Category Archives: Education

October 4, 2014 · 7:45 pm

Elementitis & Aboutitis

In his book “Making learning whole”, David Perkins talks about two widespread diseases of the educational system, namely “elementitis” and “aboutitis”.

Elementitis has to do with learning the elements of a discipline without seeing the big picture. Elementitis prevents students from understanding what is the purpose of the topic considered, rising their levels of frustration, apathy, and boredom. This disease is widely spread across the world when it comes to learning some important subjects, such as math. Students spend literally years learning topics in algebra, calculus, or statistics without immediately seeing their application. Imagine a cooking school where cooks-to-be learn how to bake doughs for years without even mention the word “pie”. It would be non-sense. But that is exactly what occurs in the math realm. High school students learn how to multiply or invert matrices, but they have no idea what matrices are for. When teaching foreign languages, some schools start teaching the elements of writing, grammar, and even style, without really practising oral skills. What is the point of doing this? I have seen hundreds of students able to understand something written in a foreign language, but totally incapable to actually speak more than a few sentences in such a language. My favourite example is how some teachers make (or pretend to make) their students learn English irregular verbs. Students have to learn by heart an endless list of verbs with three columns, such as eat-ate-eaten. Can there be a worse way to teach irregular verbs? This awful method is also applied to phrasal verbs, vocabulary, and so on. Unfortunately, math and languages are not the only examples of subjects infected with elementitis. History, chemistry, physics, or literature, to name a few more, could have been used as examples as well.

At the opposite end of the spectrum, programming applications such a Scratch from MIT give immediate feedback to programmers and a rewarding sense of utility. It is not a coincidence that Scratch has become so popular among young students. Scratch philosophy is in line with Perkins’ recommendation to avoid elementitis, namely start by playing a simple version of the whole game from the very beginning. As mentioned, this is easier said than done, but the point is there: innovation is needed to extrapolate the philosophy of Scratch to other spheres.

Aboutitis is related to the inability of students to learn a specific subject in full. There are so many subjects to learn and time is so limited, that teachers cannot go deep into any subject, they content themselves with scratching the surface. As a result, students are able to recognize the topics (“I have seen this before”, “that rings a bell”) but have not really grasped the rewarding deep insights that were awaiting them one or two steps further. Aboutitis prevents students from continuing learning on their own based on the knowledge acquired in the classroom. The building blocks on which their knowledge is based are not solid, but slippery. Sadly enough, aboutitis leads to covering the same topics again and again year after year at school, which again entails ample boredom and frustration. From my own experience, I can tell that my children, when learning English as a foreign language, have “learnt” the difference between present simple and present continuous for seven or eight consecutive years each.

Being aware of the existence of the two diseases, elementitis and aboutitis, is the first step to coming up with new, disruptive ways of teaching and learning. This transformation should occur at two levels. First, teachers in the classroom should try to teach fewer topics better and give an immediate sense of usefulness trough simple examples or experiences. Second, when designing an educational path, the whole body of knowledge to be taught should be split differently and coordinated in such a way that these two diseases are avoided. As I said, easier said than done, but for sure it is worth trying.

References
Perkins, D. “Making learning whole. How seven principles of teaching can transform education” Jossey-Bass, 2009, San Francisco, CA.

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September 5, 2012 · 5:44 pm

ZLC, a safe bet

The crisis forces us to establish priorities when deciding what to do with public money. ZLC is one of the most promising bets to achieve economic growth.

Alejandro Serrano – Heraldo de Aragón, Sep 1, 2012| Spain

Original article is aquí (in Spanish)

The years of economic bonanza drew many public and semipublic initiatives to our region geared to incentivizing economic growth in the region. Aramón, walqa, and Motorland are all examples of initiatives of this kind. In the university realm, the existing infrastructure was extended to create new campuses in Huesca and Teruel. Lastly, as a result of an agreed commitment to logistics, given our region’s potential in this industry, we witnessed the creation of PLAZA—with its younger brothers PLHUS and Platea—on the one hand, and the establishment of Zaragoza Logistics Center (ZLC) by  MIT (the world’s leading school of engineering), on the other. Years prior to joining this latter project, I already praised its creation and the commitment shown by the regional government to this project  in this newspaper (April 2004, May 2005).

Source: Heraldo de Aragón

With the advent of the economic crisis, the rules of the game have undergone a profound change. It is no longer about achieving all the goals established at a lower cost, but rather about achieving the greatest number of goals on a restrictive budget. In other words, it has become necessary to establish priorities, because it is impossible to achieve everything, which implies abandoning certain commitments viable in the boom years in order to commit firmly to others. In this regard, it would be disastrous for our region to apply a ‘one size fits all’ economy, attempting to keep all the projects running by reducing the budget of all initiatives, good ones and bad ones, by a similar percentage.

What criteria should be followed to establish which projects should be given priority? Logically, focus should be placed on those projects that are aligned with the strategy of the region, which can be inferred from the electoral program presented by the governing party last year. It stated that Aragón should be a “territory of excellence in teaching”, “achieve an optimal collaboration between companies and University”, “attract international teachers and researchers” and seek excellence “in those areas of knowledge susceptible to becoming niches of excellence comparable with the best in the world”, mentioning logistics as the first example.

In this regard,  Zaragoza Logistics Center has many of the attributes required to be part of the group of initiatives to be focused on. Some examples: The ZLC international masters course has been ranked the best logistics program in Spain in the last two years (El Mundo newspaper ranking). As to collaboration with companies, ZLC worked both with Aragon companies (two consultancy projects currently under way with Saica; seventeen alumni contracted by Transportes Carreras to date, …) and multinationals (the list is long: companies such as Adidas, Amazon, BASF, Caterpillar, Cummins, DHL, P&G or Roche have come to Zaragoza to finance projects or hire students). These brilliant students (from Aragón or from elsewhere) will eventually reach decision-making positions in the multinationals that are now hiring them. And it is then when the investment made will more than yield its return, in the form of job creation, projects or research centers.

The first significant initiative of this kind that comes off shall justify economically the existence of ZLC for more than a hundred years. But we have to give it more time. It is not fair to demand from  ZLC in 10 years of existence something that nobody has demanded from the University of Zaragoza in 450 years. ZLC is still a “start-up” in the academic sector. MIT has just celebrated its 150th birthday. Harvard Business School celebrated its centenary in 2008. Excellence in academia takes its time. It is tempting to forget priorities and focus on short-sighted, short-term projects. But if we really want our region to become a reference for others and not turn it into a piece of land with no hope, we should continue to give our firm commitment to ZLC and other similar initiatives.

To know more about ZLC please follow the link www.zlc.edu.es

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September 2, 2012 · 11:38 pm

ZLC, una apuesta segura

La crisis obliga a establecer prioridades y a decidir qué hacer con el dinero público. El ZLC supone una de las apuestas con mejores perspectivas para el futuro desarrollo de Aragón

Alejandro Serrano – Heraldo de Aragón, Sep 1, 2012| España

Accede a la noticia de Heraldo de Aragón pulsando aquí

     Read it in English

Los años de bonanza trajeron a nuestra comunidad autónoma multitud de iniciativas públicas o semipúblicas para incentivar el crecimiento económico en la región. Así nacieron por ejemplo Aramón, walqa, o Motorland. En el ámbito universitario se desdobló la infraestructura existente para crear nuevos campus en Huesca y Teruel. Finalmente, en una clara apuesta por la logística, y dado el potencial de nuestra región en este capítulo, se impulsó la creación de PLAZA—con sus hermanos menores PLHUS y Platea—por un lado y del Zaragoza Logistics Center (ZLC) de la mano del MIT (primera escuela de ingeniería del mundo) por otro. Años antes de unirme a este último proyecto ya alabé en este periódico (abril 2004, mayo 2005) la creación del mismo y la apuesta que el gobierno regional había hecho por él.

Fuente: Heraldo de Aragón

Con la llegada de la crisis económica las reglas del juego han cambiado profundamente. Ya no se trata de conseguir todos los objetivos marcados al menor coste, sino de conseguir el mayor número de objetivos dado un presupuesto restrictivo. Es decir, hay que marcar prioridades porque no todo se puede conseguir, lo que implica desechar algunas de las apuestas que podían ser sostenidas en los años de bonanza para apostar decididamente por el resto. En este sentido, sería nefasto para nuestra comunidad aplicar una economía de “café para todos”, tratando de mantener todos los proyectos en marcha reduciendo un porcentaje similar el presupuesto de todas las iniciativas, buenas y malas.

¿Cuál debe ser el criterio para marcar qué proyectos potenciar? Lógicamente es preciso seleccionar aquéllos que estén alineados con la estrategia de la región, que puede inferirse del programa electoral que el partido en el gobierno presentó el año pasado. En él se habla de que Aragón debe ser un “territorio de excelencia en la enseñanza”, “alcanzar la óptima colaboración entre empresas y Universidad”, fomentar “la atracción de docentes e investigadores internacionales” y buscar la excelencia “en aquellas áreas de conocimiento que sean susceptibles de convertirse en nichos de excelencia equiparables a los mejores del mundo”, poniendo como ejemplo de esto último en primer lugar la logística.

En este sentido, el Zaragoza Logistics Center posee muchos de los atributos requeridos para estar dentro del grupo de las iniciativas por las que apostar. Algunos ejemplos: el máster internacional de ZLC ha sido considerado como el mejor programa de logística en España en los dos últimos años (ranking del diario El Mundo). En cuanto a la colaboración con empresas, ZLC trabaja tanto con empresas aragonesas (dos proyectos de consultoría actualmente en marcha con Saica; diecisiete exalumnos contratados por Transportes Carreras hasta la fecha, …) como multinacionales (la lista es larga: empresas como Adidas, Amazon, BASF, Caterpillar, Cummins, DHL, P&G o Roche han venido a Zaragoza a financiar proyectos o a contratar alumnos). Estos alumnos brillantes (aragoneses o no) llegarán con el tiempo a los ámbitos de decisión de las multinacionales que ahora los contratan. Y es entonces cuando retornará con creces la inversión efectuada, en forma de creación de empleo, proyectos o centros de investigación.

La primera de estas iniciativas de calado que cristalice justificará económicamente  la existencia de ZLC durante más de cien años. Pero hay que darle tiempo al tiempo. No se le puede exigir a ZLC en 10 años de existencia lo que por ejemplo no se le ha exigido a la Universidad de Zaragoza en 450. ZLC es todavía una “start-up” en el sector académico. El MIT acaba de cumplir 150 años. La escuela de negocios de Harvard celebró su centenario en 2008. La excelencia en el ámbito universitario toma su tiempo. Es tentador olvidarse de las prioridades y apostar por políticas miopes y cortoplacistas. Pero si de verdad queremos hacer de nuestra comunidad un referente para otras y no convertirla en un solar sin esperanza debemos seguir apostando firmemente por ZLC y otras iniciativas similares.

Para saber más sobre ZLC www.zlc.edu.es

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June 22, 2012 · 1:05 pm

The challenge of invisibility

Why Supply Chain innovation is unlikely to be appreciated

Alejandro Serrano – . Jun 2012 | Spain

When faced with the question “What is the name of the most innovative company in the world?”, people will most likely answer “Apple, of course”. The answer seems undeniable, and there are good reasons for that: iPhone, iPad, and iWhatever are synonyms of great innovative products these days.

However, there is a bias there; when people think about innovation, the first idea that comes to mind is product innovation. However, a company can gain a huge competitive advantage trough process or supply chain innovation. I am referring to notorious cases of companies who have completely redefined the way of doing business in their industries. Some examples include Benetton, which changed the sequence of operations by delaying the time where pieces of garment were dyed (postponement), or Barilla, one of the first firms to adopt VMI (Vendor managed inventory) by managing its customers’ inventory so as to reduce order variability amplification. These changes had a large impact on the bottom line and gave these firms a clear competitive advantage in front of their competitors.

The good thing about process innovation is that it may be hard for competitors to copy: there is no “product reverse engineering” to be performed. Competitors can and will try to emulate, but success is not by any means guaranteed. Think for instance about the well-known Toyota Production System (TPS). It has been around for about 40 years and still firms are trying to adopt it with bittersweet results. Why? It is simply that most of the changes cannot be seen, they are embedded in firms’ DNA, as pointed out by Spear and Bowen*, and followers just copy the visible part of it, such as Kanban systems in the case of TPS. A more recent example is Inditex, whose flagship Zara defined a new paradigm in a mature industry by betting on speed rather than cost. With more than 5,500 stores around the world, Inditex’s founder Amancio Ortega, is today the richest man in Europe, right before Ikea’s founder, Ingvar Kamprad, another visionary who also completely changed the rules of his industry.

This may not be the case for firms launching innovative products: Samsung’s Galaxy  Tab is closely following Apple’s iPad and even introducing features which go beyond what the Apple product offers. It is true that patents help sometimes, but the fact remains that a burden of expensive legal work is usually triggered as soon as competitors start copying or including almost identical products in their portfolios. The legal battle triggered when Windows 95 was launched by Microsoft or the generic drug business in India may be good examples of this.

All in all, supply chain innovation is likely not to be appreciated as much as product innovation, but its impact on the financials and the value of firms may be larger and last longer. Also, given the relatively small efforts exerted so far on supply chain innovation, there must be indeed great opportunities available to explore, low hanging fruit to be taken by those smart people who are able to identify those opportunities.

(*) Spear, Steven and H. Kent Bowen. Decoding the DNA of Toyota. Harvard Business Review

 To learn more about Supply Chain www.zlc.edu.es

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April 12, 2012 · 9:44 am

To centralize or not to centralize, that is the question

How much safety stock to hold? A ubiquitous question in supply chain network design

Alejandro Serrano –  Apr 2012 | Spain

When a firm designs from scratch its distribution network, a repetitive question arises sooner or later: should we have a centralized or a decentralized network? In a pure centralized network, a large, central distribution center (CDC) contains all the finished-good inventory of the company. In a pure decentralized setting, inventory for each SKU is splitted among several smaller regional distribution centers (RDC).

One of the key questions mangers usually struggle with when deciding about centralizing inventories or not is how many units of each SKU should be held in each case. It is a relevant question, because holding inventory usually entails significant holding costs, both financial and material.

Let us learn the basics of how to answer that question by means of a very simple example: Assume that the demand of a typical product in the portfolio in a stable market is normally distributed, with mean 400 units/period and standard deviation 80 units/period. The market is divided into four identical regions with independent demand. The firm service level target is 95%, thus in the (assumed current) CDC there should be 532 units to maximize expected profit:

(If it is not clear to you why this formula is used, please bear with me; it will be explained in a subsequent post)

Now consider that your company is planning to change its distribution strategy from centralized to decentralized, and so you consider having  four identical RDCs, one per region. The average demand in any region will be 1/4 of total demand, or 400 / 4 = 100 units. The regional variance will be 1/4 of total variance (assuming demand independence across regions), thus the standard deviation in any region will be

The quantity per RDC is

The total inventory is 166 x 4 = 633 units. The safety stock needed, i.e., the amount of inventory to hold above the average demand is 532 – 400 = 131.5 units in the centralized case and 633 – 400 = 263 units in the decentralized case. Interestingly, the latter quantity is exactly two times the former. It is not a coincidence that 2 is the squared root of 4. In fact, when switching from 1 to n DCs, overall safety stock is multiplied by √n. For instance, had we considered 9 RDCs, total inventory to hold would have been 400 + 3 x 131.5 = 795 units.

Of course there may be more involved scenarios (e.g. constraints or demand correlations across regions) that modify the optimal solution. However, keeping in mind this simple square-root formula as a rule of thumb for safety stock will help make back-of-the envelope calculations when quick business decisions have to be made.

To know more about Supply Chain www.zlc.edu.es

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March 9, 2012 · 12:11 am

When too much is as bad as too little

Why your firm should not aim at achieving 99% service level for all products (most likely)

Alejandro Serrano – . Mar 2012 | Spain

Last week I attended a nice talk given by the head of logistics of an e-commerce retailer. His firm holds roughly 10,000 SKUs and serves daily demand to end users from a central distribution center. A characteristic of the industry in which this firm operates is that orders should be satisfied in less than 24 hours. Since suppliers’ lead-times are on the weeks or even months (some are based in Asia), the firm is forced to hold large amounts of inventory to cope with uncertain demand.

The firm has a customer service level goal as high as 99% for all products. This figure, 99%, may be judged as appropriate by some people, but rises at least two questions: 1) why 99% and not, for instance, 90% or 99.9%, and 2) why 99% for all 10,000 products.

The answer to the first question might have to do with the fact that achieving 100% service level is virtually impossible. Therefore (put yourself in the CEO’s shoes,) if you want to provide your customers with an excellent service level, a feasible, close-enough-to-100%, easy-to-remember, and popular figure is 99%. And why not, you want to keep that figure high for all 10,000 SKUs in your warehouse.

Picture: directindustry.com

A key point that is missed here is the fact that service level that maximizes expected profit should at least depend on the price and the cost of each particular item. In fact, inventory theory (or common sense) says that one should increase service level until the marginal benefit of adding an additional unit be exactly as high as the marginal cost of adding that additional unit. Note why this makes sense: given the cost of an specific item, if its market price goes up, the firm should increase the service level for that item. Why? a higher price implies a higher unit margin, and you want to capture that additional margin with higher probability, thus inventory should go up. Likewise, if given a price, the cost of an item increases, service level should be reduced to avoid a higher probability of holding too much inventory (as measured in moneys,) which is mainly driven by obsolescence, insurance, spoilage, and financial costs.

There are other factors that have an impact on service level above and beyond price and cost, such as the so-called salvage cost or goodwill cost. But as a conclusion,  and without entering into details on how to compute optimal service levels, it should be apparent that 1) There is an optimal service level that depends on the margin of the product, which may be below or above 99%; and 2) service level should be computed for each SKU, or type of SKUs, thus defining it for all items in a firm does not make by and large much economic sense.

 To know more about Supply Chain www.zlc.edu.es

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March 8, 2012 · 12:03 pm

Conocimiento y experiencias logísticas innovadoras

Sesión organizada por el IAF y las Cámaras de comercio en Zaragoza

Alejandro Serrano – Mar 2012 | España

“Conocimiento y Experiencias logísticas innovadoras”

El 19 de marzo participo en esta sesión para hablar de la “Supply Chain and Finance Initiative“, puesta en marcha desde el Zaragoza Logistics Center. Haré una breve introducción de qué es ZLC y en qué consiste esta iniciativa. Después ilustraré con un ejemplo cómo cambia el comportamiento de un comprador cuando tiene que preocuparse por el impacto de sus decisiones no sólo en la cuenta de resultados, sino también en el balance de la empresa.

Más información e inscripciones en este enlace.


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March 1, 2012 · 1:52 am

Do not call it Logistics if you mean Supply Chain

The messy shift from Logistics to Supply Chain Management

Alejandro Serrano – . Feb 2012 | Spain

The word Logistics, initially borrowed from a military context, has had to do with the ability to move materials and personnel in an efficient way from one place to another. In a business context, according to the APICS dictionary, its meaning has changed to include additional activities, such as procurement and production. Sure enough, in this context, Logistics is defined as

“The art and science of obtaining, producing, and distributing material and product in the proper place and in proper quantities.” (APICS dictionary on line, accessed in Feb 2012)

However, the CSCMP dictionary does not include production activities, but specifically mentions storage and refers also to services and information:

“The process of planning, implementing, and controlling procedures for the […] transportation and storage of goods, including services, and related information from the point of origin to the point of consumption […] This definition includes inbound, outbound, internal, and external movements.” (CSCMP. Terms and Glossary. Feb 2010)

which is closer to its original meaning, i.e., “just” transportation and storage. it seems that, as the realm of the discipline was increasing an attempt was done to adjust the word logistics to the broader context of application, what explains the broader scope of logistics according to APICS.

Enter Supply Chain

In the early eighties, logistics was not enough to refer to all the increasing types of activities performed by  “logistics” managers, and a new term was coined: “Supply Chain Management”. A strategic flavor was added and the scope was enlarged both longitudinally (from “end to end,” or E2E) and transversely (not only material flows, but also information and cash was considered.) The two aforementioned dictionaries agree on this scope, except for the CSCMP dictionary, which does not mention cash.

“The global network used to deliver products and services from raw materials to end customers through an engineered flow of information, physical distribution, and cash.” (APICS dictionary on line, accessed in Feb 2012)

“The material and informational interchanges in the logistical process stretching from acquisition of raw materials to delivery of finished products to the end user. All vendors, service providers and customers are links in the supply chain.” (CSCMP. Terms and Glossary. Feb 2010)

Since then, the two terms have coexisted, but the evolution of the word Logistics towards Supply Chain (as it can be seen in the definition of Logistics according to APICS) still creates a lot of confusion in industry and academia. In my opinion, the relationship between the two could be defined as follows: “Logistics is the portion of Supply Chain that is concerned with the activities of transportation and storage of parts,” which is line with the CSCMP definition.

Supply Chain, however, is concerned not only with transportation and storage, but with many other key processes, such as demand forecasting, planning, purchasing, collaboration (contracts), outsourcing, facility location (network design), or inventory management (how much and where to hold inventory.)

To learn more about Supply Chain www.zlc.edu.es

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February 4, 2012 · 11:02 am

Enigmatic Serranno numbers

The intriguing property of the Serranno series

Alejandro Serrano

A not-so-well extended myth confers magic  properties on Serranno numbers. These numbers make up the so-called Serranno series, which starts with two ones and subsequent numbers are calculating by adding two times the previous one to the next-to-previous one (e.g., 3 = 2 x 1 + 1, 7 = 2 x 3 + 1) . Therefore, the first ten elements of the series are:

1, 1, 3, 7, 17, 41, 99, 239, 577, 1393

An intriguing property of this series is that the ratio between two consecutive numbers is closer and closer to the so-called silver ratio. The silver ratio is usually denoted by σ (the Greek letter corresponding to ‘s’, after silver and Serranno) and its value is  σ = 1 + √  2 = 2.4142135623731…  For instance, dividing the 20th by the 19th element of the series yields a number with thirteen identical digits: 9,369,319/ 3,880,899 = 2.4142135623730….

The magic nature of Serranno numbers stems from this outstanding property, since the silver ratio is described as one of the most mysterious numbers in nature.

It is believed that Serranno followers use a drop of water as a secret symbol to identify themselves. The drop of water is made up by combining a circle and a right triangle whose edges are tanget to the circle (see figure below.)

The drop of water was chosen because it hides the silver ratio. Indeed, dividing the distance between the top corner to the bottom of the drop by the radius of the circle exactly yields the silver ratio!

The silver ratio is also related to beauty in nature. From small to large water drops, the silver ratio is supposed to confer beauty on the objects that contain it.

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Order your personalized “Enigmatic Series” here: @SerranoAlej. The kit includes a new series nominated after your name, your own golden-type ratio (choose among several precious metals), a personal geometric shape, and an entry in wikipedia.

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February 2, 2012 · 10:06 am

Fibonacci numbers: not that magic

An explanation of the “magic” property of the Fibonacci series

Alejandro Serrano

A well extended myth confers magic  properties on Fibonacci numbers. These numbers make up the so-called Fibonacci series, which starts with two ones and subsequent numbers are calculating by adding the two previous ones. Therefore, the first ten numbers of the series are: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55. An intriguing property of this series is that the ratio between two consecutive numbers is closer and closer to the so-called golden ratio (the golden ratio is usually denoted by φ and its value is   (1 + √ 5)/ 2 = 1.618033989… ) For instance, dividing the 20th by the 19th element of the series yields a number with eight identical digits: 6,765 / 4,181 = 1.61803396…,

The magic nature of Fibonacci numbers stem from this outstanding property, since the golden ratio is described as one of the most mysterious numbers in nature (see Mario Livio’s cite below.)

“Some of the greatest mathematical minds of all ages, from Pythagoras and Euclid in ancient Greece, through the medieval Italian mathematician Leonardo of Pisa and the Renaissance astronomer Johannes Kepler, to present-day scientific figures such as Oxford physicist Roger Penrose, have spent endless hours over this simple ratio and its properties. But the fascination with the Golden Ratio is not confined just to mathematicians. Biologists, artists, musicians, historians, architects, psychologists, and even mystics have pondered and debated the basis of its ubiquity and appeal. In fact, it is probably fair to say that the Golden Ratio has inspired thinkers of all disciplines like no other number in the history of mathematics.

(Mario Livio. The Golden Ratio: The Story of Phi, The World’s Most Astonishing Number, p.6

It is believed that the Pythagoreans used a 5-pointed star, called a pentagram, as a secret symbol to identify themselves. The pentagram was chosen because it hides the golden ratio everywhere.

The golden ratio is also related to beauty in nature. From sunflowers to shells, the golden ratio is supposed to confer beauty on the objects that contain it. Moreover, several dimensions in human creations are also driven by the golden number. For instance, many of the proportions of the Parthenon in Athens exhibit the golden ratio. A contemporary example is the (golden) ratio between the length and the height of a credit card.

The truth of the matter is that this “magic” property of the Fibonacci numbers can be explained with a bit of math. In fact, the numbers in the Fibonacci series can be calculated using the following expression

For instance, if n=4:

The eigenvalues of the matrix on the right-hand side of this expression turn out to be (1 + √ 5)/ 2 and  (1 – √ 5)/ 2 , i.e., φ and -0.61803… respectively. Note that φ > 1 and 0 < -0.61803… < 1. That is why, after a number of iterations, the impact of the second eigenvalue on the vector on the letf-hand side is negligible . Therefore, for n sufficiently large, only the first eigenvalue, φ, is relevant when calculating the following element of the series, and then

which explains the supposed-to-be “magic” nature of the Fibonacci numbers.

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