Studying this and related problems has led me to examining many properties of the Fibonaccis. The following result is well-known, but I hope you’ll enjoy the method of proof. Annoyingly, for this result we must set f_1 = 1, f_2 = 1 and f_{n+1} = f_n + f_{n-1} (if we had two 1’s we of course could not have a unique decomposition for all numbers).

**Theorem: The sum of the squares of the first n Fibonacci numbers is the product of the n-th and (n+1)-st Fibonacci numbers. In other words, f_1^2 + … + f_n^2 = f_n * f_{n+1}.**

A beautiful way to prove this is through the Fibonacci spiral, which provides our **third** equivalent definition for the Fibonaccis. We start with a 1×1 square. We place another 1×1 square along a side, forming a 2×1 rectangle. We now place a 2×2 square next to that, forming a 3×2 rectangle. We now add a 3×3 square, giving us a 5×3 rectangle. The pattern continues. Notice that at each stage our next choice is forced upon us, and they’re the Fibonacci numbers.

We can now prove the theorem by using one of the most powerful proof techniques in combinatorics: calculate something two different ways. Let’s say we’ve spiraled out and included the first n Fibonacci numbers. We obtain a rectangle of dimensions F_{n+1} x F_n; thus the area of this rectangle is F_{n+1} * F_n. We can, however, calculate the area another way. We built it by adding squares whose side lengths were F_1, F_2, …, F_n, and thus the area is also equal to F_1^2 + F_2^2 + … + F_n^2, which completes the proof!

I built this out of fuse beads with my daughter Kayla and my son Cameron. It was a lot of fun, and led to a nice visual proof that I can pack up and take with me to classes and schools.

For more reading / viewing, here are some clickable links:

- Professor Arthur Benjamin talks about the Fibonacci numbers (and gives this proof of the above theorem).
- Kologlu, Kopp, Miller and Wang’s paper on Fibonacci numbers and Zeckendorf decompositions.
- Survey article on Zeckendorf decompositions and the power of the combinatorial perspective (Miller and Wang).
- Talk by Miller on the Fibonacci numbers and Zeckendorf’s theorem (and generalizations).
- Perler fuse beads (note it took us four BIG baseboards to do this!).

(Photos from Akershus Castle: John Nash, Williams Trustee Joey Horn, and Frank Morgan; Morgan and Alicia Nash, with King Harald in between in the background.)

In a brilliant citation, John Rognes, chair of the Abel Committee, began with Newton and differential equations. Many processes, such as the Brownian motion of small particles, the stock market, and turbulent fluid flow require the notion of a weak, nondifferentiable solution to a differential equation, such as the generalized functions or distributions of de Rham, defined by their integrals against smooth functions. Rognes said:

“At first, a weak solution only exists in a virtual sense, through its interaction with other quantities. To become useful for applications, and to be accessible through numerical calculations with a computer, it is necessary to know that the weak solution is real, and that is generalized rates of change are actual rates of change.

“The regularity results of Nirenberg and Nash provide this kind of knowledge, with mathematical certainty.”

In my Science Lecture on “Soap Bubbles and Mathematics,” I observed that it was Nash’s famous isometric embedding theorem that initially made soap bubble theory applicable to other, curvy universes.

P.S. 27 May 2015. Just received this message from the chair of the Abel Committee:

The tragic accident that took the lives of John and Alicia Nash has left us all stunned. The president of the Norwegian Academy, and I, have posted a statement at the Abel prize webpage, but in a situation like this words do not suffice. There will be a private funeral ceremony tomorrow, Thursday, only for the closest family. A memorial service will take place later. Some of us from the Academy will come to Princeton then, to honor their memory. Louis Nirenberg was one of the last to see John and Alicia Nash. He spent most of last week very close to them, and I think the news must have been especially shocking to him. At the moment I think he is receiving an overwhelming number of calls and emails, including many requests from the press. I hope that this flood halts quickly, so that he can return to more normal conversations and interactions with friends and colleagues. Yours sincerely, John Rognes

]]>Colloquium Attendance:

Mary Gong

Goldberg Prize for Best Colloquium:

Andrew Best, Ben Hoyle

Benedict Prize for Outstanding Sophomore:

1st: David Burt, Nina Pande, Yuanchu Dang

2nd: Alex Kastner, Sarah Fleming

Witte Problem Solving Prize:

Blake Mackall

Wyskiel Award for Math Education:

Greg Ferland, Katie Bennett

Beaver Prize for Extraordinary Contribution to the Community:

Mary Gong

Morgan Prize for Accomplishment in Applied Mathematics:

Sam Petti

Kozelka Prize for Excellence in Statistics:

Nate McCue

Rosenburg prize for Excellence in Mathematics:

Jesse Freeman, Issac Loh

]]>At the start of the year, I was made chair of a committee to reimagine the Williams Course Catalog for 2015-2016. We were tasked with coming up with a “scheme and shape” for how the course catalog can be organized, and how different kinds of information can be visualized. Instead of providing a list of design ideas, we decided to offer three concrete outputs, all focusing on the 2015-2016 academic year.

Currently, courses are available in the online catalog (and as downloadable PDF files), along with a modest search engine. The design here is defined and partitioned by departments and programs, which serve in providing administrative structure and accrediting with majors and concentrations. Under this design, the courses themselves are diminished.

Our goal is simple: elevate the courses through good design. The three works below serve as (possibly, temporary) experimental works, aimed at helping reimagine the display of courses at Williams. It is important to note that the current version of the catalog (available online at the Registrar’s Website) will be unchanged in format and style for next year, enabling the ideas below to serve as supplements rather than replacements.

**NEWSPAPER SPREAD**

In a simple, 4-page newspaper-spread format (that can quickly be scanned), we showcase every course offered in 2015-2016 based on two inputs: Title and (a 140-character teaser) Description. The courses are randomly arranged, with no other information, allowing a global glimpse of the entire academic year. The newspaper was available approximately two weeks before courses was officially listed on the Registrar’s webpage. This was intentionally done, encouraging students and faculty to pursue the paper to get a glimpse of the future.

**PRINTED BOOK**

Funding for a printed version of the catalog had previously been budgeted for the 2015-2016 academic year. Printing the catalog (from the full PDF file that’s currently available online) would have yielded a bulky, almost unreadable version. By printing everything from honors programs to department policies, an opportunity cost is incurred: The more text we allow the students to see, the less emphasis is placed on courses. Thus, the current print version:

- Shows only courses offered in 2015-2016.
- Limits descriptions of courses to 500 characters.
- Does not include thesis and independent study courses.
- Only includes a short (1500 character) description of each department/program.
- Had a hard deadline of March 20, 2015 submission to the Registrar’s office.

The courses at Williams are robust and diverse, many of which do not naturally fit inside classical areas of study. To segment them as such leads to cross-listing tensions, and repeating of course descriptions in the different departments.

The heart of the new catalog is the set of courses, arranged in alphabetical order (based on title). A front matter is provided (with some basic contents from the Registrar’s office), and an index is given, based on major/concentration, pointing to the courses by their page numbers. Since the current body of students has never held a printed catalog, offering them a unique (enjoyable, readable) experience that is complemented by the online version seems ideal.

**ONLINE Course Explorer**

We also offered a distinct online version of the catalog (separate and distinctive from the current online version) which highlights methods of exploring the catalog in simple and intuitive ways. We called this the “Course Explorer”. Words in the title and description, along with other metadata tags, will be used to make connections between courses, displaying them in an interactive manner. The main entryway to this online version will be a search bar, allowing search results across disciplines and divisions to naturally appear.

Moreover, this is integrated to a “calendar” view, allowing students to manage interesting courses in a visual manner based on time conflicts across Fall and Spring semesters.

All of these versions are experiments and ideas fleshed out in working models. With something tangiable to hold onto, the hope is to use these ideas as starting points to make the Catalog more alive to both students and faculty.

]]>1. Actuary

2. Audiologist

3. Mathematician

4. Statistician

5. Biomedical Engineer

6. Data Scientist

For more information see also the original post by CarrerCast.com.

]]>Recently I had the great fortune to work with Christopher Hammond and Warren Johnson of Connecticut College. We just published a note “The James Function” (the link is to the publicly available arXiv version) in Mathematics Magazine: *We investigate the properties of the James function, associated with Bill James’s so-called “log5 method,” which assigns a probability to the result of a game between two teams based on their respective winning percentages. We also introduce and study a class of functions, which we call Jamesian, that satisfy the same a priori conditions that were originally used to describe the James function.*

I find these terrific problems to study. They’re fun, they lead to great math, and they’re of interest to many. For students here, there are a lot of great colloquium or thesis topics related to problems along these lines…..

Note: Image of Pythagoras at the Bat created by Theresa McCracken (http://mchumor.com/thekomic/math-cartoons-pg1.html).

]]>During my sabbatical at Stanford last year, I spent a part of my time having coffee with Owen Schuh, an artist in the SF bay area. The goal of this collaboration was to engage mathematics and the visual arts in a direct manner, with concrete outputs, that does not insult either field. In other words, new mathematical questions need to be formulated and new artworks need to be produced for the success of this venture. Our work should be a true collaboration, with the mathematician involved in the drawings and the artist involved in the mathematics.

In the end, we created a triptych of works, titled **Cartography of Tree Space**, involving acrylic, watercolor, and graphite on 108cm x 108cm wood panels. Our work was picked up for an inaugural gallery showing in Germany by Satellite Berlin. One of the pieces, titled “Underground”, is shown below.

The particular object of our study is a configuration space of phylogenetic trees, originally made famous by the work of Billera, Holmes, and Vogtmann. Each point in our space corresponds to a specific geometric, rooted tree with five leaves, where the internal edges of the tree are specified to be nonnegative numbers. From a global perspective, this “tree space” is made of 105 triangles glued together along their edges, where three triangles glue along each edge. This results in 105 distinct edges, and 25 distinct corners. This space of trees appears in numerous areas of mathematics, including algebraic topology, enumerative combinatorics, geometric group theory, and biological statistics.

Although the space is only two-dimensional (made of numerous triangles), the natural world for this tree space to inhabit is in four-dimensions, where the full symmetry of its structure will be made transparent. Our goal was not to describe the space in mathematical terms. Instead, we wanted to describe what it feels like to live in this tree space, to inhabit it as a world like any other world. To this end, we use the world of cartography and map making to invite the viewer to understand tree space.

The collaboration took place during an 18-month timeframe, from September 2013 until February 2015. Roughly, the first six months were spent in understanding the goals of the project and choosing a point of collaboration. The second six months were spent at coffee shops and studios, where Owen and I would meet and go through both mathematics and sketches. The final six months were focused on crafting extreme details and formulating a unifying vision to the project.

In all of this, the process behind the mathematics and the art was quite similar. Ideas were conjectured, tested, and evaluated, both visually and analytically. And there was a sense of incredible freedom to explore these worlds, with a strong instinct guiding the collaborators, as to the right road to pursue.

]]>*The past fifty years have seen computing move from a fringe activity in universities to a central part of academic life. Today’s university students never knew a world without personal computers, networks, email, and the web. Nowadays, the computers and software that we use in universities are commercial products, but this was not always the case. When the computer industry failed to meet our needs, we took the initiative. For thirty years academic computing diverged from the mainstream. We built our own state-of-the-art systems and ran them successfully. Universities led the development of timesharing and local area (campus) networks. The distributed computing, email, national networks, and the web that everybody uses today are direct descendants of systems that universities and scientific researchers built for themselves. As a student, faculty member, and administrator, I lived through many of these developments and for seventeen years I was in charge of computing at two of the leading universities, Dartmouth College and Carnegie Mellon. This talk tells the story.*

Events continue with a lunch at the Faculty House at noon and talks by the national PBK president and secretary, our visiting scholar William Arms of Cornell, the president of the NY PBK Association, and others from 4-8pm in Griffin 4 (refreshments and dinner provided). All are welcome; if possible please email Steven Miller (sjm1@williams.edu) so we can get an accurate headcount. A complete schedule of the talks is online here: http://web.williams.edu/Mathematics/sjmiller/public_html/pbk/ (videos of the talks will be posted later on YouTube and linked to this page).

]]>- iBerkshires.com: pre-build article
- iBerkshires.com: story on successful build
- Berkshire Eagle: story on successful build
- Berkshire Eagle: photographs of successful build (zipped photos superstarsuccessfulbuild2015)
- Berkshire Eagle: time lapse video of build (camera above table 1)
- WABC story: http://wamc.org/post/piece-piece-williams-students-embark-light-speed-voyage
- Lego instructions: click here

**We hope to have other events along these lines in the future — if you’re interested email Steve Miller at sjm1@williams.edu.**