The arts and mathematics have always had a close relationship. Although many people regard the arts and mathematics as distinct fields, there are several ties between them. Art is regarded as a method of expression and admiration for beauty, as well as an emotional connection. The American Mathematical Society's (AMS) Joseph Malkevitch identified various parallels between art and math.

Throughout the Renaissance, painters and mathematicians recognized one another as truth seekers. Malkevitch claims that painters like Luca Pacioli, Leonardo da Vinci, Albrecht Dürer, and M.C. Escher all employed mathematical thought to achieve their aesthetic vision. Pacioli, who taught mathematics to Leonardo da Vinci, is said to be the initiator of the double entry accounting system, among many other accomplishments. However, it is in his first textbook Summa de Arithmetica, Geometrica, proportioni et propotionalita¸ which was printed in 1494, that he explored the connection of mathematics and proportion, particularly in works of art. He also wrote the book De Divina Proportione, with beautiful illustrations of three-dimensional geometric solids and templates created by his pupil Leonardo da Vinci (Meisner). Pacioli’s main goal in writing De Divina Proportione is to reveal to artists the secret of harmonic forms. Here is a translated quote from the book:

A work necessary for all the clear-sighted and inquiring human minds, in which everyone who loves to study philosophy, perspective, painting, sculpture, architecture, music and other mathematical disciplines will find a very delicate, subtle and admirable teaching and will delight in diverse questions touching on a very secret science. (Meisner)

Da Vinci incorporated the principle of Divine proportion in his works of art, particularly in his paintings “The Annunciation”, “The Last Supper”, and “Mona Lisa.” To Da Vinci and other Renaissance masters like Botticelli and Raphael, creating works of art is synonymous to accurate proportionality, a formula first recorded by Euclid in 300 B.C. As we can see, the importance of mathematics and art is very evident in Da Vinci’s work. Not only did he use the Divine Proportion extensively in his work, he also used geometry (“The Vitruvian Man.”)

Another Renaissance man, Albrecht Dürer created a series of “Master Prints”, one of which, “Melencolia I”, has influenced generations of artists who are interested in applying math to their works of art. Dürer used 3D solids in a way that was never used before, an influence from scientist Johannes Kepler, who was seeking a method to pack spheres in the densest way possible. Dürer was a practicing mathematician and artist, and the polyhedron in the center of “Melancolia I” was studied by generations of mathematicians, and became relevant in applied sciences in the late 20th century, being used in information and communication theory, and molecular biology (Chudnovksy and Chudnovsky).

However great the Renaissance artists are and their brilliant application of mathematics in their work, this paper will specifically discuss one particular 20th century artist, M.C. Escher. Escher studied decorative arts, and in 1922 while travelling through Spain and Italy, he experienced a breakthrough in his work. He was fascinated with the decorative interlocking symmetrical designs of the Moorish-influenced Alhambra, which led him to study tessellation. Tessellation became one of the basic elements of his work, which then led to other mathematical concepts.

Preliminaries

M.C. Escher created a lot of work based on mathematics, and to guide us along that path and to better explain the mathematical concepts involved in his work, we have put together some of the terminologies related to the discipline and his works of art.

An elliptic curve can be used in artwork and architecture. Its particular used is in M.C. Escher’s lithograph entitled Print Gallery. An elliptic curve is a kind of cubic curve. In equation form, an elliptic curve is y2 z = x3 + axz2 +bz3.

A cyclic number is an (n – 1) digit integer that, when multiplied by 1, 2, 3 …., n- 1, produces the digits in a different order. It has been conjectured, but not yet proven, that an infinite number of cyclic numbers exist.

Symmetry is another concept that Escher worked with, and his works contains numerous symmetrical images. Symmetry involves more than just repeated images. Symmetry of an object in the plane is a rigid motion of the plane that leaves the object apparently unchanged. It means that if a viewer closes his or her eyes during the repositioning of a certain object, the object would look exactly the same when she opened her eyes (Tapp 3).

Symmetry is also defined as looking at the collection of all rigid plane motions, which compose of compositions of rotations, mirror reflections, and sliding translations. According to Silver, any two symmetries can be “multiplied” to produce another, possibly different symmetry.

The description of all possible symmetry groups of bounded objects is usually attributed to Leonardo da Vinci. Here is Da Vinci’s theorem on bounded objects: The symmetry group of any bounded object in the place is either infinite or isomorphic to a dihedral or cyclic group (Tapp 51). A stronger rigid version of the theorem is also true: Any bounded object in the place with a finite symmetry group is rigidly equivalent to one of the objects pictured. The corollary of Da Vinci’s symmetry theorem is: “If a bound object has exactly n rotations and zero flips, then its symmetry group is isomorphic to Cn. If a bounded object has exactly n rotations, and n flips, then its symmetric group is isomorphic to Dn.

When it comes to the symmetry of wallpaper patterns, there is also a theorem related to this. The symmetry group of any wallpaper pattern is isomorphic to the symmetry group of one of the 17 model patterns (Tapp 58). However, this theorem is not quite optimal because among the 17model patterns, certain pairs have isomorphic symmetry groups. The symmetry groups of 17 model patterns are often called “wallpaper groups”. M.C. Escher incorporated many of these patterns into his paintings. It also occurs in nature, such as in honeycombs.

In a way, symmetries behave just like numbers because they can be added or subtracted. When two rotations of symmetrical shapes or two reflections of symmetrical shapes are added or subtracted, a rotation is the result. However, if one reflection and one rotation is added or subtracted, we get a reflection.

In symmetry, addition of shapes can be repeated. Various figures such as triangles, pentagons, and three-dimensional objects such as cubes can be used. In mathematics, an overall collection of all symmetries all at one is called a Group. A Group is a collection of objects with the following properties:

If A and B are elements of the group then A ∗ B is also an element of the group.

The operation ∗ is associative, meaning that A ∗ (B ∗ C) = (A ∗ B) ∗ C for any elements A, B and C in the group.

There is an identity (a 0-element) which does nothing.

Every element has an opposite or inverse, which, when combined, gives the identity. (Mathigon)

One way of looking at symmetry is through the design of an equilateral triangle. Equilateral triangles can be rotated through 60 degrees, 120 degrees, and 360 degrees. The reflections across each of the three lines joining a vertex to the midpoint of the opposite edge can also be rotated. All of these six symmetries comprise a dihedral group (Silver).

A Tessellation is a pattern that can cover a flat surface without any gaps or overlaps. M.C. Escher created a lot of tessellations in his work. Most tessellations have translational, rotational, or reflectional symmetries. A combination of reflection and translation is called glide reflection (Mathigon). The transformations that we can do to a tessellation are called the Isometries of the Plane. The symmetry groups in tessellations that tell how a pattern is repeated.

Main Content

The Works of M.C. Escher

Prentententoonstelling (Print Gallery)

M.C. is a world-renowned Dutch graphic artist who made lithographs, woodcuts, and wood engravings. His art-work is mathematically inspired, and he explored mathematical concepts such as symmetry, perspective, hyperbolic geometry, tessellations, and many more. He was interested in curvilinear perspectives, platonic solids like spheres and cubes, and cylinders and stellated polyhedra. He was also inspired by the Möbius Strip with his work Möbius Strip I and II (1963 and 1961, respectively).

His 1948 Wood Engraving Stars are made up of cubes formed into a cage with chameleons inside. His interest in spheres and concentric circles inspired his workds Sphere Surface with Fish (1958), Sphere Spirals (1958), and Concentric Rinds (1953) wood engraving (M.C. Escher).

Dutch graphic artist Maurits Cornelis Escher created a lithograph in 1956 which he entitled Prentententoonstelling (Print Gallery in English). It is an unusual lithograph showing a man viewing a Mediterranean seaport print inside an art gallery, and among the quayside buildings, as the young man’s eyes travel from left to right, and then down, he sees that one of the buildings is the very gallery where he is standing. The Droste effect is a famous visual effect in which the image contains itself on a smaller scale, and the image realistically appears on the picture. The Droste effect is used on the Print Gallery.

It is this work of Escher that we will take a special interest on, as we delve into the mathematics behind this interesting lithograph. The lithograph is drawn on a certain elliptic curve over the field of complex numbers. There is a deduction that an idealized version of the picture repeats itself in the middle (de Smit and Lenstra, Jr., 446). In precise terms, the lithograph in question has a copy of itself which is rotated 157.63 degrees and scaled won by a factor of 22 (de Smit and Lenstra, Jr., 446).

In creating Print Gallery, Escher said that he began with the idea of the possibility of creating an annular bulge, a cyclic expansion…without beginning or end. What is a cyclic expansion? The number 142, 857 have long been recognized by those interested in number oddities as one of the most remarkable of integers. When multiplied by any number from 1 to 6 the result always consists of the same digits and, more remarkably still, these digits always appear in the same cyclic order, but with each number commencing at a different point (Lines, 141). This striking property is called a cyclic number.

Escher used curved lines to create the lithograph, which he duplicated a number of times. In the beginning, he used straight lines and attempted to expand it, however, the individual small square only retained its shape with the use of a cyclical expansion of curved lines. After several attempts, Escher created a grid that expands by a factor of 4 in each direction. With this grid, when you go clockwise going to the center, it folds unto itself, expanded by a factor of 44 = 256.

The grid rotates and scales, which could lead to many different images. Escher then needed an undistorted representation of the scene, but reduced to a factor of 256. The painstaking detail in which Escher created his work is evidenced by making four different studies, one for each corner of the lithograph. He then fitted the straight square onto the curved grid that he made.

Escher created a complex multiplicative method, a very accurate way of going back and forth between the straight and curved lines of the space he created.

Symmetry in M.C. Escher’s Work

What distinguishes M.C. Escher’s work with symmetry and pattern is that every print or drawing is not abstract. They are easily recognizable, and the viewer can identify each separate object as composed of everyday objects that we see (MacGillavry, 123).

M.C. Escher used straightedge and compass constructions to create his hyperbolic patterns. These days, symmetrical patterns can be used using computer graphics. Mathematician H.SM. Coxeter hugely influenced Escher’s work on symmetrical patterns. When Coxeter sent him a copy of his paper entitled “Crystal symmetry and its generalizations, he was shocked because he had found the long-desired solution on the problem of designing repeated patterns (Dunham 2003).

As Durnham (2003) quotes Goodman Strauss, the center of an orthogonal circular arc is external to the disk and is called its pole. The locus of all poles of arcs through a point in the disk is a line called the polar of that point.

In creating his first ever work on tessellation, Circle I, Escher used the external web of poles and polar segments, which is called the scaffolding of the tessellation.

Escher’s work Another World depicts the inside of a cube-shaped building. It shows symmetry because it contains a clever three directions of view, at 90 degree angles, which shows an apparent fourfold rotational symmetry. Another World shows the faces of the cube building with large windows separated by pillars. The axes of the pillars converge towards one point, exactly in the center of the print (MacGillavry, 124).

Escher also loves plane-filling repeating patterns, which are tessellations. Wallpaper or block printed material did not satisfy him, and instead he sought to work in constructing repeating patterns from motifs with varying sizes. Circle Limits I to IV are examples of Escher putting small motifs in the middle of the picture, which grew larger and larger towards the edges, and then largest motifs at the center, becoming smaller and smaller towards the circumference of the circle (MacGillavry 131).

Conclusion

Mathematicians have always assisted artists by providing them tools, which consists of theorems and formulas in order for them to represent their vision in various representations. Although he does not consider himself a mathematician, M.C. Escher’s work has apparent mathematical qualities because he approached them in a mathematical way.

In Escher’s Print Gallery, we learned about cyclic numbers and the multiplicative method in curved and straight lines. We also saw the possibility of placing undistorted images in curved lines, which creates a dramatic visual effect.

In Escher’s work on symmetry and tessellations appeal to our intellect rather than emotions, and through there we learn various types of symmetries.

Escher’s work is a landmark foundation in the study of how art and mathematics are harmoniously melded together. These days, a computer program can replicate and even manipulate Escher’s images, but the main contribution of Escher cannot be undermined. He laid this foundation for other artists and even scientists and mathematicians, not only for printed art, but also for architecture, sculpture, and graphic design.

Works Cited

Chudnovsky, David and Chudnovsky, Gregory. “After 500 Years, Dürer’s Art Still

Engraved on Mathematician’s Minds (Op-Ed).” Live Science, May 13, 2014. www.livescience.com/45557-durer-engraving-shaped-science-and-math.html. Accessed 11 April 2017.

De Smit, B. and Lenstra, Jr., H.W. “The Mathematical Structure of Escher’s Print

Gallery.” Notices of the AMA, April 2003, 50 (4), pp. 446 – 457. www.ams.org/notices/200304/fea-escher.pdf

Accessed 10 April 2017.

MacGillavry, Caroline H. “The Symmetry of M.C. Escher’s “Impossible” Images. Comp

& Maths, with Appls. 1986, 12B (/2), pp.123-138 ac.els-cdn.com/089812218690146X/1-s2.0-089812218690146Xmain.pdf?_tid=a084e74c-23c2-11e7-92d2 00000aab0f26&acdnat=1492470628_fee1325e312942bf317dfe388ec447c9. Accessed 10 April 2017.

Malkevitch, Joseph. “Mathematics and Art.” American Mathematical Society.

www.ams.org/samplings/feature-column/fcarc-art1. Accessed 10 April 2017.

Mathigon. “Symmetry and Groups.” Mathigon, n.d.

world.mathigon.org/Symmetry_and_Groups. Accessed 10 April 2017.

Meisner, Gary. “Da Vinci and the Divine Proportion in Art Composition.” The Golden

Number, July 7, 2014. www.goldennumber.net/leonardo-da-vinci-golden-ratio-art/ Accessed 11 April 2017.

M.C. Escher. “M.C. Escher – Biography”. www.mcescher.com/about/biography/

Accessed 10 April 2017.

Lines, M.E. A Number for Your Thoughts. Taylor and Francis, 1986. books.google.com.ph/books?id=Am9og6q_ny4C&pg=PA141&dq=cyclic+number&hl=en&sa=X&ved=0ahUKEwi4zNuWzqvTAhUJv7wKHYsADv8Q6AEIIzAA#v=onepage&q=cyclic%20number&f=false. Accessed 10 April 2017.

Silver, Daniel S. “Fearless Symmetry.” American Scientist, 2016.

www.americanscientist.org/bookshelf/pub/fearless-symmetry. Accessed 10 April

2017.

Tapp, Kristopher. Symmetry: A Mathematical Exploration. Springer, 2012.