Escher's most mathematically interesting piece
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Escher's Print Gallery 0:00
The video opens with M.C. Escher's 1956 lithograph The Print Gallery, in which a young man looks at a picture of a boat in a harbor, and the scene warps as your eye moves through a village, into a building, down a hallway, and back to the same boat picture, closing the loop on itself. Escher called this piece the most peculiar thing he had ever done. His art is beloved by mathematicians because it touches on deep mathematical ideas even though he had no formal training in math.
The blank spot in the middle 2:05
At the very center of the print there is a blank circle where the warping becomes so extreme that it is unclear what should even be drawn there. A diffusion model was asked to fill it in and failed badly, which makes sense because that spot is genuinely ambiguous. The video's goal is to show that there is nevertheless exactly one completion that fits, following the 2003 analysis by mathematicians De Smit and Lenstra.
Three steps behind the illusion 3:02
Escher's method is described in three intuitive steps. First comes a straightened-out self-similar image, called a Droste effect after the cocoa brand that popularized it, where a picture contains a smaller copy of itself, and in Escher's case the copy is 256 times smaller than the original. Second, Escher built a warped grid that distributes this zoom factor across the four corners of the square canvas. Third, he used that grid together with the straightened image to transfer content square by square from the plain grid into the warped one, so that following the warped lines around in a loop naturally recreates the zooming effect.
Demonstrating with a simpler example 5:02
To make the process concrete, a simpler custom image is used, showing a pie creature looking at a picture of the house it lives in, with a self-similar zoom factor of only 16 instead of 256. By zooming into the original image by factors of two and placing the resulting corners into a workspace, you get four rough corner pieces, and filling the gaps between them smoothly lets a viewer's gaze moving around a circle experience a continuous zoom into the self-similar copy. A naive attempt at joining these corners looks rough compared to Escher's elegant result, showing there is real craft in the smooth version.
The warped grid and mesh warp 7:03
Escher's own process, seen in a book from the Escher Museum in The Hague, involved building this warped grid by hand as step two. In his version, scaling by a factor of four occurs from one corner to the next, matching the overall 256 scale factor. Using this grid alongside the original image to transfer content piece by piece is a known graphic design technique called a mesh warp, which Escher had used before in other works, and it makes copying manageable because each small square is nearly undistorted even though the whole image is dramatically warped overall.
Why the squares stay square 11:30
A naive attempt to scale linearly from corner to corner creates conflicting pressures on the grid, so Escher curved the lines to relieve this tension. Crucially, in his final grid the tiny regions remain approximately square rather than becoming distorted parallelograms, which is not typical of most mesh warps. This property, where small squares stay square under a transformation, has a specific name in mathematics: a conformal map, a concept that shows up constantly in the study of complex numbers.
A primer on complex numbers 13:30
The video shifts into a mathematical refresher on complex numbers, where the imaginary unit i is the square root of negative one, and every point on the plane represents a real number combined with a real multiple of i. Multiplying a complex number by a constant scales and rotates the whole plane rigidly, always preserving shape, with the amount of scaling and rotation determined by where the constant lands the point at one.
Conformal maps in complex functions 17:04
More complicated functions such as squaring or cubing a complex number visibly warp a grid at large scale, yet at a small enough scale, squares remain approximately square, meaning these functions are conformal. This holds for nearly any complex function you could write down, and it traces back to the definition of a derivative: as you zoom into a point, the transformation looks like multiplication by a constant, which only rotates and scales, hence preserves shape. This is special to complex functions, since ordinary functions of two real variables typically distort tiny squares into parallelograms rather than squares.
Reframing the print gallery puzzle 21:31
With conformal maps established, the question behind the print gallery becomes whether you can build a complex function so that zooming in near the input corresponds to walking around a loop in the output. To do this, two key functions are introduced: e raised to the power z, and the natural logarithm. The exponential function is explained first, showing that increasing the imaginary part of the input causes the output to walk around a circle, completing one full rotation every time the imaginary part increases by two pi, and that multiple distinct inputs can map to the same output, a many-to-one behavior that will matter for reconstructing Escher's loop.
Taking the natural log of an image 26:03
The natural log undoes the exponential map by unraveling circles back into straight lines. Applying this to the Droste image of the pi creature means every circle of points gets straightened into a vertical line segment of height two pi, with smaller and smaller rings from the picture landing farther and farther to the left.
Why the log image repeats 27:30
The resulting log picture repeats vertically because walking up by two pi in the input just sends the exponential back around the same circle, so the image naturally tiles in that direction. Because the exponential function is many to one, its inverse, the logarithm, is naturally multi valued, so mathematicians usually restrict it to one band, called a branch cut, though for this project it helps to keep all the repeating copies.
Horizontal repetition from self similarity 30:04
The log image also repeats horizontally, but only because the original Droste picture is self similar under zooming. Multiplying a point by sixteen in the original image corresponds to shifting left by the natural log of sixteen in the log image, so a rectangle of width log sixteen and height two pi contains the entire annulus of information, and shifting that rectangle repeatedly reproduces the whole infinite zoom.
Building the Escher style function 33:04
The full construction takes a logarithm to produce the doubly periodic tiling, then rotates and scales that pattern by a carefully chosen complex constant, then applies the exponential again to unwrap it with a twist. The goal is to turn the line connecting the large pi creature and its shrunk copy into a closed loop, which means using a diagonal line in the log image and rotating it so it becomes a vertical segment of height two pi around a fixed point z naught, and the right choice of constant produces the recreated print gallery effect while naturally filling in the central hole with a spiraling pattern.
Applying the method to Escher's print gallery 38:04
For Escher's actual piece, the same four steps apply, placing the scene on a complex plane, taking its logarithm, rotating and scaling by a constant suited to a zoom factor of two hundred fifty six, and then exponentiating, which reproduces an image where walking around a loop corresponds to zooming in by that factor with no artificial hole in the middle. The whole formula collapses to raising the input to a complex power, and trying the same trick with a horizontal instead of diagonal line just swaps the roles of rotation and scaling, producing a different, less fitting pattern.
Recovering Escher's curved grid and its meaning 41:02
Applying the same log, rotate, exponentiate process to an ordinary square grid, made denser toward the center so it looks the same at every scale, produces a curved tiling nearly identical to the mesh Escher labored over by hand, and the conformal property guarantees tiny squares stay approximately square. Escher was drawn throughout his career to ideas like representing infinity in a finite space, paired with a rigid aesthetic rule such as squares staying square, and the doubly periodic pattern behind this piece belongs to a class called elliptic functions, which are central to modern number theory, the very field of the two researchers, De Smit and Lenstra, who first analyzed this work.
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