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 shapeâthe ratio of a paintingâs 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 todayâs 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 Fechnerâs 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 Ď=(1+)/2â1.618033988⌠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 Ď-round(q Ď)|. 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/Ď=Ď 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 Markowskyâs 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.
Fechnerâs 1876 work on rectangle preferences and painting aspect ratios
In volume 1 of the oft-quoted work Vorschule der Aesthetik (1876), Gustav Theodor Fechnerâphysicist, experimental psychologist, and philosopherâdiscusses 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 WeberâFechner 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 Fechnerâs 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 âexpectedâ 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 paintingâs 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 Fechnerâs 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 HĂśge. 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 âArtworkâ domain of the Wolfram Knowledgebase
One of the knowledge domains in EntityValue is âArtworkâ. 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 âArtworkâ 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, letâs 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, letâs have a look at the product, which is to say the area of the painting. (Fechnerâs 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 Fechnerâs convention, which also used height/width.
Now letâs 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, âŚ)?
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 Fechnerâs 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 xÂł-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 â has a slightly complicated form:
The resolution of the graphs from the 594 analyzed paintings was not enough to discriminate between â and 4/3, and as a result, Shortess, Clarke, and Shannon suggest that the value of the maximum of painting ratios occurs at the âplatinum constant,â 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 âplatinum constantâ is unrelated to the so-called âplatinum ratioâ 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. Letâs 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=1.2Ě
(see here, here, here, and here) or 27/20=1.35 or the so-called âmeta-golden ratio chi,â the solution of Χ²-Χ/Ď=1 with value 1.35âŚ
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, letâs 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 0pĚ
,(x) of wide paintings with 1xpĚ
(x)=p(1/x)/x². Yet at the same time, for the maxima of p(x) and , of pĚ
(x) we have the relation â1/. 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?
Letâs 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 âaspect ratioâ 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 paintingsâ 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.)
Letâs use Wikipedia again to look at the distribution of aspect ratios of some famous paintersâ 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 CĂŠzanne, preferred standard canvas sizes. (For a study of canvas sizes used by Francis Bacon, see here.)
Letâs also have a look at a more modern painter, Thomas Kincade, the âpainter of light.â 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 3/2âvery 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 âArtworkâ 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:
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 MĂźnchen (today the Alte Pinakothek):
The catalog from the Museum der bildenden KĂźnste 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 âoil on canvas,â 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 periodâfor 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 âareâ the ârealâ 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 SuĂĄrez.
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 letâs 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:
Plotting all occurring widths and lengths that are present in the data, we obtain the following graphic:
Averaging over a length scale of one centimeter, we obtain the following histogram of all widths and lengths. One notes the many pronounced peaks and discrete lengths:
A plot of the actual widths and heights of the paintings shows that many paintings are less than 140 cm in height and/or width:
A contour plot of the smoothed version of the 2D density of width-height distributions shows the two âmountain ridgesâ of wide and tall paintings:
Looking at the explicit numerical values of the common-length values shows multiples of 5 cm and 10 cm, but also many numbers that seem not to arise from potentially rounding measurement values:
The next graphic shows the most common length and width values cumulatively over time:
Plotting the widths and heights sorted by the century shows that many of the very tall spikes come from the nineteenth century. (Note the much smaller vertical scale for paintings from the twentieth century.)
For later comparison, we fit the distribution of the width of the paintings. We smooth with a bandwidth of about 5 cm to remove most of the local spikes:
We show a distribution of the ages of the paintings from this dataset:
We analyze this dataset by plotting all concrete occurring aspect ratios together with their multiplicities:
To better resolve the multiplicities of aspect ratios that are nearly identical, we plot a histogram with a bin width of 0.02:
Letâs approximate each aspect ratio with a rational number such that the error is less than 1%. What will be the distribution of the resulting denominators of the fractions approximating the aspect ratios? The following plot shows the distribution in a log-plot. It is interesting to note the relatively large fraction of paintings with a max(width/height)/min(width/height) ratio and min(width/height)/max(width/height) with denominators of 3, 4, and 7, and the relative absence of denominators 6 and 18:
For comparison, here are the corresponding plots for 20k uniformly (in [0,2]) distributed numbers:
Here are the cumulative distributions of the paintings with selected aspect ratios:
If we normalize the counts to the total number of paintings, we still see the 5/4 aspect ratio increasing over time, but most of the other aspect ratios do not change significantly:
If we do not take the measurement values for face value but assume that they are precise only up to Âą1%, we obtain quite a different picture. The following graphic shows the distribution of the paintings of a given aspect ratio interval with a given center value. Around 1500, all common aspect ratios were approximately equally popular. We see that the aspect ratios 5/4, 4/3, and 9/7 became much more common about 1600. And aspect ratios approximately equal to the golden ratio have become less popular since the thirteenth century. (This graphic is not sensitive to the Âą1% aspect ratio width; Âą0.5% to Âą5% will give quite similar results.)
So what about the denominators of the most common aspect ratios? We form all fractions with maximal denominator 16 and map all aspect ratios to the nearest of these fractions. Because of the non-uniform gaps between the selected rationals, we normalize the counts by the distance to the nearest smaller and larger rational aspect ratios. This graphic gives a view of the occurring aspect ratios that is complementary to the histogram plot. The histogram plot uses equal bins; the following plot uses non-uniform bins and adjacent minima and maxima in the histogram bins can cancel each other out. Again, the 5/4 and the 4/5 aspect ratios are global winners:
We again use the Nearest function approach to plot a detailed map of the aspect ratio distributions. The following function windowedMaximaPlot plots the distribution either as a 3D plot or as a contour plot for paintings from a sliding time window:
Here are the 3D plot and the contour plot:
The last two images show a few noteworthy features:
- Over the last 400 years, tall pictures often have an aspect ratio of approximately 1.2
- The most common aspect ratio of wide pictures changes around 1750, and a relatively wide distribution shows a few pronounced maxima, e.g. at 0.8
- Square images become more popular around 1800
Labreuche discusses the process of the standardization of canvases. In France, a first standardization happened in the seventeenth century and a second in the nineteenth century. (For a recent, more mathematical discussion, see Dinh Dang.) Simon discusses the canvas standarization in Britain.
Here are the figure, marine, and landscape sizes of the standardized canvases from nineteenth-century France. The data is in the form {width, {figure height, landscape height, marine height}}:
The aspect ratios (max(height/width, width/height)) for all canvases has the following distribution:
It is not easy to find large datasets of exact measurements of old paintings. On the other hand, various websites have tens of thousands of images of paintings in both JPG and PNG formats. Could one not just use these images for finding the aspect ratio of paintings by using the image height in pixels and the image width in pixels? Above, we saw that the majority of paintings are measured with a precision of about one centimeter. With an average painting height and width of about one meter, the resulting uncertainty is in the order of 2%. Even thumbnail images are about 100 pixels, and many images of paintings are a few hundred pixels wide (and tall). So from the literal pixel dimensions, one would again expect results to be correct in the order of (1. . .2)%. But there is no guarantee that the images were not cropped, the frame is consistently included or not included, or that boundary pixels were added. The Web Gallery of Art has, in addition to the actual measurements of the paintings, images of the paintings. After downloading the images and calculating the aspect sizes of the images, we can compare with the aspect ratios calculated from the actual heights and widths of the paintings. Here is the resulting distribution of the two aspect ratios together with a fit through a CauchyDistribution[1.003,0.019]. The mean of the two pixel dimensions is 1.036 and the standard deviation is 0.38. These numbers show that the error from using images of the paintings to determine the aspect ratios is far too large to properly resolve the observed fine-scale structure of aspect ratios:
In the data dataWGA, we also have information about the painters. Does the mean aspect ratio of the paintings change over the lifetimes of the painters? Here is the distribution of when during the paintersâ lives the paintings were made:
Interestingly, statistically we can see a pattern of the mean aspect ratio over the lifetime of a painter. The first paintings statistically have a more extreme aspect ratio. At the end of the first third of the lifetime, the aspect ratio is minimal, and at the end of the second third the aspect ratio is maximal (left graphic). The cumulative average aspect ratio shows a minimum at about 40% the lifespan of the painters (right graphic). Both graphics show max(height/width, width/height) divided by the mean of all aspect ratios. (A general discussion of creativity vs. age can be found here.)
If the reader wants to visit some of the paintings in person and wants to perform some more precise width and height measurements, let us calculate one more statistic using the Web Gallery of Art dataset. Letâs also calculate and visualize where the paintings are in the world. We take the (current) city locations of the paintings that have width and height parameters, aggregate them by city, and display the median of max(height/width, width/height) as a function of the city. Not unexpectedly, most larger collections donât deviate much from the median of 1.333. We use Interpreter to find the cities and derive their locations:
Interlude II: The importance of measuring precisely
Now let us look at the detailed width and height values. If we plot the counts of the fractional centimeters, we clearly see that the vast majority of paintings are measured within a precision of less than 1 cm. Only about 10% of all paintings have dimensions specified up to a millimeter (and some of the ones specified up to 5 millimeters are probably also rounded):
Now let us look at the detailed width and height values. As the majority of the paintings were made before the invention of the centimeter as a unit of measurement, the popular painting sizes are probably not a length that is an integer multiple of a centimeter. This means that the measured widths and heights are not the precise widths and heights of the actual paintings. The nearly homogeneous distribution of millimeters of the paintings that were measured up to the millimeter is comforting.
In many of the datasets analyzed, the widths and heights of the paintings are given as integers when measured in centimeters. (A notable exception was the Tate dataset, in which virtually every painting dimension is given to millimeter accuracy.) As most paintings are in the order of 100 cm width or height (give or take a factor of 2), for an accurate determination of the aspect ratio the rounding to integer-centimeter length will matter. How many of the observed maxima at various fractions with small denominators can be traced back to imprecise width and height values?
Letâs model this effect now. The function aspectRatioModelValue models the aspect ratio of a painting. We assume a stable distribution for the width of the paintings and assume the height to be normally distributed with a mean of 1.3xwidth. And we model only tall paintings by restricting the height to be at least as large as the width:
Now we âcut canvasesâ for tall paintings and look at the distribution of the aspect ratios. We do this twice, each time for 100,000 canvases. The top graphic shows the resulting distribution in the case of millimeter-resolution of the canvas measurements. The bottom graphic assumes that in 65% of all cases we measure up to a centimeter precision, in 25% up to half a centimeter precision, and in the resulting 10% up to millimeter precision. For each of the three computational experiments, we overlay the resulting distribution histograms:
Comparing the upper with the lower graphic shows that the aspect ratio distribution is quite smooth if all measurements are precise to the millimeter. The lower distribution shows that painting dimension measurements up to the centimeter do indeed introduce artifacts into the resulting histograms.
Looking at the pretty smooth histogram for the millimeter-precise model and the above aspect ratio histogram for the Tate collection shows that the more common occurrences of aspect ratios that are equal to simple fractions is a real effect. Yet at the same time, as the above experiment with the weights {0.65, 0.25, 0.10} shows, the mostly centimeter-precise widths and heights do artificially amplify some simple fractions, such as 6/5, 5/4, and 3/2.
An even simpler method to demonstrate the influence of rounding errors in the width/height measurements in the Web Art Gallery dataset is to modify the width and height values. For each integer centimeter measurement, we add between -5 millimeters and 5 millimeters to mimic a more precise measurement. We again use the ratio of the longest side to the smallest side of the painting:
We overlay the original aspect ratio distribution with the one obtained from the modified width and height values. We see that the maxima at some rational ratios do get suppressed, but that the global maxima keeps its position around 5/4, and the second maxima around 4/3 stays, as well as the smaller, first maximum around 6/5. At the same time, we see the peaks at 3/2 and 2 get smoothed out:
We now do the reverse with the Tate dataset: we round each width and height measurement to the nearest centimeter. Again, we plot the original aspect ratio distribution together with the modified one:
While the height of the local peaks changes, the peaks are still present, even quite pronounced.
WikiArt: another large web resource
Let us have a look at yet another large web resource, namely WikiArt. For computational purposes, it is a conveniently structured website. We have a list of more than nine hundred artists, with hyperlinks to pages of the artistsâ works. Each individual artwork (painting) in turn has a page that has conveniently structured information. For example, here is the factual information about the Mona Lisa:
We note that the above data contains style and genre. This suggests using the WikiArt dataset to look for a possible dependence of the aspect ratio on genre especially (we already quickly looked at the movements above).
There are about seven thousand paintings with width-height information in the dataset. For brevity, we encoded all data into a grayscale image:
The paintings with dimension information have the following age distribution. We see a dominance of paintings from the eighteenth and nineteenth centuries:
Based on the results obtained earlier, we expect this dataset that is mostly dominated by paintings from the last 150 years to show pronounced peaks in the aspect ratio distribution at rationals. The following distribution with grid lines at 6/5, 5/4, 4/3, and 3/2 confirms this conjecture:
The genre obviously influences whether paintings are predominantly wide, square, or tall. Here are the wide vs. square vs. tall distributions for some of the popular genres:
Now let us have a look at the distribution of the aspect ratio as a function of the genre:
Hijacking the function TimelinePlot, we show the range of the second and third quartiles of the aspect ratios:
Tall landscape paintings are much scarcer than wide landscape paintings. But even if we use the definition aspect ratioâlongest side/shortest sideâwe still see a clear dependence of the aspect ratio on the genre.
The genre frequently also influences the actual painting size. Here are the second and third quartiles in aspect ratio and area for the various genres (mouse over the opaque rectangles in the notebook to see the genre):
If we slice up each genre by the style, we get a more fine-grained resolution of the distribution of aspect ratios. We find the top genres and styles, requiring each relevant genre and style to be represented with at least 50 paintings:
The Neoclassical nude paintings stand out with the largest median aspect ratio of about 1.85:
And here is a more detailed graphic showing the median aspect ratios for all the style-genre pairs with at least five paintings. (Mouse over the vertical columns to see the genre and the aspect ratios.)
France national museumsâ collections
As we saw above, painting collections with a few thousand paintings allow us to resolve multiple maxima in the distribution in the range 1.24. . .1.33 for the aspect ratios. Now letâs look at a second large dataset.
The Joconde catalog of the French national museums covers more than half a million artifacts. A search for paintings gives about sixty-seven thousand results. Not all of them are paintings that are made for hanging on a wall; the collection also includes paintings on porcelain figures and other mediums. But one finds about thirty-one thousand paintings with explicit dimensions. As the information about the paintings comes from multiple museums, the dimensions can occur in a variety of formats. The extraction of the dimensions is a bit time consuming.
Interestingly, this time yet another maximum occurs at about 1.23.
Mapping the distribution for wide images into the one for tall images by exchanging height and width, we see that the two maxima match up very well. This makes the ratio 5/4 (or 4/5) the most common ratio:
About 11% more tall paintings than wide paintings are in the collection.
Paintings in Italian churches: tall is all
A very large database of paintings of the Catholic churches from Italy can be found here. Searching again for