Jun 172014


Today is the birthday (1898) of Maurits Cornelis Escher, usually referred to as M. C. Escher, Dutch graphic artist. He is best known for his often mathematically inspired woodcuts, lithographs, and mezzotints. These feature impossible constructions, explorations of infinity, architecture, and tessellations.

Escher was born in Leeuwarden, Friesland, in a house that forms part of the Princesseof Ceramics Museum today. He was the youngest son of civil engineer George Arnold Escher and his second wife, Sara Gleichman. 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. He briefly studied architecture, but he failed a number of subjects (partly due to a persistent skin infection) and switched to decorative arts. He studied under Samuel Jessurun de Mesquita, with whom he remained friends for years. In 1922, Escher left the school after having gained experience in drawing and making woodcuts.

In 1922, an important year of his life, Escher traveled through Italy (Florence, San Gimignano, Volterra, Siena, Ravello) and Spain (Madrid, Toledo, Granada). He was impressed by the Italian countryside and by Alhambra, a fourteenth-century Moorish castle in Granada. The intricate decorative designs at Alhambra, which were based on geometrical symmetries featuring interlocking repetitive patterns sculpted into the stone walls and ceilings, were a powerful influence on Escher’s works.

mce alhambra

In Italy, Escher met Jetta Umiker, whom he married in 1924. The couple settled in Rome where their first son, Giorgio (George) Arnaldo Escher, named after his grandfather, was born. Escher and Jetta later had two more sons: Arthur and Jan.

In 1935, the political climate in Italy (under Mussolini) became unacceptable to Escher. He had no interest in politics, finding it impossible to involve himself with any ideals other than the expressions of his own concepts through his own particular medium, but he was averse to fanaticism and hypocrisy. When his eldest son, George, was forced at the age of nine to wear a Ballila uniform in school, the family left Italy and moved to Château-d’Œx, Switzerland, where they remained for two years.

Escher, who had been very fond of and inspired by the landscapes in Italy, was decidedly unhappy in Switzerland. In 1937, the family moved again, to Uccle, a suburb of Brussels, Belgium. World War II forced them to move in January 1941, this time to Baarn, Netherlands, where Escher lived until 1970. Most of Escher’s better-known works date from this period. The sometimes cloudy, cold and wet weather of the Netherlands allowed him to focus intently on his work. For a time after undergoing surgery, 1962 was the only period in which Escher did not work on new pieces.

Escher moved to the Rosa Spier Huis in Laren in 1970, an artists’ retirement home in which he had his own studio. He died at the home on 27 March 1972, aged 73.

In his early years, Escher sketched landscapes and nature. He also sketched insects, which appeared frequently in his later work. His first artistic work, completed in 1922, featured eight human heads divided in different planes. Later around 1924, he lost interest in “regular division” of planes, and turned to sketching landscapes in Italy with irregular perspectives that are impossible in natural form.


Escher’s first print of an impossible reality was Still Life and Street, 1937. His artistic expression was created from images in his mind, rather than directly from observations and travels to other countries. Well known examples of his work include Drawing Hands, a work in which two hands are shown, each drawing the other; Sky and Water, in which light plays on shadow to morph the water background behind fish figures into bird figures on a sky background; and Ascending and Descending, in which lines of people ascend and descend stairs in an infinite loop, on a construction which is impossible to build and possible to draw only by taking advantage of quirks of perception and perspective.


He worked primarily in the media of lithographs and woodcuts, though the few mezzotints he made are considered to be masterpieces of the technique. In his graphic art, he portrayed mathematical relationships among shapes, figures and space. Additionally, he explored interlocking figures using black and white to enhance different dimensions. Integrated into his prints were mirror images of cones, spheres, cubes, rings and spirals. Escher was left-handed.

Although Escher did not have mathematical training—his understanding of mathematics was largely visual and intuitive—Escher’s work had a strong mathematical component, and more than a few of the worlds which he drew were built around impossible objects such as the Necker cube and the Penrose triangle. Many of Escher’s works employed repeated tilings called tessellations. Escher’s artwork is especially well liked by mathematicians and scientists, who enjoy his use of polyhedra and geometric distortions. For example, in Gravity, multicolored turtles poke their heads out of a stellated dodecahedron.


The mathematical influence in his work emerged around 1936, when he journeyed to the Mediterranean with the Adria Shipping Company. He became interested in order and symmetry. Escher described his journey through the Mediterranean as “the richest source of inspiration I have ever tapped.”  After his journey to the Alhambra, Escher tried to improve upon the art works of the Moors using geometric grids as the basis for his sketches, which he then overlaid with additional designs, mainly animals such as birds and lions.


His first study of mathematics, which later led to its incorporation into his art works, began with George Pólya’s academic paper on plane symmetry groups sent to him by his brother Berend. This paper inspired him to learn the concept of the 17 wallpaper groups (plane symmetry groups). Using this mathematical concept, Escher created periodic tilings with 43 colored drawings of different types of symmetry. From this point on he developed a mathematical approach to expressions of symmetry in his art works. Starting in 1937, he created woodcuts using the concept of the 17 plane symmetry groups.

In 1941, Escher summarized his findings in a sketchbook, which he labeled Regelmatige vlakverdeling in asymmetrische congruente veelhoeken (“Regular division of the plane with asymmetric congruent polygons”). His intention in writing this was to aid himself in integrating mathematics into art. Escher is considered a research mathematician of his time because of his documentation with this paper, in which he studied color based division, and developed a system of categorizing combinations of shape, color and symmetrical properties.


Around 1956, Escher explored the concept of representing infinity on a two-dimensional plane. Discussions with Canadian mathematician H.S.M. Coxeter inspired Escher’s interest in hyperbolic tessellations, which are regular tilings of the hyperbolic plane. Escher’s wood engravings Circle Limit I–IV demonstrate this concept. In 1959, Coxeter published his finding that these works were extraordinarily accurate: “Escher got it absolutely right to the millimeter.”

Escher was awarded the Knighthood of the Order of Orange Nassau in 1955. Subsequently he regularly designed art for dignitaries around the world.

In 1958, he published a book entitled Regular Division of the Plane, with reproductions of a series of woodcuts based on tessellations of the plane, in which he described the systematic buildup of mathematical designs in his artworks. He emphasized, “Mathematicians have opened the gate leading to an extensive domain.”


Overall, his early love of Roman and Italian landscapes and of nature led to his interest in the concept of regular division of a plane, which he applied in over 150 colored works. Other mathematical principles evidenced in his works include the superposition of a hyperbolic plane on a fixed 2-dimensional plane, and the incorporation of three-dimensional objects such as spheres, columns and cubes into his works. For example, in a print called Reptiles, he combined two and three-dimensional images. In one of his papers, Escher emphasized the importance of dimensionality and described himself as “irritated” by flat shapes: “I make them come out of the plane.”


Escher also studied topology. He learned additional concepts in mathematics from the British mathematician Roger Penrose (https://www.bookofdaystales.com/roger-penrose/). From this knowledge he created Waterfall and Up and Down, featuring irregular perspectives similar to the concept of the Möbius strip.


Escher printed Metamorphosis I in 1937, which was a beginning part of a series of designs that told a story through the use of pictures. These works demonstrated a culmination of Escher’s skills to incorporate mathematics into art. In Metamorphosis I, he transformed convex polygons into regular patterns in a plane to form a human motif. This effect symbolizes his change of interest from landscape and nature to regular division of a plane. His piece Metamorphosis III is wide enough to cover all the walls in a room, and then loop back on to itself.


After 1953, Escher became a lecturer at many organizations. A planned series of lectures in North America in 1962 was cancelled due to an illness, but the illustrations and text for the lectures, written out in full by Escher, were later published as part of the book Escher on Escher. 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 center and the edge of a circle.


Curl-up or Wentelteefje (original Dutch title) is a lithograph print by M. C. Escher, first printed in November 1951. This is the only work by Escher consisting largely of text. The text, which is written in Dutch, describes an imaginary species called Pedalternorotandomovens centroculatus articulosus, also known as “wentelteefje” or “rolpens”. He says this creature came into existence because of the absence in nature of wheel shaped, living creatures with the ability to roll themselves forward.


The creature is elongated and armored with several keratinized joints. It has six legs, each with what appears to be a human foot. It has a disc-shaped head with a parrot-like beak and eyes on stalks on either side.

It can either crawl over a variety of terrain with its six legs or press its beak to the ground and roll into a wheel shape. It can then roll, gaining acceleration by pushing with its legs. On slopes it can tuck its legs in and roll freely. This rolling can end in one of two ways; by abruptly unrolling in motion, which leaves the creature belly-up, or by braking to a stop with its legs and slowly unrolling backwards.

The word wentelteefje is Dutch for French toast, “wentel” meaning “to turn over”. Rolpens is a dish made with chopped meat wrapped in a roll and then fried or baked. “Een pens” means “belly”, often used in the phrase beer-belly.

There’s a good recipe for rolpens here:



Or there is the ultra-traditional version which resembles haggis in some ways – basically a stomach stuffed with meat and boiled. Recipe and images here:


However, I thought it would also be fun to follow along in the footsteps of my post on Mondrian (https://www.bookofdaystales.com/mondrian/), and create food that resembles Escher’s art.  This pizza appeals to me, taken from this site:



The problem is that this would be really hard to replicate at home (and kudos to the artist). It looks to me as if the designer created a cheese base, and then overlaid cut out pieces of pepperoni.

More promising is the use of cookie cutters with tessellating shapes. As seen in these websites:





mce11  mce10

It seems as if getting the cutters is easy enough. You need to make a cookie dough that keeps its shape while baking, and you need to make at least two contrasting colors. The last URL has a good recipe and an instructional video. Looks like a great deal of fun.



Aug 082013


Today is the birthday (1931) of Roger Penrose, mathematician, philosopher, and artist.  I am a big fan.  Some of you who read this blog regularly may wonder why I admire so many mathematicians; maybe this post will solve that puzzle.

Penrose was born in Colchester on the east coast of England, and is the brother of mathematician Oliver Penrose and of chess Grandmaster Jonathan Penrose. Penrose attended University College School and University College, London, where he graduated with a first class degree in mathematics. While an undergraduate he was already doing original research which he continued at Cambridge, taking his Ph.D. in 1958.

As a student in 1954, Penrose was attending a conference in Amsterdam when by chance he came across an exhibition of M.C. Escher’s work. Soon he was trying to conjure up impossible figures of his own and discovered the tri-bar – a triangle that looks like a real, solid three-dimensional object, but isn’t.

pen16   pen13

Together with his father, a physicist and mathematician, Penrose went on to design a staircase that simultaneously loops up and down. An article followed and a copy was sent to Escher. Completing a cyclical flow of creativity, the Dutch master of geometrical illusions was inspired to produce his two masterpieces.

penrose9  penrose7

In 1965, at Cambridge, Penrose proved that singularities (such as black holes) could be formed from the gravitational collapse of immense, dying stars. This work was extended by Stephen Hawking to create the Penrose–Hawking singularity theorems.  In 1969, he conjectured the cosmic censorship hypothesis. This proposes (rather informally) that the universe protects us from the inherent unpredictability of singularities (such as the one in the center of a black hole) by hiding them from our view behind an event horizon. Black holes have intense gravitational pull, constantly attracting matter towards their centers.  The event horizon is the boundary point beyond which nothing can escape this gravitational force. Hence we cannot know anything about what exists beyond this horizon because nothing, not even light, can escape to give us information. Proving this conjecture, and a stronger version which Penrose proposed 10 years later, are major outstanding problems within the field of general relativity.  Although the mathematics of these conjectures is beyond the comprehension of all but a few specialists, the general implications are easy enough to understand and have become part of popular culture.


Penrose is well known for his 1974 discovery of Penrose tilings. A tiling is a pattern of “tiles” (2-dimensional shapes) that can be arranged so that there are no spaces or overlaps on a flat surface. Squares and certain triangles are the simplest form of tiling (as you know from the tiles in your shower stall). Quilt patterns and tile patterns in mosques are more complicated tilings. Penrose tilings have two properties. (1) They are formed using only two different shapes of tiles. (2) They are aperiodic, meaning that you cannot copy an area of the tiling on to tracing paper and then shift the paper to another area of the tiling and have it match.  In simple, slightly inaccurate, layman’s terms, the local patterns do not repeat. They appear in nature in what are known as quasicrystals, and have become the inspiration for graphic design artists.

First Penrose Tiling

First Penrose Tiling



Mosque Tiles

Mosque Tiles



He was influential in popularizing what are commonly known as Penrose diagrams (causal diagrams). You see them a lot on the white boards in “The Big Bang Theory.”  Don’t worry, if you do not understand them; I guarantee the actors don’t have a clue what they mean either.  Incidentally, in 2010, Penrose reported possible evidence, based on concentric circles found in WMAP data of the CMB sky (don’t sweat it!), of an earlier universe existing before the Big Bang of our own present universe.


Penrose has written books on the connexion between fundamental physics and human (or animal) consciousness. In The Emperor’s New Mind (1989), he argues that the known “laws” of physics are inadequate to explain the phenomenon of consciousness.  I’ve read it at least five times.  And now perhaps you get it why so many of my heroes are mathematicians.  Admiration from afar.

In honor of Penrose tilings I thought a recipe involving cooking on a tile would be appropriate.  You can cook anything on a tile that you would grill.  Fish is especially good cooked this way. Cooking on slate tiles is common in rustic cooking in France and Spain. The traditional method, called pierrade, involves heating a thick slab of slate over an open fire, but nowadays there is a modern tabletop version using an electrically heated tile on which diners select from a platter of raw meat and cook it to their own tastes on the tile. Yawn.


What I give you here is less about a specific recipe, and more a description of the method which you can play with. You will need two things: a good hot open fire and a thick piece of slate (or bluestone). When I lived in the Catskills in New York State, I had an outdoor fire pit what was no more than a rectangular wall of cement building blocks two blocks high, with several of the blocks on the bottom layer placed sideways so that the holes provided a draft for the coals.  Nothing pleased me more, summer and winter, than to build a roaring hard wood fire, let it burn down to coals and then cook away in every possible manner. You name it; I’ve cooked it in that pit: toast, whole pig, beef stew, baked apples, scrambled eggs.  One memorable day when my son was about 8 yrs old we spent the entire day by that pit cooking breakfast, lunch, and dinner there, with s’mores (roasted marshmallow and chocolate sandwiched between graham crackers) and various things on a stick as snacks in between.

For tile cooking I used a big slab of bluestone (feldspathic sandstone) which was readily available from local quarries.  Slate is more universally available.  You can get large pavers in home improvement stores in the U.S. and Europe. You just have to be sure that they are untreated; the thicker the stone, the better. They all eventually crack, but thicker ones last longer.  First thing you need to do is make sure the tile is thoroughly clean (each time you cook with it). Do not use soap, just lots of water and a heavy brush.  Build a good bed of coals evenly spread, leaning the slate nearer and nearer to the fire to heat gradually. If placed cold directly over the coals it will crack.  Using oven mits, place the tile on a grate or fire irons directly over the coals, 6” away.  Brush the cooking surface with olive oil or cooking oil. It should take about 30 minutes to get the tile ready for cooking. A drop of water placed on the surface should dance and skitter. You are now ready to cook fish, steak, chicken, or vegetables in the same manner as you would grill them.  The tile adds a wonderful earthy flavor. Here’s a favorite of mine: herbed fish with lemon. Any firm white fish will do. I’m partial to river trout given that my house was on a trout stream.


Tiled Fish with Herbs and Lemon


4 whole fish gutted and scaled (about 1 lb each).

Lemon slices

1 cup fresh fines herbes (fresh parsley, chives, tarragon and chervil).  You can used dried, in which case you need 2 tablespoons in total.

kosher salt

black pepper

olive oil


Rinse the fish and pat dry. Season with salt and pepper inside and out. Place ¼ of the fresh herbs inside the cavity of each fish and then slot in two or three lemon slices. Lightly oil both sides of the fish.

Place the fish in the center of the tile and cook undisturbed for about 10 minutes per side, or until opaque, but still moist in the thickest part.

Serves 4.

As an accompaniment I usually cut open baking potatoes, place a knob of butter in each, and then wrap them tightly in foil. They can be cooked in the coals. Start them about 15 minutes before the fish.  Whilst eating the fish you can make baked apples by coring cooking apples, filling the cored hole with butter, brown sugar and sweet spices such as cloves or allspice, wrapping in foil, and cooking in the coals.  These usually take no more than 20 minutes to cook if the coals are still hot.