Mathematics and Art

Mary Jane SterlingOLLI – Winter, 2015

So – where’s the math?

Dürer’s Melancholia

The Flagellation of Christ by Piero della Francesca

Escher’s Ascending and Descending

Dali’s The Persistence of Memory

The Last Supper – Leonardo di Vinci

Parthenon

Wall paper

So – where’s the art?

Cubism

Galatea of the Spheres – Salvador Dali

Tile wall in Alhambra

Magic Squares

Magic Square: square array in which every row, column and diagonal add up to the same number.

9

8

7

6

5

4

3

2

1

Magic squares were known to Chinese mathematicians as early as 650 BC and to Arab mathematicians possibly as early as the seventh century AD. Chinese literature dating from as early as 650 BC tells the legend of Lo Shu or "scroll of the river Lo". According to the legend, there was at one time in ancient China a huge flood. While the great king Yu (禹) was trying to channel the water out to sea, a turtle emerged from it with a curious figure / pattern on its shell: a 3×3 grid in which circular dots of numbers were arranged, such that the sum of the numbers in each row, column and diagonal was the same: 15, which is also the number of days in each of the 24 cycles of the Chinese solar year. According to the legend, thereafter people were able to use this pattern in a certain way to control the river and protect themselves from floods.Although the early history of magic squares in Persia is not known, it has been suggested that they were known in pre-Islamic times. It is clear, however, that the study of magic squares was common in medieval Islam in Persia, and it was thought to have begun after the introduction of chess into the region. In 1514 Albrecht Dürer immortalizes a 4×4 square in his famous engraving "Melancholia I".

The date of 1514 appears in the bottom row of the magic

square, as well as above Dürer's monogram at bottom right.

Albrecht Dürer's copper plate engraving Melencolia I, 1514

Found at the Art Institute in Chicago

Sagrada Familia Church – note repeated numbers

Golden Ratio

E

D

C

B

A

Which rectangle is most “appealing”?

E

D

C

B

A

r = 2.93r = 1.62

r = 15.04

r = 1.28

r = 4.39

Mathematics and art have a long historical relationship. The ancient Egyptians and ancient Greeks knew about the golden ratio, regarded as an aesthetically pleasing ratio. They incorporated it and other mathematical relationships, such as the 3:4:5 triangle, into the design of monuments including the Great Pyramid the Parthenon, the Colesseum.

The Golden RatioThe Golden Ratio, roughly equal to 1.618, was first formally introduced in text by Greek mathematician Pythagoras and later by Euclid in the 5th century BC. In the fourth century BC, Aristotle noted its aesthetic properties. Aside from interesting mathematical properties, geometric shapes derived from the golden ratio, such as the golden rectangle, the golden triangle, and Kepler’s triangle, were believed to be aesthetically pleasing. As such, many works of ancient art exhibit and incorporate the golden ratio in their design. Various authors can discern the presence of the golden ratio in Egyptian, Summerian and Greek vases, Chinese pottery, Olmec sculptures, and Cretan and Mycenaean products from as early as the late Bronze Age. The prevalence of this special number in art and architecture even before its formal discovery by Pythagoras is perhaps evidence of an instinctive and primal human cognitive preference for the golden ratio.

Evidence of mathematical influences in art is present in the Great Pyramids, built by Egyptian Pharaoh Khufu and completed in 2560 BC. Pyramidologists since the nineteenth century have noted the presence of the golden ratio in the design of the ancient monuments. They note that the length of the base edges range from 755–756 feet while the height of the structure is 481.4 feet. Working out the math, the perpendicular bisector of the side of the pyramid comes out to 612 feet. If we divide the slant height of the pyramid by half its base length, we get a ratio of 1.619, less than 1% from the golden ratio. Debate has broken out between prominent pyramidologists over whether the presence of the golden ratio in the pyramids is due to design or chance. Of note, Rice contends that experts of Egyptian architecture have argued that ancient Egyptian architects have long known about the existence of the golden ratio. In addition, three other pyramidologists, contend that: Herodotus related in one passage that the Egyptian priests told him that the dimensions of the Great Pyramid were so chosen that the area of a square whose side was the height of the great pyramid equaled the area of the triangle.This passage, if true, would undeniably prove the intentional presence of the golden ratio in the pyramids. However, the validity of this assertion is found to be questionable. Critics of this golden ratio theory note that it is far more likely that the original Egyptian architects modeled the pyramid after the 3-4-5 triangle, rather than the Kepler’s triangle. According to the Rhind mathematical papyrus, an ancient papyrus that is the best example of Egyptian math, the Egyptians certainly knew about and used the 3-4-5 triangle extensively in mathematics and architecture. While the exact triangle the Egyptians chose to design their pyramids after remains unclear, the fact that the dimensions of pyramids correspond so strongly to a special right triangle suggest a strong mathematical influence in the last standing ancient wonder.

A Kepler triangle is a right triangle with edge lengths in geometric progression. The ratio of the edges of a Kepler triangle is linked to the golden ratio:

and can be written:

or approximately 1 : 1.272 : 1.618. The squares of the edges of this triangle (see figure) are in geometric progression according to the golden ratio.

Triangles with such ratios are named after the German mathematician and astronomer Johannes Kepler (1571–1630), who first demonstrated that this triangle is characterized by a ratio between short side and hypotenuse equal to the golden ratio.[ Kepler triangles combine two key mathematical concepts—the Pythagorean theorem and the golden ratio—that fascinated Kepler deeply, as he expressed in this quotation:

“Geometry has two great treasures: one is the theorem of Pythagoras, the other the division of a line into mean and extreme ratio. The first we may compare to a mass of gold, the second we may call a precious jewel.”

Some sources claim that a triangle with dimensions closely approximating a Kepler triangle can be recognized in the Great Pyramid of Giza.

ParthenonThe Parthenon is a temple dedicated the Greek goddess Athena, built in the 5th century BC on the Athenian Acropolis. It is contended that Phidian, the main Greek sculptor in charge of decorating the Parthenon, also knew about the golden ratio and its aesthetic properties. In fact, the Greek symbol for the Golden Ratio is named Phi (φ) because of Phidias. The golden rectangle, a rectangle whose length to width ratio is the golden ratio and considered the most pleasing to the eye, is almost omnipresent in the façade and floor plans of the Parthenon. The entire façade may be enclosed within a golden rectangle. The ratio of the length of a metope and triglyph to the height of the frieze, as well as the height of the columns and stylobate to the entire height of the temple is also the golden ratio. Phidias himself constructed many Parthenon statues that meticulously embody the golden ratio. Phidias is also notable for his contributions to the Athena Parthenos and the Statue of Zeus. As with the Pyramids however, more recent historians challenge the purposeful inclusion of the golden ratio in Greek temples, such as the Parthenon, contending that earlier studies have purposefully fitted in measurements of the temple until it conformed to a golden rectangle.

In classical architecture, a metope is a rectangular architectural element that fills the space between two triglyphs in a Doric frieze, which is a decorative band of alternating triglyphs and metopes above the architrave of a building of the Doric order. Metopes often had painted or sculptural decoration; the most famous example are the 92 metopes of the Parthenon marbles some of which depict the battle between the Centaurs and the Lapiths. The painting on most metopes has been lost, but sufficient traces remain to allow a close idea of their original appearance.

In terms of structure, metopes may be carved from a single block with a triglyph (or triglyphs), or they may be cut separately and slide into slots in the triglyph blocks as at the Temple of Aphaea. Sometimes the metopes and friezes were cut from different stone, so as to provide color contrast. Although they tend to be close to square in shape, some metopes are noticeably larger in height or in width. They may also vary in width within a single structure to allow for corner contraction, an adjustment of the column spacing and arrangement of the Doric frieze in a temple to make the design appear more harmonious.

The oldest mosque in North Africa is the Great Mosque of Kairouan (Tunisia), built by

Uqba ibn Nafi in 670 A.D. Boussora and Mazouz’s study of the mosque dimensions

reveals a very consistent application of the golden ratio in its design. Boussora and

Mazouz contend:

The geometric technique of construction of the golden section seems to

have determined the major decisions of the spatial organization. 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.

Because of urban constraints, the mosque floor plan is not a perfect

rectangle. Even so, for example, the division of the courtyard and prayer

hall is almost a perfect golden ratio.

Great Mosque of Kairouan

Notre Dame

Vitruvian Man by Di Vinci – golden ratio in body dimensions

Some scholars contend the influence of the mathematician Pythagoras on the Canon of Polykleitos. 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. Polykleitosstarts 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. From that, Polykleitos multiplies the length by radical 2 (1.14142) 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. Other proportions are less set. For example, the ideal body should be 8 heads high and 2 heads wide. However, ordinary figures are 7½ heads tall while heroic figures are 8½ heads tall.

Looking at the length of our fingers, each section — from the tip of the base to the wrist — is larger than the preceding one by roughly the ratio of phi.

Polykleitos the Elder (c.450-420B.C.) was a Greek sculptor from the school of Argos, and a contemporary of Phidian. His works and statues consisted mainly of bronze and were of athletes. 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. The influence of the Canon of Polykleitos is immense in classical Greek, Roman and Renaissance sculpture, with many sculptors after him following Polykleitos’ prescription. While none of Polykleitos’ original works survive, Roman copies of his works demonstrate and embody his ideal of physical perfection and mathematical precision.

Golden Ratio

Graham Sutherland's tapestry of Christ The King behind the altar in Coventry Cathedral.

The Fibonacci sequence starts like this: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55 and so on forever. Each number is the sum of the two numbers that precede it. It's a simple pattern, but it appears to be a kind of built-in numbering system to the cosmos. Here are 15 astounding examples of phi in nature.Leonardo Fibonacci came up with the sequence when calculating the ideal expansion pairs of rabbits over the course of one year. Today, its emergent patterns and ratios (phi = 1.61803...) can be seen from the microscale to the macroscale, and right through to biological systems and inanimate objects.

The number of petals in a flower consistently follows the Fibonacci sequence. Famous examples include the lily, which has three petals, buttercups, which have five (pictured at left), the chicory's 21, the daisy's 34, and so on. Phi appears in petals on account of the ideal packing arrangement as selected by Darwinian processes; each petal is placed at 0.618034 per turn (out of a 360° circle) allowing for the best possible exposure to sunlight and other factors.

Snail shells and nautilus shells follow the logarithmic spiral, as does the cochlea of the inner ear. It can also be seen in the horns of certain goats, and the shape of certain spider's webs.

Hurricanes

Faces, both human and nonhuman, abound with examples of the Golden Ratio. The mouth and nose are each positioned at golden sections of the distance between the eyes and the bottom of the chin. Similar proportions can been seen from the side, and even the eye and ear itself (which follows along a spiral).It's worth noting that every person's body is different, but that averages across populations tend towards phi. It has also been said that the more closely our proportions adhere to phi, the more "attractive" those traits are perceived. As an example, the most "beautiful" smiles are those in which central incisors are 1.618 wider than the lateral incisors, which are 1.618 wider than canines, and so on. It's quite possible that, from an evo-psych perspective, that we are primed to like physical forms that adhere to the golden ratio — a potential indicator of reproductive fitness and health.

Constructing a golden rectangle

EG

HG = 1.62

HG = 5.29 cm

EG = 8.56 cm

H

GE

Show construction – change length of side of original square

HG

FG = 1.62

FG = 3.27 cm

EG

HG = 1.62

HG = 5.29 cm

EG = 8.56 cm

H

GE F

HG

FG = 1.62

FG = 3.27 cm

EG

HG = 1.62

HG = 5.29 cm

EG = 8.56 cm

H

GE F

Donald in Mathemagic

LandIntro: 0 – 1:46Golden Ratio: 7:00 – 13:30

Polyhedra/Platonic Solids

Polyhedron: Three-dimensional figure with flat surfaces, straight edges (where the surfaces meet), and vertices/points (where the edges meet).

Regular polyhedron: dodecahedron

Five Platonic solids (all faces are regular polygons):

Tetrahedron: each of the four faces an equilateral triangle

Octahedron: each of the eight faces an equilateral triangle

Cube: each of the six faces a square

Dodecahedron: each of the twelve faces a regular pentagon

Icosahedron: each of the twenty faces an equilateral triangle

Only these five are possible.

A skeletal polyhedron drawn by Leonardo da Vinci to illustrate a book by Luca Pacioli.

M.C. Escher’s Stars

Euler’s Rule: for any polyhedron, V + F = E + 2

The truncated rhombohedron with a faint human skull on it. This shape is now known as Dürer’s solid; over the years, there have been numerous articles disputing the precise shape of this polyhedron.

Albrecht DürerAlbrecht 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. 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. Dürer

published another influential book on human proportions called Vier Bücher

von Menschlicher Proportion (Four Books on Human Proportion) in 1528.

Dürer's well-known engraving Melencolia I depicts a frustrated thinker sitting

by what is best interpreted as a “truncated rhomboid” or a “rhombohedron with

72-degree face angles, which has been truncated so it can be inscribed in a

sphere”.

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

Polyhedral nets:

De Divina Proportione

Written by Luca Pacioli in Milan from 1496–98,

published in Venice in 1509, De Divina

Proportione was about mathematical and

artistic. Leonardo da Vinci drew illustrations of

regular solids in De divina proportione while he

lived with and took mathematics lessons from

Pacioli. Leonardo's drawings are probably the

first illustrations of skeletonic solids, which

allowed an easy distinction between front and

back. Skeletonic solids, such as the

rhombicuboctahedron, were one of the first

solids drawn to demonstrate perspective by

being overlaid on top of each other.

Additionally, the work also discusses the use of

perspective by painters such as Piero della

Francesco, Melozzo da Forli and Marco Palmezzano.

Di Vinci

Graphic artist M.C. Escher (1898—1972) was known for his mathematically inspired work. Escher’s interest In tessellations, polyhedrons,shaping of space, and self-reference manifested itself in his work throughout his

career. 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 many more patterns. Additionally, Escher arranged the shapes to simulate images of animals and other figures. His work can be noted in Development 1 and Cycles.Escher’s was especially interested in five specific types of polyhedron, which appear many times in his work. "Regular polyhedrons" are defined as solids that have exactly similar polygonal faces, and are also known as Platonic solids. These tetrahedrons, cubes, octahedrons, dodecahedrons, and icosahedrons are especially prominent in Order and Chaos and Four Regular Solids. Here 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.

Reptiles Dodecahedron

Tetrahedral Planetoid

Stars

Flatworms

Möbius symmetry, the topological phenomenon that yields a half-twisted strip with two surfaces but only one side, has been a source of fascination since its discovery in 1858 by German mathematician August Möbius. As artist M.C. Escher so vividly demonstrated in his “parade of ants,” it is possible to traverse the “inside” and “outside” surfaces of a Möbius strip without crossing over an edge. For years, scientists have been searching for an example of Möbius symmetry in natural materials without any success. Now a team of scientists has discovered Möbiussymmetry in metamaterials – materials engineered from artificial “atoms” and “molecules” with electromagnetic properties that arise from their structure rather than their chemical composition.

Band of Moebius

Klein bottle – only “inside” or only “outside”?

Platonic solids in artThe Platonic solids and other polyhedral are a recurring theme in Western art. Examples include:A marble mosaic featuring the small stellated dodecahedron, attributed to Paolo Uccello, in the floor of the San Marco Basilica (St. Mark’s Basilica) in Venice.Leonardo da Vinci’s outstanding diagrams of regular polyhedra drawn as illustrations for Luca Pacioli's book The Divine Proportion.A glass rhombicuboctahedron in Jacopo de Barbari's portrait of Pacioli, painted in 1495. A truncated polyhedron (and various other mathematical objects) which feature in Durer’s Melancholia I.Salvador Dali’s painting The Last Supper in which Christ and his disciples are pictured inside a giant dodecahedron.

A marble mosaic featuring the small stellated dodecahedron, attributed to Paolo Uccello, in the floor of the San Marco Basilica (St. Mark’s Basilica) in Venice.

A glass rhombicuboctahedron in Jacopo de Barbari's portrait of Pacioli, painted in 1495.

The Sacrament of the Last Supper – Christ and his disciples are pictured in a giant dodecahedron.

Polyhedra, tilings, and dissections

Drawing polyhedra was an early testing ground for ideas related to perspective drawing. Renaissance artists were involved in trying to build on historical references to "Archimedean polyhedra" which were transmitted via the writings of Pappus. What constituted a complete set of convex polyhedra with the property that locally every vertex looked like every other vertex and whose faces were regular polygons, perhaps not all with the same number of sides? Perhaps surprisingly, no complete reconstruction occurred until the work of Kepler (1571-1630), who found 13 such solids, even though one can make a case for there being 14 such solids.

In more modern times polyhedra have inspired artists and mathematicians with an interest in the arts. George Hart, whose background is in computer science, is an example of a person who is contributing to the mathematical theory of polyhedra, while at the same time he uses his skills as a sculptor and artist to create original works inspired by polyhedral objects.

Hart

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

Spherical spiral – MC Escher

Perspective

Two-point perspective

Perspective Drawing

All vertical construction lines are parallel to one

another;

All horizontal construction lines are parallel to one

another;

Given the lack of a second and third vanishing point,

this perspective relies on orthogonal construction

lines;

This style of perspective, although convincing, is not

always accurate.

Using linear perspective Francesca’s Flagellation of Christ

Flagellation of Christ – Piero della Francesca

Madonna and Child with Saints – Piero della Francesca

St. Jerome in His Study - Piero della Francesca

The Annunciation – Giotti Di Bondone

Leonardo's notes on linear perspective are apparently lost, but he made great use of perspective in his own paintings, such as this study of the unfinished Adoration. Note the strict Albertian grid on the pavement.Slide 15-26: The Annunciation c. 1472 Ufizzi (Cat. # 1074)

His Annunciation shows a carefully worked out perspective framework. Incised lines beneath the paint on this wood panel show his construction. Note though that the Virgin's arm appears too long. Studies have shown that Leonardo departed from the correct perspective here for the sake of a more expressive gesture, a common practice in the Renaissance.

Annunciation Leonardo Di Vinci

Leonardo's most famous perspective painting, like all the other Last Suppers, this one is placed in a refectory or dining hall, here in the Convent of Santa Maria delle Grazie, Milan. The vanishing point is placed at Christ's right eye, where he dominates the foreground. Even Christ's arms reinforce the perspective, with his arms along the lines of the visual pyramid.

Di Vinci’s Last Supper

Escher’s Ascending and Descending

Escher’s Relativity

A much more important issue is the realism with which artists can draw on a flat piece of paper what they perceive when they look out at their 3-dimensional world. If one looks at attempts at scene representation in Egyptian and Mesopotamian art, one sees that phenomena that are associated with the human vision system are not always respected. We are all familiar with the fact that objects that are far away from us appear smaller than they actually are and that lines which are parallel appear to converge in the distance. These features, which are a standard part of the way that 3-dimensional objects are now usually represented on a planar surface, were not fully understood before the Renaissance. It is common to refer to artists as using "perspective" (or "linear perspective") to increase the realism of their representations. The issues and ideas involved in understanding perspective are quite subtle and evolved over a long time.

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

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

A variety of people whose names are known to mathematicians (but perhaps not to the general public) have contributed to a theory of perspective. Though every calculus student knows the name of Brook Taylor (1685-1731)for his work on power series, how many mathematicians know that Taylor wrote on the theory of linear perspective? On the other hand every art historian will recognize the name of Piero della Francesca (c. 1412-1492), yet how many of these art historians (or mathematicians) will be familiar with his contribution to mathematics? Similarly, Girard Desargues (1591-1661) is a well known name to geometers for his work on projective geometry (in plane projective geometry there are no parallel lines), but few people involved with art are familiar with his work. The diagram below (a portion of a "Desargues Configuration"), familiar to students of projective geometry, can be thought of as a plane drawing of an "eye" point viewing triangles which lie in different planes.

Of all the topics in the Tratatto, we are mainly interested in perspective. Leonardo had said that "perspective is the rein and rudder of painting." Invented by Brunelleschi, codified by Alberti and Piero, it was perfected by Leonardo.Slide 15-25: The Adoration of the Magi, 1481 Reti p. 224

Terme’s Gargoyles

Tesselations

Tesselations in Nature

Tilings

Alhambra

A palace and fortress complex located in Granada, Andalusia, Spain. It was originally constructed as a small fortress in 889 and then largely ignored until its ruins were renovated and rebuilt in the mid-11th century by the Moorish emir Mohammed ben Al-Ahmar of the Emirate of Granada, who built its current palace and walls. It was converted into a royal palace in 1333 by Yusuf I, Sultan of Granada.

The Alhambra tiles are remarkable in that they contain nearly all, if not all, of the seventeen mathematically possible wallpaper groups. This is a unique accomplishment in world architecture. M.C. Escher’s visit in 1922 and study of the Moorish use of of the plane.

Maurits Cornelis Escher 17 June 1898 – 27 March 1972), usually referred to as M. C. Escher, was a Dutch graphic artist. He is known for his often mathematically inspired woodcuts, lithographs, and mezzotints. These feature impossible constructions, explorations of infinity, architecture and tessellations.

He was born in Leeuwarden, Friesland. In 1903, the family moved to Arnhem, where he attended primary school and secondary school until 1918.He was a sickly child, and was placed in a special school at the age of seven and failed the second grade. Although he excelled at drawing, his grades were generally poor. He also took carpentry and piano lessons until he was thirteen years old. In 1919, Escher attended the Haarlem School of Architecture and Decorative Arts in Haarlem. He briefly studied architecture, but he failed a number of subjects (partly due to a persistent skin infection) and switched to decorative arts.

M C Escher

M.C. Escher

Rotate 180 about midpoint of the third side.

Wall paper

Egyptian Design Cloth - Tahiti

Ornamental painting Painted porcelain - China

A wallpaper group (or plane symmetry group or plane crystallographic group) is a mathematical classification of a two-dimensional repetitive pattern, based on the symmetries in the pattern. Such patterns occur frequently in architecture and decorative art. There are 17 possible distinct groups.

Wallpaper groups categorize patterns by their symmetries. Subtle differences may place similar patterns in different groups, while patterns that are very different in style, color, scale or orientation may belong to the same group.Consider the following examples:

Example A: Cloth, Tahiti Example B: Ornamental paintingExample C: Painted porcelain, China

Examples A and B have the same wallpaper group; it is called p4mm in the IUC notation and *442 in the orbifold notation. Example C has a different wallpaper group, called p4mg or 4*2 . The fact that A and B have the same wallpaper group means that they have the same symmetries, regardless of details of the designs, whereas C has a different set of symmetries despite any superficial similarities.

A proof that there were only 17 possible patterns was first carried out by Evgraf Fedorovin 1891 and then derived independently by George Polya in 1924.

Why there are exactly seventeen groups. An orbifold can be viewed as a polyhedron with face, edges, and vertices, which can be unfolded to form a possibly infinite set of polygons which tile either the sphere, the plane or the hyperbolic plane. When it tiles the plane it will give a wallpaper group and when it tiles the sphere or hyperbolic plane it gives either a spherical symmetry group or Hyperbolic symmetry group. The type of space the polygons tile can be found by calculating the Euler characteristic, χ = V − E + F, where V is the number of corners (vertices), E is the number of edges and F is the number of faces. If the Euler characteristic is positive then the orbifold has an elliptic (spherical) structure; if it is zero then it has a parabolic structure, i.e. a wallpaper group; and if it is negative it will have a hyperbolic structure. When the full set of possible orbifolds is enumerated it is found that only 17 have Euler characteristic 0.When an orbifold replicates by symmetry to fill the plane, its features create a structure of vertices, edges, and polygon faces, which must be consistent with the Euler characteristic. Reversing the process, we can assign numbers to the features of the orbifold, but fractions, rather than whole numbers. Because the orbifolditself is a quotient of the full surface by the symmetry group, the orbifold Euler characteristic is a quotient of the surface Euler characteristic by the order of the symmetry group.

The seventeen groups[edit]Each of the groups in this section has two cell structure diagrams, which are to be interpreted as follows:

Rhombus a centre of rotation of order two (180°).

Equilateral triangle a centre of rotation of order three (120°).

Square a centre of rotation of order four (90°).

Regular hexagon a centre of rotation of order six (60°).

an axis of reflection.

an axis of glide reflection.

Crystallographic short and full names

Short p2 pm pg cm pmm pmg pgg cmm p4m p4g p6m

Full p211 p1m1 p1g1 c1m1 p2mm p2mg p2gg c2mm p4mm p4mg p6mm

Here are all the names that differ in short and full notation.

The remaining names are p1, p3, p3m1, p31m, p4, and p6.

Fractals

A fractal is a natural phenomenon or a mathematical set that exhibits a repeating pattern that displays at every scale. If the replication is exactly the same at every scale, it is called a self-similar pattern. An example of this is the MengerSponge. Fractals can also be nearly the same at different levels. Fractals also includes the idea of a detailed pattern that repeats itself. Fractals are different from other geometric figures because of the way in which they scale. Doubling the edge lengths of a polygon multiplies its area by four, which is two (the ratio of the new to the old side length) raised to the power of two (the dimension of the space the polygon resides in). Likewise, if the radius of a sphere is doubled, its volume scales by eight, which is two (the ratio of the new to the old radius) to the power of three (the dimension that the sphere resides in). But if a fractal's one-dimensional lengths are all doubled, the spatial content of the fractal scales by a power of two that is not necessarily an integer. This power is called the fractal dimension of the fractal, and it usually exceeds the fractal's topological dimension.

Fractals in nature

FernCrocuses

Fractals in art and architecture

Indian and Southeast Asian temples and monuments exhibit a fractal structure: a tower surrounded by smaller towers, surrounded by still smaller towers.

Orient Station, Lisbon

Hindu Temple

Cubism

In Cubist artwork, objects are analyzed, broken up and reassembled in an abstracted form—instead of depicting objects from one viewpoint, the artist depicts the subject from a multitude of viewpoints to represent the subject in a greater context

Cubism is an early-20th-century avant-garde art movement that revolutionized European painting and

sculpture, and inspired related movements in music, literature and architecture. Cubism has been considered

the most influential art movement of the 20th century.

The movement was pioneered by George Braque and Pablo Picasso, joined by Jean Metzinger, Albert Gleizes,

Robert Delaunay, Henri Le Fauconnier, Fernand Leger and Juan Gris. A primary influence that led to Cubism

was the representation of three-dimensional form in the late works of Paul Cezanne. A retrospective of

Cézanne's paintings had been held at the Salon d’Automne of 1904, current works were displayed at the 1905

and 1906 Salon d'Automne, followed by two commemorative retrospectives after his death in 1907. In Cubist

artwork, objects are analyzed, broken up and reassembled in an abstracted form—instead of depicting objects

from one viewpoint, the artist depicts the subject from a multitude of viewpoints to represent the subject in a

greater context.

Early Futurist paintings hold in common with Cubism the fusing of the past and the present, the representation

of different views of the subject pictured at the same time, also called multiple perspective, or simultaneity, while

Constructivism was influenced by Picasso's technique of constructing sculpture from separate elements. Other

common threads between these disparate movements include the faceting or simplification of geometric forms,

and the association of mechanization and modern life.

Gray Tree, 1911

Pablo Picasso, Seated Nude, 1910