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Education & Academic

An Intriguing Identity: Connecting Distinct and Complete Integer Partitions

Number theory is a very old subject that in modern times has branched into various large areas. One of these is additive number theory, with problems like this: when is a prime the sum of two squares? Primes are part of the more classical area now called multiplicative number theory, so as this problem of Fermat’s indicates, the two areas are intimately connected. The problem I discuss in this blog is a mix of additive and multiplicative number theory, with a dash of linear algebra.

George Beck, Document & Media Systems
Education & Academic

On the Polygon Front Lines: Visualizing the Amplituhedron with the Wolfram Language

It’s rare to hear polygons mentioned in a physics class, even in higher education. This may seem unexpected given the fundamental role they play in mathematics. However, over the last few years, polygons have come to the front line in many areas of theoretical physics, helping us understand the laws of nature with their astonishing beauty.

This is particularly true in the field of particle physics, where a new geometrical object has been found to be connected to particle dynamics: the amplituhedron. It represents a novelty not only in physics but also in mathematics, generalizing the concept of a convex polygon. In this blog post, I will first discuss its relation to particle physics, and then how to visualize its geometry using the Wolfram Language.

Gabriele Dian, Visiting Scholar, Algorithms R&D SAGEX, Early Stage Researcher, Durham University, UK
Education & Academic

Announcing the Rule 30 Prizes

The Story of Rule 30 How can something that simple produce something that complex? It’s been nearly 40 years since I first saw rule 30—but it still amazes me. Long ago it became my personal all-time favorite science discovery, and over the years it’s changed my whole worldview and led me to all sorts of […]

Stephen Wolfram
Education & Academic

Building Uniform Polyhedra for Version 12

Since I started working at Wolfram, I’ve been a part of several different projects. For Version 12, my main focus was replicating models of the uniform polyhedra with the Wolfram Language to ensure that the data fulfilled certain criteria to make our models precise, including exact coordinates, consistent face orientation and a closed region in order to create a proper mesh model of each solid.

Working with visual models of polyhedra is one thing, but analyzing them mathematically proved to be much more challenging. Starting with reference models of the polyhedra, I found that the Wolfram Language made mathematical analysis of uniform polyhedra particularly efficient and easy.

But first, what really are polyhedra, and why should we care? With Version 12, we can explore what polyhedra are and how they’ve earned their continued place in our imaginations.

Keren Garcia, Algorithms R&D
Education & Academic

How I Used Last-Mover Advantage to Make Money: An Exploration of Yahtzee and Coin Flipping

This week, I won some money applying a mathematical strategy to a completely unpredictable gambling game. But before I explain how, I need to give some background on last-mover advantage.

Some time ago, I briefly considered doing some analysis of the dice game Yahtzee. But I was put off by the discovery that several papers (including this one) had already enumerated the entire game state graph to create a strategy for maximizing the expected value of the score (which is 254.59).

However, maximizing the expected value of the score only solves the solo Yahtzee game. In a competitive game, and in many other games, we are not actually trying to maximize our score—we are trying to win, and these are not always the same thing.

Jon McLoone, Director, Technical Communication & Strategy
Current Events & History

Drawing on Autopilot: Automated Plane (Geometry) Illustrations from The American Mathematical Monthly

Version 12 of the Wolfram Language introduces the functions GeometricScene, RandomInstance and FindGeometricConjectures for representing, drawing and reasoning about problems in plane geometry. In particular, abstract scene descriptions can be automatically supplied with coordinate values to produce diagrams satisfying the conditions of the scene. Let’s apply this functionality to some of the articles and problems about geometry appearing in the issues of The American Mathematical Monthly from February and March of 2019.

Dan McDonald, Lead Developer, Synthetic Geometry Project
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
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