CONCRETAGENS #510

 Por José Lino Grünewald

MINHAS METAS PARA 2021

Nos últimos tempos, tenho percebido que evoluí bastante no que se refere à definir e cumprir metas. Das que tinha planejado para 2020, alcancei quase todas, faltando apenas algumas que se tornaram um pouco inviáveis em decorrência da pandemia, como praticar mais skate e BMX.

Penso que o que torna este processo mais prazeroso é ter bom senso e listar apenas o que é possível realizar, visando evitar desapontamentos e uma possível desmotivação.

Para esse ano de 2021, pretendo "focar" em:

‣ Desenvolvimento front-end e programação, por meio dos tópicos do 100 Days of Code, dos cursos da freeCodeCamp, do React Roadmap, e dos tutoriais do HackerEarth. Além disso, também pretendo conseguir algumas certificações no HackerRank.

‣ Praticar mais xadrez no Lichess, pois estou ficando enferrujado. 

‣ Revisar e aprender matemática (geometria, álgebra linear, cálculo, estatística etc.) nos cursos da Khan Academy, e também terminar o curso Introduction to Mathematical Thinking que iniciei no Coursera, mas não consegui terminar por causa da faculdade (ESSE ANO ME FORMO!).

‣ Assistir todos os filmes mostrados no The Evolution of Cinema (1878-2017), todos os curtas da lista Um curta para dia do ano, algumas séries, animes, e também ouvir mais podcasts, como o Filosofia Pop, e alguns outros sobre ciência e tecnologia, como o FalaDev da Rocketseat.

‣ Ler alguns livros sobre idiomas e, principalmente, continuar meus estudos de inglês, francês e espanhol na plataforma Duolingo, pois gosto muito desse processo de aprendizagem por meio da gamificação.

‣ Ler mais livros técnicos sobre desenvolvimento de software e Design Generativo.

‣ Responder questões de pelo menos 50 tópicos do ESL Discussions, para praticar o livre pensar.

‣ Fazer postagens no sTuring, publicar zines, gravar vídeos para o meu canal, desenvolver projetos para o Compixelar, e concluir a implementação do PolymApp.

Se vou conseguir alcançar todas estas metas é outra história, pois estou ciente das impresivibilidades da vida, entretanto, acredito na minha capacidade para tal.

CQD

chegou iludindo
com algumas conjecturas,

prometendo resolver
os problemas.

mas a sua incompetência
sempre foi um teorema.

estamos todos agora
vivendo um absurdo.

como queríamos demonstrar,
bolsonaro é um imundo.

THE HISTORY OF MATHEMATICS AND ITS APPLICATIONS - ZACH STAR

Uma visão histórica e pragmática sobre uma das áreas do conhecimento mais importantes que existem.

O HOMEM QUE CALCULAVA - MALBA TAHAN

"O Homem que Calculava" mostra uma matemática repleta de curiosidades e aplicações práticas, abordada por meio de diversas histórias narradas de forma bastante poética pelo genial Malba Tahan. Com certeza, entrará para a minha lista de livros favoritos.

***

"- Acreditam, entretanto, os generosos amigos, ser justo incluir o meu nome entre os calculistas. Sinto-me lisonjeado com tão alta distinção. Penso, porém, que os homens são em geral bons calculistas. Calculista é o soldado que em campanha avalia com o olhar a distância de uma parasanga, calculista é o poeta que conta às sílabas e mede a cadência dos versos; calculista é o músico que aplica na divisão dos compassos as leis da perfeita harmonia; calculista é o pintor que traça as figuras segundo proporções invariáveis para atender os princípios da perspectiva; calculista é o humilde esteireiro que dispõe, um por um, os cem fios de seu trabalho – todos, enfim, ó Rei, são bons e hábeis calculistas!"

LEITURA / COMPRAR

WHAT IS THE GEOMETRY OF THE UNIVERSE?

When you gaze out at the night sky, space seems to extend forever in all directions. That’s our mental model for the universe, but it’s not necessarily correct. There was a time, after all, when everyone thought the Earth was flat because our planet’s curvature was too subtle to detect and a spherical Earth was unfathomable.

Today, we know the Earth is shaped like a sphere. But most of us give little thought to the shape of the universe. Just as the sphere offered an alternative to a flat Earth, other three-dimensional shapes offer alternatives to “ordinary” infinite space.

We can ask two separate but interrelated questions about the shape of the universe. One is about its geometry: the fine-grained local measurements of things like angles and areas. The other is about its topology: how these local pieces are stitched together into an overarching shape.

Cosmological evidence suggests that the part of the universe we can see is smooth and homogeneous, at least approximately. The local fabric of space looks much the same at every point and in every direction. Only three geometries fit this description: flat, spherical, and hyperbolic. Let’s explore these geometries, some topological considerations, and what the cosmological evidence says about which shapes best describe our universe.

FULL ARTICLE

PROGRAMAÇÃO COMPETITIVA

Recentemente, decidi iniciar uma nova série no meu canal, intitulada: Programação Competitiva - Evoluindo através da análise e resolução de desafios.

Nela, pretendo resolver os problemas de algumas plataformas de programação onde, geralmente, pratico e aprimoro as minhas habilidades.

Neste primeiro vídeo, utilizo uma abordagem um pouco mais acessível, não aplicando aquela "malícia" matemática, na resolução de um desafio de um dos meus sites favoritos sobre computação, o GeeksForGeeks.

PAPO RETO

se estou afim dela
mando o papo reta

nosso ponto de interseção
só vai ser obtido

depois de resolver
essa questão

MATEMÁTICA & COMPUTAÇÃO & ENXADRISMO

Perco a noção do tempo vendo os vídeos destes canais. Uma ótima referência para amantes de Matemática, Computação e Enxadrismo.

* * *

Numberphile: "Videos about numbers - it's that simple. Videos by Brady Haran"

* * *

Computerphile: "Videos all about computers and computer stuff."

* * *

Xadrez Brasil: "Procuro mostrar partidas incríveis e fazer as análises mesclando técnica com entretenimento, mostrando as partidas de uma forma mais jornalística."

THE SUBTLE ART OF THE MATHEMATICAL CONJECTURE

Lembro quando concluí a minha primeira demonstração, na faculdade. Apesar da dificuldade em assimilar certos assuntos da Matemática Pura, gosto muito de analisar e tentar entender todos teoremas e conjecturas que vejo pelos livros e outras leituras. É fascinante como as coisas "se encaixam".

* * *

It’s an educated guess, not a proof. But a good conjecture will guide math forward, pointing the way into the mathematical unknown.

A conjecture creates a summit to be scaled, a potential vista from which mathematicians can see entirely new mathematical worlds.

Mountain climbing is a beloved metaphor for mathematical research. The comparison is almost inevitable: The frozen world, the cold thin air and the implacable harshness of mountaineering reflect the unforgiving landscape of numbers, formulas and theorems. And just as a climber pits his abilities against an unyielding object — in his case, a sheer wall of stone — a mathematician often finds herself engaged in an individual battle of the human mind against rigid logic.

In mathematics, the role of these highest peaks is played by the great conjectures — sharply formulated statements that are most likely true but for which no conclusive proof has yet been found. These conjectures have deep roots and wide ramifications. The search for their solution guides a large part of mathematics. Eternal fame awaits those who conquer them first.

Remarkably, mathematics has elevated the formulation of a conjecture into high art. The most rigorous science cherishes the softest forms. A well-chosen but unproven statement can make its author world-famous, sometimes even more so than the person providing the ultimate proof. Poincaré’s conjecture remains Poincaré’s conjecture, even after Grigori Perelman proved that it is true. After all, Sir George Everest, the British surveyor general of India in the early 19th century, never climbed the mountain that today bears his name.

Like every art form, a great conjecture must meet a number of stringent criteria. First and foremost, it should be “nontrivial” — that is, not too easy to prove. Mathematicians will say things like “A problem is worth tackling only when it fights back,” and “If it’s not frustrating, you’re probably working on a problem that is too easy.” If a conjecture is proved within a few months, then perhaps its creator should have pondered it a bit longer before announcing it to the world.

The first effort to compose a comprehensive collection of the greatest mathematical challenges was made at the turn of the previous century by David Hilbert, who has been characterized as the last universal mathematician. Although his list of 23 problems has been very influential, in retrospect it was something of a mixed bag.

It included all-time favorites like the Riemann hypothesis — often considered the greatest of great conjectures, one that has remained the Everest of mathematics for over a century. When Hilbert was asked what would be the first thing he’d like to know after awakening from a 500-year slumber, he immediately picked this conjecture. It captures an essential intuition about the distribution of prime numbers — the atoms of arithmetic — and its establishment will have vast consequences for many branches of mathematics.

But Hilbert also listed much vaguer and more open-ended goals such as “the mathematical treatment of the axioms of physics” and “the further development of the calculus of variations.” Another of his conjectures, one concerning the relation of two polyhedra of equal volume, was solved in the same year he announced it by his student Max Dehn. While Hilbert described many towering peaks, this turned out to be more of a foothill.

The highest summits are not conquered in a single effort. Climbing expeditions carefully lay out base camps and fixed ropes, then slowly work their way to the peak. Similarly, in mathematics one often needs to erect elaborate structures to attack a major problem. A direct assault is seen as foolish and naive. These auxiliary mathematical constructions can sometimes take centuries to build and in the end often prove to be more valuable than the conquered theorem itself. The scaffold then becomes a permanent addition to the architecture of mathematics.

A wonderful example of this phenomenon is the proof of Fermat’s Last Theorem by Andrew Wiles in 1994. Fermat famously wrote his conjecture in the margin of Diophantus’ Arithmetica in 1639. Its proof required the development of more than three centuries’ worth of mathematical tools. In particular, mathematicians had to construct a very advanced combination of number theory and geometry. This new field — arithmetic geometry — is now one of deepest and far-ranging mathematical theories. It goes far beyond Fermat’s conjecture and has been used to settle many outstanding questions.

A great conjecture also has to be deep and lie at the very core of mathematics. In fact, the metaphor of scaling a summit does not adequately capture the full impact of a proof. Once the conjecture is proved, it is not so much the endpoint of an arduous journey but rather the starting point of an even greater adventure. A much more accurate image is that of a mountain pass, the saddle point that allows one to traverse from one valley into another. In fact, this is what makes the Riemann hypothesis so powerful and beloved. It unlocks many other theorems and insights, and suggests vast generalizations. Mathematicians have been busy exploring the lush valley to which it grants access, even though that valley is still, strictly speaking, hypothetical.

Furthermore, there must be substantial evidence for a conjecture. Niels Bohr famously defined a great truth by the property that its opposite is also a great truth. But this is definitely not the case for a great conjecture. Since there is generally much circumstantial evidence pointing to its truth, the negation is seen as most unlikely. For instance, the first 10 trillion cases of the Riemann hypothesis have been checked numerically using computers. Who, at this point, can still doubt its validity? But all this supporting material does not satisfy mathematicians. They demand absolute certainty and want to know why the conjecture is true. Only a conclusive proof can provide that answer. Experience shows that one can easily be fooled. Counterexamples can lie far ashore, like the one found by Noam Elkies, a mathematician at Harvard University, disproving Euler’s conjecture, a variation on Fermat’s conjecture that states that a fourth power can never be written as a sum of three other fourth powers. Who would have guessed that the first counterexample involved a number of 30 digits?

20,615,6734 = 2,682,4404 + 15,365,6394 + 18,796,7604.

The best conjectures usually have modest origins, such as Fermat’s casual note in the margin, but their implications and ramifications grow over the years. It also helps if the challenge can be stated concisely, preferably with a formula containing only a few symbols. A good conjecture should fit on a T-shirt. Goldbach’s conjecture, for instance, reads “Every even integer greater than 2 can be expressed as the sum of two primes.” This problem, formulated in 1742, remains unsolved. It became famous thanks to the novel Uncle Petros and Goldbach’s Conjecture (2000), by the Greek author Apostolos Doxiadis, not least because the publisher offered $1 million as a publicity stunt to anyone who could prove it within two years of the book’s publication. The conciseness of a great conjecture adds to its perceived beauty. One could even define mathematical aesthetics as “impact per symbol.” However, this elegant beauty can be misleading. The shortest statements can require the longest proofs, as again demonstrated by Fermat’s deceptively simple observation.

We should perhaps also add to this list of criteria the response from the famous mathematician John Conway to the question of what makes a great conjecture: “It should be outrageous.” An appealing conjecture is also somewhat ridiculous or fantastic, with unforeseen range and consequences. Ideally it combines components from distant domains that haven’t met before in a single statement, like the surprising ingredients of a signature dish.

Finally, it is good to realize that the adventure does not always end with success. Just as a mountaineer can be confronted by an unsurpassable crevasse, mathematicians can fail, too. And if they fail, they fail absolutely. There is no such thing as a 99 percent proof. For two millennia, people tried to prove that Euclid’s fifth postulate — the notorious “parallel postulate” that states roughly that two parallel lines cannot cross — can be derived from the other four axioms of planar geometry. Then, at the beginning of the 19th century, mathematicians constructed explicit examples of non-Euclidian geometry, disproving the conjecture.

This was not the end of geometry, however. In a perverse way, the refutation of a great conjecture can be even better news than its success, since the failure makes clear that our imagined map of the mathematical world is seriously wrong. Defeat can be productive, the reverse of a Pyrrhic victory. Non-Euclidean geometry proved to be an important precursor of Einstein’s curved space-time, which plays such an important role in the modern understanding of gravity and the cosmos.

Similarly, when Kurt Gödel published his famous incompleteness theorem in 1931, showing that in any reasonable mathematical system there are true statements that cannot be proved, he essentially answered in the negative one of Hilbert’s problems about the consistency of arithmetic. But the incompleteness theorem — often seen as the greatest logical achievement since Aristotle — did not herald the end of mathematical logic. Instead it induced a blossoming that even led to the development of modern computers.

So, in the end, the search for solutions to great conjectures has something else in common with climbing expeditions to the highest peaks. Only when everyone is safely home — whether the goal is reached or not — does the full extent of the adventure become clear. At that point, it is time for the heroic tales of the ascent to be told.

FONTE

CONCRETAGENS #127

Por Álvaro de Sá

AFIM

esse sentimento
crescente

que carrego
dentro de mim

representa
uma função afim

de ti

MATHEMATICS AND ART

1. Introduction

Squirrels, rabbits and birds are not artists. However, humans may admire a bird's nest as being a "work of art," and may find the patterns in the snow made by squirrel or rabbit tracks pleasing. Yet, the shape of a bird's nest may indeed be a form of communication for birds, just as "art" is a form of communication for humans. What constitutes art is a very complex and hotly debated subject. When Jackson Pollock first experimented with expressing himself by flinging paint at a canvas, many saw his activity as a form of self-indulgence rather than art. As another example, some people collect maps and some of these maps are art, but not all maps are art.

April is Mathematics Awareness Month, and this year's theme is Mathematics and Art. There are, in fact, many arts (music, dance, painting, architecture, sculpture, etc.) and there is a surprisingly rich association between mathematics and each of the arts. My goal here is to give some pointers concerning these many connections.

One mathematical connection with art is that some individuals known as artists have needed to develop or use mathematical thinking to carry out their artistic vision. Among such artists were Luca Pacioli (c. 1145-1514), Leonardo da Vinci (1452-1519), Albrecht Dürer (1471-1528), and M.C. Escher (1898-1972). Another connection is that some mathematicians have become artists, often while pursuing their mathematics.

Mathematicians commonly talk about beautiful theorems and beautiful proofs of theorems. They also often have emotional reactions to proofs or theorems. There are nifty proofs of "dull" mathematical facts and "unsatisfying" proofs of "nifty" theorems. Artists and art critics also talk about beauty. Does art have to be beautiful? Francis Bacon's paintings may or may not be beautiful to everyone, yet there are few people who have no reaction to his work. Art is concerned with communication of emotions as well as beauty. Some people may see little emotional content in many of M.C. Escher's prints but it's hard not to be "impressed" by the patterns he created. Some find Escher's prints beautiful but with a different beauty from the great works of Rembrandt. Like art itself, the issues of beauty, communication, and emotions are complex subjects, but then so is mathematics.

Joseph Malkevitch
York College (CUNY)


2. Mathematical tools for artists

Art is born of the attempt by humans to express themselves about the experience of life. Art can take the form of writing, painting, and sculpture, as well as a variety of other forms of expression. There is art in the combining of function and aesthetics in such everyday objects as plates, cutlery, and lamps. Mathematicians have been able to assist artists by creating "tools" of various kinds for them. Such tools sometimes consist of theorems which show the limitations of what artists can do. One can not attempt to represent more than 5 regular convex polyhedra in Euclidean 3-dimensional space because mathematics shows there are only 5 such regular solids. Dodecahedra can be used to put one month of a calendar on each face but the number of faces of a regular convex polyhedron in Euclidean 3-space is 4, 6, 8, 12, or 20. There can be no others. A much more important issue is the realism with which artists can draw on a flat piece of paper what they perceive when they look out at their 3-dimensional world. If one looks at attempts at scene representation in Egyptian and Mesopotamian art, one sees that phenomena that are associated with the human vision system are not always respected. We are all familiar with the fact that objects that are far away from us appear smaller than they actually are and that lines which are parallel appear to converge in the distance. These features, which are a standard part of the way that 3-dimensional objects are now usually represented on a planar surface, were not fully understood before the Renaissance. It is common to refer to artists as using "perspective" (or "linear perspective") to increase the realism of their representations. The issues and ideas involved in understanding perspective are quite subtle and evolved over a long time.

The interaction between scholars and practitioners with regard to ideas about perspective parallels the interactions between theory and application that goes on in all the arenas where mathematical ideas are put to work. An artist may want to solve a problem better than he or she did in the past and will not always be concerned with the niceties of proving that the technique used always works or has the properties that the artist wants. An analogy for a more modern situation is that if the current system used to route email packets takes on average 7.2 units of time and one discovers a way of doing the routing in 6.5 units of time on average, one may not worry that one can prove that the very best system would do the job in 6.487 units of time.

Questions about perspective are very much in the spirit of mathematical modeling questions, since in the usual approach one is concerned with the issue of the perception of, say, a scene in 3-dimensional space on a flat canvas under the assumption that the scene is being viewed by a "single point eye." Yet, we all know that humans are endowed with binocular vision! We are attacking such binocular vision questions today, because we have the mathematical tools to take such questions on, while the artist/mathematicians of the past had to content themselves with simpler approaches.

A variety of people whose names are known to mathematicians (but perhaps not to the general public) have contributed to a theory of perspective. Though every calculus student knows the name of Brook Taylor (1685-1731)

for his work on power series, how many mathematicians know that Taylor wrote on the theory of linear perspective? On the other hand every art historian will recognize the name of Piero della Francesca (c. 1412-1492), yet how many of these art historians (or mathematicians) will be familiar with his contribution to mathematics? Similarly, Girard Desargues (1591-1661) is a well known name to geometers for his work on projective geometry (in plane projective geometry there are no parallel lines), but few people involved with art are familiar with his work. The diagram below (a portion of a "Desargues Configuration"), familiar to students of projective geometry, can be thought of as a plane drawing of an "eye" point viewing triangles which lie in different planes.

Another contributor to the theory of perspective was Johann Lambert, also known for having produced results which would follow from assuming that Euclid's Fifth Postulate does not hold. Though perspective is a well-mined area, this does not stop the flow of continued thoughts on the subject. For those accustomed to work on one-point or two-point perspective, there is the monograph of D. Termes who treats one- through six-point perspective!

Related to the tool of linear perspective is the branch of geometry known as descriptive geometry. While descriptive geometry was widely taught in the 19th century, especially in schools of engineering, the subject is not widely known today. The reason in part is that computer software makes it possible for people not familiar with descriptive geometry to perform tasks which make explicit knowledge of it increasingly obsolete. Descriptive geometry provides a set of procedures for representing 3-dimensional objects in two dimensions. The 2-dimensional representation might be on either a piece of paper or a computer screen. These techniques are of great importance to engineers, architects, and designers. The design of, say, a large aircraft may involve tens of thousands of drawings. The roots of the subject lie with people such as Albrecht Dürer (1471-1528) and Gaspard Monge (1746-1818). If an artist, creative designer, sculptor, or architect can not get across his/her conception of how to manufacture or otherwise assemble a "creative" design, then the work involved might go unrealized. Descriptive geometry supports both constructive manufacture and creative design by giving procedures for showing how to represent proposed 3-dimensional creations on a flat surface.

To give some of the flavor of the issues involved, the diagram below shows in blue how parallel lines "project a triangle onto a line," while the red lines show how the same triangle is "projected" from the "eye" onto the same line:

A', B', and C' show where the vertices are moved by "parallel" projection and A'', B'', and C'' show where the vertices are moved by "conical" projection.

Here is a rather cute result which grew out of this interaction between mathematics and the needs of artists to represent three dimensions on a flat plane. The result is known as Pohlke's Theorem. Karl Wilhelm Pohlke (1810-1876) was a German painter and teacher of descriptive geometry at an art school. He formulated this result in 1853, though the first proof seems to be due to K.H.A. Schwarz (1843-1921) in 1864. The theorem is quoted in various levels of generality. Here is one version:

Theorem (Pohlke): Given three segments (no two collinear) of specified length (not necessarily the same) which meet at a point in a plane, there are three equal length line segments which meet at right angles at a point in 3-dimensional space such that a parallel projection of these segments maps them onto the three chosen line segments in the plane.

Intuitively, this means that if one wants to draw a cubical box in the plane, one can draw any triad of lines for a corner of the cube because there is some position of a cube in 3-space which maps to the given triad. Thus, in the diagram below the triad on the left can be completed to form a "cube," and there is some set of three orthogonal segments in 3-space which can be mapped using parallel projection onto the triad on the left.

Sometimes Pohlke's Theorem is referred to as the Fundamental Theorem of Axonometry. Axonometry, like descriptive geometry, deals with the problem of drawing 3-dimensional objects in the plane. Here is a recent item coauthored by Roger Penrose in this spirit.


3. Symmetry

Art critics have evolved a language for discussing and analyzing art. It turns out that some mathematics can also be useful in analyzing art. Some art or pieces of art consist of things that are pleasing to the eye because they are symmetrical in a mathematical sense. Although the study of symmetry has implicitly been done within mathematics for a long time, in some ways its systematic study is quite recent. It was Felix Klein (1849-1925) who called attention to the fact that one way of classifying different kinds of geometries is to look at the geometric transformations in each of these geometries that preserve interesting properties. In particular, it is of interest in Euclidean, spherical, or Bolyai-Lobachevsky geometry to look at the geometric transformations that are isometries. An isometry is a transformation that preserves distance.

Of course, it is important to keep in mind that many of the pieces of art that mathematicians (and others) analyze using mathematical ideas may not reflect attempts on the part of the artist to incorporate these mathematical ideas in his/her work. Just as someone who makes a cubical box may not realize that the cube is a regular polyhedron (one where all the vertices are alike and where all the faces are congruent regular polygons), a maker of a weaving may not know anything about isometries. The isometries in the Euclidean plane are translations, rotations, reflections and glide reflections. Thus, a mathematician may state which isometries are used in the design of a rug, but that does not mean that the designer of the rug invoked any mathematical thinking. Social scientists have used symmetry ideas and the conservatism of the mechanisms of cultural transmission in their studies. Archeologists and anthropologists have attempted to use symmetry ideas to date artifacts (pottery or fabrics) and to study trade and patterns of commerce.

A major tool in the analysis of symmetry has been the concept of a group.The group concept has a rich and complicated history with ties to the study of the theory of equations (attempts to show that one could not find formulas to solve quintic polynomial equations). By late in the 19th century, group theory was being used as a tool to help crystallographers understand the symmetry of crystals and other naturally occurring structures. Out of this interest grew the work that made possible the classification of tilings and patterns using group theory considerations.

One can also look at the symmetry of a single motif or ornament. Examples of such motifs (illustrated from patterns used in batiks) are shown below. Such ornaments typically have rotational symmetry and/or reflection symmetry.

A more complex pattern such as the one below can be built up from simple motifs. Such patterns have translational symmetry in one direction. Designs or patterns of this kind are known as strip, band, or frieze patterns.

The motifs used to make such frieze patterns may be isolated from one another or coalesce into a "continuous" geometric design along the strip. If a pattern has translations in two directions, then the pattern is often referred to as a wallpaper pattern.

Whereas an artist may choose to create a pattern with absolute and strict adherence in all details to have symmetry in the pattern, this is not all that common for "tribal" artists or artisans. Thus, if one looks carefully at a rug which at first view looks very symmetrical, it is common to see that at a more detailed level it is not quite totally symmetric either in the use of the design or of the colors used in different parts of the design. One can see the small liberties that are taken either because of the difficulty of making patterns exact by hand or because the artist wants consciously to make such small variations. In analyzing the symmetry of such a pattern it probably makes sense to idealize what the artist has done before applying some mathematical classification of the symmetry involved.

In the patterns shown above no color appears. We have a black design on a white background. However, in discussing the symmetry of a pattern one can study the symmetry involved if color is disregarded or by taking color into account. If you look at the batik below from a symmetry point of view you must idealize (model) what is going on to use mathematics. This batik is not infinite in either one or two directions. You must decide what colors have been used and what is the background color.

Many find it interesting to use mathematics to decide what symmetry pattern is involved for various interpretations of the whole or parts of a design. E. Fedorov (1859-1919) enumerated the seventeen 2-dimensional patterns in 1891 in a paper which did not receive wide attention because it was in Russian. P. Niggli (1888-1953) and G. Polya (1887-1985) developed the seven 1-dimensional and the seventeen 2-dimensional patterns in the 1920's; it was through this work that a mathematical approach to the analysis of symmetry patterns became more widely known. One extension of this work to color symmetry was accomplished by H. Woods in the 1930's. It turns out that there are 46 two-color types of patterns. Subsequently much work has been done with regard to studying symmetry in higher-dimensional spaces and using many colors. Recently Branko Grünbaum and Geoffrey Shephard, in a long series of joint papers and in their seminal book Tilings and Patterns, explored many extensions and facets of pattern, tilings, and their symmetries. In particular, Grünbaum and Shephard explored the interaction between symmetry and the use of a motif. This enabled them, for example, to develop a "finer" classification of the seven frieze patterns and seventeen wallpaper patterns. Unfortunately, this work is not as widely known as it should be.

Many people have been instrumental in disseminating mathematical knowledge of symmetry and pattern to scholars outside of mathematics as well as to the general public. One of the most influential and early books of this kind was Hermann Weyl (1885-1955)'s book Symmetry. Also noteworthy among these popularizers are Doris Schattschneider, Branko Grünbaum, Geoffrey Shephard, Marjorie Senechal, Michele Emmer, H. S. M. Coxeter, Dorothy Washburn (an anthropologist), Donald Crowe and Kim Williams. These individuals called attention to the use of symmetry as a tool for insight into various aspects of fabrics, ethnic designs and culture, architecture, and art, as well as to artists such as Escher whose work tantalizes people with a mathematical bent.


4. Mathematical artists and artist mathematicians

Not surprisingly, in light of the internal aesthetic qualities of mathematics, many mathematicians (and computer scientists) have chosen to express themselves not only by proving theorems but also by producing art. There are many such individuals including Helaman Ferguson, Nathaniel Friedman (also see this site), George Hart, Koos Verhoeff, and Michael Field. Complementing these career "mathematicians" who are also artists is a group of people who are not mathematicians but who have drawn great inspiration from mathematical phenomena. Some such individuals are Brent Collins, Charles Perry, and Sol LeWit (here are more samples of his work). As might be expected, there are many architects whose work has a feeling of having been influenced by "technical capability." Though perhaps having only a tangential connection with mathematics, the distinguished architect Frank Gehry has discussed how the availability of CAD (Computer Aided Design) software has made it possible for him to express himself in a way that would not have otherwise been possible. Structural engineering has many ties with mathematics. If you are not familiar with the work of Santiago Calatrava, you are in for a treat. There have also been attempts of various kinds to generate art with algorithms. Some of this work is rather interesting.

For the general public, there is one artist whose work, perhaps more than any other, is seen as having a mathematical quality. This artist was M. C. Escher. The mathematical quality of his work is apparent even though Escher did not see himself as having mathematical talent. Yet despite his lack of formal study of mathematics, Escher approached many artistic problems in a mathematical way. Doris Schattschneider has been instrumental in calling to the public's attention Escher's work and its relation to mathematics. Escher was influenced by at least one very distinguished mathematician (geometer), Harold Scott MacDonald Coxeter. Escher interacted with Coxeter about the difficulties he was having with representing "infinity" in a finite region. Coxeter responded by showing the connection to tilings of the hyperbolic plane. Coxeter has explained the details of this mathematical connection. Mathematically active until his death, Coxeter recently passed away at the age of 96 (an obituary is here). .

5. Polyhedra, tilings, and dissections

Drawing polyhedra was an early testing ground for ideas related to perspective drawing. Renaissance artists were involved in trying to build on historical references to "Archimedean polyhedra" which were transmitted via the writings of Pappus. What constituted a complete set of convex polyhedra with the property that locally every vertex looked like every other vertex and whose faces were regular polygons, perhaps not all with the same number of sides? Perhaps surprisingly, no complete reconstruction occurred until the work of Kepler (1571-1630), who found 13 such solids, even though one can make a case for there being 14 such solids. (Pappus-Archimedes missed one in ancient times. The modern definition of Archimedean solids defines them as convex polyhedra which have a symmetry group under which all the vertices are alike. Using this definition there are 13 solids, but there is little reason to believe that in ancient Greece geometers were thinking in terms of groups rather than in terms of local vertex equivalence, that is, the pattern of faces around each vertex being identical.)

In more modern times polyhedra have inspired artists and mathematicians with an interest in the arts. Inspired by polyhedra, Stewart Coffin has created a wonderful array of puzzle designs which require putting together pieces he designed made from rare woods to form polyhedra. Coffin's puzzles are remarkable for both their ingenuity as puzzles and their beauty. This beauty is a reflection of the beauty of the polyhedral objects themselves, but also the beauty of the rare woods he used to make his puzzles. Coffin showed creativity in selecting symmetrical variants of well-known polyhedra. Coffin's work, like Escher's, has been an inspiration to others. Good puzzles engender the same sense of wonder that beautiful mathematics inspires. George Hart, whose background is in computer science, is an example of a person who is contributing to the mathematical theory of polyhedra, while at the same time he uses his skills as a sculptor and artist to create original works inspired by polyhedral objects.

There is a long tradition of making precise models of polyhedra with regularity properties. It is common at mathematics conferences for geometers to feature a models room where mathematicians who enjoy building models can display the beauties of geometry in a physical form. They complement the beauty of such geometric objects in the mind's eye. The beauty of polyhedral solids in the hands of a skilled model maker results in what are, indeed, works of art. Magnus Wenninger is the author of several books about model making. His models are especially beautiful. Here is a small sample, which only hints at the variety of models that Wenninger has made over many years. His models of "stellated" polyhedra are particularly striking.

A tiling of the plane is a way of filling up the plane without holes or overlaps with shapes of various kinds. For example, one can tile the plane with congruent copies of any triangle, and, more surprisingly, with congruent copies of any simple quadrilateral, whether convex or not. Tilings are closely related to artistic designs one finds on fabrics, rugs, and wallpaper. Though there were scattered analyses at different ways of tiling the plane that date back to ancient times, there was surprisingly little in the way of a theory for tilings of the plane as compared to what was done to understand polyhedra. Kepler did important work on tilings, but from his time until the late 19th century relatively little work was done. Unfortunately, not only was work on tilings sporadic but often it was incomplete or misleading. The publication of the monumental book by Branko Grünbaum and Geoffrey Shephard, Tilings and Patterns, changed this. Many new tiling problems were addressed and solved and a variety of software tools for creating tilings (and polyhedra and playing games) of different kinds were developed. Daniel Huson and Olaf Fredrichs (RepTiles) and Kevin Lee (Tesselmania) developed very nice tiling programs but some of the locations where this software used to be available are no longer supported.

A more recent source of art inspired by mathematics has been related to dissections. A good starting place for the ideas here is the remarkable theorem known as the Bolyai-Gerwien-Wallace Theorem. It states that two (simple) polygons A and B in the plane have the same area if and only if it is possible to cut one of the polygons up into a finite number of pieces and assemble the pieces to form the other polygon. In one direction this result is straight forward: if one has cut polygon A into pieces which will assemble to form polygon B, then B's area is the same as A's area. The delightful surprise is that if A and B have the same area then one can cut up A into finitely many pieces and reassemble the pieces to get B. Where does the art come in? Given two polygons with the same area, one can ask for two extensions of the Bolyai-Gerwien-Wallace Theorem:

a. Find the smallest number of pieces into which A can be cut and reassembled to form B.

b. Find pieces with appealing properties into which A can be cut and reassembled to from B. These properties might be that all the pieces are congruent, similar, or have edges which are related by some appealing geometric transformation.

Greg Frederickson has collected together a large amount of material about how polygons of one shape can be dissected into other polygons of the same area. These dissections concentrate on dissections of regular polygons (which may be convex or "star-shaped") into other regular polygons. One might expect that the mathematical regularity of the objects leads to aesthetic solutions. This turns out to be the case.

Frederickson also describes how with a suitable mechanism, one can address how to attach the pieces from one polygon and move them so that they create the other polygon. These dissections are known as hinged dissections. The first way that comes to mind to hinge the pieces is to attach the pieces at their vertices. There are lovely examples of hinged dissections of this kind including ones that prove the Pythagorean theorem geometrically by showing how one can cut the squares on the two legs of a right triangle and assemble the pieces to form the square on the hypotenuse. However, there is another ingenious way to do the hinging. This involves hinging the edges so that the polygons that are joined along these two edges can rotate with respect to one another. This type of hinging is known as twist-hinging. Frederickson arranged for several very attractive hinged dissections to be realized physically with polygonal sections of the dissection involved to be made of beautiful woods. For their beauty these physical models draw heavily on the mathematics behind the dissections. For example, in one of the physically realized hinged dissections commissioned by Frederickson, a regular hexagon with a hole is twist-hinge dissected into a hexagram with a hole of the same area.

The mathematics behind this process is a way to dissect a hexagon with a hole into a hexagram with a hole. Based on this dissection Frederickson cleverly produced a hinge-twist dissection. This very attractive object is not very interesting as a puzzle but creates a lovely effect as one watches the unexpected transformation between the two shapes evolve as one manipulates the twist-hinged pieces.


6. Origami

Traditional origami was concerned with taking a single piece of paper and folding it into complex shapes, typically that of an animal or something representational. However, Tomoko Fusè revolutionized the world of origami from a mathematical perspective by popularizing "modular" origami. In modular origami one typically starts with congruent pieces of paper (usually squares) and folds each of these into identical "units." These units are then "woven" together to form highly symmetrical objects such as polyhedra, tilings, or boxes. By using appropriate colors, one can often construct very attractive paper models of a wide variety of polyhedra and tilings with attractive symmetry properties. The creativity involved in unit origami lies in the ingenious panels that have been developed and the way that the panels can be assembled. Fusè's books appear in the "art section" of book stores. Interestingly, for people with some experience in origami, those of Fusè's books which have not been translated into English still can be used because of the universality of the instruction system for folding origami (e.g. symbols for mountain folds, valley folds, etc.).

Parallel with the artistic aspects of origami constructions has been the development of a mathematical theory of origami. This has taken a variety of approaches. The elaborate mathematical theory of what plane figures can be drawn using the traditional Euclidean construction tools of straight edge (unmarked ruler) and compass has an origami companion. What are the shapes that can be constructed using various rules (axioms) concerning the folding of paper? Thomas Hull, Erik Demaine and others have also studied issues related to folding and origami. A major area of interest has been the study of the crease patterns (system of lines on the paper) which can be folded "flat." The mathematics needed involves ideas and methods somewhat different from what was done in the past in attempting to understand how a piece of a plane (a square of origami paper) could be transformed by a geometric transformation, because at the end of the transformation parts of the origami paper touch each other, though they do not interpenetrate other parts of the paper.

Here is an example of a spectacular proven result in this area. Suppose that after having folded a piece of paper flat, one is allowed to make one cut along a straight line with the goal of taking the pieces that are cut off and opening them up. What shaped pieces are possible in this way? The surprising answer is: Any graph consisting of vertices and straight line segments that can be drawn in the plane is possible! For example, one could cut out the shape of something prosaic like the letter "I" or the outline of a butterfly. This result was originally developed by Erik Demaine, Martin Demaine, and Anna Lubiw. Subsequently a different approach to the result was developed by Marshall Bern, Erik Demaine, David Eppstein, and Barry Hayes.

One can make many different kinds of polyhedral objects using modular origami. Here is a sample of origami models of Helena Verrill:

Origami models of polyhedra use approaches where the panels become the faces of the polyhedra, so that the challenge becomes producing panels with different numbers of sides with the same edge lengths. One can also produce polyhedra which are "pyramided." By this I mean that the solids represent convex polyhedra with pyramids erected on each face. (Those are not stellations in the usual sense that geometers use this term.) Other polyhedra like these emphasize the edges of the polyhedron and in essence serve as rigid rod models for the polyhedra. They resemble Leonardo da Vinci's drawings that demonstrated emerging techniques of drawing polyhedra in perspective.

In addition to being beautiful objects, many of the polyhedra that can be created using origami paper suggest mathematical questions of interest. Here is a simple example: One can make a cube out of six unit origami pieces. If these six pieces are all the same color, then one can make only one "type" of colored cube. Suppose that one has three panels of one color and three panels of another color. How many inequivalent cubes can one make?

The diagram below shows an origami construction based on ideas of Thomas Hull and folded by Joe Gilardi. At the mathematical level is a nested collection of tetrahedra folded from dollar bills. Many also see a work of art!

In the discussions above I have charted the tip of the iceberg of the connections between mathematics and art. These connections are good for both mathematics and art. Clearly, interest in the connections between mathematics and art will continue to grow and prosper.

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CONCRETAGENS #98

Por Philadelpho Menezes

CONCRETAGENS #86

Por Integrando Conhecimento

HOW A STRANGE GRID REVEALS HIDDEN CONNECTIONS BETWEEN SIMPLE NUMBERS

A long-standing puzzle seems to constrain how addition and multiplication relate to each other. A graduate student has gone further than anyone else in establishing the connection.

In 1983 the prolific conjecturer Paul Erdős posed a math problem: Take any set of numbers you like. These could be the whole numbers from 1 to 12, the first 10,000 prime numbers, or the dates of every birthday in your extended family. Arrange these numbers in a square grid, with your list of numbers arranged both across the bottom and up one side. Then fill in the grid with either the sums or the products of the crosswise pairs.

Erdős and his collaborator on the problem, Endre Szemerédi, were interested in the number of distinct entries in such a grid. (By “distinct,” they meant that if a number appears twice — for instance, if the number 4 appears as a product of both 1 × 4 and 2 × 2 — you only count it once.) They conjectured that the number of distinct entries in either the sum or the product grid (or both) must be at least a certain value. They even established an exact threshold that the number of distinct sums or distinct products always had to meet.

Put more generally, the conjecture says that the additive and multiplicative properties of a set of numbers somehow force each other to behave in a certain way. If we flip the problem around and look at the number of duplicate entries, it says the following: If there are a lot of duplicates in the sum grid, then there cannot be that many duplicates in the product grid (and vice versa). It’s as if the sum and product sets are magnetically charged to repel each other.

And like magnetism, the phenomenon is easy to observe but much harder to explain. “It somehow delves into the deep structure of the integers, meaning the interplay between multiplication and addition,” said Kevin Ford, a mathematician at the University of Illinois, Urbana-Champaign, who has worked on the problem.

Over the last few decades, mathematicians have clawed their way toward proving Erdős and Szemerédi’s threshold. In a paper posted last year, a graduate student at Urbana-Champaign named George Shakan came the closest yet.

Decoding a Progression

The most familiar grid in arithmetic is the times table. Write out the numbers 1 to 12 along the bottom and up the side and fill out the grid accordingly. If you compare the grid of products with the grid of sums, you might notice that there are many more distinct products than distinct sums.

Erdős and Szemerédi certainly did. At the same time they proposed their conjecture, they solved a special case of the problem. They showed that whenever the numbers used to generate the grid are consecutive (such as 1 through 12), the number of distinct sums will be 2N − 1, where N stands for the number of numbers used to generate the grid. The number of distinct products, however, will be much larger — more on the order of a constant value times N^2.

But Erdős and Szemerédi’s “sum-product” problem went far beyond the case of consecutive numbers. They proposed that no matter what set of numbers you use to generate your grid — whether the numbers are consecutive or nonconsecutive, powers of 2 or county-by-county vote totals — either the number of distinct sums, the number of distinct products, or both has to be close to N^2.

George Shakan has come the closest to proving Paul Erdős’s conjecture that connects addition to multiplication.

This basic observation has significant implications. In the field of number theory, mathematicians study different types of sequences of numbers. One is the sequence you construct by starting with a number and repeatedly adding to it multiples of some second number. So if your starting number is 2, and your second number is 3, you’d construct your sequence as: 2, 2 + (3 × 1), 2 + (3 × 2), 2 + (3 × 3), 2 + (3 × 4), and so on. Or: 2, 5, 8, 11, 14, etc.

This kind of sequence is called an arithmetic progression. It figures in many important conjectures in number theory and can arise through far more complicated procedures than the one described above. Sometimes mathematicians confront a sequence of numbers and are unable to tell straightaway whether it’s an arithmetic progression.

Yet if you use numbers from an arithmetic progression to generate a grid, the number of distinct sums is always small. And that means that if you’re handed numbers whose sum grid has few distinct entries, it’s a good bet those numbers are closely related to an arithmetic progression.

“There’s a sort of expectation that any time you have a set whose sum set is not much larger than itself, there’s an arithmetic progression around,” said Nets Katz, a mathematician at the California Institute of Technology who has worked on the sum-product problem.

Another type of numerical sequence is made by taking two numbers and repeatedly multiplying the first number by powers of the second number. So if your numbers are 2 and 3, you’d construct your sequence as: 2 × 3^0, 2 × 3^1, 2 × 3^2, 2 × 3^3, 2 × 3^4, and so on. Or: 2, 6, 18, 54, 162, etc. This kind of sequence is called a geometric progression.

In general, grids produced from geometric progressions have few distinct products, but many distinct sums, whereas grids produced from arithmetic progressions will have the opposite property: few distinct sums, but many distinct products. (For grids produced by sets of numbers chosen at random, you’d expect many distinct sums and many distinct products.)

So the sum-product conjecture, which says that no set of numbers can produce a grid with simultaneously few distinct sums and few distinct products, is really making a statement about the relationship between arithmetic and geometric progressions.

“What the conjecture is really about is asking, ‘Can a set look both like an arithmetic progression and a geometric progression at the same time?’ And the answer is, ‘No, it can’t, because these are very different objects,’” Ford said.

And one way to prove that they’re very different objects is to prove that Erdős and Szemerédi’s threshold holds for any set of numbers.


Translating the Problem

A complete solution to the sum-product problem would establish that the number of distinct sums or products is always very close to N^2. So far, mathematicians have not succeeded in proving this. Instead, they have engaged in a Lilliputian race over the last several decades: In 1997 György Elekes proved that the number of distinct products or sums must be at least N^5/4; in 2009 Jozsef Solymosi proved it must be at least N^4/3; seven years later Misha Rudnev, Ilya Shkredov and Sophie Stevens improved the exponent to 4/3 plus a little bit more. (A very little bit. Their exponent was 4/3 + 1/1509.)

But it was the Elekes result that really defined how mathematicians approach the problem. In that 1997 paper he developed a strategy that’s been followed by most everyone since. Using techniques from a field called incidence geometry, Elekes showed how the question of sums and products can be translated into a question about how lines intersect points on the plane.

The strategy has you convert the sum and product grids into grids in the plane. Take the distinct sums in your sum grid and mark them along the x-axis. Take the distinct products in your product grid and mark them along the y-axis. Then mark the points where every x-axis value intersects with every y-axis value. If your set is {1, 2, 3}, you’ll plot 30 points in a 5-by-6 grid.

Next, draw lines in the plane. The lines will be of the form y = a × (x − c), where a and c come from the original numbers used to produce the grids (in our case, the numbers 1, 2 and 3). This process will yield nine distinct lines, which traverse the space in the plane where you’ve plotted the 30 points.

The question of interest is how often those nine lines intersect the 30 points. Elekes showed that the number of distinct sums or products in a grid is inversely related to the number of intersections yielded by the point-line procedure. Therefore, if you can show that there are few intersection points, you’ve proved that the underlying set has many distinct products or sums or both.

“You’re thinking about sums and products and all of a sudden you’re drawing points and lines,” Shakan said. “It’s not obvious there should be any connection, but there is.”

In his 2018 paper, Shakan elaborated the point-line approach to wring an even larger exponent from Elekes’ method, proving that the number of distinct sums or products (or both) must be at least N^4/3+1/1055.4.

Almost a year later, Shakan’s record still stands, though a full solution to the sum-product conjecture is as far away as ever. Using Elekes’ method, mathematicians have been making progress at an incremental pace that seems unlikely ever to get them all the way there. For that, wholly new ideas are called for.

“Addition and multiplication are two of the most basic operations in math,” Shakan said. “It shows how ignorant we are, because we can’t say everything about this problem.”

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SÉTIMA ARTE

Este ano, além tentar assistir 100 obras cinematográficas (dentre os mais variados gêneros), também decidi incluir alguns curta-metragens na lista.

Comecei pelo emocionante Zero (2010), de Christopher Kezelos:

"Nascido em um mundo de números, um oprimido zero descobre que através da determinação, coragem e amor, nada pode ser realmente alguma coisa."

536 PUZZLES & CURIOUS PROBLEMS - HENRY ERNEST DUDENEY

"In his Introduction, editor Martin Gardner hails Henry E. Dudeney as "England's greatest maker of puzzles," unsurpassed in the quantity and quality of his inventions. This compilation of long-inaccessible challenges ranges from arithmetical and algebraical problems to those involving geometry, combinatorics, and topology, plus game, domino, and match puzzles. Includes answers."

 

 

COMPRAR / LEITURA

MATHEMATICIANS SEAL BACK DOOR TO BREAKING RSA ENCRYPTION

Digital security depends on the difficulty of factoring large numbers. A new proof shows why one method for breaking digital encryption won’t work. 

Kevin Hartnett - Senior Writer @ Quanta Magazine

* * *

My recent story for Quanta explained a newly proved phenomenon that might seem surprising from a naive perspective: Virtually all polynomials of a certain type are “prime,” meaning they can’t be factored.

The proof has implications for many areas of pure mathematics. It’s also great news for a pillar of modern life: digital encryption.

The main technique we use to keep digital information secure is RSA encryption. It’s a souped-up version of the encryption scheme a seventh grader might devise to pass messages to a friend: Assign a number to every letter and multiply by some secretly agreed-upon key. To decode a message, just divide by the secret key.

RSA encryption works in a similar fashion. In simplified form, it goes like this: A user starts with a message and performs arithmetic on it that involves multiplication by a very large number (hundreds of digits long). The only way to decode the message is to find the prime factors of the resulting product.1 The security of RSA encryption rests on the fact that there’s no fast way to identify the prime factors of very large numbers. If you’re not the intended recipient of a message — and if you therefore lack the right key to decode it — you could search for a thousand years with the best computers and still not find the right prime factors.

1. The prime factors of a number are the prime numbers you need to multiply together to produce that number. The prime factors of 12 are 2 × 2 × 3. The prime factors of 495 are 3 × 3 × 5 × 11.

But there is a back door, and it has to do with polynomial equations. Every number can be represented as a unique polynomial equation. While it’s hard to find the prime factors of a number, it’s easy to find the factors of a polynomial. And once you know the factors of a polynomial, you can use that information to find the prime factors of the number you started with.

Here’s how it works.

Step One: Pick a number whose prime factors you’d like to know. To take a simple example, let’s use the number 15.

Step Two: Convert 15 into binary notation:

1111.

Step Three: Turn that binary expression into a polynomial by treating the binary digits as coefficients of a polynomial:

x3 + x2 + x + 1.

(Note that this polynomial equals 15 when x = 2, because 2 is the base of binary notation.)

Step Four: Factor the polynomial:

(x2 + 1) × (x + 1).

Step Five: Plug x = 2 into each of those factors:

(22 + 1) = 5

(2 + 1) = 3.

Conclusion: 5 and 3 are the prime factors of 15.

This seems like a complicated way to find the prime factors of a small number like 15, whose factors are easy to spot straightaway. But with very large numbers — numbers with hundreds of digits — this polynomial method gives you a remarkable advantage. There’s no fast algorithm for factoring large numbers. But there are fast algorithms for factoring large polynomials. So once you convert your large number to a large polynomial, you’re very close to finding the number’s prime factors.

Does this mean RSA encryption is in trouble? Actually, no. The reason for this has to do with the new proof about polynomials. The mathematicians Emmanuel Breuillard and Péter Varjú of the University of Cambridge proved that as polynomials with only 0 and 1 as coefficients get longer, they’re less and less likely to be factorable at all. And if a polynomial can’t be factored, it can’t be used to identify the prime factors of the number it’s based on.

Breuillard and Varjú’s proof effectively slams shut the polynomial back door for breaking RSA encryption. The very large numbers used in RSA encryption correspond to very long polynomials. Breuillard and Varjú proved that it’s nearly impossible to find polynomials of that length that can be factored. Mathematicians and cryptographers alike have long suspected this is the case. But when the cybersecurity of the entire world depends on some mathematical hack not working, it’s good to have proof that it doesn’t.

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