The interatomic bond is modeled by a hyperboloid–shape structure."

"This review is a complete collection of the studies done for cooling towers and... [gives] updated and sufficient materials for the researches in this field."

"[T]here are... approaches that are closer to the field of the classical mechanics... so–called structural or discrete-continuous methods... The most straightforward example... is a modeled by the solid deformable rod... [T]he interatomic bonds are modeled as a deformable body or a construction. ...[T]hese approaches... can be implemented in standard computing packages based on the finite–element, boundary–element, or s. These methods can be considered as the bridges between the s of atomistic and continual models of the material."

"In recent years... numerical simulation... has been applied to describe the collapse of structures, e.g., the collapse of cooling towers under blasting demolition... and the collapse of the World Trade Center..."

"The collapse of three natural draft cooling towers at the Ferry bridge power station in 1965... [was] due to the inadequate design for the wind forces..."