February 12, 2020 — Ed Pegg Jr, Editor, Wolfram Demonstrations Project

New Bound for Sparse Rulers Proof using the Wolfram Language

The sparse ruler problem has been famously worked on by Paul Erdős, Marcel J. E. Golay, John Leech, Alfréd Rényi, László Rédei and Solomon W. Golomb, among many others. The problem is this: what is the smallest subset of so that the unsigned pairwise differences of give all values from 1 to ? One way to look at this is to imagine a blank yardstick. At what positions on the yardstick would you add 10 marks, so that you can measure any number of inches up to 36?

Another simple example is of size 3, which has differences , and . The sets of size 2 have only one difference. The minimal subset is not unique; the differences of also give .

Part of what makes the sparse ruler problem so compelling is its embodiment in an object inside every schoolchild’s desk—and its enduring appeal lies in its deceptive simplicity. Read on to see precisely just how complicated rulers, marks and recipes can be.

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January 9, 2020 — George Beck, Document & Media Systems

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.

An Intriguing Identity: Connecting Distinct and Complete Integer Partitions

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November 21, 2019
Gabriele Dian, Visiting Scholar, Algorithms R&D
SAGEX, Early Stage Researcher, Durham University, UK

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.

Visualizing the amplituhedron

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October 1, 2019 — Stephen Wolfram

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 science, technology, philosophy and more.

But even after all these years, there are still many basic things we don’t know about rule 30. And I’ve decided that it’s now time to do what I can to stimulate the process of finding more of them out. So as of today, I am offering $30,000 in prizes for the answers to three basic questions about rule 30.

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July 25, 2019 — Keren Garcia, Algorithms R&D

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.

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July 2, 2019 — Jon McLoone, Director, Technical Communication & Strategy

How I Used Last-Mover Advantage to Make Money

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.

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April 4, 2019 — Dan McDonald, Lead Developer, Synthetic Geometry Project

RandomInstance

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.

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March 7, 2019 — Ed Pegg Jr, Editor, Wolfram Demonstrations Project

The sqrt(χ) substitution tiling fractal

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:

Sqrt(ρ) substitution tiling
&#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]

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February 1, 2019 — Andrew Steinacher, Lead Developer, Wolfram|Alpha Scientific Content

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!

MathOverflow

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December 28, 2018 — Stephen Wolfram

Wolfram’s Spikey logo—a flattened rhombic hexecontahedron

Spikeys Everywhere

We call it “Spikey”, and in my life today, it’s everywhere:

Stephen Wolfram surrounded by Spikeys

It comes from a 3D object—a polyhedron that’s called a rhombic hexecontahedron:

3D rhombic hexecontahedron

But what is its story, and how did we come to adopt it as our symbol?

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