Mathematics and art

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Mathematics in art: Albrecht Dürer's copper plate engraving Melencolia I, 1514

Mathematics and art have a long historical relationship. The ancient Egyptians and ancient Greeks employed mathematics to plan monuments including the Great Pyramid, the Parthenon and the Colosseum, using constructs like the 3-4-5 triangle; some authors have claimed on doubtful grounds that they also used the golden ratio. There is equally a strong connection between music and mathematics: the current article concentrates on the relationship of mathematics with the visual and plastic arts rather than the performing arts such as music.

Artists have used mathematics, especially the mathematics of proportion, since at least the time of the Greek sculptor Polykleitos in the 5th century BC. In his Canon, Polykleitos prescribed a series of mathematical proportions based on the ratio 1:√2 for carving the ideal male nude, according to his concept of proportion or Symmetria. In the Italian Renaissance, Luca Pacioli wrote the influential treatise De Divina Proportione (1509), illustrated with woodcuts by Leonardo da Vinci, on the use of proportion in art. Another Italian painter, Piero della Francesca, made pioneering use of mathematics for perspective, developing ideas first put forward in Euclid's Optics in treatises such as De Prospectiva Pingendi, and applying this knowledge in his paintings. Polyhedra have appeared in Western art from the Renaissance onwards, including in Albrecht Dürer's much-analysed engraving Melencolia I.

In modern times, the graphic artist M. C. Escher made intensive use of tessellation and hyperbolic geometry, inspired by the mathematician H. S. M. Coxeter. Mathematics has inspired textile arts such as quilting, knitting, cross-stitch, crochet, embroidery and weaving.


Galileo Galilei in his Il Saggiatore wrote that "[The universe] is written in the language of mathematics, and its characters are triangles, circles, and other geometric figures."[1] Artists who strive and seek to study nature must first, in Galileo's view, fully understand mathematics. On the other hand, mathematicians have sought to interpret and analyse art through the lens of geometry and rationality. The mathematician Jerry P. King describes mathematics as an art, stating that "the keys to mathematics are beauty and elegance and not dullness and technicality", and that beauty is the motivating force for mathematical research.[2] However, this article focuses on the relationship of mathematics with the visual arts.

Artistic composition guided by mathematics[edit]

Polykleitos's Canon and symmetria[edit]

Roman copy in marble of Doryphoros, originally a bronze by Polykleitos

Polykleitos the Elder (c.450–420 BCE) was a Greek sculptor from the school of Argos, and a contemporary of Phidias. His works and statues consisted mainly of bronze and were of athletes. According to the mathematician Xenocrates, Polykleitos is ranked as one of the most important sculptors of Classical antiquity for his work on the Doryphorus and the statue of Hera in the Heraion of Argos.[3] While his sculptures may not be as famous as those of Phidias, he is better known for his approach towards sculpture. In the Canon of Polykleitos, a treatise he wrote designed to document the "perfect" anatomical proportions of the male nude, Polykleitos gives us a mathematical approach towards sculpturing the human body.[3]

The influence of the Canon of Polykleitos is immense both in Classical Greek, Roman, and Renaissance sculpture, with many sculptors after him following Polykleitos's prescription. While none of Polykleitos's original works survive, Roman copies of his works demonstrate and embody his ideal of physical perfection and mathematical precision. Some scholars contend the influence of the mathematician Pythagoras on the Canon of Polykleitos.[4] The Canon applies the basic mathematical concepts of Greek geometry, such as the ratio, proportion, and symmetria (Greek for "harmonious proportions") and turns it into a system capable of describing the human form through a series of continuous geometric progressions.[5]

Polykleitos starts with a specific human body part, the distal phalanges of the little finger, or the tip of the little finger to the first joint, and establishes that as the basic module or unit for determining all the other proportions of the human body.[5] From that, Polykleitos multiplies the length by radical 2 (1.4142) to get the distance of the second phalanges and multiplies the length again by radical 2 to get the length of the third phalanges. Next, he takes the finger length and multiplies it again by radical 2 to get the length of the palm from the base of the finger to the ulna. This geometric series of measurements progress until Polykleitos has formed the arm, chest, body, and so on.[6]

The Golden Ratio[edit]

Possible base:hypotenuse(b:a) ratios for the Pyramid of Khufu: 1:φ (Kepler’s Triangle), 3:5 (3-4-5 Triangle), and 1:4/π
Supposed ratios: Notre-Dame of Laon

The Golden Ratio, roughly equal to 1.618, has persistently been claimed[7][8][9][10][11] in modern times to have been known to the ancients in Egypt, Greece and elsewhere, without reliable evidence.[12] The claim may derive from confusion with "golden mean", which to the Ancient Greeks meant "avoidance of excess in either direction", not a ratio.[12] Pyramidologists since the nineteenth century have argued on doubtful mathematical grounds for the golden ratio in pyramid design.[a] The 5th century BCE temple, the Parthenon in Athens, has been claimed to use the golden ratio in its façade and floor plan,[15][16][17] but these claims too are disproved by measurement.[12] The Great Mosque of Kairouan in Tunisia has similarly been claimed to use the golden ratio in its design,[18] but the ratio does not appear in the original parts of the mosque.[19] The historian of architecture Frederik Macody Lund argued in 1919 that the Cathedral of Chartres (12th century), Notre-Dame of Laon (1157–1205) and Notre Dame de Paris (1160) are designed according to the golden ratio,[20] drawing regulator lines to make his case. Other scholars argue that until Pacioli's work in 1509, the golden ratio was unknown to artists and architects.[21][b]

Woodcut from De Divina Proportione showing the proportions with an equilateral triangle on a human face

In 1509, Luca Pacioli (c. 1447–1517) published De Divina Proportione on mathematical and artistic proportion, including in the human face. Leonardo da Vinci illustrated the text with woodcuts of regular solids while he studied under Pacioli. Leonardo's drawings are probably the first illustrations of skeletonic solids.[23] These, such as the rhombicuboctahedron, were among the first to be drawn to demonstrate perspective by being overlaid on top of each other. The work discusses perspective in the works of Piero della Francesca, Melozzo da Forlì, and Marco Palmezzano.[24]

Golden rectangles superimposed on the Mona Lisa

In the 1490s, Leonardo da Vinci (1452–1519) trained under Pacioli, preparing a series of drawings for De Divina Proportione. He studied Pacioli's Summa, from which he copied tables of proportions and multiplication tables.[25] In Mona Lisa and The Last Supper, Leonardo's work incorporated linear perspective with a vanishing point to provide apparent depth.[26]

Proportion: Leonardo’s Vitruvian Man

In Mona Lisa, the mismatch between the left and right backgrounds creates the illusion of perspective and depth. A Golden Rectangle whose base extends from her right wrist to her left elbow and reaches the very top of her head can be subdivided into smaller Golden Rectangles to produce a Golden Spiral. The edges of the new rectangles come to intersect the focal points of Mona Lisa: chin, eye, nose, and upturned corner of her mouth. The overall shape of the woman is a triangle with her arms as the base and her head as the tip, drawing attention to her face.[27]

In The Last Supper, Leonardo sought to create a perfect harmonic balance between the placement of the characters and the background. He did intensive studies on how the characters should be arranged at the table. The entire painting was constructed in a tight ratio of 12:6:4:3.[28] The entire piece measures 6 by 12 units. The wall in the back is equal to 4 units. The windows are 3 units and the recession of the tapestries on the side walls is 12:6:4:3.[29]

In Vitruvian Man, Leonardo expressed the ideas of the Roman architect Vitruvius.[30] Earlier artists and architects had illustrated Vitruvius' theory, but Leonardo's drawing innovates by showing the male figure twice in the same image, centring him in both a circle and a square; he suggests energy with the figure's active arms and legs. Thin lines across parts of his body indicate the architectural scheme Leonardo was using for proportion. Following Vitruvius, Leonardo is representing the body as a building.[30]


Main article: Perspective (visual)
Rays of light travel from an object to the eye. Where those rays originate from the picture plane, the object is drawn.
Piero della Francesca's Flagellation of Christ showing Piero’s usage of linear perspective

The Renaissance saw a rebirth of Classical Greek and Roman culture and ideas, among them the study of mathematics to understand nature and the arts. Two major reasons drove Renaissance artists towards mathematics. First, painters needed to figure out how to depict three-dimensional scenes on a two-dimensional canvas. Second, philosophers and artists alike were convinced that mathematics was the true essence of the physical world and that the entire universe, including the arts, could be explained in geometric terms.[31] In light of these factors, Renaissance artists became some of the best applied mathematicians of their times.

The Italian painter Paolo Uccello (1397–1475) was fascinated by perspective. A marble mosaic in the floor of the San Marco Basilica in Venice featuring the small stellated dodecahedron is attributed to him.[32]

The painter Piero della Francesca (c.1415–1492) exemplified this new shift in Italian Renaissance thinking. He was an expert mathematician and geometer, writing books on solid geometry and perspective, including De Prospectiva Pingendi (On Perspective for Painting), Trattato d’Abaco (Abacus Treatise), and De corporibus regularibus (Regular Solids).[33][34][35] The historian Vasari in his Lives of the Painters calls Piero the "greatest geometer of his time, or perhaps of any time."[36] Piero was deeply interested in perspective, as seen in his paintings including the S. Agostino altarpiece and The Flagellation of Christ. His work on geometry influenced later mathematicians and artists, including Luca Pacioli in his De Divina Proportione and Leonardo da Vinci. Piero studied classical mathematics and the works of Archimedes in the library at Urbino.[37] He was taught commercial arithmetic in "abacus schools", as witness his writings, formatted like abacus school textbooks.[38] He may have been influenced by Leonardo Pisano (Fibonacci) from whom those abacus textbooks were derived. Linear perspective was just being introduced into the artistic world. Leon Battista Alberti sums up the idea: "light rays travel in straight lines from points in the observed scene to the eye, forming a kind of pyramid with the eye as vertex."[39] A painting therefore is a cross-sectional plane of that pyramid.

However, the study of perspective precedes Piero and the Renaissance. Before perspective, artists typically sized objects and figures according to their thematic importance. Perspective was first observed in 5th century BCE. Greece; Euclid’s Optics introduced a mathematical theory of perspective. The Muslim mathematician Alhazen extended the theory of optics in his Book of Optics in 1021, but never applied it to art. Perspective burst onto the Renaissance artistic scene with Giotto di Bondone, who attempted to draw in perspective using an algebraic method to determine the placement of distant lines. In 1415, the Italian architect Filippo Brunelleschi and his friend Leon Battista Alberti demonstrated the geometrical method of applying perspective in Florence, using similar triangles as formulated by Euclid, to find the apparent height of distant objects.[40] However, Piero was the first painter to write a practical treatise on perspective in art.

In De Prospectiva Pingendi, Piero transforms his empirical observations into "vera scientia" (true science), i.e. into mathematical proofs. His treatise starts like any mathematics book in the vein of Euclid: he defines the point as "essere una costa tanto picholina quanto e possible ad ochio comprendere" (being the tiniest thing that is possible for the eye to comprehend).[31] Piero uses deductive logic to lead the reader to the perspective representation of a three-dimensional body. Piero realized that the way aspects of a figure changed with point of view obeyed mathematical laws. Piero methodically presented a graded series of perspective problems to develop the reader's understanding.[41]

Mathematical inspirations for art[edit]


The first printed illustration of a rhombicuboctahedron, by Leonardo da Vinci, published in De Divina Proportione

The Platonic solids and other polyhedra are a recurring theme in Western art. They are found, for instance, in a marble mosaic featuring the small stellated dodecahedron, attributed to Paolo Uccello, in the floor of the San Marco Basilica in Venice;[32] in Leonardo da Vinci's diagrams of regular polyhedra drawn as illustrations for Luca Pacioli's book The Divine Proportion;[32] as a glass rhombicuboctahedron in Jacopo de Barbari's portrait of Pacioli, painted in 1495;[32] in the truncated polyhedron (and various other mathematical objects) in Albrecht Dürer's engraving Melencolia I;[32] and in Salvador Dalí's painting The Last Supper in which Christ and his disciples are pictured inside a giant dodecahedron.

Albrecht Dürer (1471–1528) was a German Renaissance printmaker who made important contributions to polyhedral literature in his book, Underweysung der Messung (Education on Measurement) (1525), meant to teach the subjects of linear perspective, geometry in architecture, Platonic solids, and regular polygons. Dürer was likely influenced by the works of Luca Pacioli and Piero della Francesca during his trips to Italy.[42] While the examples of perspective in Underweysung der Messung are underdeveloped and contain a number of inaccuracies, the manual does contain a very interesting discussion of polyhedra. Dürer is also the first to introduce in text the idea of polyhedral nets, polyhedra unfolded to lie flat for printing.[43] Dürer published another influential book on human proportions called Vier Bücher von Menschlicher Proportion (Four Books on Human Proportion) in 1528.[44]

Dürer's well-known engraving Melencolia I depicts a frustrated thinker sitting by a truncated triangular trapezohedron. It has been the subject of more modern interpretation than almost any other print,[45][46] including a two-volume book by Peter-Klaus Schuster,[47] and an influential discussion in Erwin Panofsky's monograph of Dürer.[48]

The graphic artist M. C. Escher (1898—1972) was especially interested in five specific polyhedra, which appear many times in his work. The Platonic solids—tetrahedrons, cubes, octahedrons, dodecahedrons, and icosahedrons—are especially prominent in Order and Chaos and Four Regular Solids.[49] These stellated figures often reside within another figure which further distorts the viewing angle and conformation of the polyhedrons and providing a multifaceted perspective artwork.[50]


Further information: tessellation and M. C. Escher
Hyperbolic geometry in art: Circle Limit III by M.C. Escher (1959)

Escher’s interest in tessellations, polyhedrons, shaping of space, and self-reference manifested itself in his work throughout his career.[51][52] In the Alhambra Sketch, Escher showed that art can be created with polygons or regular shapes such as triangles, squares, and hexagons. Escher used irregular polygons when tiling the plane and often used reflections, glide reflections, and translations to obtain further patterns. Many of his works contain impossible constructions, made using geometrical objects that cannot exist but are pleasant to the human sight. Some of Escher's tessellation drawings were inspired by conversations with the mathematician H. S. M. Coxeter concerning hyperbolic geometry.[53]


Dalí's 1954 painting Crucifixion (Corpus Hypercubus) uses the net of a hypercube.

Salvador Dalí (1904–1989) incorporated mathematical themes in several of his later works. His 1954 painting Crucifixion (Corpus Hypercubus) depicts a crucified figure upon the net of a hypercube.[54] In The Sacrament of the Last Supper (1955) Christ and his disciples are pictured inside a giant dodecahedron.[55] Dalí's last painting, The Swallow's Tail (1983), was part of a series inspired by René Thom's catastrophe theory.[56] The Spanish painter and sculptor Pablo Palazuelo (1916–2007) focused on the investigation of form. He developed a style that he described as the geometry of life and the geometry of all nature. Consisting of simple geometric shapes with detailed patterning and coloring, in works such as Angular I and Automnes, Palazuelo expressed himself in geometric transformations.[31]


The sculptor John Robinson (1935–2007) was interested in astronomy and mathematical relationships. His works from Gordian Knot to Bands of Friendship displayed mathematical knot theory in polished bronze.[31] Other works explore the topology of the toruses. Genesis is based on Borromean rings – a set of three circles, no two of which link but in which the whole structure cannot be taken apart without breaking.[57]

The sculptor Helaman Ferguson creates complex surfaces and other topological objects.[58] His works are visual representations of mathematical objects; The Eightfold Way is based on the projective special linear group PSL(2,7), a finite group of 168 elements.[59][60]

Florentine Bargello Pattern


Mathematics has inspired textile arts including quilting,[61] knitting,[62] cross-stitch, crochet,[63] embroidery[64][65] and weaving.[66] Lorenz manifold and the hyperbolic plane have been crafted using crochet.[67][68] The American weaver Ada Dietz wrote a 1949 monograph Algebraic Expressions in Handwoven Textiles, defining weaving patterns based on the expansion of multivariate polynomials.[69] The mathematician J. C. P. Miller used the Rule 90 cellular automaton to design tapestries depicting both trees and abstract patterns of triangles.[70]

See also[edit]


  1. ^ The ratio of the slant height to half the base length is 1.619, less than 1% from the golden ratio, implying use of Kepler’s triangle (face angle 51°49’).[12][13] It is more likely that pyramids were made with the 3-4-5 triangle (face angle 53°8’), known from the Rhind Mathematical Papyrus; or with the triangle with base to hypotenuse ratio 1:4/π (face angle 51°50’).[14]
  2. ^ For example, the height and width of the front of Notre-Dame of Laon have the ratio 8/5 or 1.6, not 1.618. Such Fibonacci ratios quickly become hard to distinguish from the golden ratio.[22]


  1. ^ Galileo Galilei, The Assayer, as translated by Stillman Drake (1957), Discoveries and Opinions of Galileo, pp. 237–238.
  2. ^ King, Jerry P. (1992). The Art of Mathematics. Fawcett Columbine. pp. 8–9. ISBN 0-449-90835-6. 
  3. ^ a b Stewart, Andrew (November 1978). "Polykleitos of Argos," One Hundred Greek Sculptors: Their Careers and Extant Works". Journal of Hellenic Studies 98: 122–131. doi:10.2307/630196. 
  4. ^ Raven, J. E. (1951). "Polyclitus and Pythagoreanism". Classical Quarterly 1 (3-4): 147–. 
  5. ^ a b Tobin, Richard (October 1975). "The Canon of Polykleitos". American Journal of Archaeology 79 (4): 307–321. 
  6. ^ Lawton, Arthur J. (2013). "Pattern, Tradition and Innovation in Vernacular Architecture". Past 36. Retrieved 25 June 2015. he base figure is a square the length and width of the distal phalange of the little finger. Its diagonals rotated to one side transform the square to a 1 : √2 root rectangle. In Figure 5 this rectangular figure marks the width and length of the adjacent medial phalange. Rotating the medial diagonal proportions the proximal phalange and similarly from there to the wrist, from wrist to elbow and from elbow to shoulder top. Each new step advances the diagonal’s pivot point. 
  7. ^ Seghers, M.J. The Golden Proportion and Beauty, Plastic and Reconstructive Surgery, Vol. 34, 1964.
  8. ^ Mainzer, Klaus (1996). Symmetries of Nature: A Handbook for Philosophy of Nature and Science. Walter de Gruyter. p. 118. 
  9. ^ "Mathematical properties in ancient theatres and amphitheatres". Retrieved 29 January 2014. 
  10. ^ "Architecture: Ellipse?". Retrieved 29 January 2014. 
  11. ^ Herkommer, Mark. "The Great Pyramid, The Great Discovery, and The Great Coincidence". Retrieved 24 June 2015. 
  12. ^ a b c d Markowsky, George (January 1992). "Misconceptions about the Golden Ratio" (PDF). The College Mathematics Journal 23 (1). 
  13. ^ Taseos, Socrates G. (1990). Back in Time 3104 B.C. to the Great Pyramid. SOC Publishers. 
  14. ^ Gazale, Midhat (1999). Gnomon: From Pharaohs to Fractals. Princeton University Press. 
  15. ^ Huntley, H.E.; The Divine Proportion, Dover, New York, 1970.
  16. ^ Hemenway, Priya (2005). Divine Proportion: Phi In Art, Nature, and Science. Sterling. p. 96. 
  17. ^ Usvat, Liliana. "Mathematics of the Parthenon". Mathematics Magazine. Retrieved 24 June 2015. 
  18. ^ Boussora, Kenza; Mazouz, Said (Spring 2004). "The Use of the Golden Section in the Great Mosque of Kairouan". Nexus Network Journal 6 (1): 7–16. doi:10.1007/s00004-004-0002-y. The geometric technique of construction of the golden section seems to have determined the major decisions of the spatial organisation. The golden section appears repeatedly in some part of the building measurements. It is found in the overall proportion of the plan and in the dimensioning of the prayer space, the court and the minaret. The existence of the golden section in some parts of Kairouan mosque indicates that the elements designed and generated with this principle may have been realised at the same period. 
  19. ^ Brinkworth, Peter; Scott, Paul (2001). "The Place of Mathematics". Australian Mathematics Teacher 57 (3): 2. 
  20. ^ Chanfón Olmos, Carlos (1991). Curso sobre Proporción. Procedimientos reguladors en construcción. Convenio de intercambio Unam–Uady. México – Mérica. 
  21. ^ Livio, Mario (2002). The Golden Ratio: The Story of Phi, The World's Most Astonishing Number. Broadway Books. 
  22. ^ Smith, Norman A. F. (2001). "Cathedral Studies: Engineering or History" (PDF). Trans. Newcomen Soc. 73: 95–137. 
  23. ^ Hart, George. "Luca Pacioli's Polyhedra". Retrieved 13 August 2009. 
  24. ^ Morris, Roderick Conway (27 January 2006). "Palmezzano's Renaissance:From shadows, painter emerges". New York Times. Retrieved 22 July 2015. 
  25. ^ Calter, Paul. "Geometry and Art Unit 1". Retrieved 13 August 2009. 
  26. ^ Brizio, Anna Maria (1980). The Painter. Leonardo: The Artist. McGraw-Hill. 
  27. ^ "Matt Anderson, Jeffrey Frazier, and Kris Popendorf. Leonardo da Vinci". Retrieved 13 August 2009. 
  28. ^ Melissa Kimich. "Leonardo Da Vinci's Connection with Mathematics" (PDF). Retrieved 2011-12-19. 
  29. ^ Turner, Richard A. (1992). Inventing Leonardo. Alfred A. Knopf. 
  30. ^ a b "Vitruvian Man". 2006-10-18. Retrieved 2009-08-13. 
  31. ^ a b c d Emmer, Michelle, ed. (2005). The Visual Mind II. MIT Press. 
  32. ^ a b c d e Hart, George W. "Polyhedra in Art". Retrieved 24 June 2015. 
  33. ^ della Francesca, Piero. De Prospectiva Pingendi, ed. G. Nicco Fasola, 2 vols., Florence (1942).
  34. ^ della Francesca, Piero. Trattato d'Abaco, ed. G. Arrighi, Pisa (1970).
  35. ^ della Francesca, Piero. L'opera "De corporibus regularibus" di Pietro Franceschi detto della Francesca usurpata da Fra Luca Pacioli, ed. G. Mancini, Rome, (1916).
  36. ^ Vasari, G. (1878). G. Milanesi, ed. Le Opere, volume 2. p. 490. 
  37. ^ Heath, T.L. (1908). The Thirteen Books of Euclid's Elements. Cambridge University Press. p. 97. 
  38. ^ Grendler, P. "What Piero Learned in School: Fifteenth-Century Vernacular Education," in Piero della Francesca and His Legacy, ed. M.A. Lavin, University Press of New England (1995), 161–176.
  39. ^ Alberti, Leon Battista. On Painting, ed. Martin Kemp, trans. Cecil Grayson, London-New York (1991).
  40. ^ Vasari. Lives of the Artists. Chapter on Brunelleschi
  41. ^ Peterson, Mark. "The Geometry of Piero della Francesca". In Book I, after some elementary constructions to introduce the idea of the apparent size of an object being actually its angle subtended at the eye, and referring to Euclid's Elements Books I and VI, and Euclid's Optics, he turns, in Proposition 13, to the representation of a square lying flat on the ground in front of the viewer. What should the artist actually draw? After this, objects are constructed in the square (tilings, for example, to represent a tiled floor), and corresponding objects are constructed in perspective; in Book II prisms are erected over these planar objects, to represent houses, columns, etc.; but the basis of the method is the original square, from which everything else follows. 
  42. ^ Panofsky, E. (1955). The Life and Art of Albrecht Durer. Princeton. 
  43. ^ George W. Hart. "Dürer’s Polyhedra". Retrieved 13 August 2009. 
  44. ^ Dürer, Albrecht (1528). Hierinn sind begriffen vier Bucher von menschlicher Proportion. Nurenberg: Retrieved 24 June 2015. 
  45. ^ Schreiber, P. (1999). "A New Hypothesis on Durer's Enigmatic Polyhedron in His Copper Engraving 'Melencolia I'". Historia Mathematica 26: 369–377. 
  46. ^ Dodgson, Campbell (1926). Albrecht Dürer. London: Medici Society. p. 94. 
  47. ^ Schuster, Peter-Klaus (1991). Melencolia I: Dürers Denkbild. Berlin: Gebr. Mann Verlag. pp. 17–83. 
  48. ^ Panofsky, Erwin; Klibansky, Raymond; Saxl, Fritz (1964). Saturn and melancholy. Basic Books. 
  49. ^ Escher, M.C. (1988). The World of MC Escher. Random House. 
  50. ^ Escher, M.C.; Vermeulen, M.W.; Ford, K. (1989). Escher on Escher: Exploring the Infinite. HN Abrams. 
  51. ^ "Tour: M.C. Escher – Life and Work". Retrieved 2009-08-13. 
  52. ^ "MC Escher". 2007-11-01. Retrieved 2009-08-13. 
  53. ^ Roberts, Siobhan (2006). 'Coxetering' with M.C. Escher. King of Infinite Space: Donald Coxeter, the Man Who Saved Geometry (Walker). p. Chapter 11. 
  54. ^ "Crucifixion (Corpus Hypercubus)". Metropolitan Museum of Art. Retrieved 26 June 2015. 
  55. ^ Livio, Mario. "The golden ratio and aesthetics". Retrieved 26 June 2015. 
  56. ^ King, Elliott (2004). Ades, Dawn, ed. Dali. Milan: Bompiani Arte. pp. 418–421. 
  57. ^ "John Robinson". 2007. Retrieved 13 August 2009. 
  58. ^ "Helaman Ferguson web site". Retrieved 13 August 2009. 
  59. ^ The Eightfold Way: A Mathematical Sculpture by Helaman Ferguson, William P. Thurston
  60. ^ "MAA book review of ''The Eightfold Way: The Beauty of Klein's Quartic Curve''". 1993-11-14. Retrieved 13 August 2009. 
  61. ^ Ellison, Elaine; Venters, Diana (1999). Mathematical Quilts: No Sewing Required. Key Curriculum. 
  62. ^ Belcastro, Sarah-Marie. "Adventures in Mathematical Knitting". American Scientist. Retrieved 24 June 2015. 
  63. ^ Taimina, Daina (2009). Crocheting Adventures with Hyperbolic Planes. A K Peters. ISBN 1-56881-452-6. 
  64. ^ Snook, Barbara. Florentine Embroidery. Scribner, Second edition 1967.
  65. ^ Williams, Elsa S. Bargello: Florentine Canvas Work. Van Nostrand Reinhold, 1967.
  66. ^ Grünbaum, Branko; Shephard, Geoffrey C. (May 1980). "Satins and Twills: An Introduction to the Geometry of Fabrics". Mathematics Magazine 53 (3): 139–161. doi:10.2307/2690105. JSTOR 2690105. 
  67. ^ Henderson, David; Taimina, Daina (2001). "Crocheting the hyperbolic plane" (PDF). Mathematical Intelligencer 23 (2): 17–28. doi:10.1007/BF03026623. .
  68. ^ Osinga, Hinke M,; Krauskopf, Bernd (2004). "Crocheting the Lorenz manifold". Mathematical Intelligencer 26 (4): 25–37. doi:10.1007/BF02985416. 
  69. ^ Dietz, Ada K. (1949), Algebraic Expressions in Handwoven Textiles (PDF), Louisville, Kentucky: The Little Loomhouse 
  70. ^ Miller, J. C. P. (1970). "Periodic forests of stunted trees". Philosophical Transactions of the Royal Society of London. Series A, Mathematical and Physical Sciences 266 (1172): 63–111. doi:10.1098/rsta.1970.0003. JSTOR 73779. 

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