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Mathematics

Education & Academic

Shattering the Plane with Twelve New Substitution Tilings Using 2, φ, ψ, χ, ρ

Similar Triangle Dissections

Version 12 of the Wolfram Language introduces solvers for geometry problems. The documentation for the new function GeometricScene has a neat example showing the following piece of code, with GeometricAssertion calling for seven similar triangles:

[Wolfram_Notebook_Download]
&#10005 o=Sequence[Opacity[.9],EdgeForm[Black]];plasticDissection=RandomInstance[GeometricScene[{a,b,c,d,e,f,g},{ a=={1,0},e=={0,0},Line[{a,e,d,c}], p0==Polygon[{a,b,c}], p1==Style[Polygon[{b,d,c}],Orange,o], p2==Style[Polygon[{d,f,e}],Yellow,o], p3==Style[Polygon[{b,f,d}],Blue,o], p4==Style[Polygon[{g,f,b}],Green,o], p5==Style[Polygon[{e,g,f}],Magenta,o], p6==Style[Polygon[{a,e,g}],Purple,o], GeometricAssertion[{p0,p1,p2,p3,p4,p5,p6},"Similar"]}],RandomSeeding->28]
Ed Pegg Jr, Scientific Information Editor, Scientific Information Group
Education & Academic

The Data Science of MathOverflow

New Archive Conversion Utility in Version 12

Soon there will be 100,000 questions on MathOverflow.net, a question-and-answer site for professional mathematicians! To celebrate this event, I have been working on a Wolfram Language utility package to convert archives of Stack Exchange network websites into Wolfram Language entity stores.

The archives are hosted on the Internet Archive and are updated every few months. The package, although not yet publicly available, will be released in the coming weeks as part of Version 12 of the Wolfram Language—so keep watching this space for more news about the release!

Andrew Steinacher, Lead Developer, Wolfram|Alpha Scientific Content
Education & Academic

The Story of Spikey

Spikeys Everywhere We call it “Spikey”, and in my life today, it’s everywhere: It comes from a 3D object—a polyhedron that’s called a rhombic hexecontahedron: But what is its story, and how did we come to adopt it as our symbol?

Stephen Wolfram
Education & Academic

Interning at Wolfram: My Regeneration as a Theoretical Scientist

How does it feel to be an intern at Wolfram?

Most undergraduate college students chase opportunities for internships in New York, Miami, Seattle and particularly San Francisco at very young but large high-tech companies like Uber, Pinterest, Quora, Expedia and similar internet companies. These companies offer the best salaries, perks, bosses, coworkers, catered lunches and other luxurious amenities available in such large cities. You would seldom hear about any of these people pursuing opportunities in small, lesser-known towns like Ames, Iowa, or Laramie, Wyoming—and Champaign, Illinois, where Wolfram Research is based, is one of those smaller towns.

Many students want to go into computer science, as it’s such a rapidly developing field. They especially want to work in those companies on the West Coast. If you’re in a different field, like natural science, you might think there’s nothing beyond on-campus research for work experience. At Wolfram Research, though, there is.

Parik Kapadia, Algorithms R&D
Current Events & History

As of Today, the Fundamental Constants of Physics (c, h, e, k, NA) Are Finally… Constant!

This morning, representatives of more than 100 countries agreed on a new definition of the base units for all weights and measures. Here’s a picture of the event that I took this morning at the Palais des Congrès in Versailles (down the street from the Château):

An important vote for the future weights and measures used in science, technology, commerce and even daily life happened here today. This morning’s agreement is the culmination of at least 230 years of wishing and labor by some of the world’s most famous scientists. The preface to the story entails Galileo and Kepler. Chapter one involves Laplace, Legendre and many other late-18th-century French scientists. Chapter two includes Arago and Gauss. Some of the main figures of chapter three (which I would call “The Rise of the Constants”) are Maxwell and Planck. And the final chapter (“Reign of the Constants”) begins today and builds on the work of contemporary Nobel laureates like Klaus von Klitzing, Bill Phillips and Brian Josephson.

I had the good fortune to witness today’s historic event in person.

Michael Trott, Chief Scientist, Wolfram|Alpha Scientific Content
Education & Academic

Prepare for AP Calculus and More with Wolfram U

Today I am proud to announce a free interactive course, Introduction to Calculus, hosted on Wolfram's learning hub, Wolfram U! The course is designed to give a comprehensive introduction to fundamental concepts in calculus such as limits, derivatives and integrals. It includes 38 video lessons along with interactive notebooks that offer examples in the Wolfram Cloud—all for free. This is the second of Wolfram U's fully interactive free online courses, powered by our cloud and notebook technology.

This introduction to the profound ideas that underlie calculus will help students and learners of all ages anywhere in the world to master the subject. While the course requires no prior knowledge of the Wolfram Language, the concepts illustrated by the language are geared toward easy reader comprehension due to its human-readable nature. Studying calculus through this course is a good way for high-school students to prepare for AP Calculus AB.

Devendra Kapadia, Manager of Calculus and Algebra, Algorithms R&D
Education & Academic

Mathematics Genealogy Project: Computational Exploration in the Wolfram Language

The Mathematics Genealogy Project (MGP) is a project dedicated to the compilation of information about all mathematicians of the world, storing this information in a database and exposing it via a web-based search interface. The MGP database contains more than 230,000 mathematicians as of July 2018, and has continued to grow roughly linearly in size since its inception in 1997.

In order to make this data more accessible and easily computable, we created an internal version of the MGP data using the Wolfram Language’s entity framework. Using this dataset within the Wolfram Language allows one to easily make computations and visualizations that provide interesting and sometimes unexpected insights into mathematicians and their works. Note that for the time being, these entities are defined only in our private dataset and so are not (yet) available for general use.
Aaron Enright, Senior Data Scientist, Wolfram|Alpha Socioeconomic Content Eric Weisstein, Senior Researcher, Wolfram|Alpha Scientific Content
Education & Academic

Big O and Friends: Tales of the Big, the Small and Every Scale in Between

One of the many beautiful aspects of mathematics is that often, things that look radically different are in fact the same—or at least share a common core. On their faces, algorithm analysis, function approximation and number theory seem radically different. After all, the first is about computer programs, the second is about smooth functions and the third is about whole numbers. However, they share a common toolset: asymptotic relations and the important concept of asymptotic scale.

By comparing the “important parts” of two functions—a common trick in mathematics—asymptotic analysis classifies functions based on the relative size of their absolute values near a particular point. Depending on the application, this comparison provides quantitative answers to questions such as “Which of these algorithms is fastest?” or “Is function a good approximation to function g?”. Version 11.3 of the Wolfram Language introduces six of these relations, summarized in the following table.

Itai Seggev, Senior Kernel Developer, Algorithms R&D
Education & Academic

Getting to the Point: Asymptotic Expansions in the Wolfram Language

Asymptotic expansions have played a key role in the development of fields such as aerodynamics, quantum physics and mathematical analysis, as they allow us to bridge the gap between intricate theories and practical calculations. Indeed, the leading term in such an expansion often gives more insight into the solution of a problem than a long and complicated exact solution. Version 11.3 of the Wolfram Language introduces two new functions, AsymptoticDSolveValue and AsymptoticIntegrate, which compute asymptotic expansions for differential equations and integrals, respectively. Here, I would like to give you an introduction to asymptotic expansions using these new functions.
Devendra Kapadia, Manager of Calculus and Algebra, Algorithms R&D
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