Albrecht Altdorfer's paradox landscape

Few paintings can be safely said to be unique and original in as many ways as Albrecht Altdorfer’s “Battle of Issus” (1529). First, it could well be the biggest crowd ever depicted on a single work of art: the armies of Alexander the Great and the Persian king Darius are literally thousands of miniature human figures worked in exhaustive detail, forming smaller battle fronts here and there. The Greek warriors are shown dressed and armed like German knights on an Alpine landscape of mountains, castles and churches while, not surprisingly for a time when even Vienna was threatened by the Ottomans, the Persians are depicted as turbaned Turks. Alexander himself, in bright armour, is shown chasing Darius who is fleeing for his life.
It is another of Altdorfer’s novelties that various statistics of the battle appear on banners and shields, not very different than the basketball game statistics on our TV screens today. A huge tablet hangs high over the heads of the soldiers, against a magnificent, turbulent sky, bearing an inscription in Latin:
“The defeat of Darius by Alexander the Great, following the deaths of 100000 Persian soldiers and more than 10000 Persian horsemen. King Darius’ mother, wife and children were taken prisoner, together with about 1000 fleeing horse – soldiers.”
The whole scene is viewed from a “bird’s eye” view, offering a vast panorama of the land, the sea and the sky far in the distance. Issus was an ancient settlement close to present day Iskenderun, Turkey, not far from the Turkish – Syrian border. Altdorfer took much of his information on the eastern Mediterranean geography from “Schedelsche Welt Chronik” (Schedel’s World Chronicle, 1493), a work by the Nurnberg physician Hartmann Schedel. In this work Issus was identified with an area near Tarsus, some 100 km west of the Iskenderun area. The “Battle of Issus”, viewed as a map, is basically accurate. Altdorfer has paid particular attention in depicting the geography of the eastern Mediterranean region as faithfully as possible. The Mediterranean Sea, the island of Cyprus, the northern part of the African continent, the river Nile and its presumed sources deep in the horizon and finally the Red Sea are features easily recognizable in the painting.
Above the battlefield the huge, tumultuous sky and the cosmic, eternal battle between light and darkness makes the human battle below seem petty and almost comically microscopic. On the upper left part of the sky the lunar crescent can be seen while diametrically, in the lower right, where the west is supposed to be, the solar disk struggles with a vortex of clouds. The line of the horizon is obviously curved, denoting the round shape of the earth and making the composition resemble a satellite image: the “Battle of Issus” may thus be not only the first painting ever depicting a curved horizon but also the first painting ever depicting in a cosmic scale at the same time the Earth, the Moon and the Sun.
Much has been written about the meaning and symbolism of the “Battle of Issus”, the anachronisms and inaccuracies present in the picture (e.g. the presence of women in the midst of the battle), especially in the historical context of the time and in the context of the Altdorfer’s presumed own beliefs. However, from our mathematical or quantitative point of view, the painting presents also many small challenges for direct measurements and calculations and provides interesting information about Altdorfer’s understanding of the world and its dimensions. Several features of the composition are particularly striking when examined quantitatively from a scientific point of view and may even seem puzzling or paradoxical.
First, a question is raised about the relative sizes of objects, figures and geographical features as depicted in the painting. It seems extraordinary that ships sailing even south of Cyprus, halfway between Cyprus and the Nile, can be observed from the painter’s point of view, supposedly located in southern Asia Minor. Even more remarkable is the fact that the length of these ships, sailing behind the island of Cyprus, is about 1/30 of the apparent length of Cyprus itself. A clear indication about the distances as depicted in the painting is provided by the sizes of the solar and lunar disks. It seems that Altdorfer had not realized that the apparent size of the moon and the sun are almost identical: his sun’s diameter is almost half than his crescent’s diameter. It must be noted however that the crescent has the correct orientation with respect to the sun which shows that the painter had a good understanding of the relation between the two celestial bodies. Using the method described in a previous essay (“Sowing suns and reaping distances in Van Gogh’s paintings”, September 2008) and assuming that the biggest ship sailing south of Cyprus in the painting has an actual size of 30 m, the distance between the painter and the ship is estimated to be a little more than 13 km, which is anywhere between 2 – 11% of the actual distance. To put it differently, had Altdorfer depicted this distance correctly, the ship’s length should be anywhere between 270 m and 1500 m (note that the largest supertankers today have a length of 400 m approximately). Just as distances are greatly underestimated, heights are greatly exaggerated to make a more vivid and spectacular relief. It seems therefore that Altdorfer painted lengths and distances liberally and rather arbitrarily without any quantitative concern.
Second, it is questionable whether it is even possible for anyone to see so far and so deep in the horizon as Altdorfer suggests in his painting, regardless of how perfect her or his vision may be. The obvious reason is the curvature of the earth, the very curvature Altdorfer so nicely rendered in the horizon line. It is not difficult to prove that an observer standing at a height h meters from sea level can see a point on sea level only if the distance of this point from the observer does not exceed A=sqrt(h^2+2hR) meters, where R is the earth radius. Assuming that h=9000 m (approximately equal to the highest peak on earth) and R=6400 km, we get a distance A=340 km, which is the maximum distance an observer could possibly look into the horizon towards a point at sea level under perfect conditions on planet Earth (given that the highest peak would be appropriately located). In “Battle of Issus” the landscape stretches from Asia Minor to the Red Sea and beyond, at a distance of at least 1100 km and probably much more than that. The elevation in the Iskenderun region or the Tarsus area, where Issus is supposed to have been, does not exceed 3000 m by the most generous estimation and therefore the most distant visible point at sea level could not have been further away than 200 km. Though Altdorfer has made an effort to depict a round earth, the magnificent landscape view he offers is actually impossible or paradoxical and would correspond better to a cylindrical shaped planet, curved across one direction and flat across another, rather than a spherical one.
Third, the curvature of the horizon indicates a specific assumed size of the earth which can be calculated by relating the lengths of the corresponding chord and versine (i.e. the line segment drawn perpendicular to a chord, between the midpoint of that chord and the circumference of the circle). The horizon arc chord and its versine can be directly measured and their ratio q (chord/versine) is found to be approximately equal to 45. It is straightforward to calculate the arc length L of the horizon:

L=(2pi-4cotanq)R

where R is the earth’s radius, cotan is the inverse tangent function and pi=3.14159… is the well known ratio of the diameter over the circumference of any circle. Taking again R=6400 km, the arc length L is found to be approximately equal to 570 km. This is a result almost half than expected and certainly not enough to include the river Nile, the Red Sea and even large parts of land on the west and on the east at one picture. However, it can be considered a remarkable and delicate intuitive touch by the painter, within a reasonable degree of accuracy.
It seems that in the “Battle of Issus” Altdorfer had no concern whatever about the quantitatively accurate depiction of relative sizes, lengths and distances or about the plausibility of the offered landscape. From that point of view, scientific truth is not quantitatively present in the painting and is not one of the artist’s considerations, making thus the “Battle of Issus” a paradox image.

Sowing suns and reaping distances in Van Gogh's paintings

In 1888 Vincent Van Gogh painted two pictures that later became known as “the sower”. They both depict a lonely figure moving across a plowed field. A wide movement of the right hand indicates that sowing is taking place. However, the real protagonist of the scene is the sun in the horizon, a huge, godlike, bright disk flooding the sky with light, inverting colors and painting the sky yellow and the land violet.
Next year, while Van Gogh lived in an asylum for the mentally ill in Saint Remy, France, he painted two similar pictures that later became known as “the reaper”. It is not a rare instance to find celestial bodies in Van Gogh’s paintings, the sun, moon or stars, even turbulent galaxies. But the “sowers” and the “reapers” are exceptional: the viewer is left with the impression that the event of interest here is not the sowing or reaping that happens to take place on the foreground, but instead the cosmically important movement of the sun on the horizon. The presence of the human figure seems almost coincidental.
In a previous essay (“Leopold Bloom’s little finger”) I made the point that the sun, against our intuition and our impression from everyday experience, actually looks very small in the sky. This apparent size of the sun is expressed in degrees, varies slightly during the year and its measure is barely 0,533 degrees on average. Thus, the solar disk looks about the same as an “o” in these lines, when viewed from a distance of 25 cm, very different than Van Gogh’s suns.
So what exactly is that huge, bright disk on the “sowers” and the “reapers”?
In each painting (at least those not abstract enough so as to be completely disconnected from reality), two characteristic distances are relevant: the theoretical distance P of the painter from the subject and the distance V of the viewer from the painting. Once these two distances are in some way determined, the theoretical distance of the viewer from the subject is immediately implied.
In general, we are accustomed to consider the distance V of the viewer from the painting as arbitrary; however, this might not be always correct. Obviously, the size of the painting itself suggests to the viewer an acceptable range of distances at which he or she should be positioned. In some cases, a specific distance is imposed by special circumstances such as in Michelangelo Buonarotti’s “Cappella Sistina”: this magnificent masterpiece is painted on the ceiling, thus compelling the artist to adjust his work to a fixed distance from the viewer. In other cases, the painter himself, by the specific nature of his work, imposes a certain distance to the viewer. A good example is George Seurat’s “La Parade de Cirque”, where the image consists of a large number of small colored dots, only in primary colors. At close enough distance, the picture is unclear and obscure but at larger distances, the tiny dots appear to mix and compose a clear image, with a wide range of hues. Thus the mixing of the primary colors takes place in the eye and not in the palette. This is a technique similar to the modern printing techniques (mimicked by Roy Lichtenstein’s work).
In most cases, it is impossible to determine distances P and V with some accuracy, unless there are in the picture some kind of visual clues to be used as reference. It turns out that the solar or lunar disk can be such a visual clue, at least when the picture includes another object of known or assumed dimensions, e.g. a human figure. Taking in account that the apparent sizes of the sun and the moon are both approximately equal to 0,5 degrees, it is not difficult to prove that, in order to estimate the distance P, first we need to measure the diameter d of the sun or moon (e.g. in cm) as depicted on the painting (a non distorted reproduction of any measurable size can be used). Then we need to measure also the size s of the object of known or assumed real dimensions, as depicted on the painting (e.g. we can measure on the reproduction the height of a human figure in cm). The theoretical distance P of the painter from the subject is given by the formula:

P=110h/r

where r is the ratio s/d and h is the assumed real size of the object.
Applying this method to Van Gogh’s “sower” of November 1888, we arrive to an amazing result: the artist observes the sower from a very large distance, approximately 350 m. What is actually depicted in the painting is a tiny detail of the landscape, not wider than 2,5 degrees, leaving to the viewer a very narrow “window” through which to observe the sower and the solar disk.
In order to estimate the other distance of interest, namely the distance V at which the viewer should be positioned away from the painting, we can use the approximate formula

V=720D/π

where D is the exact diameter of the solar or lunar disk on the painting and π is the well known ratio of the length over the diameter of any circle (approximately equal to 3,14). This formula is derived by simply demanding the painted sun to have an apparent size equal to 0,5 degrees and thus to coincide with the apparent size of the sun on the sky. Measurements on the November 1888 “sower” yield a distance V approximately equal to 22,4 m. Note that D can be easily calculated by measuring directly the diameter d of the solar disk on any non distorted scaled reproduction of length x and using the formula

D=Ld/x

where L is the real length of the painting (in this case 40 cm).
The basic technique painters use to create a depth illusion is relied upon the perspective lines, i.e. straight lines that seem to converge to a point on the horizon. The relative sizes of the various depicted objects are adjusted accordingly, so as to promote this illusion. A realistically rendered scene is governed by the rules of linear perspective, i.e. the rules of properly constructing perspective lines. However, an important part in the creation of the depth illusion has the so called “atmospheric” perspective, based on the observation that objects at large distances acquire a bluish or “cooler” hue and this is why mountains in the distance seem to have a purple color. The reason is obvious: objects positioned very far away from the observer are actually viewed through a thick “lens” of air, altering the colors. In addition, the outline and the details of the object are blurred while at the same time its contrast is reduced. An example of this technique can be found in Claude Monet’s “Impression – Soleil levant” (1873) where the atmospheric perspective is the only visual clue to create the illusion of depth and distance.
It is interesting, that both Monet in “Impression - Soleil levant” and Van Gogh in the “sowers” and the “reapers” have also used the solar disk as a visual clue to suggest depth and distance. It seems that the specific, great distance implied by the solar disk is exactly the reason for the absence of any detail in the human figure of the “sower” (November 1888). The mere size of the sun, as depicted on the painting, positions the painter 350 m away from the sower. The painting of a detailed and clear human figure in front of a huge solar disk would seem to Van Gogh an impossible, paradox image, simply because the size of the sun implies a distance not compatible with detailed representation. Similar conclusions can be derived for the other three paintings of the “sower” and “reaper” quartet.
In the “Raising of Lazarus” (1890), another Van Gogh’s painting depicting a solar disk, the theoretical distance of the painter and the human figures can be estimated to be 35 m approximately. The detail of the human figures is much increased compared to the “sowers” and “reapers”, as expected for such a smaller distance.
It is known that Van Gogh had been especially interested in perspective and had experimented with perspective frames. We can therefore reasonably assume that he had absolutely realized that the depiction of the solar disk creates a self evident distance between the painter and the subject. Contrast, colors, degree of detail are all qualitative features varying with distance and thus ought to be accordingly adjusted when the solar disk is present. In the “sowers” and the “reapers”, and especially in the November 1888 version of the “sower”, Van Gogh narrows his view to focus on a distant detail of the landscape and uses the sun as a primary clue of perspective. The bright disk on the sky becomes the means through which the painter positions himself against his subject and strengthens the depth illusion. This technique appears in tens of his paintings and reveals a remarkable intuition and ability to create distances simply by painting bright, colored disks in the sky.

Leopold Bloom's little finger

James Joyce himself once claimed that he has put in “Ulysses” so many enigmas and puzzles that “it will keep the professors busy for centuries arguing over what I meant, and that’s the only way of insuring one’s immortality”. It is rather uncertain whether Joyce had in mind professors of Mathematics but it seems that "Ulysses" is actually packed with tiny enigmas and puzzles relevant to Mathematics and Science.

Leopold Bloom is the central character of the so called “novel to end all novels”, a more than 700 page long book. The whole plot, loosely connected to Homer's "Odyssey", unfolds in a single day, June 16, 1904, and follows Bloom's day long odyssey in the city of Dublin, his wandering, his encounters, his memories and his thoughts in the form of an "internal monologue". This technique reveals the character’s thoughts, expressed in first person and unprocessed, often without cohesion. Bloom is a rather complex character, haunted by the death of his infant son and the affair of his wife Molly, an opera singer, with her manager. The text itself is of immense complexity and frequent ambiguities while adopting at the same time several different styles of expression in different parts. Often the reader is presented with such difficulties that comprehension seems beyond grasp.
In the "Lestrygonians" chapter and during one of his wanderings in Dublin, Bloom stands before the window of "Yeates and Son", an optical store. Triggered by what he sees, his mind strays from cooking and food to his old glasses that need to get fixed, the Germans who are “making their way everywhere”, including Goetz lenses, and the possibility of getting a new pair of glasses at the railway lost property office.
“Astonishing the things people leave behind in trains and cloak rooms. What do they be thinking about? Women too. Incredible. Last year travelling to Ennis had to pick up that farmers’ daughter’s bag and hand it to her at Limerick station. Unclaimed money too.”
And then Bloom suddenly starts thinking of experiments and even carries out a couple:
“There’s a little watch up there on the roof of the bank to test those glasses by.
His lids came down on the lower rims of his irides. Can’t see it. If you imagine it’s there you can almost see it. Can’t see it.
He faced about and, standing between the awnings, held out his right hand at arm's length toward the sun. Wanted to try that often. Yes: completely. The tip of his little finger blotted out the sun's disk."
What Bloom does here is exactly what painters do when using their thumb or a pencil to measure the relative size of their subject. And his subject is the sun. Bloom confirms his suspicion that the apparent size of the sun is so small that a little finger at arm's length could completely cover the solar disk. Anyone who has tried to take a photograph of a person with a magnificent sunset in the background must have arrived, often with surprise, to similar conclusions: in the end the sun appears tiny in the picture and the sunset not so magnificent. So what is the apparent size of the sun after all and how is it measured?
Any object of any size, no matter how small, can completely cover the solar disk, if it is located at a distance small enough from our eyes (this is exactly what the brim of a hat does). However, for every specific object there is a specific distance such as the object just barely covers the sun. For example, a soccer ball (which has a diameter of approximately 22 cm) just barely covers the sun when put at a distance of 25 m from the observer. In this case we can say that the ball and the sun have the same apparent size i.e. they look the same although their actual sizes are very different. An orange has the same apparent size as the sun when put approximately 10 m from the observer. These examples express well the fact that, perhaps against our intuition, the apparent size of the sun is actually very small. One can convince himself or herself by trying to hit with a stone an orange located at a distance of 10 m. The target seems hopelessly small.
These conclusions can be generalized for any object of size s cm. It can be seen that such an object has the same apparent size as the sun when put at a characteristic distance of about 110s m from the observer. Obviously, the bigger an object is, the further away it must be put in order to have the same apparent size as the sun. Then, it takes approximately 720 such objects to complete a circle with radius the characteristic distance 110s m. For example it takes 720 soccer balls to complete a full circle with a radius of 25 m. It also takes 720 oranges to complete a full circle with a radius of 10 m.
The ancient Greek scientists had arrived to similar conclusions. In his work "Βίοι και γνώμαι των εν φιλοσοφία ευδοκιμησάντων" (Life and opinions of the distinguished in philosophy), Diogenes Laertius (probably third century A.C.), a biographer of ancient Greek scientists, states that the famous Thales of Miletus is "mentioned by some as the first to determine the size of the sun to be the one seven hundred and twentieth of the solar circle". In other words it takes 720 suns to complete a full circle in the horizon and therefore, we conclude, it takes 720 objects with apparent size equal to the sun's to complete a full circle with radius the distance of these objects from the observer.
Taking in account the well known elementary school fact that a full circle consists of 360 degrees, it is obvious that the apparent size of the sun in the sky is approximately 360/720=0,5 degrees. This result is amazing: half a degree is the measure of the angle made by the minute hand of an analogue watch in just 5 seconds. This angle is so small that the respective motion of the minute hand is unperceivable to the human eye. To understand this better one may imagine himself or herself standing at the centre of a huge, imaginary clock with its face on the ground and its hands extending to the horizon, as far as human eyes can see. Each second, the second hand of such a clock moves defining a certain angle. Within this angle on the horizon line, there is enough space to accommodate 12 solar disks, next to each other. It is extraordinary that the apparent size of our sun, a star the presence of which we perceive so bright and imposing in the sky, is after all measured using almost insignificant subdivisions of a unit so small as a degree.
However, it must be noted that the apparent size of the sun is not constant. Due to the elliptic shape of the earth's orbit around the sun, sometimes the earth comes a little closer and sometimes goes a little further away from the sun. The result is a slight variation, about 3%, of the sun’s apparent size during the year. Its average value has been determined to be 0,533 degrees.