Paintings of the great masters are among the most beautiful human artifacts ever produced. They are treasured and admired, carefully preserved, sold for hundreds of millions of dollars, and, perhaps not coincidentally, are the prime target of art thieves. Their composition, colors, details, and themes can fascinate us for hours. But what about their outer shape—the ratio of a painting’s height to its width?
In 1876, the German scientist Gustav Theodor Fechner studied human responses to rectangular shapes, concluding that rectangles with an aspect ratio equal to the golden ratio are most pleasing to the human eye. To validate his experimental observations, Fechner also analyzed the aspect ratios of more than ten thousand paintings.
We can find out more about Fechner with the following piece of code:
By 1876 standards, Fechner did amazing work, and we can redo some of his analysis in today’s world of big data, infographics, numerical models, and the rise of digital humanities as a scholarly discipline.
After a review of the golden ratio and Fechner’s findings, we will study the distribution of the height/width ratios of several large painting collections and the overall distribution, as well as the most common aspect ratios for paintings. We will discover that the trend over the last century or so is to become more rationalist.
Prelude: The golden ratio, a beautiful construction in mathematics
The golden ratio ϕ=(1+
)/2≈1.618033988… is a special number in mathematics. Its base 2 or base 10 digit sequences are more or less random digit sequences:
Its continued fraction representation is as simple and beautiful as a mathematical expression can get:
Or, written more explicitly:
Another similar form is the following iterated square root:
Although just a simple square root, mathematically the golden ratio is a special number. For instance, it is the maximally badly approximable irrational number:
Here is a graphic showing the sequence q *|q ϕ-round(q ϕ)|. The value of the sequence terms is always larger than 1/5^½:
Furthermore, we can show the approximation to the golden ratio that one obtains by truncating the continued fraction expansion:
A visualization of the defining equation 1+1/ϕ=ϕ is the ratio of the length of the following line segments:
Here are a wide and a tall rectangle with aspect ratio, golden ratio, and 1/(golden ratio):
Not surprisingly, this mathematically beautiful number has been used to generate aesthetically beautiful visual forms. This has a long history. Mathematically described already by Euclid, da Vinci made famous drawings that are based on the golden ratio.
The Wolfram Demonstrations Project has more than 90 interactive Manipulates that make use of the golden ratio. See especially Mona Lisa and the Golden Rectangle and Golden Spiral.
The golden ratio is also prevalent in nature. The angle version of the golden ratio is the so-called golden angle, which splits the circumference of a circle into two parts whose lengths have a ratio equal to the golden ratio:
The golden angle in turn appears, for instance, in phyllotaxis models:
For a long list of occurrences of the golden ratio in nature and in manmade products, see M. Akhtaruzzaman and A. Shafie.
However, the universality of the golden ratio in art is often overstated. For some common myths, see Markowsky’s paper.
Later, we will also encounter the square root of the golden ratio. If we allow for complex numbers, then another, quite simple continued fraction yields the square root of the golden ratio as a natural ingredient of its real and imaginary parts:
The name golden ratio seems to go back to Martin Ohm, the younger brother of the well-known physicist Georg Ohm, who used the term for the first time in a book in 1835.
Fechner’s 1876 work on rectangle preferences and painting aspect ratios
In volume 1 of the oft-quoted work Vorschule der Aesthetik (1876), Gustav Theodor Fechner—physicist, experimental psychologist, and philosopher—discusses the relevance of the golden ratio to human perception.
Today, Fechner is probably best known for the subjective sensation law jointly named after him, the Weber–Fechner law:
In chapter 14.3 (volume 1) of his book, Fechner discusses the aesthetics of the size (aspect ratio) of rectangles. Carrying out experiments with 347 probands, each given 10 rectangles of different aspect ratios, the rectangle that was most often considered pleasing by his experimental audience was the one with an aspect ratio equal to 34/21, which deviates from the golden ratio by less than 0.1%. Here is the today-still-cited but rarely reproduced table of Fechner’s results:
Chapter 33 in volume 2 discusses the sizes of paintings, and Chapter 44 of volume 2 contains a forty-one-page detailed analysis of 10,558 total images from 22 European art galleries. Interestingly, Fechner found that the typical ratio of painting heights and widths clearly deviated from the “expected” golden ratio.
Fechner carried out a detailed analysis of 775 hunting and war paintings, and a coarser analysis on the remaining 9,783 paintings. Here are the results for hunting and war paintings (Genre), landscapes (Landschaft), and still life (Stillleben) paintings. In the table, h indicates the painting’s height and b the width. And V.-M. is the ratio h/b or b/h:
Here in the twenty-first century, we can repeat this analysis of the aspect ratios of paintings.
For detailed discussions and modified versions of Fechner’s experiments with humans, see the works of McManus (here and here), McManus et al., Konecni, Bachmann, Stieger and Swami, Friedenberg, Ohta, Russel, Green, Davis and Jahnke, Phillips et al., and Höge. Jensen recently analyzed paintings from the CGFA database, but the discretized heights and width values used (from analyzing the pixel counts of the images) did not allow resolution of the fine-scale structure of the aspect ratios, especially the occurrence of multiple, well-resolvable maxima. (See below for the analysis of a test set of images.)
While Fechner did a detailed analysis of quantitative invariants (e.g. mean, median) of the aspect ratios of paintings, he did not study the overall shape of the aspect ratio distribution, and he also did not study the distribution of the local maxima in the distribution of the aspect ratios.
An easy start: analyzing entities from the “Artwork” domain of the Wolfram Knowledgebase
One of the knowledge domains in EntityValue is “Artwork”. Here we can retrieve the names, artists, completion dates, heights, and widths of a few thousand paintings. Paintings are conveniently available as an entity class in the “Artwork” domain of the Wolfram Knowledgebase:
Here is a typical example of the retrieved data:
Paintings come in a wide variety of height-to-width aspect ratios, ranging from very wide to quite tall. Here is a collage of 36 thumbnails of the images ordered by their aspect ratio. Each thumbnail of a painting is embedded into a gray square with a red border:
The majority of the paintings have aspect ratios between 1/4 and 4. Here are some examples of quite wide and quite tall paintings:
We can get an idea about the most common topics depicted in the paintings by making a word cloud of words from the titles of the paintings:
Now that we have downloaded all the thumbnails, let’s play with them. Considering their colors, we could embed the average value of all pixel colors of the image thumbnails in a color triangle:
Before analyzing the aspect ratios h/b in more detail, let’s have a look at the product, which is to say the area of the painting. (Fechner’s aforementioned work devoted a lot of attention to the natural area of paintings too.)
We show all paintings in the aspect ratio area plane. Because paintings occur in greatly different sizes, we use a logarithmic scale for the areas (vertical axis). We also add a tooltip for each point to see the actual painting:
And here is a histogram of the distribution of the height/width aspect ratios.
Starting now, following the Wolfram Language definition of aspect ratio, I will use the definition aspect ratio=height/width rather than the sometimes-used definition aspect ratio=width/height. As we saw above, this convention also follows Fechner’s convention, which also used height/width.
Now let’s analyze the histogram of the aspect ratios in more detail. Qualitatively, we see a trimodal distribution. For wide paintings (width>height) we have an aspect ratio less than 1, for square paintings we have an aspect ratio of about 1, and for tall paintings (height>width) we have an aspect ratio greater than 1. The tall and the wide paintings both have a global peak, and some smaller local peaks are also visible.
The trimodal structure for wide, square, and tall paintings was to be expected. Two natural questions that arise when looking at the above distribution are:
1) what are the positions of the local peaks?
2) what is the approximate overall shape of the distribution (normal, lognormal, …)?
In 1997, Shortess, Clarke, and Shannon analyzed 594 paintings and took a closer look at the point where the maximum of the distribution occurs. In agreement with Fechner’s 1876 work, they found that 1.3 seems to be the local maximum for the distribution of max(h/b,b/h). Again, 1.3 is clearly different from the golden ratio and the authors suggest either the Pythagorean number (4/3) or the so-called plastic constant as the possible exact value for the maximum.
The plastic constant is the positive real solution of x³-x-1=0:
The plastic constant was introduced by Dom Hans van der Laan in 1928 as a special number with respect to human aesthetics for 3D (rather than 2D) figures. If explicitly expressed in radicals, the plastic constant ℘ has a slightly complicated form:
The resolution of the graphs from the 594 analyzed paintings was not enough to discriminate between ℘ and 4/3, and as a result, Shortess, Clarke, and Shannon suggest that the value of the maximum of painting ratios occurs at the “platinum constant,” a constant whose numerical value is approximately 1.3. Their paper also did not resolve any fine-scale structure of the height/width distribution. (Note: this “platinum constant” is unrelated to the so-called “platinum ratio” used in numerical analysis.)
(There is an interesting mathematical relation between the golden ratio and the plastic constant: the golden ratio is the smallest accumulation point of Pisot numbers, and the plastic constant is the smallest Pisot number; but we will not elaborate on this connection here.)
If we use a smaller bin size for the bins of the histogram, at least two maxima for both tall and wide paintings become visible:
If we show the cumulative distribution function, we see that the absolute number of paintings that are square is pretty small. The square paintings correspond to the small vertical step at aspect ratio=1:
Next, let us take all tall paintings and show the inverse of their aspect ratios together with the aspect ratios of the wide paintings. The two global maxima at about 0.8 map reasonably well into each other, and so does the secondary maxima at about 0.75:
Graphing smoothed distributions of the aspect ratios of wide paintings and the inverse of the aspect ratios for tall paintings shows how the maxima map into each other:
A quantile plot shows the similarity of the distributions for wide and tall paintings under inversion of the aspect ratios:
Will it be possible to resolve the maxima numerically and associate explicit numbers with them? Here are the above-mentioned constants and three further constants: the square root of the golden ratio, 5/4, and 6/5:
Among all possible constants, we added the square root of the golden ratio because it appears naturally in the so-called Kepler triangle. Its side lengths have the ratio 1:sqrt(golden ratio):golden ratio:
The Pythagorean theorem is also important for the square root of the golden ratio. The Kepler triangle becomes the defining equation for the golden ratio:
Shortess et al. included 4/3 as the Pythagorean constant because this number is the ratio of the smaller two edges of the smallest Pythagorean triangle with edge length 3, 4, 5 (3²+4²=5²).
And the rational 6/5 was included because, as we will see later, it often occurs as an aspect ratio of paintings in the last 200 years.
The distribution of the painting aspect ratios together with the selected constants shows that the largest peak seems to occur at the sqrt(golden ratio) value and a second, smaller peak at 1.32… 1.33.
Here is a list of potential constants that potentially represent the position of the maxima. We will use this list repeatedly in the following to compare the aspect ratio distributions of various painting collections. Let’s start with some visualizations showing these aspect ratios:
The next graph shows the six constants on the number line. The difference between the plastic constant and 4/3 is the smallest between all pairs of the six selected constants:
Here are wide rectangles with aspect ratios of the selected constants:
And for better comparison, the next graphic shows the six rectangles laid over each other:
And here is the above graphic overlaid with the positions of the constants at the horizontal axis:
Various other fractions with small denominators will be encountered in selected painting datasets below, and various alternative rationals could be included based on aesthetically pleasing proportions of other objects, such as 55/45=11/9=1.2̅ (see here, here, here, and here) or 27/20=1.35 or the so-called “meta-golden ratio chi,” the solution of Χ²-Χ/ϕ=1 with value 1.35…
Because the resolution of a histogram is a bit limited, let us carefully count the number of paintings that are a certain aspect ratio plus or minus a small deviation. To do this efficiently, we form a Nearest function:
Again, we clearly see two well-separated maxima, the larger one nearer to the square root of the golden ratio than to the plastic constant or the Pythagorean number:
Interlude I: Features of the probability distribution of aspect ratios
Before looking at more painters and paintings, let’s have a more detailed look at the distribution of the aspect ratios.
The most commonly used means are all larger than the tallest maximum for tall images:
Here are the means for the wide paintings:
What is the ratio of taller to wider paintings? Interestingly, we have nearly exactly as many tall paintings as wide paintings:
The averages for the paintings viewed as a rectangles (meaning the aspect ratios (max(height, width)/min(height,width)) have means that are very similar to the tall paintings:
As above in the plot of the two overlaid histograms, the distribution of tall paintings agrees nearly exactly with the distribution of wide paintings when we invert the aspect ratio. But what is the actual distribution for tall (or all) paintings (question 2) from above? If we ignore the multiple peaks and use a more coarse-grained view, we could try to fit the distribution of the tall paintings with various named probability distributions, e.g. a normal, lognormal, or heavy-tailed distribution.
We restrict ourselves to paintings with aspect ratios less than 4 to avoid artifacts in the fitting process due to outliers:
Using SmoothKernelDistribution allows us to smooth over the multiple maxima and obtain a smooth distribution (shown on the left). A log-log plot of the hazard function (f(a)/(1-F(a))) together with the function 1/a gives the first hint that we expect a heavy-tailed distribution to be the best approximation:
Here are fits with a normal and a lognormal distribution:
And here are some heavy-tailed distributions:
As the height/width ratios have a slow-decaying tail, the normal, lognormal, and extreme value distribution are a poor fit. The range of aspect ratios between about 1.4 and 2 shows this most pronounced:
The four heavy-tailed distributions show a much better overall fit:
If we quantify the fit using a log-likelihood ratio statistic, we see that the truncated heavy-tailed distributions perform the better fits:
The distribution for the aspect ratio has a curious property: we saw above that the distributions of the wide and tall paintings appropriately match after an appropriate mapping. This means their maxima agree, at least approximately. But by mapping the distribution p(x) of tall paintings with 0
p̅,(x) of wide paintings with 1xp̅(x)=p(1/x)/x². Yet at the same time, for the maxima of p(x) and , of p̅(x) we have the relation ≈1/. Interestingly, for the parameters found for the stable distribution fit, this property is fulfilled within two percent. Here we quantify this difference in maxima position for the beta prime distribution. (The results for stable distributions are nearly identical.)
The aspect ratio through the ages, for movements and painters
Now, a natural question is: how reproducible is the trimodal distribution across the ages, across painting genres, and across artists?
Let’s look at time dependence by grouping all aspect ratios according to the century in which the paintings were completed. We see that at least since the fourteenth century, tall paintings have frequently had an aspect ratio of about 1.3, wide paintings an aspect ratio of about 0.76, and that square paintings became popular only relatively recently. We also see that for tall paintings the distribution is much flatter in the sixteenth, seventeenth, and eighteenth centuries as compared with the nineteenth century (we will see a similar tendency in other painting datasets later):
The median of the aspect ratios of all paintings decreased over the last 500 years and is slightly higher than 1.3. (here we define “aspect ratio” as the ratio of the length of the longer side to the length of the smaller side). The mean also decreased and seems to stabilize slightly above 1.35:
For comparison, here are the distributions of the paintings’ areas (in square meters) over the centuries:
The median area of paintings has been remarkably stable at a value slightly above 2 square meters over the last 450 years:
What about the aspect ratios across artistic movements? WikiGallery has visually appealing pages about movements. We import the page and get a listing of movements and how many paintings are covered in each movement:
But unfortunately, width and height information is available for only a fraction of the paintings. Importing all individual painting pages and extracting the height and width data from the size of the thumbnail images allows us to make at least some quantitative histograms about the distribution of the aspect ratios for each movement.
The overwhelming majority of movements shows again strong bimodal distributions with aspect ratio peaks around 1.3 and 0.76. (The movements are sorted by the total number of paintings listed on the corresponding Wiki pages.)
Let’s use Wikipedia again to look at the distribution of aspect ratios of some famous painters’ works.
Although the total number of paintings is now much smaller per histogram, again the bimodal (ignoring the square case) distributions are visible. And again we see clear maxima at tall paintings with aspect ratios of about 1.3 and wide paintings with aspect ratios of about 0.76:
We see again relatively strongly peaked distributions. Some painters, for example Cézanne, preferred standard canvas sizes. (For a study of canvas sizes used by Francis Bacon, see here.)
Let’s also have a look at a more modern painter, Thomas Kincade, the “painter of light.” Modern paintings use standardized materials and come in a set of sizes and aspect ratios that result much more from standardization of canvases and paper rather than aesthetics. So this time we do not analyze the textual image descriptions, but rather the images themselves, and extract the pixel widths and heights. Even for thumbnails, this will yield an aspect ratio in the correct percent range:
In addition to our typical maximum around 1.3, we see a very pronounced maximum around 3/2—very probably a standardization artifact:
Analyzing five old German museum catalogs
The above histograms indicate at least two maxima for tall paintings, as well as two maxima for wide paintings, with the larger peak very near to the square root of the golden ratio. As we do not know what exactly was the selection criterion for artwork included in the “Artwork” domain of Entity, we should test our conjecture on some independent collections of paintings.
An easily accessible source for widths and heights of paintings are museum catalogs. Various older catalogs, similar to the ones used by Fechner, are available in scanned and OCR forms. Examples are:
Beschreibendes Verzeichnis der Gemälde im Kaiser Friedrich-Museum, 1906
Katalog der Gemälde-Sammlung der Kgl. Älteren Pinakothek in München, 1886
Verzeichnis der Gemälde-Sammlung im Kgl. Museum der bildenden Künste zu Stuttgart, 1891
Katalog der Königlichen Gemäldegalerie zu Dresden, 1896
Katalog der Königlichen Gemäldegalerie zu Cassel, 1913
It is straightforward to directly import the OCR test versions of the catalogs. While the form of giving the heights and widths varies from catalog to catalog, within a single catalog the employed description formatting is quite uniform. As a result, specifying the string patterns that allow you to extract the heights and widths is pretty straightforward after having looked at some example descriptions of paintings in each catalog:
The catalog from the Kaiser-Friedrich Museum (today the Bode Museum):
The catalog from the Pinakothek München (today the Alte Pinakothek):
The catalog from the Museum der bildenden Künste zu Stuttgart (today the Staatsgalerie Stuttgart):
The catalog from the Gemäldegalerie Dresden (today the Gemäldegalerie Alte Meister Dresden):
The catalog from the Gemäldegalerie zu Cassel (today the Neue Galerie Kassel):
Qualitatively, the results for the aspect ratios are very similar for the five museums:
We join the data of the five catalogs and add grid lines for the above-defined six constants:
Again, we clearly see two global maxima in the aspect ratio distribution. For tall paintings we obtain a relatively flat maximum, without clearly resolved local minima.
(The archive.org website has various even older painting catalogs, e.g. of the Schloss Schleissheim, the catalog of the collection of Berthold Zacharias, the collection of the National Gallery of Bavaria, and more. The aspect ratio distribution of the paintings of these catalogs is very similar to the five we analyze here.)
The Kress collection: four large PDF files
A famous painting collection is the Kress collection. The individual images are distributed across many museums in the US. But fortunately (for our analysis), the details of the paintings that are in the collection are available in four detailed catalogs, available as PDF documents totaling 900 pages of detailed descriptions of the paintings. (Much of the data analyzed in this blog refers nearly exclusively to Western art. For measurable aesthetic considerations of Eastern art, see, for instance, the recent paper by Zheng, Weidong, and Xuchen.)
After importing the PDF files as text and extracting the aspect ratios, we have about 700 data points. (From now on, in the following, we will not give all code to import the data from various sites to analyze the aspect ratios; the times to download all data are sometimes too large to be quickly repeated.)
This time, we also have a local maxima near sqrt(2) as well as the golden ratio.
Current gallery collections: Metropolitan, Art Institute of Chicago, Hermitage, National Gallery, Rijks, and Tate
To confirm the existence of well-defined maxima in the aspect ratio distributions and their locations, let us now look at the distribution of selected famous art museums worldwide
The Metropolitan museum of art has a fantastic online catalog. Searching for paintings of the type “oil on canvas,” we can extract their aspect ratios.
This time, the global maximum seems to be a bit smaller than 1.27:
The Art Institute of Chicago has a handy search that allows you to find paintings by period—for instance, paintings made between 1600 and 1800. Accumulating all the data gives about 1,200 data points, and the global maxima seems very near to the root of the golden ratio:
The State Hermitage Museum has an easy-to-analyze website that has information about more than 3,400 paintings from its collection. Analyzing the aspect ratios shows again two distinct maxima for tall images:
As a fourth collection, we analyze the paintings from the National Gallery. The distribution is visibly different from previous graphics:
The relatively unusual distribution goes together with the following age distribution. We see many more 500-year-old paintings as compared to other collections:
The Rijks Museum in Amsterdam is another extensive collection of old paintings. Here is the aspect ratio distribution of 4,600 paintings from the collection:
As a sixth example of analyzing current collections, we have a look at the paintings of the Tate collection. Many of the 8,000+ paintings from the Tate collection are relatively new. Here is a breakdown of their creation years:
The aspect ratio distribution, when overlaid with our constants from above, shows a good (but not perfect) match:
But with an overlay of the rationals 6/5, 5/4, 9/7, 4/3, and 3/2, we see a good approximation of the local maxima for the tall paintings. (We use a slightly smaller bin size for better resolution in the following graphic.)
Using the better-resolving Nearest-based counts of paintings within a small range shows that the maxima of the wide as well as the tall paintings occur at the rationals 6/5, 5/4, 9/7, 4/3, 3/2, and their inverses. (We use an aspect ratio window of size 0.01.)
There is little dependency of the peak positions on the window size used in Nearest:
Note that we showed grid lines at rational numbers in the above plot. Within 1% of 9/7, we find the square root of the golden ratio and fractions such as 14/11. So deciding which of these numbers “are” the “real” position of the maxima cannot be answered with the precision and amount of data available:
There is one thing unique about the Tate collection, and that one thing is especially relevant for this project. Here are two examples of its data:
Note the very precise measurements of the painting dimensions, up to millimeters. This means this is a dataset whose detailed aspect ratio distribution curve has a lot of credibility with respect to the exact values of the curve maxima.
An aspect ratio exception: the National Portrait Gallery collection
The National Portrait Gallery has tens of thousands of portrait paintings.
The individual web pages are easily imported and dimensions are extracted:
Not unexpectedly, portraits have on average a much more uniform aspect ratio than landscapes, hunting events, war scenes, and other types of paintings. This time, we have a much more unimodal distribution. The following histogram uses about 45k aspect ratios:
Zooming into the region of the maximum shows that a large fraction of portrait paintings have an aspect ratio of about 6/5. A secondary maximum occurs at 5/4 and a third one at 4/3:
While the golden ratio seems to be relevant for the center part of the human face (see e.g. here, here, and here), most portraits show the whole head. With an average height/width ratio of the human face (excluding ears and hair) of 1.48, the observed maximum at 1.2 seems not unexpected. For a more detailed investigation of faces in paintings, see de la Rosa and Suárez.
The Web Gallery of Art: a convenient database ready to use
So far, the datasets analyzed have not allowed us to uniquely resolve the position of the maxima. There are two reasons for this: the datasets do not have enough paintings, and the measurements of the paintings are often not precise enough. So let’s take a larger collection. The Web Gallery of Art, a Hungarian website, offers a downloadable tabular dataset of paintings as a CSV file.
The file uses a semicolon as the separator, so we extract the columns manually rather than using Import:
The following data is available:
And here is how a typical entry looks. The dimensions are in the form height x width:
The majority of listings of artworks are, fortunately, paintings:
Extracting the paintings with dimension data (not all paintings have dimension information), we have 18.6k data points:
<img src="http://blog.wolfram.com/data/uploads/2015/11/Extracting-the-paintings-with-dimension-data.png" alt="E