Gradiva Prize, National Association for the Advancement of Psychoanalysis; Book of the Year, Times (London, England).
Project and content management for Contemporary Authors volumes
WORK TITLE: Mathematics + Art
WORK NOTES:
PSEUDONYM(S):
BIRTHDATE: 6/24/1943
WEBSITE:
CITY:
STATE:
COUNTRY:
NATIONALITY:
http://www.sva.edu/faculty/lynn-gamwell * http://press.princeton.edu/titles/10537.html * https://www.theguardian.com/science/alexs-adventures-in-numberland/2015/dec/02/why-the-history-of-maths-is-also-the-history-of-art
RESEARCHER NOTES:
LC control no.: n 80088509
LCCN Permalink: https://lccn.loc.gov/n80088509
HEADING: Gamwell, Lynn, 1943-
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373 __ |a School of Visual Arts (New York, N.Y.) |2 naf
374 __ |a College teachers |2 lcsh
375 __ |a female
377 __ |a eng
670 __ |a His Cubist criticism, c1980: |b t.p. (Lynn Gamwell) CIP galley (Lynn Wissing Gamwell) CIP data sheet (b. 1943)
670 __ |a Mathematics + art, 2016: |b title page (Lynn Gamwell) jacket (Lynn Gamwell is a lecturer in the history of art, science, and mathematics at the School of Visual Arts in New York)
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PERSONAL
Born June 24, 1943.
EDUCATION:University of Illinois, Chicago, B.A.; Claremont McKenna College, M.F.A.; University of California, Los Angeles, M.A., Ph.D.
ADDRESS
CAREER
School of Visual Arts, New York, NY, lecturer in history of art, science, and mathematics. Curator of art exhibitions.
AWARDS:Gradiva Prize, National Association for the Advancement of Psychoanalysis; Book of the Year, Times (London, England).
WRITINGS
SIDELIGHTS
Lynn Gamwell was born on June 24, 1943. Her writings often explore the relationship of art to other disciplines and cultural phenomena, including mathematics, religion, and the sciences. After completing graduate school in art history, she spent much time studying the history of various sciences, and she brings this knowledge to her writing and teaching.
Madness in America
Madness in America: Cultural and Medical Perceptions of Mental Illness before 1914, a collaboration with Nancy Tomes, is based on an art exhibition and deals with, among other things, visual expressions of these perceptions. Gamwell and Tomes note the many biases that informed the diagnosis and treatment of mental illness in the nineteenth century. Women were considered more susceptible than men, southerners saw mental illness among African Americans in the North as a sign that blacks could not manage their lives outside the structure of slavery, and wealthy mental patients were provided much better conditions than poor ones. The authors offer numerous individual case histories as well. The book’s many illustrations include photographs of keys made by patients in the hope of escape from asylums, reproductions of maps one man created to depict his emotional state, and advertisements for corsets intended to cure illness through electric shock.
Some critics found the book both enlightening and disturbing. Gamwell and Tomes have produced “a grimly fascinating account” of the “sometimes harrowing history of America’s treatment of mental illness,” remarked an American Heritage contributor, adding: “The case histories and illustrations … could form their own compelling book.” William Beatty, writing in Booklist, pronounced Madness in America a “sound and attractive volume.”
Exploring the Invisible
In Exploring the Invisible: Art, Science, and the Spiritual, Gamwell lays out numerous connections between art and her other topics, demonstrating that they all informed one another throughout history, as she chronicles artistic trends and scientific advances. She includes a variety of illustrations as evidence of these relationships. Caspar David Friedrich’s 1818 painting Wanderer above a Sea of Fog depicts a time when humans were still mystified by natural phenomena but were about to make many discoveries, while an image of a nebula captured by the Hubble Space Telescope shows the beautiful visuals made accessible by science.
Several reviewers thought Exploring the Invisible a successful effort. Gamwell shows that “art, science, and religion are entwined in a dance, each affecting the others,” related Nadine Dalton in Library Journal. Skeptical Inquirer critic Austin Dacey added that the “sumptuously illustrated” volume is no less than “the definitive introduction to the territory between science and art.”
Mathematics + Art
Mathematics + Art: A Cultural History discusses how mathematics has informed art and how artists have expressed mathematical concepts. Gamwell covers a broad range of eras and cultures—prehistoric times, ancient Greece and Egypt, the Middle Ages, the Enlightenment, and the present day. She offers a variety of examples. American mathematician Robert Bosch’s diagram of the optimal route for a traveling salesman visiting five thousand cities looks like a Celtic knot. British artist Simon Thomas bases his sculptures on geometric forms and mathematical formulas. Karl Gerstner created artworks with units of color and shape arranged according to mathematical principles. Mathematicians Reza Sarhangi and Robert Fathauer crafted an image of a heptagon inspired by tenth-century Islamic scholar Abū al-Wafā’ Būzjānī’s work in geometry.
“As a lecturer at the School of Visual Arts in Manhattan, I wrote this book for my students, such as Maria, who told me she was never good at history because she couldn’t remember dates, and for Jin Sug, who failed high school algebra because he couldn’t memorize formulae,” Gamwell explained in a Guardian blog post. “I hope they will read this book and discover that history is a storybook and that math is about captivating ideas.”
Several critics deemed Gamwell’s book rather captivating. “By the time I was halfway through this book, I had accumulated ten pages of notes,” remarked Frank Swetz on the Mathematical Association of America’s Web site. “While some of my comments were for this review, the majority, were for my own reference: information to pursue, new wonderful facts I had not previously known: so much information, so many issues to consider or reconsider in light of new revelations and so many inspiring and intriguing images.” Danny Yee, the blogger behind Danny Reviews, noted that the volume “is, above all, fun to browse in, probably for people from quite a wide range of backgrounds, and it makes a lovely coffee table book, which provides some substance for those drawn in by the illustrations and which may bring people into contact with new ideas and inspire novel connections.” Ian Alexander, writing online at Obsessed by Nature, called Mathematics + Art “a beautiful, magnificent, and rather large book” that “makes a serious attempt to cover the ground comprehensively.” He thought Gamwell should have devoted more space to Islamic art, but he found the work overall “fascinating, beautiful, intriguing, and stimulating.” Swetz summed up Mathematics + Art as “an intellectual tour de force of art history and its interaction with mathematics that will draw most readers, including me, back for further reading and study.”
BIOCRIT
PERIODICALS
American Heritage, September, 1995, review of Madness in America: Cultural and Medical Perceptions of Mental Illness before 1914, p. 92.
Booklist, June 1, 1995, William Beatty, review of Madness in America, p. 1702.
Library Journal, December, 2002, Nadine Dalton, review of Exploring the Invisible: Art, Science, and the Spiritual, p. 116.
Skeptical Inquirer, November-December, 2006, Austin Dacey, review of Exploring the Invisible, p. 61.
ONLINE
Danny Reviews, http://dannyreviews.com/ (June, 2016), Danny Yee, review of Mathematics + Art: A Cultural History.
Guardian Online (London, England), https://www.theguardian.com (December 2, 2015), Lynn Gamwell, “Why the History of Maths Is Also the History of Art.”
Mathematical Association of America Web site, http://www.maa.org/ (December 11, 2015), Frank Swetz, review of Mathematics + Art.
Obsessed by Nature, http://www.obsessedbynature.com/ (February 22, 2016), Ian Alexander, review of Mathematics + Art.
School of Visual Arts Web site, http://www.sva.edu/ (March 22, 2017), brief biography.
Education
BA, University of Illinois, Chicago; MFA, Claremont McKenna College; MA, PhD, University of California, Los Angeles
Curatorial Work Includes
"Sigmund Freud Antiquities: Fragments from a Buried Past," Freud Museum, London; "Madness in America: Cultural and Medical Perceptions of Mental Illness Before 1914," University of Pennsylvania; "Art After Einstein," New York Academy of Sciences; "Sacred Geometry and Secular Science," Loyola University Museum of Art
Books Include
Exploring the Invisible: Art, Science, and the Spiritual; Searching for Certainty: Art, Mathematics, and the Mystical (note from Trudy--can't find book by this title); editor, Dreams 1900-2000: Art, Science, and the Unconscious Mind
Awards Include
Gradiva Prize, National Association for the Advancement of Psychoanalysis; Book of the Year, London Times
Quoted in Sidelights: art, science, and religion are entwined in a dance, each affecting the others
Gamwell, Lynn. Exploring the Invisible: Art, Science, and
the Spiritual
Nadine Dalton
Library Journal.
127.20 (Dec. 2002): p116.
COPYRIGHT 2002 Library Journals, LLC. A wholly owned subsidiary of Media Source, Inc. No redistribution permitted.
http://www.libraryjournal.com/
Full Text:
Princeton Univ. 2002. c.344p. permanent paper, index. LC 2002025106. ISBN 0-691-08972-8. $49.95. FINE ARTS
The director of the art museum at SUNY at Binghampton and adjunct science professor at the School of Visual Arts, Gamwell attempts to
enumerate what we've suspected all along: art, science, and religion are entwined in a dance, each affecting the others. Text and images flow
nicely from epoch to epoch, as Gamwell illustrates the zeitgeists that created some of the world's great ideas. One of the first images in the book
is a painting by Caspar David Friedrich, Wanderer Above a Sea of Fog, which perfectly illustrates the essence of life on the brink of the modern
scientific era. From there, the reader moves through various art movements and scientific discoveries, culminating in (of course) an image of a
cone nebula from the Hubble Space Telescope. Following the text are notes, a chronology of events, a broad list of suggestions for further
reading, and a functional index. Small problems of perception occur, such as listing the diagnosis of anorexia nervosa in the "spiritual" realm, and
there is a lack of spiritual emphasis in general; however, these issues do not detract from the book as a whole. Recommended for academic and
larger public libraries.--Nadine Dalton Speidel Cuyahoga Cty. P.L., Parma, OH
Dalton, Nadine
Source Citation (MLA 8th
Edition)
Dalton, Nadine. "Gamwell, Lynn. Exploring the Invisible: Art, Science, and the Spiritual." Library Journal, Dec. 2002, p. 116. General OneFile,
go.galegroup.com/ps/i.do?
p=ITOF&sw=w&u=schlager&v=2.1&id=GALE%7CA95917294&it=r&asid=92e8a5530c94d2775d34300addf3ccfe. Accessed 4 Mar. 2017.
3/4/2017 General OneFile - Saved Articles
http://go.galegroup.com/ps/marklist.do?actionCmd=GET_MARK_LIST&userGroupName=schlager&inPS=true&prodId=ITOF&ts=1488663552372 2/6
Gale Document Number: GALE|A95917294
---
Quoted in Sidelights:
sumptuously illustrated
the definitive introduction to the territory between
science and art
3/4/2017 General OneFile - Saved Articles
http://go.galegroup.com/ps/marklist.do?actionCmd=GET_MARK_LIST&userGroupName=schlager&inPS=true&prodId=ITOF&ts=1488663552372 3/6
Exploring the Invisible: Art, Science, and the Spiritual
Austin Dacey
Skeptical Inquirer.
30.6 (November-December 2006): p61.
COPYRIGHT 2006 Committee for the Scientific Investigation of Claims of the Paranormal
Full Text:
Exploring the Invisible: Art, Science, and the Spiritual. Lynn Gamwell, Foreword by Nell deGrasse Tyson. Princeton University Press Princeton,
2005. 344 pp. Hardcover, $23.95. This sumptuously illustrated book can rightly claim to be the definitive introduction to the territory between
science and art, in particular visual art. The author, who curates the Gallery of Art and Science at the New York Academy of Sciences and teaches
science at the School of Visual Arts in New York City, may well be world's most qualified guide to that territory. As Robert Rosenblum of New
York University puts it, "[n]obody before has dared to tackle this huge topic. Art historians don't know enough about science; scientists don't
know enough about art. Miraculously, probably uniquely, Lynn Gamwell seems equally at home in both areas." Skeptics should not be spooked
by the word spiritual. Gamwell uses it to mean a mystic's engagement with the unknown or ineffable. The book includes 156 color and 208 blackand-white
illustrations.--A.D.
Dacey, Austin
Source Citation (MLA 8th
Edition)
Dacey, Austin. "Exploring the Invisible: Art, Science, and the Spiritual." Skeptical Inquirer, Nov.-Dec. 2006, p. 61. General OneFile,
go.galegroup.com/ps/i.do?
p=ITOF&sw=w&u=schlager&v=2.1&id=GALE%7CA154238681&it=r&asid=87870285a1f00857e716f550b1103753. Accessed 4 Mar.
2017.
Gale Document Number: GALE|A154238681
---
Quoted in Sidelights:
sound and attractive volume
3/4/2017 General OneFile - Saved Articles
http://go.galegroup.com/ps/marklist.do?actionCmd=GET_MARK_LIST&userGroupName=schlager&inPS=true&prodId=ITOF&ts=1488663552372 4/6
Madness in America: Cultural and Medical Perceptions of
Mental Illness Before 1914
William Beatty
Booklist.
91.19-20 (June 1, 1995): p1702.
COPYRIGHT 1995 American Library Association
http://www.ala.org/ala/aboutala/offices/publishing/booklist_publications/booklist/booklist.cfm
Full Text:
Gamwell, Lynn and Tomes, Nancy. June 1995. 192p. index. illus. Cornell, $39.95 (0-8014-3161-1). Galley.
362 2,0973 Psychiatry--U.S--History [parallel] Mental illness--U.S.--Public opinion [parallel] Social psychiatry--U.S.--History [OCLC]
Gamwell and Tomes, sound and attractive volume is based on an art exhibition, and features about 200 illustrations, many in color and some quite
graphic. The text surveys Native American approaches to and techniques for dealing with mental illness and colonial and nineteenth-century
medical and lay views of it, giving considerable attention to phrenology Much of this material is not widely known, and brief accounts of
journalist Nellie Bly's ten days as a patient in the Blackwell Island asylum and of the unusual paintings of currency by Ralph Albert Blakelock are
especially interesting. Camwell and Tomes also describe attitudes and practices related to the mental illnesses of women and blacks and explore
the growth of asylum medicine and the struggle between scientific neurologists and standpat asylum directors.
Source Citation (MLA 8th
Edition)
Beatty, William. "Madness in America: Cultural and Medical Perceptions of Mental Illness Before 1914." Booklist, 1 June 1995, p. 1702. General
OneFile, go.galegroup.com/ps/i.do?
p=ITOF&sw=w&u=schlager&v=2.1&id=GALE%7CA17218316&it=r&asid=d505964e7926281f1affe4071d0a34f2. Accessed 4 Mar. 2017.
Gale Document Number: GALE|A17218316
---
Quoted in Sidelights:
a grimly fascinating account
sometimes harrowing history of America's treatment of mental illness
The case histories and illustrations … could form their own compelling book.
3/4/2017 General OneFile - Saved Articles
http://go.galegroup.com/ps/marklist.do?actionCmd=GET_MARK_LIST&userGroupName=schlager&inPS=true&prodId=ITOF&ts=1488663552372 5/6
Madness in America: Cultural and Medical Perceptions of
Mental Illness before 1914
American Heritage.
46.5 (Sept. 1995): p92.
COPYRIGHT 1995 American Heritage Publishing
http://www.americanheritage.com/
Full Text:
by Lynn Gamwell and Nancy Tomes, Cornell University Press, 192 pages, $39.9S. CODE: COR-2
* In 1865 a New York newspaper reported patients at New York City Lunatic Asylum "tripping the light fantastic toe" during a "lunatic ball," a
common activity in the more enlightened institutions of the nineteenth century. These surreal dancing parties were just one of many experiments
in the uncertain, sometimes harrowing history of America's treatment of mental illness that the authors of Madness in America, Lynn Gamwell
and Nancy Tomes, have compiled in a grimly fascinating account. The early medical community, guided partly by the Enlightenment belief that a
loss of reason equaled a loss of humanity, treated patients by subjecting them to physical abuse and displaying them in cages. Later on such harsh
methods became rarer, but developing scientific theories were so riddled with social prejudices and half-formed ideas about evolution that they
caused their own kinds of damage. An 1840 census, for example, found many insane blacks in the North but none in the South --clear evidence,
pro-slavery advocates argued, that blacks could not function as free people. We gained the terms highbrow and lowbrow from the theory that base
and unstable instincts "were apparent in [the] sloping forehead, or low brow" of non-Caucasians. Class biases thrived in asylums where the
working classes labored while their wealthier counterparts were pampered, and gender biases were rampant, such as the theory that hysteria was
an inherently female condition. Until ideas about the forces of the unconscious mind slowly emerged, in the late 1800s, the distortions of the sane
world reflected themselves exactly in the worlds of the mentally ill.
The case histories and illustrations (many previously unpublished) that flesh out the text could form their own compelling book. They include ads
for electric corsets to cure hysteria, a sheet-music cover for "Maniac Waltzes," and a collection of keys made by would-be escapees. One patient's
haunting maps of his mental travels are reproduced, as is "landscape money" by the painter Ralph Blakelock, who suffered a nervous breakdown
because of financial trouble and thereafter always carried a wad of bills that were actually landscapes painted on currency-sized cloth. All are
telling examples of the struggle to understand a world of illnesses still revealing itself today.
Source Citation (MLA 8th
Edition)
"Madness in America: Cultural and Medical Perceptions of Mental Illness before 1914." American Heritage, Sept. 1995, p. 92+. General
OneFile, go.galegroup.com/ps/i.do?
p=ITOF&sw=w&u=schlager&v=2.1&id=GALE%7CA17170125&it=r&asid=1fcecc6a00c3c3733b914a4ff2c80b2e. Accessed 4 Mar. 2017.
Gale Document Number: GALE|A17170125
Quoted in sidelights: As a lecturer at the School of Visual Arts in Manhattan, I wrote this book for my students, such as Maria, who told me she was never good at history because she couldn’t remember dates, and for Jin Sug, who failed high school algebra because he couldn’t memorize formulae. I hope they will read this book and discover that history is a storybook and that math is about captivating ideas.
Why the history of maths is also the history of art
In her new book Mathematics and Art, historian Lyn Gamwell explores how artists have for thousands of years used mathematical concepts - such as infinity, number and form - in their work. Here she choses ten stunning images from her book that reveal connections between maths and art.
Detail of Karl Gerstner’s Polychrome of Pure Colors. For full credit see full image below.
Detail of Karl Gerstner’s Polychrome of Pure Colors. For full credit see full image below.
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Lynn Gamwell
Wednesday 2 December 2015 01.55 EST Last modified on Wednesday 22 February 2017 13.00 EST
When I was a graduate student in art history, I read many explanations of abstract art, but they were invariably inadequate and misleading. So after completing my PhD, I went on to learn the history of biology, physics, and astronomy, and to publish a book detailing how modern art is an expression of the scientific worldview.
Yet many artworks also express the mathematics and technology of their times. To research Math and Art I had to learn maths concepts like calculus, group theory and predicate logic. As a novice struggling to understand these ideas, I was struck with the poor quality and confusing content of illustrations in most educational books. So I vowed to create for my book a set of cogent math diagrams that are crystal-clear visualizations of the abstract concepts.
As a lecturer at the School of Visual Arts in Manhattan, I wrote this book for my students, such as Maria, who told me she was never good at history because she couldn’t remember dates, and for Jin Sug, who failed high school algebra because he couldn’t memorize formulae. I hope they will read this book and discover that history is a storybook and that math is about captivating ideas.
Here are ten images followed by descriptions:
Eric J. Heller (American, born 1946), Transport VI, ca. 2000. Digital print. Courtesy of the artist.
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Eric J. Heller (American, born 1946), Transport VI, ca. 2000. Digital print. Courtesy of the artist.
Throughout history, scientists have discovered mathematical patterns in nature, such as the paths taken by electrons as they flow over the hills and valleys of tiny “landscapes” that are measured in microns (one micron equals one millionth of a meter). Paths of electrons in this digital print were recorded by Eric J. Heller, who studies rogue waves (freak waves, killer waves) on large and small scales. When a wave of electrons flows through a computer, a freak wave in a semiconductor can suddenly threaten the smooth functioning of the device.
Jim Sanborn (American, b. 1945), Kilkee County Clare, Ireland, 1997. Large-format projection, digital print, 30 × 36 in. (76.2 × 91.4 cm). Courtesy of the artist.
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Jim Sanborn (American, b. 1945), Kilkee County Clare, Ireland, 1997. Large-format projection, digital print, 30 × 36 in. (76.2 × 91.4 cm). Courtesy of the artist.
Western mathematics proceeds by increasing abstraction and generalization. In the Renaissance the Italian architect Filippo Brunelleschi invented linear perspective, a method to project geometric objects onto a “picture plane” from a given viewpoint. Three centuries later the French mathematician Jean-Victor Poncelet generalized perspective into projective geometry for planes that are tipped or rotated. Then in the early twentieth century the Dutchman L.E.J. Brouwer generalized Poncelet’s projective geometry to projections onto surfaces that are stretched or distorted into any shape—so-called rubber-sheet geometry—provided that the plane remains continuous (with no holes or tears), which is the subject of this photograph. The contemporary artist Jim Sanborn created it by projecting a pattern of concentric circles onto a large rock formation at night from about 1/2 mile away. He then took this photograph with a long exposure at moonrise.
Reza Sarhangi (Iranian-born American, b. 1952) and Robert Fathauer (American, b. 1960), Būzjānī’s Heptagon, 2007. Digital print, 13 × 13 in. (33 × 33 cm). Courtesy of the artists.
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Reza Sarhangi (Iranian-born American, b. 1952) and Robert Fathauer (American, b. 1960), Būzjānī’s Heptagon, 2007. Digital print, 13 × 13 in. (33 × 33 cm). Courtesy of the artists.
Knowledge of ancient Greek mathematics, such as Euclid and Ptolemy, was lost to the medieval West, but Islamic scholars preserved their writings in Arabic translations. In the ninth century, caliphs established the House of Wisdom in Baghdad as a place for scholars to acquire and translate foreign texts in mathematics and philosophy. Ptolemy’s thirteen-volume work is known today by the name they gave it, Almagest (Arabic for “the greatest”).
Two contemporary mathematicians, Reza Sarhangi and Robert Fathauer, pay homage to the Islamic mathematician Abū al-Wafā’ Būzjānī (AD 940¬–98), who worked at the House of Wisdom, where he wrote a practical text, On Those Parts of Geometry Needed by Craftsmen. He showed how to construct a regular heptagon (a polygon with seven equal sides and angles), which is in the center portion of this print. Around the perimeter of the heptagon Sarhangi and Fathauer wrote Buzjani’s name seven times in Farsi, the language of Persia (modern Iran).
Robert Bosch (American, b. 1963), Knot? 2006. Digital print, 34 × 34 in. (86.3 × 86.3 cm). Courtesy of the artist.
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Robert Bosch (American, b. 1963), Knot? 2006. Digital print, 34 × 34 in. (86.3 × 86.3 cm). Courtesy of the artist.
With the development of railroads in the nineteenth century, the topic of finding an optimal route for a journey was of practical interest. The topic entered the mathematics literature in 1930, when the Viennese mathematician Karl Menger described it as the “messenger problem” (das Botenproblem) of finding an optimal delivery route. It was soon dubbed the “travelling salesman’s problem”: given a list of cities and the distances between each pair, find the shortest route that visits each city once and returns to the city of origin
The American mathematician Robert Bosch drew this continuous line based on the solution to a 5000-city instance of the travelling salesman problem. From a distance, the print appears to depict a black cord against a grey background in the form of a Celtic knot. But on close inspection the apparent “grey” is actually a continuous white line moving on top of a black background. The white line never crosses over itself—it is a network rather than a knot—and so the punny answer to the title is “Not.”
Karl Gerstner (Swiss, b. 1930), Polychrome of Pure Colors, 1956-58. Printer’s ink on cubes of Plexiglas, 1 1/4 × 1 1/4 in. (3 × 3 cm). ea., fixed in a chrome-plated metal frame, 18 7/8 × 18 7/8 in. (48 × 48 cm) ea. Courtesy of the artist.
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Karl Gerstner (Swiss, b. 1930), Polychrome of Pure Colors, 1956-58. Printer’s ink on cubes of Plexiglas, 1 1/4 × 1 1/4 in. (3 × 3 cm). ea., fixed in a chrome-plated metal frame, 18 7/8 × 18 7/8 in. (48 × 48 cm) ea. Courtesy of the artist.
In 1905, Albert Einstein discovered the symmetry of mass and energy—mass can be converted into energy, and vice versa (E = mc2). Then in the early decades of the twentieth century, physicists and mathematicians, including Einstein, gathered in Zurich and employed group theory in their exploration of the symmetry of nature.
Swiss artists such as Gerstner created patterns that resonate with these mathematical descriptions of nature in terms of symmetry. Like the mathematicians, these artists established basic aesthetic building-blocks—units of color and form—and arranged them using rules that preserve proportion and balance.
In 1956 Gerstner devised a modular system—a movable palette with 196 hues in 28 groups—for experimenting with progressions that link form with color. Gerstner’s palette of 196 squares has 28 groups with 7 squares each. Shown here are four of myriad possible arrangements, which the artist describes using the mathematician’s terms: groups, permutations, algorithms, and invariance.
Karl Gerstner (Swiss, b. 1930), Polychrome of Pure Colors, 1956-58. Printer’s ink on cubes of Plexiglas, 1 1/4 × 1 1/4 in. (3 × 3 cm). ea., fixed in a chrome-plated metal frame, 18 7/8 × 18 7/8 in. (48 × 48 cm) ea. Courtesy of the artist.
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Karl Gerstner (Swiss, b. 1930), Polychrome of Pure Colors, 1956-58. Printer’s ink on cubes of Plexiglas, 1 1/4 × 1 1/4 in. (3 × 3 cm). ea., fixed in a chrome-plated metal frame, 18 7/8 × 18 7/8 in. (48 × 48 cm) ea. Courtesy of the artist.
Karl Gerstner (Swiss, b. 1930), Polychrome of Pure Colors, 1956-58. Printer’s ink on cubes of Plexiglas, 1 1/4 × 1 1/4 in. (3 × 3 cm). ea., fixed in a chrome-plated metal frame, 18 7/8 × 18 7/8 in. (48 × 48 cm) ea. Courtesy of the artist.
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Karl Gerstner (Swiss, b. 1930), Polychrome of Pure Colors, 1956-58. Printer’s ink on cubes of Plexiglas, 1 1/4 × 1 1/4 in. (3 × 3 cm). ea., fixed in a chrome-plated metal frame, 18 7/8 × 18 7/8 in. (48 × 48 cm) ea. Courtesy of the artist.
Karl Gerstner (Swiss, b. 1930), Polychrome of Pure Colors, 1956-58. Printer’s ink on cubes of Plexiglas, 1 1/4 × 1 1/4 in. (3 × 3 cm). ea., fixed in a chrome-plated metal frame, 18 7/8 × 18 7/8 in. (48 × 48 cm) ea. Courtesy of the artist.
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Karl Gerstner (Swiss, b. 1930), Polychrome of Pure Colors, 1956-58. Printer’s ink on cubes of Plexiglas, 1 1/4 × 1 1/4 in. (3 × 3 cm). ea., fixed in a chrome-plated metal frame, 18 7/8 × 18 7/8 in. (48 × 48 cm) ea. Courtesy of the artist.
Karl Gerstner (Swiss, b. 1930), Color Spiral Icon x65b, 2008. Acrylic on aluminum, diameter 41 in. (104 cm). Collection of Esther Grether, Basel, Switzerland.
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Karl Gerstner (Swiss, b. 1930), Color Spiral Icon x65b, 2008. Acrylic on aluminum, diameter 41 in. (104 cm). Collection of Esther Grether, Basel, Switzerland.
Scientific insights into the deepest levels of the natural world are explanations based on symmetry, which the artist Karl Gerstner symbolizes with this circular “icon” for the secular age of science and technology. The most symmetrical geometric form is a sphere (all points equidistant from a point in three-dimensional space). In the late twentieth century, scientists concluded that the universe began in perfect symmetry as a point that exploded into a sphere of plasma. As the infant universe expanded, the primordial sphere cooled, and matter condensed from the plasma to form the first particles, then atoms, gas clouds, and stars. At some point the original symmetry of the universe was broken; the resulting asymmetries appear to be the result of random shifts analogous to mutations during evolution. Today physicists are recreating samples of this primordial spherical plasma to determine the degree to which the universe retains traces of its original symmetry.
Simon Thomas (British, b. 1960), Planeliner, 2005. Bead blasted stainless steel, 23 5/8 in. (60 cm) diam. × 2 1/4 in. (5.55 cm) high. Courtesy of the artist.
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Simon Thomas (British, b. 1960), Planeliner, 2005. Bead blasted stainless steel, 23 5/8 in. (60 cm) diam. × 2 1/4 in. (5.55 cm) high. Courtesy of the artist.
Simon Thomas is a young British artist whose work, such as this sculpture, is a visualization of a mathematical formula. He studied visual arts at the Royal College of Art in London in the 1980s and went on to create sculpture with striking geometric patterns, serving as artist-in-residence at the University of Bristol, both in the department of physics (1993–95) and mathematics (2002).
Quoted in Sidelights:
is, above all, fun to browse in, probably for people from quite a wide range of backgrounds, and it makes a lovely coffee table book, which provides some substance for those drawn in by the illustrations and which may bring people into contact with new ideas and inspire novel connections.
Mathematics and Art: A Cultural History
Lynn Gamwell
Princeton University Press 2016
A book review by Danny Yee © 2016 http://dannyreviews.com/
Gamwell is up front in her introduction about mathematics inspiring art much more than the other way around, and Mathematics and Art is distinctly asymmetrical. The larger part of the text describes people, schools and trends in the history of art which were influenced by or connected in some way to mathematics, with a smaller part devoted to the history of mathematics and related sciences. (There is also a little exposition of the latter, much of it in separate boxes, on for example Kepler's laws and the double-slit experiment.) And more than half the space is devoted to glorious full colour reproductions, almost entirely of art works, with only a little space used for diagrams illustrating aspects of mathematics or physics. The captions to these photographs are often quite substantial, sometimes offering mini-biographies of artists as well as discussions of specific works. There are also some fun, if slightly random, quotes in the wide margins.
The first seven chapters are named after topics —"Arithmetic and Geometry", "Proportion", "Infinity", "Formalism", "Logic", "Intuitionism" and "Symmetry" — but are roughly chronological, taking the story down to the early 20th century. The last six chapters are also roughly chronological: "Utopian Visions after World War I", "The Incompleteness of Mathematics", "Computation", "Geometric Abstraction after World War II", "Computers in Mathematics and Art", and "Platonism in the Postmodern Era". The illustrations are mostly from the period being covered, but are sometimes used to illustrate it. So Boticelli's 1480 Saint Augustine in his Study appears in chapter one, alongside a marginal quote from Wittgenstein, to accompany a discussion of Augustine's attitude to mathematics. And a diagram from Kepler's 1611 On the six-cornered snowflake appears in the penultimate chapter, to illustrate a discussion of the proof of Kepler's sphere-packing conjecture.
I describe two chapters in more detail to illustrate the approach. "Formalism" begins with Euclidean and non-Euclidean geometries, Helmholtz's "learned geometry" (looking at the psychological foundations of space perception), Hilbert's formalisation of geometry, and the modern mathematical notion of "formalism" (the term was coined in 1912 by Brouwer as an insult directed at Hilbert). A brief look at formalism in Russian linguistics and literature (Khlebnikov) leads into the extensively illustrated second half of the chapter, which is devoted to Russian constructivist art (Tatlin and Rodchenko) and Unism in Poland (Strzeminski and Kobro).
"Of the three approaches to the foundations of mathematics — formalism, logicism, and intuitionism — formalism has had by far the most profound and lasting impact on the visual arts, first on Russian Constructivism in the 1910s, then throughout the West in the 1920s, then globally after 1945."
"But people did not forget that for a brief moment after October 1917 art had become an instrument of social change; Rodchenko and Tatlin had applied abstract color and form to practical tasks to make the world a better place."
"Far from being only a formalist, Hilbert used formalism as a tool to provide a foundation for the modern version of Plato's view of mathematics as the cognition of perfect, timeless abstract objects."
"Computers in Mathematics and Art" touches on computer proofs "by exhaustion" (the four-colour theorem and sphere packing), visualization, knots, networks (graphs), origami, recursive algorithms (including cellular automata), and fractal geometry and its applications. Gamwell manages decent introductions to these topics, not attempting too much but trying to give non-mathematicians some feel for them. Some of the photographs directly illustrate these topics, showing for example Julia and Mandelbrot sets or mountains and crystals (exemplifying the fractal geometry of nature), but more are of works of art inspired (more or less directly) by these ideas, from the Book of Kells through to recent work.
"Robert Bosch (American, b. 1963), Knot? 2006. Digital print, 34 x 34 in.
"The American mathematician Robert Bosch drew this continuous line based on the solution to a 5000-city instance of the travelling salesman problem. From a distance, the print appears to depict a black cord against a grey background in the form of a Celtic knot. But on close inspection the apparent 'grey' is actually a continuous white line moving on top of a black background. The white line never crosses itself — it is a network rather than a knot — and so the answer to the title is 'Not.'"
I skipped over almost all the explanations of mathematics, computer science, and physics, and only a little of the history of science was new to me, but it's always nice to get new angles on familiar topics, for example the correspondence between the geometer Coxeter and the artist Escher. The potted histories of different artistic movements were what I enjoyed most: 6 pages (half illustrations) on "Early-Twentieth Century Meta-Art" (covering de Chirico and Magritte), for example, and 26 pages (two-thirds illustrations) spread over two chapters on Swiss Concrete Art (Max Bill, Andreas Speiser, Verena Loewensburg, Karl Gerstner and others).
In a few places Gamwell goes off on tangents. There are seventeen pages on quantum mechanics, arguing for the De Broglie-Bohm interpretation and against the Copenhagen interpretation, which she explains as a strand in German idealism. There are only two small illustrations in this, of two of Antony Gormley's Quantum Cloud sculptures, and the connection to art seems a little tenuous. There's also a bit on music, with seven pages on Schoenberg and serialism and computer music.
Gamwell sensibly refrains from attempting to fit any kind of meta-narrative to the book or even the individual chapters. The links between art and mathematics she describes are mostly local, without larger-scale structure, and it would be hard to make a case for mathematical ideas driving long-term changes in art or aesthetics.
Given how much Gamwell covers and how multi-disciplinary Mathematics and Art is, there were bound to be some errors in it, and I found a good number, albeit mostly minor, just in the material I was familiar with. There's a confusion between the number of possible sequences for a protein and the number of possible structures into which it can fold; there's a reference to "Chinese, the oldest living language" that will make linguists blanch; Charles Darwin may have declared in the Origin of Species that "animals have innate drives for survival and reproduction", but that's hardly the notable idea in the work; Hermann Weyl's 1952 book Symmetry was based on lectures delivered in 1951 and is not the same as a much shorter 1938 article with the same title; and so forth.
Mathematics and Art is not designed for learning mathematics or science from. It works better for learning about artists and schools of art, but its selection and coverage are probably too idiosyncratic for use as a reference. Mathematics and Art is, above all, fun to browse in, probably for people from quite a wide range of backgrounds, and it makes a lovely coffee table book, which provides some substance for those drawn in by the illustrations and which may bring people into contact with new ideas and inspire novel connections.
June 2016
Quoted in Sidelights: a beautiful, magnificent, and rather large book.
makes a serious attempt to cover the ground comprehensively
a fascinating, beautiful, intriguing, and stimulating book.
BOOK REVIEW
BOOK REVIEW: MATHEMATICS + ART BY LYNN GAMWELL
22 FEBRUARY 2016 IAN ALEXANDER
Mathematics and Art by Lynn Gamwell, 2015
Mathematics and Art by Lynn Gamwell, 2015
Lynn Gamwell’s Mathematics + Art, A Cultural History (Princeton, 2016) is a beautiful, magnificent, and rather large book. Given its size, its cover price ($50) is very reasonable. The topic is an enormous one, ranging from the ancient to the ultra-modern.
Gamwell makes a serious attempt to cover the ground comprehensively. The book begins with Arithmetic and Geometry (two huge areas in themselves), and a glorious image from a Bible moralisée of 1208-1215 of God the Geometer, measuring out the world – it looks rather like a geode in section, actually – with a pair of dividers.
God the Geometer (Wikimedia Commons)
God the Geometer (Wikimedia Commons)
Page 1 mentions “Mankind’s ape-like ancestors” and talks about the first symmetrical tools; 300,000 years ago, hand axes started to have elegant symmetry. Clearly Gamwell’s intention is to cover the interaction of mathematics + art in the whole span of human history and prehistory: it’s nothing if not ambitious.
The text sweeps rapidly through time, so that on page 3 we reach 3000 BC and the ancient foundations of recorded mathematics, with I, II, III tracking quantities; the Egyptians introduced ∩ for 10, so 12 was ||∩. Soon we are in Ancient Greece and the theorems of Thales (the base angles of an isosceles triangle are equal…), and what the sculptor Polykleitos wrote about the perfect proportions of the human body in the 5th century BC. Attention switches to the elements and the Platonic solids (with a forward reference to Kepler’s depiction in Harmonices Mundi, 1619), Democritus’s mechanical universe, and Euclid’s Elements – all in the first chapter, and I haven’t even mentioned the detailed treatment of the birth of modern physics and Newton’s law of universal gravitation, which Gamwell actually explains with the famous inverse square law equation.
Julia_set_fractal (Wikimedia Commons)
Julia Set detail (Wikimedia Commons / Joshi1983)
The book, in other words, is big, and dares to boldly go where others fear to tread (Steven Hawking wrote in his A Brief History of Time that each equation halved the number of readers: Gamwell has plenty, and explains the symbols of formal logic, too.) She can cover the plans of Gothic cathedrals, the mysteries of perspective from the Italian Renaissance, Zeno’s Paradox (can Achilles catch that tortoise?), modern art from Mondrian to Henry Moore, Bauhaus to Bourbaki. It’s kaleidoscopic, and if you wanted a coffee-table book then you could just flick through it and enjoy the Mandelbrot and Julia sets, gloriously illustrated.
But of course readers expect and deserve more. The chapters cover Arithmetic and Geometry; Proportion; Infinity; Formalism; Logic; Intuitionism; Symmetry; Utopian visions after World War I; The Incompleteness of Mathematics; Computation; Geometric Abstraction after World War II; Computers in Mathematics and Art; and Platonism in the Postmodern Era. This is visibly a huge scope – all of mathematics, all of art, and all of their intersection (to coin a phrase from set theory).
But wait a minute: all of art? It’s certainly all of the time during which art has been created, bone flute (Hohle Fels cave, c. 42,000 years ago) and Lascaux cave paintings (ca 15,000 BC) included. The discussion of art cheerfully scoots about from Iceland to Renaissance Italy; from Russia to China to Japan; from a Hungarian-born Argentine artiss (Gyula Kosice) to the American hand-blown glass and steel sculptures of Josiah McElheny. The reader grapples with fractals and their recursive algorithms; formalist mathematics and constructivism; Klein bottles and the odd behaviour of electrons in quantum mechanics. Gödel, Escher and Bach do their Hofstadter-esque dance of self-reference.
The Elephant in the Room
Missing from the book: Islamic Art. (Girih strapwork tiling, Green Mosque, Turkey)
Missing from the book: Islamic Art. Girih strapwork tiling, Green Mosque, Turkey. (WIkimedia Commons)
So what is missing? The 556 pages barely so much as blink in the direction of Islamic Art, of the dazzling complexity and virtuosity of its geometric designs and decorations, of its centuries-long contribution to mathematics – even the words algorithm and algebra come from Al-Khwarizmi’s name and his book of pioneering mathematics. Nothing. Zilch. Nix. Or as the Arabs would say, Zifr. Well, they invented it.
Did Gamwell simply not know about Islamic tessellations? Of Escher’s inspiration in the Alhambra? Of Girih strapwork all over the minbar pulpits of Egypt, all over the turquoise domes of Persia? Of the dazzling Zellige tilework of Morocco? Of the lustre tiles of Tunisia? Of the inlaid geometric stonework of the Umayyad Mosque in Damascus? Of the airy Jali stone screens of the Mughal palaces of India? It seems not. Her index includes “Arabic numerals”, but she did not follow up that broad clue.
Gamwell has written a fascinating, beautiful, intriguing, and stimulating book. It is sometimes rather too academically picky; sometimes a bit too thorough in explanation, but then you may need more than me on some topics. It is perhaps a bit too much focussed on the twentieth century – after all, why that century, not all the others? Recentism is no reliable guide. But the glaring gap, or as a pretentious art critic would say, the lacuna of all lacunae, is the extraordinary lack of coverage of the whole of the Islamic world. Try a look in the Index – you won’t find Arabia (apart from Arabic numerals), or Morocco, or Syria, or Iran (or even Persia), or Moghul/Mughal. It’s just not there.
I think this matters, and matters terribly. If George Bush and Tony Blair had it in their blood that art, science, mathematics, medicine, poetry, music, pottery, metalwork, masonry, glass, carpets, and gardens all flourished in the Islamic world, for century after century, from the Moroccan Maghreb (“The West”) to Turkey, Egypt, Persia, Afghanistan, all of the Indian subcontinent, Indonesia – would they have acted as ignorantly as they did? And more importantly now, will our future leaders be any better informed, or will they treat Sunni and Shia alike as ignorant savages? For what it’s worth, Daesh / ISIS / ISIL is not Islam, it’s a stupid and wicked splinter group, and has nothing whatsoever to do with a great cultural heritage. Especially, it should not colour our attitudes to Muslims and Islam.
Some of the white space in Mathematics + Art can be filled by other books, such as Luca Mozzati’s Islamic Art, or Eric Broug’s Islamic Geometric Designs.
All the same, I felt sufficiently engaged at the lack of coverage to do something about it. I brought two Wikipedia articles, Mathematics and art, and Islamic geometric patterns, to “Good Article” status, pretty much rewriting them from scratch in the process. To make the whole area a bit easier to navigate, I also rewrote the navigation template on Islamic art. Together, these are seen – and perhaps read – by around 100,000 people a year, and of course they help to inform blogs and social media postings, so maybe they will have some effect. If you can suggest ways of reaching more people with this sort of knowledge, I’d be happy to hear from you.
Quoted in Sidelights:
By the time I was halfway through this book, I had accumulated ten pages of notes. While some of my comments were for this review, the majority, were for my own reference: information to pursue, new wonderful facts I had not previously known: so much information, so many issues to consider or reconsider in light of new revelations and so many inspiring and intriguing images. This is the beauty and power of this book: it is an intellectual tour de force of art history and its interaction with mathematics that will draw most readers, including me, back for further reading and study.
Mathematics + Art: A Cultural History
Lynn Gamwell
Publisher:
Princeton University Press
Publication Date:
2015
Number of Pages:
576
Format:
Hardcover
Price:
49.50
ISBN:
9780691165288
Category:
Monograph
MAA REVIEW TABLE OF CONTENTS
[Reviewed by
Frank Swetz
, on
12/11/2015
]
A large, yellow, mailing envelope was delivered to my doorstep. When I picked it up, I realized that it was a book, one of the heaviest books I have recently handled. Upon opening the parcel, I found a beautiful tome, a book of “coffee-table” appearance and quality: Mathematics + Art: A Cultural History, by Lynn Gamwell, a lecturer in the history of art, science, and mathematics at New York’s School of Visual Arts.
The book abounds with colorful illustrations, line drawings and numerous sidebars. The contents of the sidebars clarify and amplify the mathematical concepts referenced in the text. They are very good. The chapter headings point the direction in which the discussion will proceed: 1. “Arithmetic and Geometry”; 2. “Proportion”; 3. “Infinity”; 4. “Formalism”; … 13. “Platonism in the Postmodern Era”. While the title of the book designates an ambitious undertaking, Gamwell’s objective is to examine art as inspired by mathematics and not the inverse situation. The “cultural” settings within which these creative movements take place is the art scene itself.
Within the immense task of surveying the history of mathematical influence in the conception and creation of the visual arts — painting, sculpture, graphic design, photography… — just where does one begin? Gamwell’s Chapter I: “Arithmetic and Geometry”, seeks to provide an historical basis for mathematics and associate its place in an emerging Greek philosophical scaffolding. Since it is mainly the Western world that the ensuing discussions will concern, the scaffolding is Greek thought, specifically Platonism.
The reader is told that mathematicians are Platonists, theorizing about realities they can never fully understand: “shadows on a wall”. Perhaps. But, on second thought, wouldn’t a consideration of the Greek philosophical dichotomy between “arithmetike”, the theoretical, and “logistike”, the applied, be more fitting in appraising both the development of mathematics and its interactions with the processes of artistic creation?
In the concise survey of the first chapter, the reader traverses eleven millennia of human progress, from the symmetric design of Paleolithic spear points, through the geometry of soaring Gothic Cathedrals, to the cosmic speculations of Johannes Kepler and Isaac Newton: a long span. During this interval, when generalizations converge to focus on particular events, factual errors occur. On page 11, it is noted that during a visit to “peninsular Italy”, Plato sees Dion in Syracuse. Syracuse is in Sicily, an island off the mainland and a kingdom unto itself. On page 32, the Islamic mathematician Al’Khowarizmi (ca. 825) is credited with the introduction of letters for variables in algebraic computation. Not in my readings of his works!
Occasionally, similar errors appear in other sections of the book, for example: in a sidebar on page 124, the British mathematician, John Wallis (1616–1703) is mentioned for introducing the symbol “∞” to represent infinity and naming it “lemniscate”. Yes, he adopted the symbol from existing Roman numerals, but did not name the geometric curve. Jacob Bernoulli did this in 1694. On page 127, the reader is told that “Galileo invented the telescope”, which he did not; and on page 192, the Chinese Communists are credited with initiating the Chinese Revolution of 1911. The Communist Party did not exist as an entity in China until 1921. The Revolution of 1911 was nationalistic in nature and promoted mainly by disenchanted students. With such a great scope of material being considered, however, some errors are likely to occur.
While the narrative of the first chapter may disturb an historian of mathematics, the following chapters pick up intellectual and conceptual momentum. The author is apparently more comfortable discussing post-nineteenth century philosophical trends and artistic developments. Beginning with the second chapter, “Proportions”, the reader is transported to a rich and challenging environment of visual and conceptual information. The proportions considered are based on Western models via Vitruvius, Dürer and Da Vinci. Personally, I would also like to have seen some examination of other non-Western, systems of proportion, for example, those evident in Egyptian, Buddhist and Hindu art. The “Golden Proportion” is debunked as an intuitive component of artistic composition. I was pleased to learn that the controversial Spanish artist Salvador Dali (1904–1989), in his rendering of the painting, “The Last Supper”, was influenced by the fifteenth century works of Luca Pacioli and Leonardo da Vinci. Although I have viewed this particular painting several times, I did not identify the segment of a dodecahedron [the Platonic universe] arching over the subjects. Dali certainly was influenced by mathematics.
Throughout the book, in several instances, the artists, themselves, will credit their techniques to mathematical influences: the Russian formalist Aleksandr Rodchenko (1891–1956) credits Greg Cantor (1845–1918) and his concept of the continuum for inspiring his 1918 series of paintings “Black on White” and in the 1940s, the Swiss formalist painters Paul Lohse (1902–1988) and Max Bill (1908–1994) openly used group theory in their compositions.
The examination of the development of art is definitely stronger for the post-eighteenth era. Chapter 12, “Computers in Mathematics and Art” examines knots, networks and fractal geometry; while the following and last chapter, “Platonism in the Postmodern Era” returns the reader to the issue of philosophical directions established at the beginning of the narrative. Most mathematicians I know just do mathematics: solve the problem, probe the theory; and do not devote much, or any, contemplation as to the existence of their “worlds”.
By the time I was halfway through this book, I had accumulated ten pages of notes. While some of my comments were for this review, the majority, were for my own reference: information to pursue, new wonderful facts I had not previously known: so much information, so many issues to consider or reconsider in light of new revelations and so many inspiring and intriguing images. This is the beauty and power of this book: it is an intellectual tour de force of art history and its interaction with mathematics that will draw most readers, including me, back for further reading and study. The high quality of the illustrations and design, as well as the quantity of information provided, makes this book a true bargain at the stated price. I realize now why it was so heavy.
Frank Swetz, Professor of Mathematics and Education, Emeritus, The Pennsylvania State University, is the author of several books on the history of mathematics. His research interests focus on societal impact on the development, and the teaching and learning, of mathematics.