A class of illusion in which an object which is physically
unrealizable is apparently depicted.
More than 100 papers have been written about impossible figures (Kulpa 1987), and Escher made extensive use of them in some of his well-known drawings.
Dodik et al. (2025) introduced the mescher representation, in which an oriented triangle mesh has screen-space vertex coordinates and a signed relative-depth value on each oriented edge. In the language of discrete
exterior calculus, local consistency requires the oriented depth changes to sum
to zero around every face, so is a discrete closed form.
Its discrete Hodge decomposition is
where
assigns an absolute depth to each vertex and is harmonic. The mesh can be globally embedded with those
local depth cues iff; a nonzero is therefore a cohomological
obstruction to global embeddability. This develops a connection between impossible
figures and cohomology studied earlier by Penrose
(1992).
Cerf, C. "A Family of Impossible Figures Studied by Knot Theory." https://www.mi.sanu.ac.rs/vismath/cerf/.Cowan,
T. M. "The Theory of Braids and the Analysis of Impossible Figures."
J. Math. Psych.11, 190-212, 1974.Cowan, T. M. "Supplementary
Report: Braids, Side Segments, and Impossible Figures." J. Math. Psych.16,
254-260, 1977.Cowan, T. M. "Organizing the Properties of Impossible
Figures." Perception6, 41-56, 1977.Cowan, T. M.
"Turning the Penrose Triangle Inside Out." J. Math. Psych.26,
252-262, 1982.Cowan, T. M. and Pringle, R. "An Investigation
of the Cues Responsible for Figure Impossibility." J. Exper. Psych. Human
Perception Performance4, 112-120, 1978.Dodik, A.; Yu, I.;
Chandra, K.; Ragan-Kelley, J.; Tenenbaum, J.; Sitzmann, V.; and Solomon, J. "Meschers:
Geometry Processing of Impossible Objects." ACM Trans. Graphics44,
70:1-70:10, 2025. https://doi.org/10.1145/3731422.Ernst,
B. Adventures
with Impossible Figures. Stradbroke, England: Tarquin, 1987.Fineman,
M. The
Nature of Visual Illusion. New York: Dover, pp. 119-122, 1996.Harris,
W. F. "Perceptual Singularities in Impossible Pictures Represent Screw
Dislocations." South African J. Sci.69, 10-13, 1973.Jablan,
S. "Impossible Figures." https://www.mi.sanu.ac.rs/vismath/jablan/impos.htm.Knuth,
D. E. §7.2.2.3 in The Art of Computer Programming, Vol. 4.
Pre-Fascicle 7A, p. 14, Dec. 5, 2024.Kulpa, Z. "Are Impossible
Figures Possible?" Signal Processing5, 201-220, 1983.Kulpa,
Z. "Putting Order in the Impossible." Perception16, 201-214,
1987.Penrose, R. "On the Cohomology of Impossible Figures."
Leonardo25, 245-247, 1992. https://doi.org/10.2307/1575844.Rakov,
D. "Impossible Objects." http://www.rakov.de/ARTE.HTM.Rausch,
J. "Impossible Objects." http://www.johnrausch.com/PuzzleWorld/cat/io000.htm.Robinson,
J. O. "Impossible Figures." In The
Psychology of Visual Illusion. New York: Dover, p. 176, 1998.Sugihara,
K. "Classification of Impossible Objects." Perception11,
65-74, 1982.Terouanne, E. "Impossible Figures and Interpretations
of Polyhedral Figures." J. Math. Psych.27, 370-405, 1983.Terouanne,
E. "On a Class of 'Impossible' Figures: A New Language for a New Analysis."
J. Math. Psych.22, 24-47, 1983.Thro, E. B. "Distinguishing
Two Classes of Impossible Objects." Perception12, 733-751, 1983.Wilson,
R. "Stamp Corner: Impossible Figures." Math. Intell.13,
80, 1991.
on each oriented edge. In the language of discrete
exterior calculus, local consistency requires the oriented depth changes to sum
to zero around every face, so 
is a discrete closed form.
Its discrete Hodge decomposition is
assigns an absolute depth to each vertex and 
is harmonic. The mesh can be globally embedded with those
local depth cues iff 
; a nonzero 
is therefore a cohomological
obstruction to global embeddability. This develops a connection between impossible
figures and cohomology studied earlier by Penrose
(1992).
