What Do Math And Drawing Have In Common

Math Sensation Month - April 2003 Mathematics and Fine art
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Mathematics and Art -- So Many Connections

Doris Schattschneider, Moravian Higher

The theme Mathematics AND Art may seem strange to those who are more used to thinking Mathematics OR Art, but, in fact, there are many connectors to fill the blank in

Mathematics ___ Art and its twin Art___ Mathematics. Indeed, the many interconnections between mathematics and art provide a wealth of material from which organizers of Math Awareness Month events tin select. The list of sources on the Math Awareness spider web site is a bully place to start. In this cursory essay, I'll highlight a few of the possible ways to fill up in the blanks in a higher place. I hope that this will stimulate y'all to explore many others.

Mathematics produces art

At the nigh practical level, mathematical tools have ever been used in an essential way in the creation of fine art. Since ancient times, the lowly compass and straightedge, augmented by other simple draftsmen's and craftsmen'south tools, accept been used to create beautiful designs realized in the architecture and decoration of palaces, cathedrals, and mosques. The intricate Moorish tessellations in tile, brick, and stucco that adorn their buildings and the equally intricate tracery of Gothic windows and interiors are a attestation to the imaginative use of ancient geometric noesis. [R1], [R4], [R21]

During the Renaissance, several artists used uncomplicated grids and mathematically-based devices to accurately portray scenes on a flat surface, co-ordinate to the principles of linear perspective. Several of Dürer's engravings requite a glimpse of these techniques. The symbiosis of art and mathematics during these times as linear perspective and projective geometry were developing is one of the almost hitting examples of art and mathematics evolving most simultaneously in new directions. [R7]

Today's mathematical tools are more sophisticated, with digital technology fast becoming a primary choice. In the easily of an artist, computers can produce art, powered by unseen complex internal mathematical processes that provide their magical abilities. Mathematical transformations provide the means past which an image or form in one surface or space is represented in another. Art is illusion, and transformations are important in creating illusion. Isometries, similarities, and affine transformations can transform images exactly or with purposeful distortion, projections can represent 3 (and higher)-dimensional forms on 2-dimensional moving picture surfaces, even curved ones. Special transformations can distort or unscramble a distorted image, producing anamorphic fine art. All these transformations can be mathematically described, and the use of guiding grids to assist in performing these transformations has been replaced today largely by computer software. Compasses, rulers, grids, mechanical devices, keyboard and mouse are physical tools for the creation of art, only without the power of mathematical relationships and processes these tools would have little creative power.

Mathematics generates art

Design is a fundamental concept in both mathematics and art. Mathematical patterns can generate artistic patterns. Frequently a coloring algorithm can produce "automated art" that may be as surprising or aesthetically pleasing as that produced by a homo paw. Colored versions of the Mandelbrot fix and Julia sets are hit examples of this: each is generated by the recursive equation z_n = z_{n -1} ⁱⁱ + C. In the case of the Mandelbrot set the equation is iterated for each point C in the complex plane, where z₀ = 0 and the point C is colored according to rules based on whether the iterated values eventually exceed two and the number of iterations after which this occurs. [P15] Other fractals, as well equally images based on attractors, are too produced past iteration and coloring according to rules. The intricacy of these images, their symmetries, and the endless (in theory) continuance of the designs on e'er-smaller scales, makes them spellbinding. [P10]

Much more mundane mathematical patterns tin besides provide surprising art. For example, begin with an array of numbers (such as a big data ready, a sequence, a modular operation table, or Pascal's triangle) and color the numbers in the array according to some rule. Often surprising patterns -- even fine art -- emerges. Recursive algorithms practical to geometric figures can generate attractive self-like patterns. Begin with a curve, a closed figure, or a simple spatial form, employ an algorithm to alter that figure by adding to (or subtracting from) specified parts of that figure, so echo the algorithm recursively. Many nonperiodic tilings (such every bit the Penrose tilings) can also be generated automatically, beginning with a small patch of tiles and then applying a recursive "inflation" algorithm.

Transformations and symmetry are also fundamental concepts in both mathematics and art. Mathematicians actually define symmetry of objects (functions, matrices, designs or forms on surfaces or in space) by their invariance under a grouping of transformations. Conversely, the application of a group of transformations to simple designs or spatial objects automatically generates beautifully symmetric patterns and forms. In 1816, Brewster's newly-invented kaleidoscope demonstrated the power of the laws of reflection in automatically generating eye-catching rosettes from jumbles of colored shards between two mirrors. [P3] Today computer programs use symmetry groups to generate rosettes, borders, wallpaper designs [R11], and Escher-like circle-limit designs such as featured on the 2003 Math Awareness Month poster. [R6] Each of these designs begins with a small fragment or motif (chosen judiciously or randomly) whose transformed images fill up out the full design. Periodic tessellations, whether geometric or Escher-like, can be automatically generated by figurer programs [R12] or by hand, post-obit recipes that employ isometries.

Art illuminates mathematics

When mathematical patterns or processes automatically generate art, a surprising opposite effect tin occur: the art often illuminates the mathematics. Who could take guessed the mathematical nuggets that might otherwise be subconscious in a torrent of symbolic or numerical information? The process of coloring allows the information to have on a visual shape that provides identity and recognition. Who could estimate the limiting shape or the symmetry of an algorithmically produced fractal? With visual representation, the mathematician can exclaim "now I run across!"

Since periodic tessellations tin exist generated by groups of isometries, they can be used to illuminate abstract mathematical concepts in group theory that many find hard to grasp in symbolic class: generators, cosets, stabilizer subgroups, normal subgroups, conjugates, orbits, and group extensions, to proper name a few. [R20]

In the examples to a higher place, illumination of mathematics is a serendipitous event of fine art created for other reasons. Merely there are examples in which the artist'due south main purpose is to express, even embody mathematics. Several prints by Chiliad.C. Escher are the result of his attempts to visually express such mathematical concepts as infinity, duality, dimension, recursion, topological morphing, and self-similarity. [R16] Perhaps the most striking examples of art illuminating mathematics are provided by the paintings of Crockett Johnson and the sculptures of Helaman Ferguson. From 1965 to 1975, Johnson produced over 100 abstract oil paintings, each a representation of a mathematical theorem. [P14] Ferguson'due south sculptures celebrate mathematical form, and take been termed "theorems in bronze and stone." Each begins with the thought of capturing the essence of a mathematical theorem or relationship, and is executed past harnessing the full ability of mathematically-driven and mitt-guided tools. [P9]

Mathematics inspires art

Patterns, designs, and forms that are the "automatic" product of purely mathematical processes (such as those described in "Mathematics generates art") are unremarkably too precise, too symmetrical, as well mechanical, or too repetitive to hold the art viewer's attention. They can be pleasing and interesting, and are fun to create (and provide much "hobby-fine art") just are mostly devoid of the subtlety, spontaneity, and deviation from precision that creative intuition and inventiveness provide. In the hands of an creative person, mathematically-produced art is only a starting time, a skeleton or a template to which the artist brings imagination, training, and a personal vision that can transform the mathematically perfect to an epitome or form that is truly inspired.

Wallpaper patterns and tessellations tin be pleasing from a decorative betoken of view; few would be viewed as art. [P24] (I feel [P20] is an exception). Escher did not view his tessellations as fine art, only as fragments to be an integral office of his complex prints. Makoto Nakamura's art also employs this technique. [P17] Jinny Beyer, a designer and quilt creative person, uses her artistic intuition and color sense to plough tessellations into art. [P2] Kaleidoscopic designs are the inspiration for quilted art by Paula Nadelstern; her utilise of colour and composition subtly break mathematical rules. [P16]

Dick Termes uses photography and grids to guide his projections of images onto the surface of a sphere, but his "Termespheres" conduct his personal interpretation. [P23] Anamorphic artists István Orosz [P18] and Kelly Houle [P13] are guided past mathematical rules of transformation as they create mysterious distortions of images on the picture show plane, only also utilise their intuition and imagination, checking with a mirrored cylinder as the work develops.

Pure mathematical grade, often with high symmetry, is the inspiration for several sculptors who create lyrical, scenic works. With adept eye and hand, relying on their experience with wood, stone, statuary, and other tactile materials, the artists deviate, exaggerate, subtract, overlay, environs, or otherwise change the grade into something new, ofttimes dazzlingly beautiful. With the appearance of digital tools to create sculpture, the possibilities of experimentation without devastation of material or of producing otherwise impossible forms infinitely extends the sculptor's abilities. [P4], [P5], [P12], [P19], [P21], [P22]

Mathematics constrains fine art

We frequently hear of "artistic freedom" or "artistic license," which imply the rejection of rules in order to have freedom of expression. Nonetheless many mathematical constraints cannot exist rejected; artists ignorant of these constraints may labor to realize an idea simply to find that its realization is, indeed, impossible. Euler's theorem (five + f = e + 2) and Descartes' theorem (the sum of the vertex defects of every convex polyhedron is 720°) govern the geometry of polyhedra. Other theorems govern the topology of knots and surfaces, aspects of symmetry and periodicity on surfaces and in infinite, facts of ratio, proportion, and similarity, the necessity for convergence of parallel lines to a signal, and and then on. Rather than circumscribed fine art or requiring art to accommodate to a narrow set of rules, an understanding of essential mathematical constraints frees artists to employ their total intuition and inventiveness within the constraints, even to push the boundaries of those constraints. Constraints demand not exist negative -- they tin show the often limitless realm of the possible.

Voluntary mathematical constraints can serve to guide artistic creation. Proportion has always been fundamental in the aesthetic of fine art, guiding limerick, design, and form. Mathematically, this translates into the observance of ratios. Whether these exist canons of human proportion, architectural design, or even symbols and letter fonts, ratios connect parts of a blueprint to the whole, and to each other. Repeated ratios imply cocky-similarity, hardly a new topic despite its recent mathematical attending. One of the earliest recorded notices of it is in Euclid'due south Prop. 30, Book 6, the division of a segment in extreme and hateful ratio (too known as the golden cut, or golden section). A segment AB is to be divided internally by signal E and then that the ratio of the whole AB to the part AE equals the ratio of the (larger) part AE to the (smaller) part EB. [P8] This geometric task produces the common ratio AB/AE = (one + 5)/2, known equally the gold ratio, denoted as (or ). The ratio has many unique, almost magical mathematical backdrop (for example, ² = + 1, and ane/ = - one), and it is these properties, likewise as connections to the Fibonacci sequence, that accept fascinated artists and architects, enabling them to produce designs and compositions with special properties. Other ratios and special geometric constructions (root rectangles, reciprocal rectangles, and grids of similar figures) besides guide limerick and blueprint. [R10], [P11]

Art engenders mathematics

It is to be expected that in the execution of an artwork, mathematical questions volition ascend that the artist (or fabricator) must answer. This goes with the territory. In many instances, artists volition struggle to answer the questions on their own in order to reach the answer in a style that makes sense to them. Escher did this in seeking to answer the question "How can I create a shape that will tile the plane in such a mode that every tile is surrounded in the same way?" [R15] Sometimes these questions need the attention of trained mathematicians, engineers, or software designers and provide interesting practical problems to solve. The intricate textile patterns of designer Jhane Barnes result from close collaboration with mathematician Bill Jones and computer software designer Dana Cartwright of Designer Software. [R14], [P1], [P6]

There are too frequent instances where finished works of fine art propose purely mathematical questions, ones that the artist never imagined, nor needed to consider. Folk art from other times and other cultures is a rich source for mathematical questions. Celtic knots and art from African cultures are two examples. [R5], [R8] Modern sculptures can also pb to mathematical questions. [R3] Escher's tessellations and some prints have been the source of several mathematical challenges, most not even so settled. Ii of these mathematical questions seek to understand the relationships between local and global symmetry. [R9], [R19].

A most mathematical artist

I want to end this essay with a flake more about the work of the Dutch graphic artist Thousand.C. Escher (1898-1972), who is possibly the near astonishing recent case of an artist whose work contains a multitude of connections between mathematics and art. [P7] Escher was non mathematically trained, and even struggled with mathematics as a school student. Nonetheless he did non refuse mathematics, simply instead figured out in his own manner, using various (more often than not pictorial) sources, the mathematics that he needed in society to realize his ideas and visions. Escher historic mathematical forms: polyhedra equally decoration, stars, or living structures, mvbius bands, knots, and spatial grids. He used (and sometimes fused) diverse geometries in his work -- Euclidean in his tessellations, hyperbolic in his Circumvolve Limit serial, projective in depicting scenes in linear perspective, spherical in prints and his carved spheres. He employed topological distortions and transformations, strange or multiple perspectives, and visual recursion. He explored the topic of symmetry and tessellation in the plane, on the sphere, and in the Poincari deejay, developing his own "layman's theory" of classification of types of planar periodic tilings and symmetric coloring of them, anticipating mathematician's and crystallographer's later studies of these topics. [R15] He asked and answered, in his ain way, combinatorial geometric questions. [R17] He depicted abstract mathematical concepts in visual metaphors. And though Escher's work gained him the adoration of mathematicians and scientists, he felt isolated as an creative person. Today in that location are many artists whose work is directly or indirectly inspired past Escher'south work. While he has left united states his own legacy, others are continuing to explore some of the paths he blazed and also are striking out on new paths from these.[R18]

Resources

Iii recent books contain collections of essays on mathematics and art. Ivars Peterson [R13] showcases a wide choice of art and artists of today exemplifying the strong symbiosis betwixt art and mathematics. Two others, [R2] and [R18], incorporate thoughtful commentaries and discussions too as essays and fine art by contemporary artists. Several books and web sites provide text, ideas, issues and projects for courses focused on art and mathematics. Many of these are listed on the 2003 Mathematics Awareness Calendar month spider web site, which also contains essays by Mark Frantz and Paul Calter, who teach courses on mathematics, art, and architecture. There are several organizations that are defended to fostering interaction between art, mathematics, and science. Nearly hold annual conferences at which artists and mathematicians (and many others) gather to exhibit, lecture, discuss, and mingle; ofttimes proceedings (print or electronic) publish the presentations. Web sites for several of these are listed under Organizations below.

Organizations

[O1] Digital Sculpture: http://www.intersculpt.org/

[O2] Dutch Society for Arts and Mathematics: http://www.arsetmathesis.nl (cheque the galerij)

[O3] Leonardo/ISAST: http://mitpress2.mit.edu/e-journals/Leonardo (International Order for the Arts, Sciences, and Engineering science; check the Gallery)

[O4] ISAMA: http://www.isama.org/ (International Society of the Arts, Mathematics, and Architecture)

[O5] Nexus Network Journal: http://www.nexusjournal.com/ (Architecture and Mathematics)

[O6] Bridges: http://www.sckans.edu/~bridges (Almanac International conference on mathematical connections in art, music, and scientific discipline; nerveless papers from its annual conferences are printed)

[O7] Visual Mathematics: http://www.mi.sanu.air-conditioning.yu/vismath (electronic journal of ISIS-Symmetry)

[08] Aesthetic Computing http://www.cise.ufl.edu/~fishwick/aescomputing

References: Books, manufactures, software

[R1] Bourgoin, J., Arabic and Geometrical Pattern and Design (plates), New York: Dover, 1973 (orig. 1879).

[R2] Bruter, Claude P., ed., Mathematics and Fine art: Mathematical Visualization in Art and Teaching, Heidelberg: Springer, 2002.
http://www.springer.de/cgi/svcat/search_book.pl?isbn=3-540-43422-4

[R3] Coxeter, H.S.Grand., "Symmetric Combinations of Three or 4 Hollow Triangles," Math. Intelligencer, v. 16 (1994) 25-30. See also Burgiel, H., Franzblau, D.Southward. and Gutschera, K.R., "The Mystery of the Linked Triangles," Mathematics Magazine, v. 69 (1996) 94-102.

[R4] Critchlow, Keith, Islamic Patterns: An analytical and cosmological arroyo, New York: Schocken books, 1976. Paperback reprint, London: Thames and Hudson, 1999.

[R5] Cromwell, Peter R., "Celtic Knotwork: Mathematical Art," Math. Intelligencer, five. 15 (1993) 36-47.

[R6] Dunham, Douglas, "Transformation of Escher Hyperbolic Patterns," Visual Mathematics, v. 1, no. i, 1999.
http://members.tripod.com/vismath/pap.htm - n11 Too see Dunham'south essay about the 2003 Mathematics Awareness Month web site.

[R7] Field, J.V., The Invention of Infinity: Mathematics and Art in the Renaissance, Oxford: Oxford Academy Pr., 1997.

[R8] Gerdes, Paulus, Geometry from Africa: Mathematical and Educational Explorations, MAA, 1999.

[R9] Grünbaum, Branko, "Mathematical Challenges in Escher'south Geometry," in One thousand.C. Escher: Art and Science, H.Due south.Thou. Coxeter, M. Emmer, R. Penrose, and Thou.L. Teuber, eds, Amsterdam: Northward-Kingdom of the netherlands, 1986, pp. 53-67.

[R10] Kappraff, Jay, Connections, The Geometric Bridge Between Art and Science, New York: McGraw-Colina, 1991. 2d ed., Singapore: World Scientific Publ. Co., 2002. http://www.nexusjournal.com/reviews_v4n4-Jablan.html

[R11] Lee, Kevin, Kaleidomania!, Key Curriculum Press.
http://www.keycollege.com/catalog/titles/kaleidomania.html

[R12] Lee, Kevin, Tessellation Exploration, Tom Snyder Productions.
http://www.tomsnyder.com/free_stuff/free_download.asp?PS=TESEXP

[R13] Peterson, Ivars, Fragments of Infinity: A Kaleidoscope of Math and Fine art, New York: Wiley, 2001.
http://www.isama.org/volume/fragments/biblio.html

[R14] Ross, Teri, Math + Technology = Technique: The Jhane Barnes School of Textile Design.
http://world wide web.techexchange.com/thelibrary/jhanebarnes.html

[R15] Schattschneider, Doris, Visions of Symmetry: Notebooks, Periodic Drawings, and Related Work of Chiliad.C. Escher, New York: W.H. Freeman, 1990.

[R16] Schattschneider, Doris, "Escher's Metaphors," Scientific American, 5. 271 no. 5 (November 1994) 66-71.

[R17] Schattschneider, Doris, "Escher'southward Combinatorial Patterns," Electronic J. of Combinatorics, four (no. two) (1997), #R17.
http://www.combinatorics.org/Volume_4/wilftoc.html

[R18] Schattschneider, Doris and Emmer, Michele, eds. Yard.C. Escher's Legacy: A Centennial Commemoration (with CD Rom), Heidelberg: Springer, 2003.
http://www.springer.de/cgi-bin/search_book.pl?isbn=iii-540-42458-X

[R19] Schattschneider, Doris, and Dolbilin, Nikolai, "One Corona is Enough for the Euclidean Plane," In Quasicrystals and Detached Geometry (J. Patera, editor). Fields Plant Monographs, Vol. 10, AMS, Providence, RI, 1998, pp. 207-246.

[R20] Senechal, Marjorie, "The Algebraic Escher," Structural Topology, 5. 15 (1988) 31-42.

[R21] Sykes, Mabel, Sourcebook of Problems for Geometry, Palo Alto: Dale Seymour Publ, 2000 (orig. 1912).

People and terms

[P1] Jhane Barnes: http://www.jhanebarnes.com/

[P2] Jinny Beyer: http://world wide web.jinnybeyer.com/(check the Quilt Gallery)

[P3] David Brewster: http://www.brewstersociety.com/brewster_bio.html

[P4] Paul Calter: http://www.sover.internet/~pcalter (meet Calter's essay on the MAM web site)

[P5] Brent Collins: http://www.cs.berkeley.edu/~sequin/SCULPTS/collins.html

[P6] Designer Software: http://www.weavemaker.com/

[P7] Yard.C. Escher: http://www.mcescher.com/

[P8] Euclid: http://aleph0.clarku.edu/~djoyce/java/elements/elements.html

[P9] Helaman Ferguson: http://www.helasculpt.com/

[P10] Fractal art: Googling on "fractal art" produces over 170,000 hits. Many sites take beautiful images.

[P11] Golden ratio: Googling on the words golden ratio, gold section, or divine proportion will produce over 360,000 hits; at that place is much good, merely also much erroneous and fabricated information on this topic.

[P12] George Hart: http://www.georgehart.com/sculpture/sculpture.html (see Hart's essay on the MAM web site)

[P13] Kelly Chiliad. Houle: http://world wide web.kellymhoule.com/

[P14] Crockett Johnson: http://www.ksu.edu/english language/nelp/regal/art.html

[P15] Mandelbrot and Julia sets: Google produces 8000 hits on the paired names. http://www.geocities.com/fabioc gives an elementary word and pictures

[P16] Paula Nadelstern: http://www.paulanadelstern.com/ (check the Quilt Gallery)

[P17] Makoto Nakamura: http://www18.large.or.jp/%7Emnaka/work.html

[P18] István Orosz: http://world wide web.geocities.com/SoHo/Museum/8716/index.html

[P19] Charles O. Perry: http://www.charlesperry.com/ (see Perry'southward essay on the MAM spider web site)

[P20] Marjorie Rice: http://members.aol.com/tessellations

[P21] John Robinson: http://www.johnrobinson.com/

[P22] Rinus Roelofs: http://www.rinusroelofs.nl/

[P23] Dick Termes: http://www.termespheres.com/

[P24] Tessellation: Googling on "tessellation" produces over 35,000 hits. Many contain examples of original creations.

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Mathematics Awareness Month is sponsored each yr by the Joint Policy Board for Mathematics to recognize the importance of mathematics through written materials and an accompanying affiche that highlight mathematical developments and applications in a particular expanse.

Source: https://ww2.amstat.org/mam/03/essay3.html

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