Mathematics and the machines

The Chicago Tribune, Opinion Editorial (September 21, 2023)

Over the past 50 years, math has embraced the power of machines, making mathematicians valued and coveted in nearly every sector of the job market. But with the rise of AI, we are finding ourselves ill-prepared for the heavy price and heartbreaking consequences of this dangerous partnership.

2.5D SIGNAGE FROM SHEET MATERIAL WITH ORTHOGONAL CUTS AND FOLDS

Proceedings of the ASME International Design Engineering Technical Conferences and Computers and Information in Engineering Conference (August 2023)

Erik D. Demaine (MIT), Martin L. Demaine (MIT), Perla Myers (USD), Alfonso Parra Rubio (MIT)

Motivated by making signs with depth from bendable but thick sheet material, we analyze 2.5D structures manufacturable from a rectangular sheet by orthogonal cuts and folds and develop two fonts that enable textual signs within this family of designs.

Unfoldings and Nets of Regular Polytopes

Computational Geometry: Theory and Applications 111 (2023) 1 - 10

Matt Harvey (University of Virginia, Wise)

Over a decade ago, it was shown that every edge unfolding of the Platonic solids was without self-overlap, yielding a valid net.  We consider this property for polytopes in higher dimensions. In particular, three classes of regular polytopes exist: simplex, cube, orthoplex. It was recently proven that all unfoldings of the n-cube yield nets.  We extend this property to the n-simplex and the 4-orthoplex but demonstrate its failure for any orthoplex of higher dimension.

Colorful Graph associahedra

Journal of Combinatorics 14 (2023) 369 - 398

Mia Smith (Michigan)

Given a graph G, the graph associahedron is a simple convex polytope whose face poset is based on the connected subgraphs of G. With the additional assignment of a color palette, we define the colorful graph associahedron, show it to be a collection of simple abstract polytopes, and explore its properties.

CURATING MATHEMATICS FOR THE 21st CEntury

Notices of the American Mathematical Society 69 (2022) 1004 - 1007

With the tremendous growth in data acquisition and analysis, this century has brought about a phenomenal resurgence in mathematics through a computational lens.   The pure versus applied partition has given way to theory versus computation. A great benefit to this process-based approach to mathematics curation is that it is discipline independent: every subfield of mathematics can play the game. Moreover, our need for data and its acquisition will only increase in the 21st century and we need to be prepared for the consequences that come with technological entanglements.  

Visualizing and unfolding nets of 4-Polytopes

Symposium on Computational Geometry:  Media (2022)

Matt Harvey (University of Virginia, Wise), Sam Zhang (University of Colorado, Boulder)

Over a decade ago, it was shown that every edge unfolding of the Platonic solids was without self-overlap, yielding a valid net. Recent work has extended this property to their higher-dimensional analogs: the 4-cube, 4-simplex, and 4-orthoplex. We present an interactive visualization that allows the user to unfold these polytopes by drawing on their dual 1-skeleton graph.

UNfolding the simplex and orthoplex

European Workshop on Computational Geometry (2022)

Matt Harvey (University of Virginia, Wise)

Over a decade ago, it was shown that every edge unfolding of the Platonic solids was without self- overlap, yielding a valid net. We consider this property for regular polytopes in higher dimensions, notably the simplex, the cube, and the orthoplex. It was recently proven that all unfoldings of the n-cube yield nets. We show that this property holds for the n-simplex and the 4-orthoplex but fails for any orthoplex of higher dimension.

A vacation from useful math

The Los Angeles Times, Opinion Editorial (August 22, 2021)

Mathematicians have done a good job of showcasing the usefulness of mathematics. Ironically, this very feature of math is its greatest weakness, for touting the usefulness of math for building spaceships makes one excited about the spaceships, not the math. Happily, we can unlock mathematical pleasures by seeking out the unexplored, the unknown, the undiscovered.

MAge merlin’s unsolved mathematical mysteries

MIT Press (hardcover 2020, paperback 2021)

Matt Harvey (University of Virginia, Wise)

Most people think of mathematics as a set of useful tools designed to answer analytical questions, beginning with simple arithmetic and ending with advanced calculus. But mathematics is filled with intriguing mysteries that take us to the edge of the unknown.

This richly illustrated, story-driven volume presents sixteen of today's greatest unsolved mathematical puzzles, all understandable by anyone with elementary math skills. These intriguing mysteries are presented to readers as puzzles that have time-traveled from Camelot, preserved in the notebook of Merlin, the wise magician in King Arthur's court.

2020 Choice Award: Outstanding Academic Title in Mathematics

Flexing on Topology, or, Contrapposto Architecture

Proceedings of the Association of Collegiate Schools of Architecture 109 (2021)

Shannon Starkey (San Diego)

Taking topological thinking filtered through Gilles Deleuze, Paul Virilio, Greg Lynn and others harnessed emerging computer technology to produce infinite variations of doubly-curved, amorphous spheres. In this context, flexible polyhedra are both incredibly simple geometric forms in a world of complex-curved topological spheres; and much more complex, capable of flexibility without abandoning geometry or rigidity and without cutting-edge technologies.

At the Intersection of Unsolved Mathematics, Visual Art and Religious Thought : UNFOLDING HUMANITY

preprint, 2020

Susie Babka, Diane Hoffoss (San Diego)

Art, religious thought, and mathematics meet not in collections of information but in ways to enhance the mode or process of knowing, and Unfolding Humanity demonstrates that making and the embodiment of abstract ideas are a way to knowledge. In other words, these diverse fields study the same things and can ask the same questions but approach these through different methodologies.

Unfolding cubes : nets, packings, partitions, chords

Electronic Journal of Combinatorics 27 (2020) P4.41

Kristin DeSplinter (Utah), Jordan Readyhough (Columbia), Bryce Wimberly (Trident Analysis)

Every ridge unfolding of an n-cube is without self-overlap, yielding a valid net. The results are obtained by developing machinery that translates cube unfolding into combinatorial frameworks. Moreover, the geometry of the bounding boxes of these cube nets are classified using integer partitions, as well as the combinatorics of path unfoldings seen through the lens of chord diagrams.

Makerspaces on the continuum

International Journal of Engineering Education 36 (2020) 1184 - 1195

Gordon Hoople, Joel Alejandro Mejia, Diane Hoffoss (San Diego)

The project at the center of this study, a two-ton interactive metal sculpture, was completed simultaneously in two makerspaces: a university machine shop (embedded in a formal academic space) and a community- based arts space (an informal space). Using a phenomenographic approach, we examined how students experienced these spaces and the ways in which the characteristics of both environments may complement, hinder, or support student learning.

Nets of higher-dimensional cubes

Proceedings of the Canadian Conference on Computational Geometry (2020)

Kristin DeSplinter (Utah), Jordan Readyhough (Columbia), Bryce Wimberly (Trident Analysis)

We show that every ridge unfolding of an n-cube is without self-overlap, yielding a valid net. The results are obtained by developing machinery that translates cube unfolding into combinatorial frameworks. The bounding boxes of these cube nets are also explored using integer partitions.

GIving Good talks

Notices of the American Mathematical Society 66 (2019) 1647 - 1650

Mathematicians are like rock stars: after recording an album, they need to go on tour. Like an album, a paper conveys a polished, finished product, with all the notes perfectly in place. A talk, on the other hand, is akin to a concert performance, highlighting the essential parts of our mathematics through the brushstrokes of intuition and personality. Giving good talks involves not just a command of words and images but speech, body movement, control of time, and a strong emphasis on storytelling.

THE PHYLOGENETICS OF BEER

Math Horizons 26 (2019) 10 - 13

Lia Hebert (Axiom Recovery), Sophia Raglione (Qualcomm)

Although beer is quite ancient, with recorded history dating back as early as 4000 BC, the modern classification of beer styles began in 1977 with the publication of Michael Jackson’s seminal book The World Guide to Beer. This classification includes two main types of beer, ales and lagers, along with numerous substyles including pilsners, stouts, porters, and wheat beer. In this study, we compare and classify beer not based on classical labels, but rather using actual data and phylogenetic techniques.

UNFOLDING HUMANITY : MATHEMATICS AT BURNING MAN

Notices of the American Mathematical Society 66 (2019) 572 - 575

Diane Hoffoss (San Diego)

A two-ton interactive sculpture came to life at Burning Man 2018, the world’s most influential large-scale sculpture showcase. Rising 12 feet tall with an 18-foot wingspan in the Nevada desert, the unfolding dodecahedron is fully lined with massive mirrors, alluding to a possible shape of our universe.  The unfolding exterior points to the 500-year-old work of Albrecht Dürer, and the tantalizing open problem of discovering a geometric unfolding for every convex polyhedron.

Split network polytopes and network spaces

Proceedings of the 31st Conference on Formal Power Series and Algebraic Combinatorics, Séminaire Lotharingien de Combinatoire (82) 2019

Cassandra Durell, Stefan Forcey (Akron)

Phylogenetics begins with reconstructing biological family trees from genetic data. Since Nature is not limited to tree-like histories, we use networks to organize our data, and have discovered new polytopes, metric spaces, and simplicial complexes that help us do so. Moreover, we show that the space of phylogenetic trees dually embeds into the Balanced Minimal Evolution polytope, and use this result to find a complex of faces within the Symmetric Traveling Salesman polytope.

Cartography of Tree spacE

MIT Leonardo Journal 52 (2019) 279 - 283

Owen Schuh (San Francisco . Philadelphia . Vermont)

How can vibrant, contemporary art be produced that deals with vibrant, contemporary mathematics? To address this, a collaboration began between an artist and a mathematician, revolving around recent problems in phylogenetics and the space of evolutionary trees. The result was two-fold: First, a triptych of paintings was created, using acrylic, graphite, watercolor, and metal leaf, that focused on different navigations within this tree space. And second, a novel set of open mathematics problems was discovered solely from this investigation. 

A MATH PROBLEM FOR PI DAY

The Washington Post, Opinion Editorial (March 14, 2018)

Today's dominant narrative casts mathematics and the sciences as stewards of difficult ideas, with the humanities and arts relegated to the simpler struggles. This assumption is not only flawed but also completely backward, for there is a missing part to this story: the hidden dimension of complexity.

Chosen by the Washington Post as one their favorites of 2018.

Geometric Realizations of the 3D AssociahedrON

Symposium on Computational Geometry:  Media (2018)

Daniel Johnson (Harvey Mudd), Justin Lee (Boston University), Jackson Warley (UMass, Amherst)

The associahedron is a convex polytope whose 1-skeleton is isomorphic to the flip graph of a convex polygon.  There exists an elegant geometric realization of the associahedron, using the remarkable theory of secondary polytopes, based on the geometry of the underlying polygon. We present an interactive  application that visualizes this correspondence in the 3D case.

STAR UNFOLDING OF BOXES

Symposium on Computational Geometry:  Media (2018)

Yu Xuan Hong (Pomona) , Dani Demas (Amazon)

Given a convex polyhedron, the star unfolding of its surface is obtained by cutting along the shortest paths from a fixed source point to each of its vertices. We present an interactive application that visualizes the star unfolding of a box, such that its dimensions and source point locations can be continuously toggled by the user.

A Space of phylogenetic Networks

SIAM Journal on Applied Algebra and Geometry 1 (2017) 683 - 705

Samantha Petti (Georgia Tech)

A classic problem in computational biology is constructing a phylogenetic tree given a set of distances between n species.  In most cases, a tree structure is too constraining.  We consider a circular split network, a generalization of a tree in which multiple parallel edges signify divergence. A geometric space of such networks is introduced, forming a natural extension of the work by Billera, Holmes, and Vogtmann on tree space.  We explore properties of this space, and show a natural embedding of the compactification of the real moduli space of curves within it.

Visualizing Scissors Congruence

Symposium on Computational Geometry:  Media (2016)

Zivvy Epstein, Dima Smirnov (Pomona)

Consider two simple polygons with equal area.  The Wallace-Bolyai-Gerwien theorem states that these polygons are scissors congruent, able to be dissected into finitely many congruent polygonal pieces.  We present an interactive application that visualizes this constructive proof.

2018 Popular Science + NSF Visualization of the Year Award.

Navigation in Tree Spaces

Advances in Applied Mathematics 67 (2015) 75 - 95

Jack Morava (Johns Hopkins)

The orientable cover of the moduli space of real genus zero algebraic curves with marked points is a compact aspherical manifold tiled by associahedra, which resolves the singularities of the space of phylogenetic trees. The resolution maps planar metric trees to their underlying abstract representatives, collapsing and folding an explicit geometric decomposition of the moduli space into cubes. This decomposition endows the resolving space with an interesting canonical pseudometric.

Convex Polytopes from Nested Posets

European Journal of Combinatorics 43 (2015) 229 - 248

Stefan Forcey, Stephen Reisdorf, Patrick Showers (Akron)

Motivated by the graph associahedron KG, a polytope whose face poset is based on connected subgraphs of G, we consider the notion of associativity and tubes on posets. This leads to a new family of simple convex polytopes obtained by iterated truncations. These generalize graph associahedra and nestohedra, even encompassing notions of nestings on CW-complexes, but fall in a different category altogether than generalized permutohedra.

Polyhedral Covers of Tree Space

SIAM Journal of Discrete Mathematics 28 (2014) 1508 - 1514

Daoji Huang (Cornell), Dominic Spadacene (Michigan)

We construct the space of phylogenetic trees from local gluings of classical polytopes, the associahedron and the permutohedron. Its homotopy is also reinterpreted and calculated based on polytope data.

Skeletal Configurations of Ribbon Trees

Discrete Applied Mathematics 170 (2014) 46 - 54

Howard Cheng (Arizona), Brian Li (NIH), Andrej Risteski (Princeton)

The straight skeleton construction creates a straight-line tree from a polygon. Motivated by moduli spaces from algebraic geometry, we consider the inverse problem of constructing a polygon whose straight skeleton is a given tree. We prove there exists only a finite set of planar embeddings of a tree appearing as straight skeletons of convex polygons. The heavy lifting of this result is performed by using an analogous version of Cauchy's arm lemma.

Diagonalizing the Genome II : Toward Possible Applications

arxiv preprint 2012

Jack Morava (Johns Hopkins)

We construct a related (stacky) resolution of a space of real quadratic forms, and suggest, perhaps without much justification, that systems of oscillators parametrized by such objects may may provide useful models in genomics. 

The Shape of Associativity

Canadian Mathematical Society Notes 44 (2012) 12 - 14

Associativity is ubiquitous in mathematics. Unlike commutativity, its more popular cousin, associativity has for the most part taken a backseat in importance. But over the past few decades, this concept has blossomed and matured. We start with a brief look at three fields of mathematics where this has transpired, and then explore the visualization of associativity.

Deformations of Associahedra and Visibility Graphs

Contributions to Discrete Mathematics 7 (2012) 68 - 81

Rahul Shah (Williams), Xuan Cheng Shao (MIT), Ezra Winston (Bard)

Given a arbitrary polygon P with holes, we construct a polytopal complex analogous to the associahedron based on convex diagonalizations of P. This polytopal complex is shown to be contractible, and a geometric realization is provided based on the theory of secondary polytopes.  We then reformulate a combinatorial deformation theory and present an open problem based on visibility which is a close cousin to the carpenter's rule statement of computational geometry.

What makes a Tree a Straight Skeleton?

Proceedings of the Canadian Conference on Computational Geometry (2012)

Oswin Aichholzer (Graz), Howard Cheng (Arizona), Thomas Hackl (Graz), Stefan Huber (Salzburg), Brian Li (Williams), Andrej Risteski (Princeton)

Let G be a cycle-free connected straight-line graph with predefined edge lengths and fixed order of incident edges around each vertex. We address the problem of deciding whether there exists a simple polygon P such that G is the straight skeleton of P. We show that for given G, such a polygon P might not exist, and if it exists it might not be unique.

Moduli Spaces of Punctured Poincaré Disks

Tamari Memorial Festschrift: Progress in Mathematics Volume 299 (2012) 

Ben Fehrman (Chicago), Tim Heath (Columbia), Ashiti Vashist (Michigan)

The Tamari lattice and the associahedron provide methods of measuring associativity on a line. The real moduli space of marked curves captures the space of such associativity.  We consider a natural generalization by considering the moduli space of marked particles on the Poincaré disk, extending Tamari's notion of associativity based on nesting. A geometric and combinatorial construction of this space is provided, which appears in Kontsevich's deformation quantization, Voronov's swiss-cheese operad, and Kajiura and Stasheff's open-closed string theory.

DIScrete and Computational Geometry

Princeton University Press (2011)

Joe O’Rourke (Smith)

This book covers traditional topics such as convex hulls, triangulations, and Voronoi diagrams, as well as more recent subjects like pseudotriangulations, curve reconstruction, and locked chains. It also touches on more advanced material, including Dehn invariants, associahedra, quasigeodesics, Morse theory, and the recent resolution of the Poincaré conjecture. Connections to real-world applications are made throughout, and algorithms are presented independently of any programming language, along with numerous exercises and unsolved problems.

Triple Infinity

Esopus Magazine (Fall 2011)

Allison Shotz (artist), Priyamvada Natarajan (Yale)

An artist, a mathematician, and a cosmologist walk into a bar, and discuss the nature of infinity.

Pseudograph Associahedra 

Journal of Combinatorial Theory, Series A 118 (2011) 2035 - 2055

Mike Carr (Brandeis), Stefan Forcey (Tennessee State)

Given a simple graph G, the graph associahedron KG is a simple polytope whose face poset is based on the connected subgraphs of G. This paper defines and constructs graph associahedra in a general context, for pseudographs with loops and multiple edges, which are also allowed to be disconnected.  We then consider deformations of pseudograph associahedra as their underlying graphs are altered by edge contractions and edge deletions.

Deformations of Bordered Surfaces and Convex Polytopes

Notices of the American Mathematical Society 58 (2011) 530 - 541

Tim Heath (Columbia), Cid Vipismakul (UT Austin)

We consider the moduli space of Riemann surfaces with boundary and marked points. Such spaces appear in open-closed string theory, particularly with respect to holomorphic curves with Lagrangian submanifolds. We consider a combinatorial framework to view the compactification of this space based on the pair-of-pants decomposition of the surface, relating it to the well-known phenomenon of bubbling. Our main result classifies all such spaces that can be realized as convex polytopes.

Particle Configurations and Coxeter Operads

Homotopy and Related Structures 4 (2009) 83 - 109

Zan Armstrong (Williams), Mike Carr (Michigan), Eric Engler (Williams), Ananada Leininger (MIT), Michael Manapat (Berkeley)

There exist natural generalizations of  the real moduli space of Riemann spheres based on manipulations of Coxeter complexes. These novel spaces inherit a tiling by the graph-associahedra convex polytopes. We obtain explicit configuration space models for the classical infinite families of finite and affine Weyl groups using particles on lines and circles. A Fulton-MacPherson compactification of these spaces is described, which in then used to define the Coxeter operad.

Shape Deformation in Continuous Map Generalization

GeoInformatica 13 (2009) 203 - 221

Jeff Danciger (Stanford), John Mugno (Maryland), Don Sheehy (Carnegie Mellon), Rachel Ward (Princeton)

Given a collection of regions on a map, we seek a method of continuously altering the regions as the scale is varied. This is formalized and brought to rigor as well-defined problems in homotopic deformation. We ask the regions to preserve topology, area-ratios, and relative position as they change over time. A solution is presented using differential methods and computational techniques.

A Realization of Graph Associahedra

Discrete Mathematics 309 (2009) 271 - 276

Given any finite graph, we offer a simple realization of its corresponding graph associahedron polytope using integer coordinates.

Marked Tubes and the Graph Multiplihedron

Algebraic and Geometric Topology 8 (2008) 2081 - 2108

Stefan Forcey (Tennessee State)

Given a graph G, we construct a convex polytope whose face poset is based on marked subgraphs of G. Not only does this yield a natural generalization of the multiphihedron, but features of this polytope appear in works related to quilted disks, bordered Riemann surfaces, and operadic structures. Certain examples of graph multiplihedra are related to Minkowski sums of simplices and cubes and others to the permutohedron.

Juggling Braids and Links

The Mathematical Intelligencer (2007) 15 - 22

John Mugno (Maryland)

Using a simplistic model of juggling based on physics, a natural map is constructed from the set of periodic juggling patterns to links. We then show that all topological links can be juggled.

Compatible Triangulations and Point Partitions by Series-Triangular Graphs

Computational Geometry: Theory and Applications 34 (2006) 195 - 202

Jeff Danciger (UC Santa Barbara), Don Sheehy (Princeton)

We introduce series-triangular graph embeddings and show how to partition point sets with them.  This result is then used to prove an upper bound on the number of Steiner points needed to obtain compatible triangulations of point sets.  The problem is generalized to finding compatible triangulations for more than two point sets and we show that such triangulations can be constructed with only a linear number of Steiner points added to each point set.

Coxeter Complexes and Graph Associahedra

Topology and its Applications 153 (2006) 2155 - 2168

Mike Carr (Michigan)

Given a graph G, we construct a simple, convex polytope whose face poset is based on the connected subgraphs of G. This provides a natural generalization of the Stasheff associahedron and the Bott-Taubes cyclohedron. Moreover, we show that for any simplicial Coxeter system, the minimal blow-ups of its associated Coxeter complex has a tiling by graph associahedra. These spaces are natural generalizations of the Deligne-Mumford compactification of the real moduli space of curves.

Continuous Foldability of Polygonal Paper

Proceedings of the Canadian Conference on Computational Geometry (2004)

Erik L. Demaine (MIT), Joe Mitchell (Stony Brook), Joe O'Rourke (Smith)

We prove that any given well-behaved folded state of a piece of paper can be reached via a continuous folding process starting from the unfolded paper and ending with the folded state.

Combinatorial Equivalence of Real Moduli Spaces

Notices of the American Mathematical Society 51 (2004) 620 - 628

The Riemann moduli space of spheres with marked points has become a central object in mathematical physics. There is a Deligne-Knudsen-Mumford compactification of this space which allows collisions of points of the configuration space. The real spaces, unlike their complex counterparts, have an inherent tiling, which allows an intuitive, combinatorial construction. Along the way, we provide a new construction of the associahedron from products of simplices.

A Space of Cyclohedra

Discrete and Computational Geometry 29 (2003) 61 - 75

The real points of the Deligne-Knudsen-Mumford moduli space of marked points on the sphere have a natural tiling by associahedra. We extend this idea to construct an aspherical space tiled by cyclohedra. We explore the structure of this space, coming from blow-ups of hyperplane arrangements, as well as discuss possibilities of its role in knot theory and mathematical physics.

Cellular Structures determined by Polygons and Trees

Annals of Combinatorics 5 (2001) 71 - 98

Ronald C. Read (Waterloo)

The polytope structure of the associahedron is decomposed into two categories, types and classes. The classification of types is related to integer partitions, whereas the classes present a new combinatorial problem.  We solve this, and incorporate the results into properties of moduli spaces. Connections are discussed with relation to classic combinatorial problems as well as to other sciences.

Tessellations of Moduli Spaces and the Mosaic Operad

Contemporary Mathematics 239 (1999) 91 - 114

We construct a new (cyclic) operad of mosaics defined by polygons with marked diagonals. Its underlying (aspherical) spaces are the compactification of the real moduli space of curves, naturally tiled by Stasheff associahedra. We describe them as iterated blow-ups and show that their fundamental groups form an operad with similarities to the operad of braid groups.