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# Date Archive: 2013 March

## From Close to Perfect—A Triangle Problem

RootApproximant can turn an approximate solution into a perfect solution, such as for a square divided into fifty 45°-60°-75° triangles. A square can be divided into triangles, for example by connecting opposite corners. It's possible to divide a square into seven similar but differently sized triangles or ten acute isosceles triangles. Classic puzzles involve cutting a square into eight acute triangles, or twenty 1 - 2 - √5 triangles. The last image uses 45°-60°-75° triangles, but one triangle has a flaw. It's easy to divide a square with similar right triangles. Can a square be divided into similar non-right triangles? In his paper "Tilings of Polygons with Similar Triangles" (Combinatorica, 10(3), 1990 pp. 281–306), Laczkovich proved exactly three triangles provided solutions, with angles 22.5°-45°-122.5°, 15°-45°-120°, and 45°-60°-75°. I read his paper to try to make an image for the 45°-60°-75° case, but his construction was complex, and seemed to require thousands of triangles, so I tried to find my own solutions. All my attempts had flaws, such as the last image above, so I made a contest out of it: \$200, minus a dollar for every triangle in the solution.

## The Mathematics of Queues

Waiting in line is a common, though not always pleasant, experience for us all. We wait patiently to be served by the next free teller at a bank, clear the security check at an airport, or be answered by technical support when we call a phone service provider. At a more abstract level, these waiting lines, or queues, are also encountered in computer and communication systems. For example, every email you send is broken up into a series of packets. Each packet is then sent off to its destination by the best available route to avoid the queues formed by other packets in the network. Hence, queues play an important role in our lives, and it seems worthwhile to spend some time understanding their dynamics, with a view to answering questions such as, "How many tellers does your bank need to provide good customer service?" or "How can you speed up the security check?" or "On average, how long will you have to wait for technical support?" My purpose in writing this post is to give a gentle introduction to queueing theory, which attempts to answer such questions, using new functions that are available in Mathematica 9. Queueing theory has its origins in the research of the Danish mathematician A. K. Erlang (1878–1929). While working for the Copenhagen Telephone Company, Erlang was interested in determining how many circuits and switchboard operators were needed to provide an acceptable telephone service. This investigation resulted in his seminal paper "The Theory of Probabilities and Telephone Conversations," which was published in 1909. Erlang proved that the arrivals for such queues can be modeled as a Poisson process, which immediately made the problem mathematically tractable. Another major advance was made by the American engineer and computer scientist Leonard Kleinrock (1934–), who used queueing theory to develop the mathematical framework for packet switching networks, the basic technology behind the internet. Queueing theory has continued to be an active area of research and finds applications in diverse fields such as traffic engineering and hospital emergency room management.

## Registration Is Open for the Mathematica Summer Camp 2013

It’s that time of year again! Time to apply for the Mathematica Summer Camp 2013! The camp is being held at Bentley University in Waltham, Massachusetts, July 7–19. Students will have the opportunity to learn Mathematica’s computing language, work with Wolfram mentors, and interact with other students with similar interests. By the end of camp, each student will have created his or her very own Mathematica program! Last year the camp was a great success, and students worked on a variety of projects, from modeling diseases to stereographic projection of platonic solids.
Announcements & Events

## Using Mathematica Enterprise Edition to Create Professional Apps, Tools, and Reports

For more than two decades, Mathematica users have been using our technology to solve some of their most difficult problems. And when they find solutions, they need to communicate them to managers, colleagues, and clients. Like many other organizations, we also need to effectively communicate concepts when we design new technologies, and we need to make decisions quickly and efficiently. In the past, our own technology lacked a means of distributing results that could be viewed with a free document player, in which users could enter their own data, and that could update interactively and in real time. We made great strides in addressing all of those issues with the introduction of the Computable Document Format (CDF). CDF is a computation-powered knowledge container that supports all sorts of applications, dashboards, and reports.
Announcements & Events

## Register Now for the First European Wolfram Technology Conference!

Our first ever European Wolfram Technology Conference will be held June 11–12 in Frankfurt, Germany (pre-conference training on June 10 in Friedrichsdorf). Join Wolfram developers and experts as we look at how combined computation expertise across all our technologies—Wolfram|Alpha, Computable Document Format, Wolfram SystemModeler, Wolfram Workbench, and of course Mathematica—can empower you and your organization in research, development, deployment—and progress.
Computation & Analysis

## Mathematica’s Role in Powering Energy Saving Solutions

Using Mathematica and other Wolfram technologies, Joseph Hirl, founder of Agilis Energy, has developed a new approach to energy analytics that is helping building owners and energy equipment suppliers around the world cut energy consumption and costs. At the core of the company's success is its Mathematica-based dynamic energy analysis application, which gives the full picture of a building's performance, measures the impact of potential operational changes, and quantifies the results. About Mathematica's role in the development of the tool and the Agilis business, Hirl says, "The flexibility of Mathematica is tremendous. Our ability to build and develop this program with a lean staff has allowed us to build out a substantial business." The application, which has now been used at more than 800 sites in at least 12 different industries, begins with data streams, including high-interval smart meter data as well as Mathematica's built-in WeatherData. It then applies sophisticated statistics and dynamic visualization functionality to generate what Hirl calls an "MRI of a building," a dynamic interface with a simulation of the building's energy use and demand and forecasting and benchmarking tools.