Author Archives: andrewqross

Differential & Integral Calculus – The Math of Change

Most will remember their first experience with calculus. From limits to derivatives, rates of changes, and integrals, it was as if the heavens had opened up and the beauty of mathematics was finally made clear. There was, in fact, more to the world than routine numerical manipulation. Numbers and symbols became the foundational building blocks with which theories could be written down, examined, and shared with others. The language of mathematics was emerging and with it a new realm of thinking. For me, calculus marked the beginning of an intellectual awakening and with it a new way of thinking. It is therefore perhaps worthy to examine the early development of our modern calculus and to provide a more concrete historical context.

The method of exhaustion. Image: Margaret Nelson, illustration for New York Times article “Take it to the Limit” by Steven Strogatz.

The distinguishing feature of our modern calculus is, undoubtedly, its unique ability to utilize the power of infinitesimals. However, this power was only realized after more than a millennium of intense mathematical debate and reformation. To the early Greek mathematicians, the notion of infinity was but a paradoxical concept lacking the geometric backing necessary to put it on a rigorous footing. It was this initial struggle to provide both a convincing and proper proof for the existence and usage of infinitesimals that led to some of the greatest mathematical development this world has ever seen. The necessity for this development is believed to be the result of early attempts to calculate difficult volumes and areas of various objects. Among the first advancements was the use of the method of exhaustion. First used by the Greek mathematician Eudoxus (c. 408-355 BC) and later refined by the Chinese mathematician Liu Hui in the 3rd century AD[1], the method of exhaustion was initially used as a means of “sandwiching” a desired value between two known values through repeated application of a given procedure. A notable application of this method was its use in estimating the true value of pi through inscribing/circumscribing a circle with higher degree n-gons.[1] With the age of Archimedes (c. 287-212) came the development of heuristics – a practical mathematical methodology not guaranteed to be optimal or perfect, but sufficient for the immediate goals.[2] Followed by advancements made by Indian mathematicians on trigonometric functions and summations (specifically work on integration), the groundwork for modern limiting analysis began to unfold and thus the relevance for infinitesimals in the mathematical world.

Isaac Newton. Image: Portrait of Isaac Newton by Sir Godfrey Kneller. Public domain.

By the turn of the 17th century, many influential mathematicians including Isaac Barrow, René Descartes, Pierre de Fermat, Blaise Pascal, John Wallis, and others had already been applying the results on infinitesimals to the study of tangent lines and differentiation.[2] However, when we think of modern calculus today the first names to come to mind are almost certainly Isaac Newton and Gottfried Leibniz. Before 1650, much of Europe was still in what historians refer to as the Hellenistic age of mathematics. Prior to the contributions of Newton and Leibniz, European mathematics was largely an “informal mass of various techniques, methods, notations, and theories.”[2] Through the creation of a more structured and algorithmic approach to mathematics, Newton and Leibniz succeeded in transforming the heart of the mathematical system itself giving rise to what we now call “the calculus.”

Both Newton and Leibniz shared the belief that the tangent could be defined as a ratio but Newton insisted that it was simply the ratio between ordinates and abscissas (the x and y coordinates respectively in the plane in regular Euclidean geometry).[2] Newton further added that the integral was merely the “sum of the ordinates for infinitesimals intervals in the abscissa” (i.e., the sum of an infinite number of rectangles).[4] From Leibniz we gain the well-known “Leibniz notation” still in use today. Leibniz denoted infinitesimal increments of abscissas and ordinates as dx and dy and the sum of infinitely many infinitesimally thin rectangles as a “long s” which today constitutes our modern integral symbol ò.[2] To Leibniz, the world was a collection of infinitesimal points and that infinitesimals were ideal quantities “less than any given quantity.”[3] Here we might draw the connection between this description and our modern use of the greek letter e (epsilon) – a fundamental tool in modern analysis in which assertions can be made by proving that a desired property is true provided we can always produce a value less than any given (usually small) epsilon.

From Newton, on the other hand, we get the groundwork for differential calculus which he developed through his theory on Fluxionary Calculus first published in his work Methodus Fluxionum.[2] Initially bothered by the use of infinitesimals in his calculations, Newton saught to avoid using them by instead forming calculations based on ratios of changes. He defined the rate of generated change as a fluxion (represented by a dotted letter) and the quantity generated as a fluent. He went on to define the derivative as the “ultimate ratio of change,” which he considered to be the ratio between evanescent increments (the ratio of fluxions) exactly at the moment in question – does this sound familiar to the instanteous rate of change? Newton is credited with saying that “the ultimate ratio is the ratio as the increments vanish into nothingness.”[2/3] The word “vanish” best reflects the idea of a value approaching zero in a limit.

The derivative of a function.

The derivative of a function.

Contrary to popular belief, Newton and Leibniz did not develop the same calculus nor did they conceive of our modern Calculus. Both aimed to create a system in which one could easily manage variable quantities but their intial approaches varied. Newton believed change was a variable quantity over time while for Leibniz change was the difference ranging over a sequence of infinitely close values.[3] The historical debate has therefore been, who invented calculus first? The current understanding is that Newton began work on what he called “the science of fluents and fluxions” no later than 1666. Leibniz on the other hand did not begin work until 1673. Between 1673 and 1677, there exists documented correspondence between Leibniz and several English scientists (as well as Newton himself) where it is believed that he may have come into contact with some of Newton’s   unpublished manuscripts.[2] However, there is no clear consensus on how heavily this may have actually influenced Leibniz’s work. Eventually both Newton and Leibniz became personally involved in the matter and in 1711 began to formally accuse each other of plagiarism.[2/3] Then in the 1820’s, following the efforts of the Analytical society, Leibnizian analytical calculus was formally accepted in England.[2] Today, both Newton and Leibniz are credited for independently developing the foundations of calculus but it is Leibniz who is credited with giving the discipline the name it has today: “calculus.”

The applications of differential and integral calculus are far reaching and cannot be overstated. From modern physics to neoclassical economics, there is hardly a discipline that does not rely on the tools of calculus. Over the course of thousands of years of mathematical development and countless instrumental players (e.g. Newton and Leibniz), we now have at our disposal some of the most advanced and beautifully simple problem solving tools the world has ever seen. What will be the next breakthrough? The next calculus? Only time will tell. What is certain is that the future of mathematics is, indeed, very bright.

Works Cited

[1]Dun, Liu; Fan, Dainian; Cohen, Robert Sonné (1966). “A comparison of Archimdes’ and Liu Hui’s studies of circles”. Chinese studies in the history and philosophy of science and technology 130. Springer. p. 279. ISBN 0-7923-3463-9., Chapter , p. 279

[2]“History of Calculus.” Wikipedia. Wikimedia Foundation, n.d. Web. 14 Mar. 2015.

[3]“A History of the Calculus.” Calculus History. N.p., n.d. Web. 14 Mar. 2015.

[4] Valentine, Vincent. “Editor’s Corner: Voltaire and The Man Who Knew Too Much, Que Sera, Sera, by Vincent Valentine.” Editor’s Corner: Voltaire and The Man Who Knew Too Much, Que Sera, Sera, by Vincent Valentine. ISHLT, Sept. 2014. Web. 15 Apr. 2015.

Leonardo of Pisa – The Great Fibonacci

Figure 1-Fibonacci. Image: Public domain, via Wikimedia Commons.

Most mathematically inclined people are familiar with the famous and unique Fibonacci sequence. Defined by the recurrence relation (*) Fn=Fn-1+Fn-2 with initial values F1=1 and F2=1 and (or sometimes F0=1 and F1=1), the Fibonacci sequence is an integer sequence (1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, …) with many remarkable mathematical and real world applications. However, it seems that few are as well informed on the man behind this sequence as they are on the sequence itself. Did you know that Fibonacci didn’t even discover the sequence? Of course not! Predating Fibonacci by almost a century, the so called “Fibonacci sequence” was actually the brainchild of Indian mathematicians interested in poetic forms and meter who, through studying the unique arithmetic properties of certain linguistic sequences and syllable counts, derived a great deal of insight into some of the most fascinating mathematical patterns known today. But with a little bit of time (few hundred years), some historical distortion, inaccurate accreditation[1], and a healthy dose of blind western ethnocentrism and voila! Every high school kid in America now thinks there is a connection between Fibonacci and pizza. Or is it Pisa? (That’s a pun, laugh.) While often given more credit than deserved for the “discovery” of the sequence, Fibonacci was nonetheless an instrumental player in the development of arithmetic sequences, the spread of emerging new ideas, and in the advancement of mathematics as a whole. We thus postpone discussion of Fibonacci’s sequence – don’t worry, we shall return – to examine some of the other significant and often overlooked contributions of the “greatest European mathematician of the middle ages.”[1]

Born around the year 1175 in Pisa, Italy, Leonardo of Pisa (more commonly known as Fibonacci) would have been 840 years old this year! (Can you guess the two indexing numbers between which Fibonacci’s age falls?[2]) The son of a customs officer, Fibonacci was raised in a North African education system under the influence of the Moors.[3] Fibonacci’s fortunate upbringing and educational experience allowed him the opportunity to visit many different places along the Mediterranean coast. It is during these travels that historians believe Fibonacci may have first developed an interest in mathematics and at some point come into contact with alternative arithmetic systems. Among these was the Hindu-Arabic number system – the positional number system most commonly used in mathematics today. It appears that we owe a great deal of respect to Fibonacci for, prior to introducing the Hindu-Arabic system to Europe, the predominant number system relied on the far more cumbersome use of roman numerals. It is interesting to note that while the Hindu-Arabic system may have been introduced to Europe as early as the 10th century in the book Codex Vigilanus, it was Fibonacci who, in conjunction with the invention of printing in 1482, helped to gain support for the new system. In his book Liber abbaci[4], Fibonacci explains how arithmetic operations (i.e., addition, subtraction, multiplication, and division) are to be carried out and the advantages that come with the adoption of such a system.

Figure 2-Golden spiral. Image: Weisstein, Eric W. "Golden Spiral." From MathWorld--A Wolfram Web Resource.

Figure 2-Golden spiral. Image: Weisstein, Eric W. “Golden Spiral.” From MathWorld–A Wolfram Web Resource.

Whereas the number system most familiar to us uses the relative position of numbers next to each other to represent variable quantities (i.e., the 1’s, 10’s, 100’s, 1000’s, … place), Roman numerals rely on a set of standard measurement symbols which, in combination with others, can be used to express any desired quantity. The obvious problem with this approach is that it severely limits the numbers that can be reasonably represented by the given set of symbols. For example, the concise representation of the number four hundred seventy eight in the Hindu-Arabic system is simply 478 in which “4” is in the hundreds place, “7” is in the tens place, and “8” is in the ones place. In the Roman numeral system, however, this same number takes on the form CDLXXVIII. As numbers increase arbitrarily so does the complexity of their Roman numeral representation. The adoption of the Hindu-Arabic number system was, in large part, the result of Fibonacci’s publications and public support for this new way of thinking. Can you imagine trying to do modern mathematical analysis with numbers as clunky as MMMDCCXXXVIII??? Me either. Thanks, Fibonacci!

Fibonacci’s other works include publications on surveying techniques, area and volume measurement, Diophantine equations, commercial bookkeeping, and various contributions to geometry.[4] But among these works nothing stands out more than that of Fibonacci’s sequence – yes, we have returned! Among the more interesting mathematical properties of Fibonacci’s sequence is undoubtedly its connection to the golden ratio (shall be defined shortly). To illustrate, we look momentarily at the ratios of several successive Fibonacci numbers. Beginning with F1=1 and F2=1 we see that the ratio F2/F1=1. Continuing in this manner using the recurrence relation (*) from above or any suitable Fibonacci table we find that F3/F2=2, F4/F3=3/2, F5/F4=5/3,F6/F5=8/5, F7/F6=13/8, F8/F7=21/13, … As the indexing number tends to infinity, the ratio of successive terms converge to the value 1.6180339887… (the golden ratio) denoted by the Greek letter phi. We may thus concisely represent this convergent value by the expression as the lim n–> infinity (Fn+1/Fn). Studied extensively, the golden ratio is a special value appearing in many areas of mathematics and in everyday life. Intimately connected to the concept of proportion, the golden ratio (sometimes called the golden proportion) is often viewed as the optimal aesthetic proportion of measurable quantities making it an important feature in fields including architecture, finance, geometry, and music. Perhaps surprisingly, the golden ratio has even been documented in nature with pine cones, shells, trees, ferns, crystal structures, and more all appearing to have physical properties related to the value of (e.g., the arrangement of branches around the stems of certain plants seem to follow the Fibonacci pattern). While an interesting number no doubt, we must not forget that mathematics is the business of patterns and all too often we draw conclusions and make big picture claims that are less supported by evidence and facts than we may believe. There is, in fact, a lot of “woo” behind the golden ratio and the informed reader is encouraged to be weary of unsubstantiated claims and grandiose connections to the universe. It is also worth mentioning that, using relatively basic linear algebra techniques, it is possible to derive a closed-form solution of the n-th Fibonacci number.

Figure 3-Computing the 18th Fibonacci Number in Mathematica.

Figure 3-Computing the 18th Fibonacci Number in Mathematica.

Omitting the details (see link for thorough derivation), the n-th Fibonacci number may be computed directly using the formula Fn=((φ)(n+1)+((-1)(n-1)/(φ)^(n-1))/((φ2)+1).[5] While initially clunky in appearance, this formula is incredibly useful in determining any desired Fibonacci number as a function of the indexing value n. For example, the 18-th Fibonacci number may be calculated using F18=((φ)(18+1)+((-1)(18-1)/(φ)^(18-1))/((φ2)+1)=2584. Comparing this value to a list of Fibonacci numbers and to a Mathematica calculation (see picture above), we see that the 18-th Fibonacci number is, indeed, 2584. Without having to determine all previous numbers in the sequence, the above formula allows us to calculate directly any desired value in the sequence saving substantial amounts of time and processing power.

From the study of syllables and poetic forms in 12th-century India to a closed-form solution for the n-th Fibonacci number via modern linear algebra techniques, our understanding of sequences and the important mathematical properties they possess is continuing to grow. Future study may reveal even greater mathematical truths whose applications we cannot yet conceive. It is thus the beauty of mathematics and the excitement of discovery that push us onward, compel us to dig deeper, and to learn more from the world we inhabit. Who knows, you might even be the next Leonardo of Pizza – errrrr Pisa. What patterns will you find?
[1] French mathematician Edouard Lucas (1842-1891) was the first to attribute Fibonacci’s name to the sequence. After which point little is ever mentioned of the Indian mathematicians who laid the groundwork for Fibonacci’s research.

[2] Answer: n=15 –> 610 and n=16 –> 987.

[3] Medieval Muslim inhabitants of the Maghreb, Iberian Peninsula, Sicily, and Malta.[2]

[4] Translation: Book of Calculation[3]


 [1] Knott, Ron. Who Was Fibonacci? N.p., 11 Mar. 1998. Web. 27 Apr. 2015.

[2] “Moors.” Wikipedia. Wikimedia Foundation, n.d. Web. 27 Apr. 2015.

[3] Leonardo Pisano – page 3: “Contributions to number theory”. Encyclopædia Britannica Online, 2006. Retrieved 18 September 2006.

[4] “Famous Mathematicians.” The Greatest Mathematicians of All Time. N.p., n.d. Web. 28 Apr. 2015.

 [5] Grinfeld, Pavel. “Linear Algebra 18e: The Eigenvalue Decomposition and Fibonacci Numbers.” YouTube. YouTube, 2 Dec. 2014. Web. 28 Apr. 2015.

 Figure 1: Fibonacci. Digital image. Wikimedia Foundation, n.d. Web. 27 Apr. 2015.

Figure 2: Golden Spiral. Digital image. Mathworld. Wolfram, n.d. Web. 1 May 2015.

Figure 3: Ross, Andrew Q. Closed-Form Computation of Fibonacci. Digital image. Mathematica, 28 Apr. 2015. Web. 28 Apr. 2015.

History, Development, and Applications of Fractal Geometry

Fractal [frak-tl], noun

  1. A geometric or physical structure having an irregular or fragmented shape at all scales of measurement between a greatest and smallest scale such that certain mathematical or physical properties of the structure, as the perimeter of a curve or the flow rate in a porous medium, behave as if the dimensions of the structure (fractal dimensions) are greater than the spatial dimensions.[i]

It was the work of philosopher and mathematician Gottfried Leibniz in the 17th century that first provided groundwork for the early development of fractal geometry.[ii] However, the nature of recursive self-similarity (a core tenet in the study of many fractal-like objects) and the mathematical “monsters”[1] to which this emerging concept gave birth, consequently delayed meaningful research for roughly two centuries.[iii] Nearly two hundred years elapsed before Karl Weierstrass, in 1872, presented the first definition of a function whose graph by today’s standards would be considered fractal. That is, Weierstrass had shown it was possible to define a function with the non-intuitive property that it could be both everywhere continuous and nowhere differentiable.[iv] Subsequent works including those of Georg Cantor (specifically “Cantor sets”), Felix Klein, and Henri Poincaré were crucial in laying a foundation on which much of the modern mathematical investigation of fractal geometry unfolded. Without the aid of modern computational and graphical tools, however, much of the early research in fractal geometry was severely limited. It would be more than a half-century until, in the 1960’s, equipped with the work of his predecessors (i.e., Helge von Koch, Wacław Sierpiński, Pierre Fatou, Gaston Julia, Felix Hausdorff, and Paul Lévy to name a few) did the French-American mathematician Benoit Mandelbrot succeed in uniting hundreds of years of mathematical research by coining the word “fractal”[2] and illustrating his mathematical definition with the aid of remarkable computer-generated visualizations.[v] Most notable among Mandelbrot’s demonstrations was his use of infinite recursion to define what is known today as the Mandelbrot set. Mathematically speaking, the Mandelbrot set is defined to be the set of values of c in the complex plane for which the orbit of 0 under iteration of the complex polynomial zn+1=zn2+c remains bounded. Equivalently, a complex number c is part of the Mandelbrot set if, when starting with 0=0 and applying the iteration repeatedly, the absolute value of zn remains bounded however large n gets.[vi] This seemingly innocuous procedure is responsible for generating the beautiful, infinitely detailed Mandelbrot set pictured below.

Mandelbrot set. Image: Wolfgangbeyer, via Wikimedia Commons.

This figure, and others like it, is generated with the aid of modern computational tools capable of carrying out a myriad of successive iterations far greater in number than the computational limitations of “by-hand” human calculation. The study of infinite descent is also seen in other areas of mathematics such as algebraic geometry (specifically Fermat’s Last Theorem). In these fields, infinite descent is a tool by which the existence of certain special triangles may be determined. For example, infinite descent may be used to prove that no right-angled triangle having integral side lengths will have a square area. The mysterious nature of objects when examined at their infinite limits is, indeed, quite peculiar.

In addition to producing stunning artistic images, fractal geometry has also found diverse applications in fields such as structural engineering, medicine, telecommunications, urban planning, and more.[viii] Iterative methods have been used to create high-strength construction cables by interweaving a series of thinner wires into thicker ones that, in turn, are used in the next iteration to make even larger and stronger cables than before. This simple procedure, usually no more than a few iterations for most industrial purposes, bears a resemblance to a fractal pattern and allows us to make use of the special properties that fractal geometry offers.

Iterative construction techniques used to create high-strength construction cable. Image: Bernard S. Jansen and Jonathan Wolfe, via Fractal Foundation.

As medical knowledge continues to improve, the usefulness of fractals in curing and identifying health concerns becomes increasingly more apparent. Biomimicry, the concept of deriving inspiration for human designs from the natural world, is currently being employed in an attempt to solve the problem of fluid transport by mimicking the fractal patterns of our blood vessels and lungs.viii

Fractal heat exchanger etched in silicon and designed by Deb Pence at Oregon State University, via Fractal Foundation.

According to researchers at Oregon State University, the above figure can be etched into silicon chips, allowing for a cooling fluid (such as liquid nitrogen) to uniformly flow across the surface of the chip, keeping it cool. Researchers say this fractal pattern was derived from human blood vessels and provides a simple low-pressure system to easily cool sensitive computer chips.viii In this case, the relationship between biomimicry and fractals becomes quite clear. The natural question to ask, then, is to what extent may biomimicry assist researchers in addressing biological and natural phenomena? The future is certainly promising.

While classical Euclidian geometry and other related areas of mathematics are often used to understand and predict natural phenomena, these traditional modes of thinking may prove insufficient in answering some of the more complex questions that arise in nature (some of which have been discussed here). After all, as Benoit Mandelbrot himself once said, “Clouds are not spheres, mountains are not cones, coastlines are not circles, and bark is not smooth, nor does lightning travel in a straight line.”[ix] It is the heart of fractal geometry that attempts to analyze and understand many of these more complex phenomena.

[1] These “monsters” were mathematical problems of immense complexity that, according to Leibniz and others, were believed to be beyond the scope of contemporary geometric knowledge.

[2] From the Latin word frāctus meaning “broken” or “fractured.”

[i] “fractal.” Unabridged. Random House, Inc. 07 Feb. 2015. <>.

[ii] Pickover, Clifford A. (2009). The Math Book: From Pythagoras to the 57th Dimension, 250 Milestones in the History of Mathematics. Sterling Publishing Company, Inc. p. 310. ISBN 978-1-4027-5796-9. Retrieved 2011-02-05.

[iii] Gordon, Nigel (2000). Introducing fractal geometry. Duxford, UK: Icon. p. 71. ISBN 978-1-84046-123-7.

[iv]Edgar, Gerald (2004). Classics on Fractals. Boulder, CO: Westview Press. ISBN 978-0-8133-4153-8.

[v]Albers, Donald J.; Alexanderson, Gerald L. (2008). “Benoît Mandelbrot: In his own words”. Mathematical people : profiles and interviews. Wellesley, MA: AK Peters. p. 214. ISBN 978-1-56881-340-0.

[vi]“Mandelbrot Set Explorer: Mathematical Glossary”. Retrieved 2007-10-07.

[vii] Mandelbrot Set. Digital image. Mandelbrot Set. Wikipedia, 30 Jan. 2015. Web. 7 Feb. 2015.

[viii] “Fractal Applications.” Fractal Foundation Online Course. Fractal Foundation, 30 Apr. 2003. Web. 07 Feb. 2015.

[ix] Mandelbrot, Benoit. “THE FRACTAL GEOMETRY OF NATURE – Introduction.” THE FRACTAL GEOMETRY OF NATURE – Introduction., 11 June 2001. Web. 07 Feb. 2015.