How (and why) to take a logarithm of an image

Whenever I'm making one of these videos, there's sometimes a special moment where the act of animating involves solving a whole bunch of little technical puzzlers , and then the underlying math I'm trying to explain clicks for me in a way that it hadn't before once I see it alive on screen . The best versions of those moments often tell me when a video is going to be one of my favorites , and putting together the end of this piece right here was one such time when I got that feeling. . Our story doesn't actually start in the math classroom today . We begin in the art room . Imagine standing in a gallery, looking at a picture of a boat in a harbour , and the whole world warps as your gaze shifts upwards and to the right , where you see a village on this waterfront . Among the tightly clustered building s, the world warps even more as your gaze shifts as your gaze shifts downwards to the entrance of one building, leading to a hallway full of artwork . And at the end of this hall, here you are again, staring at a picture of a boat . This is M.C. Escher's 1956 lithograph , the Print Gallery, or in Dutch, Prentendentunstelling. . In a letter that he wrote to his son, describing his creation of a young man looking with interest at a print that features himself , Escher called this the most peculiar thing I have ever done , which for him is saying a lot . Escher's art is widely loved around the world , frequently featuring paradoxical themes or uniquely satisfying geometric patterns with some kind of character . This love is especially pronounced among mathematicians , since his art often touches on surprisingly deep concepts within math , despite the fact that Escher himself had no formal training in the field . The Print Gallery offers a perfect example of this unexpected depth . In 2003, the mathematicians De Smit and Lenstra offered a delightful analysis of what's really going on in this piece and the mind-bending self-contained loop that Escher managed to achieve . One of my main goals with this video is to offer a visual unpacking of their analysis , aiming as always to give you a feeling that you could have rediscovered this yourself