escher esxhibition: student's workbook (4º eso)

ART & MATHS : M.C. ESCHER Bilingual Cross-Curricular Activity - 4° ESO

Belen Pena & Luis M. Rodriguez Departamentos de Historia y Matematicas

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INTRODUCTION

ESCHER 'S ARTWORK : ART & MATHS

RESOURCES

VIDEOS (SPANISH)

FURTHER READINGS

ACTIVITIES:

ART ASSIGNMENTS HISTORY

MATHS ASSIGNMENTS MATHEMATICS

OTHER RESOURCES:

DOWNLOAD FROM IES ALBAYZIN WEBSITE

REFERENCES

AKNOWLEDGEMENTS



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Mauritis Cornelis Escher (1898-1972) is one of the world´s most famous graphic artists and

an outstanding example of such an artist and designer.

He was never a good student during his childhood days. After failing his high school exams,

Mauritis (nicknamed Mauk) was enrolled in the School for Architecture and Decorative Arts in

Haarlem; after one week, he informed his father that he would study graphic art instead of

architecture.

Although Escher occasionally produced watercolours and sculptures, he was first and

foremost a graphic artist. During his lifetime, he mad 448 lithographs, woodcuts and wood

engravings and over 2000 drawings and sketches.

Apart from this, he also illustrated books, designed tapestries, postage stamps and murals.

And in 1958, he published an illustrated book entitled Regular Division of the Plane, with

reproductions of a series of woodcuts based on tessellations of the plane.

In July 1969 he finished his last work, a woodcut called Snakes, in which snakes wind

through a pattern of linked rings which fade to infinity toward both the centre and the edge of

a circle.

Escher’s work shows how art can be enhanced by math, and vice versa. Despite having no

formal training in Mathematics, Escher created artwork that followed certain mathematical

principles, and was influenced by developments in science and mathematics. His works

included exploration of the three dimensional world, perspective, abstract mathematical

solids, approaches to infinity and also tessellations (arrangements of closed, regular and

irregular, shapes that completely cover the plane without overlapping and leaving gaps).

While certain schools of art history may not teach Escher as a great artist, his

popularity gives him enormous educational leverage to teach topics such as photo-realism,

hyper realism, optical art, lithography, illustration, and surrealism.

So, creating highly imaginative artwork that marries the world of art and mathematics,

he would become a famous artist.



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Escher didn´t like being called an artist; he preferred to be known as a graphic artist

because printmaking fascinated him.

The printmaking methods he used most often were mezzotint (one intaglio

technique), linocut, wood engraving and woodcut (these three ones are relief techniques), and

lithography (a planographic technique). But he worked primarily in the media of lithographs

and woodcuts; the few mezzotints he made are considered to be master pieces of the

technique.

Judging from the complexity of some of his prints, there can be little doubt that he was

indeed a master of his craft.

In his graphic art, he portrayed mathematical relationships among shapes, figures and

space. A large part of his popularity is due to his depictions of impossible worlds, and his

prints on this theme are based on his research into perspective (the system to represent depth

on a flat surface).

But undoubtedly, tessellations was Escher´s primary interest. His fascination with

tessellations began when he briefly visit the Alhambra in 1922. He was impressed by the

Moorish tilings decorating the Alhambra and the way in which the geometric figures on the

plane were repeating.

It was only during the second visit in 1936 that he began a more serious and

theoretical study into tessellations. And so, the Alhambra became one of the sources of

inspiration for Escher´s designs. Unlike the Islamic designs, Escher´s tilings resemble

recognizable objects, usually living beings.

It is quite common to describe Escher’s works with regard to mathematics. Thus, we can

easily find on Internet many descriptions of his artwork where the role of mathematics is

pointed out or even emphasized.

“Maurits Cornelis Escher, who was born in Leeuwarden, Holland in 1898, created

unique and fascinating works of art that explore and exhibit a wide range of

mathematical ideas”.1

1 http://www.mathacademy.com/pr/minitext/escher/index.asp



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“His engravings and drawings have been admired for years by many artists, mathematicians

and intellectuals throughout the world”.2

But, why is mathematics so important to a better understanding of Escher’s works?

In order to clarify this relationship we can try to answer this tricky question by focusing on

either the artist or the artwork. If we first consider the artist, we will realize mathematics is

essential mainly in two different ways:

As a tool

To represent the world, his world: continuous, dual and dialectic (black

and white, good and evil, limit and unlimited), relativist, cyclic, magic,

staggering, bewildering.

To represent the conflict between reality and imagination.

To represent the mystery of perception.

As a magic structure under reality

“The laws of mathematics are not merely human inventions or creations. They

simply ‘are;’ they exist quite independently of the human intellect. The most

that any(one) ... can do is to find that they are there and to take cognizance of

them.”

M.C. Escher

If we now concentrate on the second focus of attention to answer the initial question we will

find two remarkable aspects, too.

Escher’s artwork has been considered by some mathematicians as a way of visualizing

difficult mathematics concepts or principles.

Escher’s artwork is full of mathematics objects and references: Regular Polyhedra,

Archimedean Solids, Regular Division of the Plane (Reflections, Translations and

Rotations), Topological objects (Möbius Strip), Non-Euclidean and Fractal Geometry,

Perspective laws….

2 M.C.Escher Infinite Universes leaflet. Parque de las Ciencias de Granada.



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1) EXPOSICIÓN M.C. ESCHER. UNIVERSOS INFINITOS Español. Visita

obligada. Consultar la biografía (http://www.eschergranada.com/es/mc-

escher/biografia ); explica toda la exposición en un repaso de sus obras, agrupadas

más por temas que por épocas ( aquí tienes los enlaces de cada sección:

Representación figurativa /Paisajes naturales y artificiales, el Mediterráneo/

Metamorfosis/ Cruce de mundos/ Formas tridimensionales y matemáticas/

Geometrización del plano/ Perspectivas y arquitecturas ). Dedica además un apartado

especial a su relación con la Alhambra y a sus visitas a España.

http://www.eschergranada.com/es

2) M.C. ESCHER - INFINITE UNIVERSES. La exposición en inglés – muy útil

para responder tus tareas y preparar la visita. http://www.eschergranada.com/en.

Enlaces a las secciones: Natural and Artificial Landscapes. The Mediterranean/

Metamorphosis / Crossing Worlds / Three-dimensional and Mathematical Forms /

Geometrisation of the Plane / Perspectives and Architectures

3) Escher mindscapes - National Gallery of Canada Presentación flash muy

buena en lo referente a los temas y técnicas. Base de partida para estos apartados en

nuestra actividad, así como para el estudio d las citas. Muy recomendable galería de

imágenes agrupadas por etapas cronológicas, no temáticas (Cada obra incluye

nombre, fecha y técnica empleada…). Atrévete a escuchar los vídeos con las

entrevistas realizadas al autor. Inglés.

http://cybermuse.gallery.ca/cybermuse/youth/escher/home/home01_e.jsp

4) The Official M.C. Escher Website. Breve biografía. Visitar galería de

imágenes –por etapas cronológicas-. Inglés. http://www.mcescher.com/

5) M. C. Escher – Wikipedia. Para toma de contacto inicial. Español:

http://es.wikipedia.org/wiki/Maurits_Cornelis_Escher

6) M. C. Escher – Wikipedia. Para toma de contacto inicial. Algo más completa e

interesante que la versión española. Inglés:

http://en.wikipedia.org/wiki/M._C._Escher

http://www.eschergranada.com/

http://www.eschergranada.com/es/mc-escher/biografia

http://www.eschergranada.com/es/mc-escher/biografia

http://www.eschergranada.com/es/exposicion/ambitos/itemlist/category/7-ambito-1







http://www.eschergranada.com/es

http://www.eschergranada.com/en



http://www.eschergranada.com/es/exposicion/ambitos/ambitos/itemlist/category/14-ambito-4




http://www.mcescher.com/

http://www.mcescher.com/

http://es.wikipedia.org/wiki/Maurits_Cornelis_Escher

http://en.wikipedia.org/wiki/M._C._Escher



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7) M.C. Escher Brief Biography Biografía muy detallada, por épocas de su

vida. Inglés. http://users.erols.com/ziring/escher_bio.htm

8) Biography of M.C. Escher biografía completa, algo más sencilla que la anterior.

Inglés. http://im-possible.info/english/articles/escher/escher.html

9) Mini-biografía de M.C. Escher | Microsiervos (Arte y Diseño) brevísima

biografía e introducción al análisis de su obra. Español.

http://www.microsiervos.com/archivo/arte-y-diseno/biografia-mc-escher.html



nuestra actividad, así como para el estudio d las citas. Muy recomendable galería de





11) Técnicas gráficas –español. Tipo diccionario. http://tecnica.z0ro.com/index.htm.

Ver en el apartado técnicas una clasificación global

http://tecnica.z0ro.com/tecnica.htm

12) Printmaking – Wikipedia inglés. Básico para conocer y diferenciar las técnicas

de impresión , así como algunos de los más importantes aristas impresores agrupados

por técnicas para comparar con Escher. La versión en español no es más completa y

puede induciros a confusión y malinterpretaciones, más que aclararos; pero tiene un

cuadro de artistas muy completo. http://en.wikipedia.org/wiki/Printmaking

13) Grabado – Wikipedia español. Básico para conocer de qué hablamos en lo

referente a técnicas de grabado y tipos http://es.wikipedia.org/wiki/Grabado

14) Engraving – Wikipedia inglés. Básico para obtener ideas claras y un vocabulario

esencial antes de moverte en otras páginas de consulta (te interesa hasta el apartado

3.1) http://en.wikipedia.org/wiki/Engraving

15) Apuntes sobre técnicas y tecnología del grabado. Español. Materias

primas, procedimientos, técnicas… Exhaustivo pero no complicado, interesante para

aclarar y diferenciar conceptos, con cierto nivel.

http://www.uchile.cl/cultura/grabadosvirtuales/apuntes/grabado.html#1.2

16) What is a print. Moma presentación flash muy buena para entender y explorar

las técnicas. Incluye un vocabulario breve y completo; para cada técnica una galería de

imágenes útil para estudiar otros artistas y sobre todo, no olvides apretar la flecha de

comenzar, se abre una de las más claras y amenas explicaciones de cada proceso

http://users.erols.com/ziring/escher_bio.htm

http://im-possible.info/english/articles/escher/escher.html

http://www.microsiervos.com/archivo/arte-y-diseno/biografia-mc-escher.html


http://tecnica.z0ro.com/index.htme

http://tecnica.z0ro.com/tecnica.htm

http://en.wikipedia.org/wiki/Printmaking

http://es.wikipedia.org/wiki/Grabado

http://en.wikipedia.org/wiki/Engraving

http://www.uchile.cl/cultura/grabadosvirtuales/apuntes/grabado.html#1.2



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artístico, además fácil (=Press arrow to start). Más que recomendable, obligada.

Inglés. http://www.moma.org/interactives/projects/2001/whatisaprint/print.html



nuestra actividad, así como para el estudio de las citas. Muy recomendable galería de





18) Perspectiva - Wikipedia. Ideas claras y tipos de perspectivas. Evolución en el

tiempo. Español http://es.wikipedia.org/wiki/Perspectiva

19) Perspective visual- Wikipedia inglés (más simple y con menos información) http://en.wikipedia.org/wiki/Perspective_%28visual%29

20) Perspective graphical- Wikipedia inglés. Muy recomendable: definición e

ideas claras; historia y evolución, obras teóricas, explicación de temas y esquemas de

visuales muy útiles. Recomendable.

http://en.wikipedia.org/wiki/Perspective_%28graphical%29

21) Aproximación a la Perspectiva Conceptos esenciales de perspectiva, en

formato diccionario; útil para definir y aclarar términos usados en las actividades y

páginas de consulta. Español.

http://www.imageandart.com/tutoriales/morfologia/perspectiva.htm

22) Tipos de perspectiva Una revisión sencilla y clara de la perspectiva a lo largo de

la Historia del Arte. Muy resolutiva para algunas actividades. Español.

http://www.profesorenlinea.cl/artes/Perspectiva_Tipos.htm

23) La perspectiva lineal. Una ventana abierta al mundo. Sencilla y

elemental introducción a la perspectiva. Español.

http://aprendersociales.blogspot.com/2009/02/la-perspectiva-lineal.html

24) Perspectiva en la pintura Breve y concisa, pero completa visión de la

perspectiva y su evolución en la historia del arte. Español

http://www.cossio.net/actividades/pinacoteca/p_02_03/perspectiva.htm

25) Perspective Muy bueno para aclarar tipos y para tratar brevemente periodos, en

especial en artistas modernos. Inglés.

http://www.op-art.co.uk/history/perspective.php

http://www.moma.org/interactives/projects/2001/whatisaprint/print.html


http://es.wikipedia.org/wiki/Perspectiva

http://en.wikipedia.org/wiki/Perspective_%28visual%29

http://en.wikipedia.org/wiki/Perspective_%28graphical%29

http://www.imageandart.com/tutoriales/morfologia/perspectiva.htm

http://www.profesorenlinea.cl/artes/Perspectiva_Tipos.htm

http://aprendersociales.blogspot.com/2009/02/la-perspectiva-lineal.html


http://www.op-art.co.uk/history/perspective.php



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26) Perspective Drawing Lessons muy bueno para diferenciar lineal de aérea;

para introducción a la geometría en arte y para ejemplos de perspectiva en función de

la posición del espectador. Inglés, no demasiado fácil.

http://www.artyfactory.com/perspective_drawing/perspective_index.htm

27) Anamorfosis – Wikipedia breve y claro. Interesantes diagramas y lista de

autores que la han usado en sus obras. Ingles

http://en.wikipedia.org/wiki/Anamorphosis

28) Anamorfosis – Wikipedia En español ; demasiado técnico. Procura visitar la

página dedicada a Julian Beever. http://es.wikipedia.org/wiki/Anamorfosis.

29) Perspective http://www.artgraphica.net/free-art-lessons/wetcanvas/basic-

perspective-for-artists/basic-perspective-for-artists.htm

Ver el contenido convertido en Presentación de Power Point 1, 2 & 3 point

perspective.pptx (descárgalo de esta página web del IES: (Download:

http://www.iesalbayzin.org/index.php/departamentos-didacticos/65-dpto-geografia-e-historia)

30) Mathematics and art. para comprender la estrecha relación entre el arte y las

matemáticas, en especial en lo referido a la proporción y perspectiva, de Grecia a

nuestros días. Inglés. http://en.wikipedia.org/wiki/Mathematics_and_art

31) The mathematical art of M.C. Escher. Una completa y sencilla introducción

a los temas matemáticos que más interesan a Escher . Inglés.

http://www.mathacademy.com/pr/minitext/escher/index.asp

32) The mathematics behind the art of M.C. Escher. Una página muy Buena

para trabajar y comprender las teselaciones en la obra de Escher. Aborda la relación

entre el autor y la Alhambra. Unas excelentes animaciones para comprender los

procesos geométricos de creación de teselaciones. Inglés.

http://www.math.nus.edu.sg/aslaksen/gem-projects/maa/0203-2-03-

Escher/main.html#Introduction

33) Polyhedra and Art. Una página donde podrás rastrear la relación entre el arte y

los poliedros a lo largo de la historia. De especial interés el enlace hacia la página de

Escher donde se comenta uno de sus grabados “Stars”. Inglés.

http://www.georgehart.com/virtual-polyhedra/art.html

34) Archimedean Solids. Se trata de una página en la que podrás informarte acerca

de lo que es un poliedro arquimediano. Este tipo de poliedros fueron utilizados por

Escher en algunas de sus obras. Inglés. http://en.wikipedia.org/wiki/Archimedean_solid

http://www.artyfactory.com/perspective_drawing/perspective_index.htm

http://en.wikipedia.org/wiki/Anamorphosis

http://es.wikipedia.org/wiki/Julian_Beever

http://es.wikipedia.org/wiki/Anamorfosis

http://www.artgraphica.net/free-art-lessons/wetcanvas/basic-perspective-for-artists/basic-perspective-for-artists.htm

http://www.artgraphica.net/free-art-lessons/wetcanvas/basic-perspective-for-artists/basic-perspective-for-artists.htm

http://www.iesalbayzin.org/index.php/departamentos-didacticos/65-dpto-geografia-e-historia

http://en.wikipedia.org/wiki/Mathematics_and_art

http://www.mathacademy.com/pr/minitext/escher/index.asp

http://www.math.nus.edu.sg/aslaksen/gem-projects/maa/0203-2-03-Escher/main.html#Introduction

http://www.math.nus.edu.sg/aslaksen/gem-projects/maa/0203-2-03-Escher/main.html#Introduction

http://www.georgehart.com/virtual-polyhedra/art.html

http://en.wikipedia.org/wiki/Archimedean_solid



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35) Impossible Constructor (1.25). Aquí podrás descargarte un sencillo programa

para crear tus propias figuras imposibles. Se trata de un programa que trabaja por

medio de cubos en perspectiva y que genera formas como la siguiente:

http://imp-world.narod.ru/programs/index.html

http://www.youtube.com/watch?v=HVvdAiPrvk8&feature=related

http://www.youtube.com/watch?v=fiDfFfR108U&feature=related

http://www.youtube.com/watch?v=lN1zBjIVF68&feature=related

http://www.youtube.com/watch?v=HVvdAiPrvk8

http://www.youtube.com/watch?v=h9eqeBsNMBo

36) El espejo mágico de M.C. Escher. Bruno Ernst. Ed.: Evergreen, Köln 1994.

Este libro es un clásico en la bibliografía sobre Escher. Se trata de un texto autorizado

por el propio Escher, muy válido para conocer las opiniones del propio artista.

http://imp-world.narod.ru/programs/index.html

http://www.youtube.com/watch?v=HVvdAiPrvk8&feature=related

http://www.youtube.com/watch?v=fiDfFfR108U&feature=related

http://www.youtube.com/watch?v=lN1zBjIVF68&feature=related

http://www.youtube.com/watch?v=HVvdAiPrvk8

http://www.youtube.com/watch?v=h9eqeBsNMBo



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MC Escher. Art & Maths. Art Assigments ∞Resources

(Download: http://www.iesalbayzin.org/index.php/departamentos-didacticos/65-dpto-geografia-e-historia )

A. With the biographical data which appear in SLIDES 3 to 6, design a timeline about MC

Escher life. Incorporate the information about the chronological periods (SLIDE 7) in

order to organize them. Identify the timeline date with a specific colour related to

Escher artistic evolution.

For further information visit the websites on your WORKBOOK (in sections

Generales and Vida y Obra): Resources number 1, 2, 3, 4 and 7.

“What is really so fascinating about graphic processes? What is that strange power of attraction that keeps its hold on the graphic artist? There are, I believe, three elements that are an inherent part of this fascination: 1. desire for multiplication; 2. beauty of the craft; 3. forced limitations resulting from the technique."

(Escher in Escher, 1989)

MC Escher. Art & Maths. Art Assigments Resources


A. Observe the two pictures included in SLIDE 9. In both of them you can see Escher

working in his workshop. Try to find out which technique is employed in each image or

in which stage of the artistic process is it in.

Visit the websites in section Técnicas de Grabado e Impresión in your WORKBOOK.

I especially recommend you (in order of preference) numbers 16, 17 (English) and 11,

15 (Spanish).





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A. Play close attention to the SLIDES in section Escher´s Tecniques from a to f (SLIDES 10

to 15). Every image show a different printmaking technique used by Escher

(exemplified by different Works) with the exception of letters e and f which refer to

the same technique. Can you identify the represented technique in each SLIDE?

See your WORKBOOK websites for further information numbers 1 and 4 (look for

the Picture Gallery). Number 3 can also be useful although it may need some more

time to be loaded.

B. Design a comparative table of the 5 graphic printmaking techniques more frequently

used by Escher. Show in each technique the year in which it was invented, the matrix

used (the material used for the engraving), the media (material on which it is printed),

the tools which had been used for the engraving and a brief summary of the process.

You will find especially useful the website (Flash presentation) What Is a Print from the

MOMA.

You may find further information in the websites of your WORKBOOK numbers 12,

14 and 15.

Read the article What´s Perspective by Jim Elkins (see below, at the end of activity 12,

) and answer the following

questions:

A. Define the words which appeared highlighted in turquoise.

B. In the tradition of Western Art, which period, work and author can we claimed the first

linear perspective?

C. Over which elements of representation has no power linear perspective?

D. According to the majority of art critics, who destroys the concept and use of

perspective in the 20th Century? What’s the article writer’s opinion?

E. Value from 1 to 10 the difficulty of the text considering 1 the minimum and 10 the

maximum. Indicate if your evaluation comes from text references and artistic

vocabulary, to the difficulty of vocabulary in general, to the difficulty of understanding

the expressions and sentence structures. How many words did you have to look up in

the dictionary (not including those in activity A)?




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In the twentieth century artists began to play with perspective by drawing impossible

objects. These objects included stairs that always go up or cubes where the back meets the

front. Such works were popularized by artist M. C. Escher and mathematician Roger Penrose.

Although referred to as impossible objects such objects as the Necker Cube and the Penrose

triangle can be built using anamorphosis. When viewed at a certain angle such sculptures

appear as the so-called impossible objects. http://en.wikipedia.org



A. Look for SLIDE 17 and identify the name which corresponds to each of the represented

images. Why are they called Impossible Objects?

B. The image of stairs is recurrent in Escher’s work (see SLIDE 18). Name at least 4 works

from Escher (title and date) in which the stairs theme is emphasized.

- Are they common examples of representing a stair? Why? In your opinion, what is

the sense of this recurrent theme?

C. It is commonly assumed the influence of G.B. Piranesi (discovered in his Italian

period) in Escher´s artwork

-Elaborate a brief file on Piranesi (Time-space location, artistic style, techniques and

main Works)

- Observe the Piranesi image on SLIDE 18. Which Escher’s work does it remind you?

Which technical element do they share?


(Download: http://www.iesalbayzin.org/index.php/departamentos-didacticos/65-dpto-geografia-e-historia)

A. In SLIDE 20 (Picture Gallery 1) there are three photographs showing linear

perspective. Which differences can you notice? Analyse the vanishing points, the

horizon line and the viewer’s point of view (also eye or dark point, is the place

from which the image is observed. It would be as the viewer’s eye)

B. Image 1 represents The Last Supper by Leonardo da Vinci. Observe the perspective

diagram. Which is the vanishing point? Where does it go to? Does it refer to a

symbolic or spatial necessity? Reason your answer out.

Visit the websites in section Perspectiva in your WORKBOOK. I especially

recommend you (in order of preference) numbers 20, 26 (English) and numbers

22, 18, 25 (Spanish).





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C. Analyse and compare the images in SLIDE 21 (Picture Gallery 2). What kind of

perspective do they show? What do they have in common and which differences

can you notice? Reason your answers out.


recommend you (in order of preference) numbers 29, 26 (English) and number 22

y 18 (Spanish).

La realidad tiene tres dimensiones, alto, ancho y profundo, pero un cuadro sólo dos, lo alto

y lo ancho. Este es el principal problema de todo pintor: ¿ cómo conseguir dar la ilusión de

profundidad en un cuadro?. La respuesta es mediante el engaño a nuestro sentido de la vista.

Es en el Renacimiento cuando los pintores florentinos comienzan a investigar en serio la

perspectiva como una ciencia, con sus leyes y sus principios matemáticos. Mantegna, Ghiberti,

Massaccio y otros establecieron ciertos principios necesariamente observables para reproducir

la distancia. Estos principios fueron posteriormente perfeccionados por Leonardo, Miguel

Angel, Giorgione y Rafael.

Pero, ¿cuáles son los engaños necesarios para lograr la tridimensionalidad en un plano?

Fue Leonardo, precisamente, quien en su "Tratado de la pintura" definió a la perspectiva como

la "ciencia de las líneas de la visión", dividiéndola en tres partes: lineal, de color y menguante.

Perspectiva lineal. El cuadro se estructura como si mirásemos una pirámide desde dentro

de su base. Vemos así un punto de fuga imaginario al fondo sobre el que convergen una serie

de líneas de fuga, a veces imaginarias y a veces reales (pavimentos, techos, personajes, etc.)

Perspectiva menguante. A medida que aumenta la distancia, disminuye la nitidez, los

contornos se van haciendo borrosos y desdibujados, al igual que ocurre en la realidad.

Perspectiva de color. En este caso, cuanto más lejos aparece representado un objeto, más

tenues son sus colores. Existe también en el mundo real un desvaimiento de los tonos al

aumentar la lejanía. (Vemos las montañas azules desde lejos).

Además de estas tres perspectivas generales hay otros recursos añadidos para subrayar la

tridimensionalidad como, por ejemplo, el punto de vista alto (perspectiva caballera) que

aumenta el campo visual y por tanto la sensación de profundidad. También la alternancia de

planos iluminados y otros en penumbra; o un fondo ilimitado e infinito; o disminuir el tamaño

de los objetos progresivamente según se alejan del espectador, etc.

A la perspectiva que toma en consideración las tres citadas anteriormente se la suele

conocer como perspectiva aérea.





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SLIDE 22 – PICTURE GALLERY 3 SLIDE 23 - PICTURE GALLERY 4

1 MANTEGNA 1 VELÁZQUEZ

2 PIERO della FRANCESCA 2 RAFAEL de SANZIO

3 MASACCIO 3 PONTORMO

SLIDE 25 – PICTURE GALLERY 6 SLIDE 26 - PICTURE GALLERY 7 SLIDE 27- PICTURE GALLERY 8

1 JAN van EYCK 1 MASTER of FLÉMALLE 1 JAN van EYCK

2 ROGER van der WEYDEN 2 MASTER of FLÉMALLE 2



A. The Works which are showed in SLIDE 22 belong the great Italian painters of the

Renaissance, who are pioneer in linear perspective. Which elements do these Works

have in common from the perspective point of view? Which differences can you notice

among them? You have to analyze the vanishing points, the horizon lines and the eye

view.

B. Look for and define in Spanish the words escorzo and perspectiva caballera. Explain

their relationship with the Works in this slide.

C. SLIDE 23 shows three different ways of representing the 3D space in the picture.

Identify the Works, period and style they belong to and the type of perspective which

has been used in each case.

D. In SLIDE 24 you can see Las Meninas by Velázquez. Explain in Spanish how Velázquez

creates the space in the painting and what kind of perspective he uses. Interpret and

explain the coloured lines on top. Which spatial concept does each color correspond

to?

E. SLIDE 25, 26 Y 27 (Picture Gallery 6, 7 y 8) belong to the Early Flemish Painters School

(Northern Renaissance). These painters will also be interested in the 3D space

representation on the plane and they will approach its study from different premises

to those used by the Italian painters and, therefore, with different results.

- Analyse in SLIDES 25 and 26 the elements on which the author leans to create space

and depth in the picture (they can be architectural, furniture, wall decoration, floor,

ceiling…)

- Locate the vanishing points and the horizon line. Is it the same perspective as the one

used by the Italian painters? If not, identify which type does it belong to. Reason your

answer out.

F. SLIDE 27 represents a famous picture by Jan Van Eyck. Which one are we referring to?

Observe the orthogonal lines and locate the vanishing point. What kind of perspective

do you find? Does the element in the middle of the picture coincide with the main




16

vanishing point? Which symbolic value is given to the main vanishing point in this

work? What does it represent?

La derisoire effervescence des comprimes by

Francois Boucq


A. Visit the PPT Presentation Perspective Comic by F. Boucq in the IES Albayzín website

and answer the following questions:

1. In SLIDE 6 you can read: “perspective is an obsolete conception of the

representation of reality. … nobody is interested in perspective anymore”. Do you

think this is a true sentence in contemporary art? Reason your answer out.

2. In SLIDE 6 you can read: “… I have been a Sunday afternoon painter for twenty-five

years and, for me, perspective is much more than a simple technique, it is

something connected with universal laws!!...” . What does the expression Sunday

afternoon painter mean? Do you think it has a pejorative sense? Why? Which are

the “universal laws” the author refers to?

3. Why is there a connection between perspective and the expression Sunday

afternoon painter, and the lack of perspective and artistic creation in

contemporary art?

4. Which three basic rules of perspective are implicitly mentioned in the dialogues in

SLIDE 7?

5. Mr. Ferdinand, the man with the plough, says in SLIDE 9 “I can´t draw

hyperboloids”. Which type of perspective does he refer to? Justify your answer.

6. In SLIDE 16 you can read: “… I wanted to send the rules of perspective to hell and

that’s why the bananas are hard to recognize”. Now look at the picture with the

bananas: What is the meaning of this sentence?

7. Try to find images in the Presentation with perspectives which are equivalent to

those on pictures 2, 3 and 4 in SLIDE 20 of the PPT Presentation MC Escher. Art

& Maths. Art Assigments Resources

Curvilinear perspective involves the representation of space using vanishing curves

rather than vanishing lines. As these curves seem to converge at both ends, the horizontal and

vertical transversals create two vanishing points each with a fifth created by the orthogonals

parallel to the direction of view. Hence the name five point perspective or spherical perspective




17

for some of these projection systems.

Historical Uses of Curvilinear Perspective: From the 16th to the 20th centuries,

perspective theorists explored the problem of anamorphic or geometrically distorted images,

which can be rectified back to a normal perspective image when viewed using an appropriately

curved mirror. These studies often overlapped with the problems of projective distortions in

two dimensional images, specifically the difference in perspective view straight ahead and the

view obliquely to either side.

FROM: http://www.handprint.com/HP/WCL/tech10.html#index

Curvilinear perspective is a graphical projection used to draw 3D objects on 2D surfaces. It

was formally codified in 1968 by the artists and art historians André Barre and Albert Flocon in

the book La Perspective curviligne.

In 1959, Flocon had acquired a copy of Grafiek en tekeningen by M. C. Escher who strongly

impressed him with his use of bent and curved perspective, which influenced the theory

Flocon and Barre were developing. They started a long correspondence, in which Escher called

Flocon a "kindred spirit".

Examples of approximated (not necessarily systematically constructed, but emulated

through an empirical method) five-point perspective can also be found in several mannerist

paintings such as the famous self-portrait of Parmigianino seen through a shaving mirror as

well as in the curved mirror in Jan van Eyck's Arnolfini's Wedding.

http://en.wikipedia.org

A. In SLIDE 28 you can find three examples of Escher’s interest in curvilinear perspective

which he would develop in his hyperbolic tessellations (=regular tilings of the

hyperbolic plane) together with his mathematician friend Coxeter, an expert in

Hyperbolic Geometry. This interest in curvilinear perspective is not new in Art History.

We have chosen Jan van Eyck and Parmigianino and in picture 1 in SLIDE 26 you can

see a work by Master of Flémalle. Can you identify these three pictures? Which style or

artistic school do the authors belong to?

- Look for information and write an abstract on Parmigianino’s work and his interest in

the distortion of curvilinear perspective.

B. Try to find some other examples (five at least) of curvilinear perspective in Escher’s

work apart from those which appear in this Slide.

Anamorphosis is a distorted projection or perspective requiring the viewer to use special

devices or occupy a specific vantage point to reconstitute the image. There are two main types

of anamorphosis: Perspective (oblique) and Mirror (catoptric).

Examples of perspectival anamorphosis date to the early Renaissance (15th Century)

Examples of mirror anamorphosis occurred at the time of the late Renaissance (16th

Century).

http://www.handprint.com/HP/WCL/tech10.html#index

http://en.wikipedia.org/wiki/Mannerism

http://en.wikipedia.org/wiki/Parmigianino

http://en.wikipedia.org/wiki/Jan_van_Eyck



18

During the 17th century, Baroque trompe l'oeil murals often used this technique to

combine actual architectural elements with an illusion. When standing in front of the art work

in a specific spot, the architecture blends with the decorative painting.

Cinemascope, Panavision, Technirama and other widescreen formats use anamorphosis to

project a wider image from a narrower film frame. http://en.wikipedia.org

Anamorphic images have been distorted so that they appear flat or undistorted (veridical)

when they are (a) viewed from a direction that is not perpendicular to the image plane;

(b) viewed in a curved mirror or other highly reflective object; or (c) painted on a curved or

faceted surface (i.e., the image plane is not a plane). http://en.wikipedia.org



Hans Holbein the Younger is well known for incorporating this type of anamorphic trick.

His painting The Ambassadors is the most famous example for anamorphosis (mirror

anamorphosis), in which a distorted shape lies diagonally across the bottom of the frame.

Viewing this from an acute angle transforms it into the plastic image of a skull.

C. Search for information on the Internet about the meaning of the skull in Holbein’s

picture.

D. Which Escher’s work shows a skull reflected on a naturally curve surface?

E. Escher uses the perspectival anamorphosis , in the Baroque tradition of trompe l'oeil.

What does trompe l ´oeil mean (explain in Spanish)?. Find out an Escher’s work

showing this visual stunt. You can find at least two examples included in the Art

Resources PPT Presentation-.

- How can you explain the anamorphosis (visual trick) in each example you have

chosen?.


recommend you number 27 (English).

Este recurso del cuadro dentro del cuadro utilizado ya antes del Renacimiento convierte al

cuadro incluido en un objeto tan real como el resto de objetos que forman parte de la obra

total. No hay distinción alguna entre la realidad y la ilusión, entro lo viviente y lo pintado. A

partir del siglo XVI se le va a añadir un atributo más a dicho recurso: el cuadro incluido

consigue que el espectador lo destaque del resto de la imagen representada.

http://vailima.blogia.com/2004/111101-el-cuadro-dentro-del-cuadro.php




19

“….otro procedimiento, mucho más sutil, de introducir un espacio dentro de otro, un exterior

en un interior: el del cuadro dentro del cuadro, es decir, la presencia en la pared del fondo de

un interior de otro cuadro colgado, que representa algo fuera de él. Tal sistema comprende

tres soluciones, (…) a veces difíciles de distinguir: el cuadro (o tapiz) colgado sobre el falso

muro pintado; el hueco abierto en este muro, puerta o ventana a otra estancia o al aire libre;

en fin, el espejo, que introduce en el espacio fingidamente real del cuadro lo que se halla

frente a él (…).En los tres casos nos hallamos ante una derivación de la veduta (…). Esta

práctica alcanza entre los españoles del Siglo de Oro, especialmente en Velázquez, tal

perfección….”

Julián Gállego, Visión y Símbolos en la Pintura Española del Siglo de Oro, 1984



A. Escher uses this resource in his work. Find some examples of it in his production

(some of them are in this presentation). Do you think this resource is connected with

his interest in representing infinite universes? In which way? Reason your answer out.

B. In SLIDE 30 (Picture Gallery 11) there are some examples of Spanish painters. One of

them is Velázquez. Who is the author of image 1?

C. Find in this presentation some other Works by Velázquez of a picture within a picture.

Explain the within picture and read carefully the attached text by Julián Gállego.

En la crítica literaria, negación del significado

normal de los conceptos y favorecimiento de otro tipo de interpretaciones

SLIDE 31 – PICTURE GALLERY 12 SLIDE 32 - PICTURE GALLERY 13 SLIDE 33- PICTURE GALLERY 14

1 CEZANNE 1 MIRO 1 MONDRIAN

2 VAN GOGH 2 KIRCHNER 2 TURNER

3 VAN GOGH 3 PICASSO 3 CEZANNE

4 CEZANNE 4 ROTHKO 4 GAUGIN

5 5 BRAQUE 5




20



Observe carefully the Works which are included in SLIDES 31 to 33 (Picture Gallery 12

to 149). Design a table with the works in a chronological order. Mention the author,

style or artistic period of the work and name of it.

Do you think the rules of linear perspective are fulfilled in these Works? Justify your

answer. Which works maintain a certain concept of linear perspective in these pictures

(Write number and author)?.

Pay close attention to the four Works in SLIDE 31 y numbers 2, 3 and 4 in SLIDE 33

(Picture Gallery 12 and 14). Analyse the type of representation of the space which has

been used by the authors. Consult the table in SLIDE 19.

Which of these Works represent the elimination of perspective? Reason your answer

out. Does Cubism mean the elimination or the intensification of the concept of 3D in

painting? Justify your answer.


recommend you (in order of preference) numbers 25 and 20 (English) and numbers

18 y 22 (Spanish).

Another form of anamorphic art is often called "Slant Art." Examples are the sidewalk chalk

paintings of Kurt Wenner, Manfred Stader and Julian Beever where the chalk painting, the

pavement and the architectural surroundings all become part of an illusion.



A. What is the meaning of epigone? In which sense can we name these chalk artists as

epigones of Escher? What is the relationship between Escher’s art and these Street

artists / sidewalk chalk painters?








21

A. In this section (SLIDES 35 to 41) some quotes have been selected on two main

subjects: the relationship between Escher and Maths and Escher and his artistic

creation. Choose one relevant quotation for each subject. Justify your choice and the

meaning of the quotation within the artistic production of Escher (SLIDE 35 with

Spanish quotations cannot be chosen for this exercise).

FROM: www.artic.edu/aic/

http://www.artic.edu/aic/education/sciarttech/2d.html

What Is Perspective?

Simple one-point perspective

drawing

The Chemistry and

Physics of Color

What Is Perspective?

The Basics of Perspective Linear perspective is a mathematical system for projecting the three-dimensional world

onto a two-dimensional surface, such as paper or canvas. In brief, this type of

perspective begins with a horizon line, which defines the farthest distance of the

background and a central vanishing point. To this vanishing point, orthogonals may be

drawn from the bottom of the picture plane, which defines the foreground of the space.

The orthogonals, vanishing point, and horizon line establish the space in which the artist

may arrange figures, objects, or architecture such that they appear to exist in three

dimensions. Once these basic elements have been set in place, the artist may add further

elements to create a more complicated, yet more realistic, space. For example, to

http://www.artic.edu/aic/index.php






http://www.artic.edu/aic/education/sciarttech/index.html

http://www.artic.edu/aic/education/sciarttech/2d1.html



22

represent a square-tiled floor, the artist chooses another point on the horizon line, called

the distance point, and draws a line through the orthogonals to a point at the bottom of

the picture plane. The points at which this line bisects the orthogonals establish the

points at which horizontal lines, called transversals, may be placed. These lines

represent the perspectively correct regression of the square tiles into space (see

diagram). These elements of linear perspective link the science of three-dimensional

geometry with the art of illusionistic representation.

Art-Historical Meanings of Perspective In the 13th and 14th centuries, before linear perspective was discovered, artists

occasionally employed something called reverse perspective, in which parallel lines

splay rather than converge as they approach the horizon line. One of the rules set forth

in an early artists' manual is that elements above the eye of the viewer tend downward

(like roofs), while elements below the viewer’s eye tend upward (like tables). While

arbitrary tilting of lines upward and downward can create unusual effects, this is

generally considered to be a significant step in the progression toward the rational

application of linear perspective.

It was not until the Renaissance that artists began to refine this science. Linear

perspective soon emerged as the tool for artists to capture the world around them in a

remarkably illusionistic manner (this was the same time that cartographers were

mapping the surface of the earth using a similar system of mathematical projection).

Masaccio's (1401–28) Trinity (1427–28), considered to be the first accurately

perspectival painting in the Western tradition, introduced the

relationship between linear perspective and subject matter in art.

The painting is divided into three levels: the figure of God

stands on a tomb above and behind Christ crucified on the cross;

the Virgin Mary and St. John the Baptist stand at the base of the

cross; and two donors, who commissioned the painting, kneel on

either side of the cross at the lowest level. The figures are

harmoniously organized underneath a barrel-vaulted ceiling. The

figure of Christ and the barrel vault are rendered as if seen from

below. The figures of Mary and John are rendered again as if

from below, but just lower than Christ. The two donors, given

the lowest vanishing point, are rendered as if seen, directly in

front of the viewer’s eye. In this way, the painting provides the

illusion that the viewer is looking at sculptural forms that exist in three dimensions and

rise vertically in space. Yet the accurate, illusionistic representation of space in this

work has no obvious theological meaning. It has much more to do with the artist

exhibiting his skill—injecting his own thoughts into this religious image.

Unusual Kinds of Perspective Some artists of the Renaissance were not as concerned with putting their subjects in

perfect perspective as they were with making religious statements. For example, in

pictures that include the Virgin Mary the vanishing point is often intentionally placed on

Mary's womb to indicate her place as the mother of Christ. This placement of the

vanishing point has religious significance and may not be related to the intention to



23

create a rational perspectival space. Further, if the artist chooses a short distance

between the distance point and the vanishing point then the perspective will appear

warped. The distance between the distance point and the vanishing point should, in

theory, correspond exactly to the distance between the viewer’s eye and the picture

plane. When the distance between the points is small, the viewer must place his or her

eye at this same distance from the painting directly in front of the distance point in order

to see the work with no distortion whatsoever. When the viewer stands back from the

work, the space in the image will appear distorted.

Anamorphosis (from the Greek, “something without form”) involves stretching an

ordinary linear-perspective image in one or more directions to obscure its original form.

To achieve this, the artist draws a grid over the original image and then translates the

image point by point to a grid that has been stretched. If the viewer looks at the image

directly, it appears formless and amorphous. In order to recognize the image, the eye of

the viewer must be positioned from a particular spot, generally off to the side, and from

this point the image appears in linear perspective.

Curvilinear perspective is an alternate to linear perspective. Although technically all

straight lines are curved, curvilinear lines are suppressed in Western painting—that is,

straight lines are represented as straight rather than arced. In the 19th century, a group

of artists made an attempt to return curvilinear perspective to painting, but the idea was

short lived because it presented a philosophical problem. When observing lines in the

real world, such those of as walls and buildings, the lines appear curved. (Think of

standing in front of a long wall, and looking left and right: The top of the wall seems to

curve up from either side.) It follows that a wall in a painting, drawn with straight lines,

can also seem curved. Therefore, if those curves are represented in painting they will

seem doubly curved. This tension between reality and the representation of reality in

painting posed a challenge to the painters who employed this technique.

Perspective and Nature There are some elements of representation over which linear perspective has no power,

such as landscapes, faces, and organic forms, for this type of perspective only describes

linear things. In his Landscape with Saint John on Patmos (1640), Nicolas Poussin

(1593/4–1665) used linear perspective to

demonstrate his knowledge of geometry. In the

ruins in the foreground, a cube, a three-

dimensional rectangle, and a cylinder are

represented in perfect perspectival form, each

illustrating a different aspect of foreshortening.

Yet the landscape in which the objects and the

figure of Saint John exist does not adhere to the

rules of linear perspective. The pathway recedes

into the distance giving an illusion of depth, but

nature twists and turns and rises and falls in its unpredictable, organic way.



24

Modern Art and Linear Perspective Since the Renaissance, painters have reworked and refined linear perspective. The

American 19th-century realist Thomas Eakins (1844–1916) created remarkably accurate

outdoor scenes, with shadows painted so precisely that art historians have been able to

determine, based on their knowledge of where the works

were painted, the exact date and time of day he painted them.

Some critics have argued that perspective was destroyed by

modern artists such as Pablo Picasso in the early-20th

century. In works such as his portrait of Daniel-Henry

Kahnweiler (1910), Picasso sought to break up the picture

plane and divide the forms into individual geometric pieces.

Yet early modern artists did not actually overthrow

perspective; they borrowed from it, elaborated upon it, and

redefined it for the viewer. That linear perspective is still

very much a part of representation today is evident in video

games, which employ the most exact perspective in the

Western tradition. In computer software, all figures and

objects are drawn using a perfect geometric grid. Regardless of the vantage point from

which the player views figures in a game, all figures, objects, and elements in the

settings adhere to the established rules of representation. Perspective is both an exacting

art and an exacting science that is still very much all around us.

Adapted from a lecture titled “What Is Perspective?” by Jim Elkins.

www.jameselkins.com.

http://www.philamuseum.org/exhibitions/exhibits/eakins/index.shtml

http://www.artic.edu/aic/education/sciarttech/lecturers.html

http://www.jameselkins.com./



25

Along the Grain = al hilo, a fibra Anamorphosis = anamorfosis

Atmospheric Perspective= perspectiva aérea

Background = fondo

Barrel-Vault = bóveda de cañón Canvas = lienzo Donors = donante, comitente End-Grain Wood

Engrave = grabado Etching = aguafuerte, grabado al ácido

Foreground = primer plano Foreshortening = acortar

Horizon Line = línea del horizonte Intaglio = talla dulce o grabado a buril (huecograbado)

Linear Perspective Linocut = linoleografía

Lithography= litografía Mezzotint = mezzotinta, manera negra o inglesa

Planographic = planográfico Reverse Perspective

Tiles = azulejos , baldosas Tiling = alicatar, enlosar, poner azulejos

Vanishing Point = punto de fuga Wood Engraving = xilografía (a testa, a contrafibra)

Woodcut = xilografía (a fibra, al hilo)



26

Escher was very interested in visual aspects of Topology, a branch of mathematics just coming

into full flower during his lifetime. The Möbius strip is perhaps the prime example, and Escher

made many representations of it. It has the curious property that it has only one side, and one

edge. Thus, if you trace the path of the ants in Möbius Strip II, you will discover that they are

not walking on opposite sides of the strip at all – they are all walking on the same side.

Escher’s Topological Images

Balcony

Print Gallery

Möbius Strip II

Print Gallery, (grid-paper sketch)



27

Another very remarkable lithograph, called Print Gallery, explores both the logic and the

topology of space. Here a young man in an art gallery is looking at a print of a seaside town

with a shop along the docks, and in the shop is an art gallery, with a young man looking at a

print of a seaside town . . . but wait! What's happened?

All of Escher's works reward a prolonged stare, but this one does especially. Somehow, Escher

has turned space back into itself, so that the young man is both inside the picture and outside

of it simultaneously. The secret of its making can be rendered somewhat less obscure by

examining the grid-paper sketch the artist made in preparation for this lithograph. Note how

the scale of the grid grows continuously in a clockwise direction. And note especially what this

trick entails: A hole in the middle. A mathematician would call this a singularity, a place where

the fabric of the space no longer holds together. There is just no way to knit this bizarre space

into a seamless whole, and Escher, rather than try to obscure it in some way, has put his

trademark initials smack in the centre of it.

As we have seen Escher was very interested in Topology. Now we will try to clarify the

meaning of this branch of mathematics.

Topological Equivalence

Someone once said that topologist is a person who does not know the

difference between a doughnut and a coffee cup. Two geometric figures are

said to be topologically equivalent if one figure can be elastically twisted

(torcida), stretched (estirada), bent (doblada), or shrunk (encogida) into the

other figure without puncturing (perforar) or ripping (rasgar) the original

figure. If a doughnut is made of elastic material, it can be stretched, twisted,

bent, shrunk, and distorted until it resembles a coffee cup with a handle, as

shown in the picture below.

In topology, figures are classified according to their genus. The genus of an

object is determined by the number of holes that go through the object. A

cup and a doughnut each have one hole and are of genus 1 (and are therefore

topologically equivalent). Notice that the cup handle is considered a hole, whereas the

opening at the rim of the cup (borde de la taza) is not considered a hole.



28

The following chart illustrates the genus of several objects.

Marble. Genus 0

Doughnut.

Genus 1

Strainer. Genus 3 or more.

Bowling ball.

Genus 0

Coffee cup.

Genus 1

Kettle. Genus 2

Scissors. Genus 2



29

Activity 1:

Give the name and the genus of the following objects. If the object has a genus larger than 5,

write “larger than 5”.

Name:

Genus:

Name:

Genus:

Name:

Genus:

Name:

Genus:

Name:

Genus:

Name:

Genus:

Name:

Genus:

Name:

Genus:

Name:

Genus:

Activity 2: Jordan Curves

A Jordan Curve is a topological object that can be thought of as a circle twisted out of shape.

Like a circle, it has an inside and an outside. To get from one side to the other, at least one line

must be crossed. Consider the following Jordan curve; are points A and B inside or outside the



30

curve? Could you establish a general rule to know whether a point is inside or outside the

Jordan curve?

Activity 3: Möbius Strip

If you place a pencil on one surface of a sheet of paper and do not remove it from the sheet,

you must across the edge to get to the other surface. Thus, a sheet of paper has one edge and

two surfaces. The sheet retains these properties even when crumpled into a ball. The Möbius

strip, also called a Möbius band, is a one-sided, one-edged surface. You can construct one by:

a) Taking a strip of paper

b) Giving one end a half twist

c) Taping the ends together



31

The Möbius strip has some very interesting properties. To better understand these properties,

perform the following experiments.

Experiment 1: Take a sheet of paper, a strip of paper and construct a paper ring as shown

in the picture.

Could you tell how many edges and how many sides these different surfaces have?

Surface Number of edges Number of sides

Sheet of paper

Strip of paper

Ring of paper

Hints:

How to count the edges: Start colouring an edge at one point with your felt-tip pen, if you

colour the entire edge and never have to lift the pen from the paper then the paper has one

edge. A pointy vertex does not divide an edge into two parts.

How to count the sides: Start colouring one side, fill it with colour but don't cross over any

sharp edges. When you are done, one side will be coloured the other will not. So, the strip has

2 sides.

A simpler way to test for the number of sides is to draw a line along one side. If any point can

be reached from the line without crossing an edge then that point is on the same side as the

line. Draw a line on one side of the paper, points on the other side cannot be reached without

crossing an edge, this means the paper has two sides.



32

Experiment 2: Make a Möbius strip using a strip of paper and tape as illustrated above.

Check a Möbius band is a one-sided, one-edged surface.

Experiment 3: Make a Möbius strip. Use scissors to make a small slit in the middle of the

strip. Starting at the slit, cut along the strip, keeping the scissors in the middle of the strip.

Continue cutting and observe what happens.

Experiment 4: Make a Möbius strip. Make a small slit at a point about one-third of the

width of the strip. Cut along the strip, keeping the scissors the same distance from the edge.

Continue cutting and observe what happens.

Impossible Constructions are two-dimensional shapes whose equivalent three-dimensional

constructions are impossible. This means we can draw these impossible constructions on a

sheet of paper but we can’t build a three-dimensional model of them.

How can such a disturbing event like that happen?

Explaining this strange event involves to think of perspective rules. Two-dimensional

impossible constructions are possible because when we draw them we break some of the

perspective rules. For instance:



33

Parallel line segments in 3D world are also parallel in their 2D representations, and the

other way round.

Every 2D representation of the 3D world is made using the same type of coordinate

trihedral.

Activity 1:

How could you interpret this coordinate trihedral? How many possibilities can you see?

Maybe if you look carefully at the following picture you will realize the different points

of view. To represent a trihedral in 2D is always ambiguous.

Juan Muñoz, Wasteland. 1986



34

Activity 2:

The following chart contains several 2D representations. Some of them are possible

constructions in a 3D world, others are not. Could you identify or classify the possible and

impossible constructions? Could you explain your answer according to the perspective rules

explained above?



35

Activity 3: “His fantastical structures, which couldn’t possibly exist in the real

world, are optical illusions that play with perspective.”

Could you recognize any of the critiria explained above to create impossible constructions in

these Escher’s artwork?

http://www.google.es/imgres?q=escher+images&hl=es&qscrl=1&nord=1&rlz=1T4SUNA_esES258ES283&biw=1280&bih=617&tbm=isch&tbnid=t5zIn3N4dKt50M:&imgrefurl=http://en.wikipedia.org/wiki/M._C._Escher&docid=_eenC7rIxRskPM&imgurl=http://upload.wikimedia.org/wikipedia/en/thumb/e/e8/Escher_Waterfall.jpg/250px-Escher_Waterfall.jpg&w=250&h=319&ei=OIa1TpqgFsm68gOSyOShBQ&zoom=1



36

In addition to being wonderfully engaging art, the work of Maurits Cornelius Escher also

displays some of the more beautiful and intricate aspects of mathematics. In 1936 Escher,

became obsessed with tessellations, that is, with creating art that used objects to cover a

plane so as to leave no gaps. Symmetry became a cornerstone of Escher’s

famous tessellations.

Escher kept a notebook in which he kept background information for his

artwork. In this notebook, Escher characterized all possible combinations

of shapes, colours and symmetrical properties of polygons in the plane.

By doing so, Escher had unwittingly developed areas of a branch of

mathematics known as crystallography years before any mathematician

had done so!!

These pictures have been created by Escher using the rules of transformational geometry. One of the targets of this activity will be to discover that rules.

3 This activity has been developed using different materials selected from the following bibliographic sources: AAVV, A

Survey of Mathematics with Applications, Eighth Edition (Pearson Education, 2009). Recursos digitales de la Editorial ANAYA

(digital.com).



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We will now introduce a type of geometry called transformational geometry. In

transformational geometry we study various ways to move a geometric figure without altering

the shape or size of the figure. When discussing transformational geometry, we often use the

term rigid motion.

The act of moving a geometric figure from some starting position to some ending position

without altering its shape or size is called a rigid motion (or transformation).

When discussing rigid motion of two-dimensional figures, there are four basic types of rigid

motions: Reflections, Rotations, Translations, and Glide Reflections. We call these four types of

rigid motions the basic rigid motions in a plane.

A reflection is a rigid motion that moves a geometric figure to a new position such that (tal

que) figure in the new position is a mirror image of the figure in the starting position. In two

dimensions, the figure and its mirror image are equidistant from a line called the reflection

line or the axis of reflection.

A translation (or glide) is a rigid motion that moves a geometric figure by sliding (deslizar) it

along a straight line segment in the plane. The direction and length of the line segment

completely determine the translation. A concise way to indicate the direction and the distance

that a figure is moved during the translation is with a translation vector.



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A rotation is a rigid motion performed by rotating a geometric figure in the plane about a

specific point, called the rotation point or the centre of rotation. The angle through which the

object is rotated is called the angle of rotation.

We will measures angles of rotation using degrees. In mathematics, generally,

counterclockwise angles have positive degree measures and clockwise angles have negative

degree measures.

A glide reflection is a rigid motion formed by performing a translation (or glide) followed by a

reflection.

As a summary of the basic rigid motion in a plane we can bear in mind the following image:



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Activity 1:

Construct the reflection of polygon ABCDE, shown in Figure a, about line e and the

reflection of polygon ABCD, shown in Figure b, about point O.

Activity 2:

Given the shapes shown in Figure a and b, and translation vector ⃗ , construct the translated

shapes (A’B’C’D’E’).

Activity 3:

Use the given figure and rotation point O to construct the indicated rotations

a) A 30º rotation of point A about point O.



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b) A 90º rotation of segment AB about point O.

c) A 30º rotation of trapezoid ABCD about point O.

If you compare this rigid motion to the one in the exercise 1b, what would you say

about it?

Activity 4:

Construct the reflection of polygon shown below about line e.



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Activity 5:

Construct the reflection of polygon shown below about point O.

Activity 6:

Given triangle OAB, where ( ) ( ) y ( ):

a) Plot all the points and draw the triangle.

b) Construct the translation of triangle OAB using ⃗ ( ) as a translation vector.

c) Determine the coordinates of the three vertices of triangle O’A’B’.

Activity 7:

John has to study this composition that he has found out in an art exhibition, could you help

him to answer the following questions?



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What rigid motion would you use to transform tile 1 into tile 2? And tile 1 into tile 3? And tile 1

into tile 6?

Activity 8:

Construct a glide reflection of square ABCD using vector ⃗ and reflection line e.

Activity 9:

Determine whether the following Escher’s work have been created using reflection,

translation, rotation or glide reflection.

You can use a tracing paper (a transparent sheet placed over the original) to answer

this question.



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http://www.moma.org/interactives/projects/2001/whatisaprint/flash.html



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Todas las fuentes de información usadas en la realización de este proyecto aparecen

recogidas en el apartado inicial de , en las notas de pie de página y en el

apartado correspondiente a .

Todas las imágenes empleadas en la elaboración de esta actividad interdisciplinar

(actividades o Presentaciones con Power Point) han sido tomadas de Internet, en su mayoría

proceden de las páginas web recomendadas.

Los autores de esta actividad quieren agradecer su inestimable colaboración a:

D. Rafael Moreno y los alumnos de T.I.C. de 2º de Bachillerato, por su cooperación

en la Presentación La derisoire effervescence des comprimes

D. Javier Paños, por proporcionar la idea y el material original base de dicha

presentación

D. Francisco Julio, por su asesoramiento y supervisión en la redacción final del texto

en inglés, y muy especialmente, por la traducción de las actividades correspondientes

al apartado Art Assignments



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Snakes (1969), last M.C. Escher´s artwork Woodcut in orange, green and black, printed from 3 blocks

escher esxhibition: student's workbook (4º eso)

Documents