Such paintings are a tour d'force.
Norman Rockwell painted a similar self portrait in a humorous vein for a Saturday Evening Post cover:
Norman Rockwell - Triple Self-Portrait, 1960
Many of old master works of the past were intentionally done as visual puzzles. The most famous of all was Deigo Velazquez' huge painting
Las Meninas, a group portrait of the young princess, daughter of King Phillip of Spain and her retinue.
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Velazquez includes himself in the painting prominently, but as one looks more closely at the painting, the King and Queen are also in it, a mysterious figure in the far background, and increasingly more questions arise as to how the painter managed to accomplish painting the picture as he did. Were mirrors used? How many, and where were they placed? Who are all the people in the painting looking at? Who is the artist actually painting on the huge canvas in front of him? We can't see. All that's visible is the back of the canvas.
It's a painting within a painting within a painting, and very subtle tricks abound. The room is full of paintings and each is a clue, as are all the figures.
This painting has been written about more than any other for 300 years now, and some of the many questions about it are still hotly discussed by art historians. Some of the questions have never been resolved.
In it's day, paintings like this were common among royalty and those who could afford them. They are the equivalents of cliff-hanger movies that never have an obvious conclusion, and were intentionally painted to arouse discussion about them. Velazquez' work became the most famous example that showed the possibilities of what was capable in an easel painting.